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How to Write a Great Hypothesis

Hypothesis Definition, Format, Examples, and Tips

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Amy Morin, LCSW, is a psychotherapist and international bestselling author. Her books, including "13 Things Mentally Strong People Don't Do," have been translated into more than 40 languages. Her TEDx talk,  "The Secret of Becoming Mentally Strong," is one of the most viewed talks of all time.

Verywell / Alex Dos Diaz

• The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis.

• Operationalization

Hypothesis Types

Hypotheses examples.

• Collecting Data

A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study. It is a preliminary answer to your question that helps guide the research process.

Consider a study designed to examine the relationship between sleep deprivation and test performance. The hypothesis might be: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

At a Glance

A hypothesis is crucial to scientific research because it offers a clear direction for what the researchers are looking to find. This allows them to design experiments to test their predictions and add to our scientific knowledge about the world. This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

• Forming a question
• Performing background research
• Creating a hypothesis
• Designing an experiment
• Collecting data
• Analyzing the results
• Drawing conclusions
• Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. At this point, researchers then begin to develop a testable hypothesis.

Unless you are creating an exploratory study, your hypothesis should always explain what you  expect  to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore numerous factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment  do not  support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk adage that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

• Is your hypothesis based on your research on a topic?
• Can your hypothesis be tested?
• Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the  journal articles you read . Many authors will suggest questions that still need to be explored.

How to Formulate a Good Hypothesis

To form a hypothesis, you should take these steps:

• Collect as many observations about a topic or problem as you can.
• Evaluate these observations and look for possible causes of the problem.
• Create a list of possible explanations that you might want to explore.
• After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method ,  falsifiability is an important part of any valid hypothesis. In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that  if  something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

The Importance of Operational Definitions

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

Operational definitions are specific definitions for all relevant factors in a study. This process helps make vague or ambiguous concepts detailed and measurable.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in various ways. Clearly defining these variables and how they are measured helps ensure that other researchers can replicate your results.

Replicability

One of the basic principles of any type of scientific research is that the results must be replicable.

Replication means repeating an experiment in the same way to produce the same results. By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. For example, how would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

To measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming others. The researcher might utilize a simulated task to measure aggressiveness in this situation.

Hypothesis Checklist

• Does your hypothesis focus on something that you can actually test?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate the variables?
• Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

• Simple hypothesis : This type of hypothesis suggests there is a relationship between one independent variable and one dependent variable.
• Complex hypothesis : This type suggests a relationship between three or more variables, such as two independent and dependent variables.
• Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
• Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
• Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative population sample and then generalizes the findings to the larger group.
• Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the  dependent variable  if you change the  independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

A few examples of simple hypotheses:

• "Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
• "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."​
• "Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."
• "Children who receive a new reading intervention will have higher reading scores than students who do not receive the intervention."

Examples of a complex hypothesis include:

• "People with high-sugar diets and sedentary activity levels are more likely to develop depression."
• "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

Examples of a null hypothesis include:

• "There is no difference in anxiety levels between people who take St. John's wort supplements and those who do not."
• "There is no difference in scores on a memory recall task between children and adults."
• "There is no difference in aggression levels between children who play first-person shooter games and those who do not."

Examples of an alternative hypothesis:

• "People who take St. John's wort supplements will have less anxiety than those who do not."
• "Adults will perform better on a memory task than children."
• "Children who play first-person shooter games will show higher levels of aggression than children who do not." 

Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

Descriptive Research Methods

Descriptive research such as  case studies ,  naturalistic observations , and surveys are often used when  conducting an experiment is difficult or impossible. These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a  correlational study  can examine how the variables are related. This research method might be used to investigate a hypothesis that is difficult to test experimentally.

Experimental Research Methods

Experimental methods  are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually  cause  another to change.

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Thompson WH, Skau S. On the scope of scientific hypotheses .  R Soc Open Sci . 2023;10(8):230607. doi:10.1098/rsos.230607

Taran S, Adhikari NKJ, Fan E. Falsifiability in medicine: what clinicians can learn from Karl Popper [published correction appears in Intensive Care Med. 2021 Jun 17;:].  Intensive Care Med . 2021;47(9):1054-1056. doi:10.1007/s00134-021-06432-z

Eyler AA. Research Methods for Public Health . 1st ed. Springer Publishing Company; 2020. doi:10.1891/9780826182067.0004

Nosek BA, Errington TM. What is replication ?  PLoS Biol . 2020;18(3):e3000691. doi:10.1371/journal.pbio.3000691

Aggarwal R, Ranganathan P. Study designs: Part 2 - Descriptive studies .  Perspect Clin Res . 2019;10(1):34-36. doi:10.4103/picr.PICR_154_18

Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

Part Five: Evaluating Deductive Logic

Chapter Eleven: If–Then Arguments

“Contrariwise,” continued Tweedledee, “if it was so, it might be; and if it were so, it would be: but as it isn’t, it ain’t. That’s logic.” —Lewis Carroll, Through the Looking-Glass

• Forms of If–Then Arguments
• Evaluating the Truth of If–Then Premises
• If–Then Arguments with Implicit Statements
• Bringing It All Together

If–then arguments , also known as conditional arguments or hypothetical syllogisms, are the workhorses of deductive logic. They make up a loosely defined family of deductive arguments that have an if–then statement —that is, a conditional —as a premise. The conditional has the standard form If P then Q. The if portion, since it typically comes first, is called the antecedent ; the then portion is called the consequent .

These arguments—often with implicit premises or conclusions—are pressed into service again and again in everyday communication. In The De-Valuing of America, for example, William Bennett gives this brief if–then argument:

If we believe that good art, good music, and good books will elevate taste and improve the sensibilities of the young—which they certainly do—then we must also believe that bad music, bad art, and bad books will degrade.

The if–then premise—lightly paraphrased—is this:

If good art, good music, and good books elevate taste and improve the sensibilities of the young, then bad music, bad art, and bad books degrade taste and degrade the sensibilities of the young.

The second premise—set off in the original by dashes—is:

Good art, good music, and good books elevate taste and improve the sensibilities of the young.

And the implicit conclusion is this:

Bad music, bad art, and bad books degrade taste and degrade the sensibilities of the young.

Whether the argument is sound depends on whether the logic of the argument is successful and whether the premises are true. We now look at each of these two categories of evaluation.

11.1 Forms of If–Then Arguments

The arguments of this chapter are deductive, so the success of their logic is entirely a matter of form. The form of Bennett’s argument in the preceding paragraph is the most common and the most obviously valid. It is normally termed affirming the antecedent ; a common Latin term for this form is modus ponens, which means “the method (or mode, from modus ) of affirming (or propounding, from ponens ).”

• If P then Q.

Almost as common is the valid form denying the consequent ; the Latin term for this is modus tollens, which means “the method of denying.”

This is the form of my argument if I say to you, “If you decide to adopt that puppy, then you’re going to be stuck at home for a long time. But you could never accept that—you live for your trips. This pup’s adorable, but it’s not for you.”

Each of these two valid forms may be contrasted with an invalid form that unsuccessfully mimics it. The invalid form that is tempting due to its similarity to affirming the antecedent is the fallacy of affirming the consequent ; its structure is this:

• If P then Q .

I’ve committed this fallacy if I argue, “If you decide to adopt that puppy, then you’re going to be stuck at home for a long time. Fortunately you hate to sleep in any bed other than your own. So, this pup’s for you!” After all, you may love staying at home but also have a severe allergy to dog hair; the conclusion surely does not follow.

And deceptively similar to denying the consequent is the fallacy of denying the antecedent ; this invalid form is as follows:

I made this mistake in the following argument: “If you decide to adopt that puppy, then you’re going to be stuck at home for a long time. But, knowing you, of course you’re not going to decide to adopt the puppy. So, it follows that you’re not going to be such a homebody anymore.” If you pass on the puppy because of your asthma, that has no bearing on your travel plans one way or the other. Again, the conclusion does not follow.

Recall that when you find these fallacious forms, there is normally no need to apply the principle of charity in your paraphrase. The ease with which such mistakes are made (thus earning each fallacy a name of its own) is usually reason to think that the arguer might have been truly mistaken in his or her thinking, and thus is reason to clarify the argument in the invalid form. [1]

Another form of argument, a valid one, that belongs to the if–then family is often termed transitivity of implication. This form of argument links if–then statements into a chain, as follows:

• If Q then R.
• ∴ If P then R.

I’ve given an argument of this form if I contend, “If you decide to adopt that puppy, then you’re going to be stuck at home for a long time. But if you’re stuck at home for a long time, you’d better fix your toaster oven. So, if you decide to adopt that puppy, you’d better fix your toaster oven.” There is no limit to the number of if–then links that this chain could contain and still be valid. [2]

Incidentally, Lewis Carroll’s argument at the chapter’s opening presents some interesting evaluative possibilities. Here is one reasonable paraphrase:

• If it is, then it is.
• ∴ It is not.

On the one hand, it has the form of the fallacy of denying the antecedent, which is invalid; on the other hand, it has the form of denying the consequent, which is valid. Further, it also has the form of repetition—looking only at 2 and C —which is also valid. The solution to the puzzle is that it is valid—not because the two valid forms outnumber the one invalid one, but because we should charitably suppose that the valid form is the one that was intended. Charity, unfortunately, cannot prevent us from noting that whatever the form, this argument probably commits the fallacy of begging the question. And if it does, then it does.

If–Then Arguments

EXERCISES Chapter 11, set (a)

Create a brief argument that takes the specified form.

Sample exercise. Transitivity of implication.

Sample answer. If I run out of gas I’ll be late. And if I’m late I’ll get fired. So, if I run out of gas I’ll get fired.

• Affirming the antecedent
• Affirming the consequent
• Denying the consequent
• Denying the antecedent
• Transitivity of implication

11.1.1 Stylistic Variants for If–Then

If–then arguments, as we have seen, make crucial use of statements of the form If P then Q as premises. Using the terminology of Chapter 6, the expression if–then is the logical constant of such statements, while P and Q are the variables— sentential variables, you will recall—replaceable by declarative sentences as the content of the argument.

These constants are anything but constant in ordinary language; a wide variety of everyday English expressions are used to express if–then. In the structuring phase of the clarifying process, it is important that you translate them into standard constants. This helps bring the structure of the argument close to the surface and makes it much easier to tell whether the argument is logically successful.

All of the expressions listed below—and many more—can be used as stylistic variants for if–then. More precisely, each of the expressions can be translated, for logical purposes, into If P then Q.

Stylistic Variants for if P then Q

Q if P. P only if Q. Only if Q, P. Assuming P, Q. Q assuming P. Supposing P, Q. Q supposing P. Given P, Q. Q given P. That P is a sufficient condition for that Q. That Q is a necessary condition for that P.

This list includes some of the most obvious variants, but it is not comprehensive. Unexpected variants for if–then statements show up with regularity. A politician says, for example, “Vote for my bill and I’ll vote for yours.” This can be taken as a stylistic variant for, “If you vote for my bill, then I’ll vote for yours.” A story about new television shows says, “ With good summer ratings, the series will end up on the fall schedule of NBC.” This translates into, “If the series gets good summer ratings, then it will end up on the fall schedule of NBC.” And language watcher Thomas Middleton, complaining in the Los Angeles Times about a tendency he has noticed among teens to use expressions like “and then my friend goes so-and-so” instead of “and then my friend said so-and-so,” presses his point thus:

The ability to say things . . . is consummately precious, and to describe “saying” as “going” is to debase this glorious gift. It is to treat speech as though it were no more than, as Random House says, making a certain sound—like a cat’s purr.

This passage, it seems, translates into something like the following:

If someone describes “saying” as “going,” then that person debases the gift of speech and treats it as though it were no more than making a sound.

Be very careful, however, with words like with, and, and to ; they are rarely stylistic variants for if–then. It is only when they are used in these distinctive kinds of contexts that they should be taken this way.

EXERCISES Chapter 11, set (b)

Translate the stylistic variant in each of the following if–then statements.

Sample exercise. As long as history textbooks make white racism invisible in the 19th century, students will never be able to analyze racism intelligently in the present.—James Loewen, Lies My Teacher Told Me

Sample answer. If history textbooks make white racism invisible in the 19th century, then students will never be able to analyze racism intelligently in the present.

• Oxygen is necessary for combustion.
• The governor agreed Tuesday to a legislative compromise for ending the community colleges’ financial troubles—but only if lawmakers can find another $121 million.
• Teachers should assign passages and require students to summarize the passages in their own words. Do that consistently, and students will not only learn to write a lot better, they will also learn to analyze, evaluate, sort out, and synthesize information.
• Those who are not willing to give to everyone else the same intellectual rights they claim for themselves are dishonest, selfish, and brutal. —from Robert Ingersoll, Ingersoll: The Immortal Infidel
• “To address kids in masses, you have to be an entertainer, which I’m not,” Dr. Seuss said, sounding a little like the Grinch.— Los Angeles Times
• “Strip a woman’s body of its breasts and hips, of all of its nurturing curves, and replace it with enough stringy, sinewy muscle, and a lot of people will simply not know what to make of what you have left.” — Pumping Iron II: The Women

EXERCISES Chapter 11, set (c)

Clarify each of the if–then arguments. Then state whether the argument is valid and provide the name of the valid or invalid form.

Sample exercise. “I submit that the author is thoroughly wrong to criticize analogical argumentation, that if argument by analogy were really as weak as he allows we would not use it as extensively as we do.”—book review in Teaching Philosophy

Sample answer.

• If argument by analogy is as weak as the author allows, then we do not use argument by analogy as extensively as we do.
• [We do use argument by analogy as extensively as we do.]
• ∴ Argument by analogy is not as weak as the author allows. Valid, denying the consequent.
• Universal mandatory screening for AIDS can be justified on the basis of beneficence when a therapeutic intervention is available or when an infectious state puts others at risk merely by casual contact. However, neither is the case with AIDS. Thus, there is no demonstrable public health benefit that justifies universal mandatory screening.—N. F. McKenzie, ed., The AIDS Reader (Even though an extremely charitable reading of this argument might suggest otherwise, go ahead and clarify it as a fallacy.)
• “The prolonged study of ethics does not by itself make you a better person. If it did, philosophy professors would in general be better people than average. But they aren’t.”—William Bennett, The De-Valuing of America
• “If the North Koreans are smart—and we know they are smart—they will move in the direction of reform.”—Daryl Plank, Korea expert and visiting fellow at Washington’s Heritage Foundation
• “If, instead of offering the occasional high-profile prize of $35 million, New York awarded 350 prizes of $100,000, making not one multimillionaire, but a great number of $100,000 winners, it would create an environment where far more people would know, or know of, large prize winners. More people will buy tickets if they know large prize winners. So experimenting with such a format should reverse the current negative trend.” —letter to the editor, New York Times
• “Ladies and Gentlemen, I’ll be brief. The issue here is not whether we broke a few rules or took a few liberties with our female party guests. We did. But you can’t hold a whole fraternity responsible for the behavior of a few sick perverted individuals. For if you do, then shouldn’t we blame the whole fraternity system? And if the whole fraternity system is guilty, then isn’t this an indictment of our educational institutions in general? I put it to you, Greg: Isn’t this an indictment of our entire American society? Well, you can do what you want to us, but we’re not going to sit here and listen to you badmouth the entire United States of America.”—Eric “Otter” Stratton, in the film Animal House (This is a complex argument with the subconclusion If the fraternity is guilty, then the entire American society is guilty. )
• “And if, for example, antiabortionism required the perverting of natural reason and normal sensibilities by a system of superstitions, then the liberal could discredit it—but it doesn’t, so he can’t.”—Roger Wertheimer, Philosophy and Public Affairs

11.1.2 Singular Inferences

One common variation on the preceding forms is worth our attention. Note the remark made by legendary heavyweight boxing champ Joe Frazier to the short-lived and less legendary title holder, Jimmy Ellis:

You ain’t no champ. You won’t fight anybody. A champ’s got to fight everybody.

This provides several opportunities for following the rules of paraphrasing arguments—a stylistic variant for if–then, the need to follow the principle of charity (because of the rather extreme words everybody, anybody, and even what Frazier means by being a champ), wording to be matched, and emptiness to be avoided (because of the word you ). The result of clarifying it is something like this:

• If any person deserves to be the heavyweight boxing champion, then that person will fight all worthy contenders.
• Jimmy Ellis will not fight all worthy contenders.
• ∴ Jimmy Ellis does not deserve to be the heavyweight boxing champion.

This looks very much like denying the consequent—that is, it seems to depend on this form:

But the Q of premise 1 and the Q of premise 2 do not really match, nor do the P of premise 1 and the P of C. For there is no mention of Jimmy Ellis anywhere in premise 1, yet Jimmy Ellis is the subject of premise 2 and of the conclusion. This certainly does not harm the argument’s logic, however, since Jimmy Ellis is included—as a single person—among those encompassed by the term any person in the first premise. So, for practical purposes, we can continue to call this form denying the consequent, but with a slight difference. It will be identified as singular denying the consequent

The same modification is permitted for every form of sentential logic that we cover, assuming two things hold. First, there must be a universal statement as a premise—that is, a premise with a term like all, none, anything, or nothing, to mention a few examples. If any person deserves to be the heavyweight boxing champion, then that person will fight all worthy contenders is universal, since it applies to any person. Second, there must be a conclusion in which a single instance is specified that is encompassed by the universal term. Jimmy Ellis will not fight all worthy contenders provides an example, since Jimmy Ellis is encompassed by any person. All the if–then forms mentioned above can be modified in this way. Singular affirming the antecedent and singular transitivity of implication are also valid forms, while the fallacy of singular affirming the consequent and the fallacy of singular denying the antecedent are invalid ones.

Singular If–Then Arguments

EXERCISES Chapter 11, set (d)

Clarify the following arguments as examples of singular if–then arguments. Then state whether the argument is valid and provide the name of the valid or invalid form.

Sample exercise. “ Q : You mentioned that Bundy was mentally ill? A : Sane people do not go round killing dozens of women, and the person that the state of Florida strapped in the electric chair was a man who was severely mentally ill.”—I. Gray and M. Stanley, eds., A Punishment in Search of a Crime: Americans Speak Out Against the Death Penalty

• If anyone is sane, that person does not kill dozens of women.
• [Bundy killed dozens of women.]
• ∴ Bundy was not sane. Valid, singular denying the consequent.
• If someone is in Birmingham, then that person is in Alabama. And if someone is in Alabama, then that person is in America. So, if I am in Birmingham, then I am in America.
• I conceded that speeding is sufficient reason for getting pulled over by a police officer. But I wasn’t speeding. So I should not have been pulled over.
• One argument for the immorality of adultery might go something like this: it involves the breaking of a promise, and it is immoral to break a promise.—from a lecture by philosopher Richard Wasserstrom
• I refer you to the verdict in the English Court sustaining Whistler’s contention that a man did not wholly own a picture by simply buying it. So, although I may have sold my painting to him, I have a right to protect my picture from the vandalism of his cleaning it.—from American artist Albert Pinkham Ryder

11.2 Evaluating the Truth of If–Then Premises

If–then statements usually propose a special connection between the if-clause and then-clause. Identifying the specific nature of the connection is usually the key to judging the truth of such a statement and to successfully defending that judgment. [3]

Sometimes the proposed connection is causal, as in the case of the statement If you push the ignition button, then the car will start. Pushing the button would cause the car to start. But in other cases the proposed connection is broadly logical ; the if-clause does not cause the then-clause but is offered as counting toward or even guaranteeing its truth. [4] Consider the statement that became a book and movie title— If it’s Tuesday, this must be Belgium. It’s being Tuesday cannot cause this to be Belgium, but could presumably (combined with other statements about the itinerary) count in favor of the belief that this is Belgium. Or consider the statement If there is intelligent life on other planets, then we are not alone in the universe. That there is intelligent life elsewhere in the universe is just what we mean by not being alone in the universe. So the connection here is also a logical one—and in this case it is such a tight connection that we can safely call it self-evidently true.

Whether the proposed connection is causal or logical, it is helpful to think of the if-clause as not being offered as alone sufficient for the then-clause. When we use if–then statements we are typically allowing for other relevant factors as well. We have simply picked out the if-clause for special mention because it is the one factor that happens to be most important in the context. These implicit assumptions about other relevant factors are termed secondary assumptions (or auxiliary hypotheses).

Return to the if–then statement If you push the ignition button, then the car will start. Behind such a statement there usually are implicit secondary assumptions about many other factors that contribute to the starting of the car—but that are presumed to be already in place, and thus do not merit mention. They may include assumptions about the specific situation, such as these:

There is a functioning engine in the car. There is gas in the tank and the starter battery is not dead (if it has an internal combustion engine). The battery pack is charged (if it has an electric engine). The key fob is nearby. The ignition system is not defective.

They may also include more general assumptions about the relation between the if-clause and then-clause, such as this:

Ignition systems are designed to start properly functioning cars.

And they may include even broader principles that guide much of our reasoning, such as this:

The laws of nature will not suddenly change.

When you judge an if–then statement to be true, a good way to defend your judgment is to identify the secondary assumptions that are most likely to be in question, given the circumstances, and to point out their truth. You might say, for example,

I judge this premise to be very probably true because this is what ignition systems are designed to do, and there is no reason to think that this car is out of fuel or is defective in some other way.

Thus a connection between if-clause and then-clause is affirmed.

Alternatively, if you judge the if–then statement to be false, a good defense is to point out that a secondary assumption is false; for example,

This premise is probably false, since the headlights were left on all day and the battery is dead by now.

Thus you have denied one of the secondary assumptions and shown that the connection between if-clause and then-clause is severed.

The same strategy works well for if–then statements in which the connection is broadly logical rather than causal. Consider If you are reading this book, then you understand English. One important secondary assumption is This book is written in English. Another is Reading something just means that you understand it. (You might wonder whether this, or the earlier life on other planets example, should count as a secondary assumption, since it is part of the very meaning of the terms used—what we have in preceding chapters called self-evidently true. It will nevertheless make good practical sense in this text for us to count it so.) So here is an exemplary defense of the statement:

This premise is certainly true, since the book is written in English, and part of what it means to read something is to understand it.

Again, its truth is defended by pointing out the cords that connect if-clause to then-clause.

Consider, finally, If New York City were in Quebec, then it would still be in the United States. There is, unfortunately, no way of knowing what secondary assumptions are supposed to connect this if-clause and then-clause. Is New York City to be located further north, or Quebec further south? And, on either scenario, what historical events would have caused such a difference—and would they, perhaps, have resulted in Quebec’s being included within the United States? There is simply not enough information to decide. The best evaluation of this premise, then, would be something like this:

I can’t decide whether this premise is true or false. There is no way of knowing whether New York City is to be located further north, Quebec further south, or what relevant historical events might have led to it.

The daughter of Rudolf Carnap, one of the great philosophers and logicians of the 20th century, tells of asking her father, when she was a young child, “If you were offered a million dollars, would you be willing to have your right arm amputated?” “I don’t know,” he replied. “Would I be given an anesthetic?” Lack of information about relevant secondary assumptions can sometimes make it impossible for any of us, even Carnap, to say any more than “can’t decide” in evaluating if–then statements.

EXERCISES Chapter 11, set (e)

For each of the following if–then statements, list the most plausible and relevant secondary assumptions (or explain why you cannot do so). Then provide a judgment of the premise’s truth by reference to your list. (They are not provided with any context, so you will have to use your imagination.)

Sample exercise. If there were more solar panels available, then air pollution would decrease.

Sample answer. Secondary assumptions: Consumers will buy and install more solar panels if they become available. Solar panels produce less air pollution than the more conventional forms of energy production. Probably false, since, at least currently, in much of the world consumers do not have enough economic incentive to convert to solar energy.

• If there were more classical music on the radio, then there would be more appreciation of classical music among the public.
• If George W. Bush was the 45th president, then Barack Obama was the 46th.
• If it rains tomorrow, then you should take your umbrella to work.
• If people recycled more, the environment would be in better shape.
• If I were two feet taller, I would have played in the NBA.
• If it’s Tuesday, this must be Belgium.
• If the stock market rises next year, then Alphabet stock will rise in value.
• If all private ownership of guns were made illegal, then violence in our country would drop dramatically.
• If you can do 20 pushups, then you are in good shape.
• If you are planning to go to medical school, then you can expect to take several science courses.

11.2.1 The Retranslation Mistake

Consider the statement If New York City is in the state of New York, then it is in the United States. It is certainly true, but it is tempting to defend that judgment by more or less repeating the if–then statement in slightly different words, as follows:

My view is that the premise is certainly true, since New York City has to be in the United States, given that it is in the state of New York.

You have said nothing that goes beyond the premise itself, thus nothing that would be enlightening to the reasonable objector over your shoulder. You have merely retranslated the if–then constant back into one of its stylistic variants! Be careful to avoid this sort of defense. (I’ll leave it as an exercise for you to identify the simple secondary assumption that provides the crucial connection for this if–then statement.)

11.2.2 Truth Counterexamples

It can be especially tempting to ignore mention of secondary assumptions when the if-clause is clearly true and the then-clause is clearly false. These are the most straightforward cases, for if you know that the if-clause is true and the then-clause is false, you know that the if–then statement is false. The if–then statement has vividly failed to deliver on its promise.

But even here it is better, if possible, to show the severed connection between the two by identifying the false secondary assumption. Take, for example, If New York City is in the state of New York, then it is in Canada. You might defend your judgment as follows:

I consider the premise to be certainly false since, based on my experience in my own travels and based on the testimony of every authority I’ve ever encountered, New York City is in the state of New York and it is not in Canada (but in the United States).

But this makes no mention of any connection between the if-clause and then-clause. If there is supposed to be one, it is the assumption that the state of New York is itself in Canada. And your defense is stronger if you include the rejection of this assumption, as follows:

Further, the state of New York is wholly located within the United States, not Canada.

There can be, however, exceptions to this rule. One exception applies when it is a universal if–then statement that is false. Universal if–then statements, recall, are if–then statements with a universal term like anything, anyone, nothing, or nobody in the if-clause. An example we have already seen is If any person deserves to be the heavyweight boxing champion, then that person will fight all worthy contenders. A property—such as deserving to be champ —is applied universally—to any person —rather than to a single instance. When such statements are false, the method of truth counterexample can be a simple and effective way of defending that judgment. This method identifies a single instance in which the if-clause is obviously true and the then-clause is obviously false.

A newspaper story on the homeless, for example, contains the line, “No one is poor by choice.” This is a stylistic variant of the universal if–then statement, “If anyone is poor, then it is not by choice.” Yet the same newspaper, on the facing page, has a story about Mother Theresa’s religious order, stating, “These nuns have voluntarily taken an oath of poverty.” Here we have a ready-made truth counterexample. The nuns are instances of the if-clause’s truth—they are poor—and at the same time are instances of the then-clause’s falsity—their poverty is by choice. Thus armed, your defense of your judgment of the universal if–then statement can be stated simply as follows: “The premise is certainly false, since certain orders of nuns are poor by choice.”

EXERCISES Chapter 11, set (f)

Provide a truth counterexample for each of these false universal if–then statements. If necessary, first translate stylistic variants into the standard constant.

Sample exercise. Only animals that can fly are endowed with wings.

Sample answer. If any animal has wings, then it can fly. Certainly false; the ostrich has wings but cannot fly.

• No major-party presidential or vice presidential candidate has been a female.
• If any substance is made of metal, then it is attracted to a magnet.
• Everything reported in the newspaper is true.
• Nice guys finish last.
• What goes up must come down.

11.2.3 The Educated Ignorance Defense

Another occasion for ignoring secondary assumptions—also occurring under the true if-clause/false then-clause scenario—is when the evidence for the truth of the if-clause and the evidence for the falsity of the then-clause are each stronger than the evidence for the truth of the if–then statement. In these cases, even though you may not know which secondary assumption is at fault, it can be reasonable to say that the premise is false “because some secondary assumption—not yet identified—is mistaken.” This we will term the educated ignorance defense (“Ignorance” because you admit ignorance regarding which secondary assumption is faulty; “educated” because you nevertheless have good evidence that the if-clause is true and the then-clause is false.)

Return, for example, to our car-starting example. Imagine that when you pick up your car after extensive repairs your mechanic says to you, “If you push the ignition button, then the car will start.” He has extremely good reasons to believe this is true. He has checked out all of the systems—in short, his experience and expert judgment support the truth of any secondary assumption that might be reasonably questioned. You push the ignition button. But the car does not start.

Something has to give. There are three statements for which you apparently have very good evidence:

If you push the ignition button, then the car will start. You push the ignition button. (The if-clause is true.) The car will not start. (The then-clause is false.)

They cannot all be true at the same time. You will probably quickly give up on the truth of the if–then statement, not knowing what went wrong but knowing quite well that you pushed the button and the car didn’t start. But the mechanic, who has especially good reasons to believe the if–then statement—he did the work, and he has his reputation to think about—will probably start off by doubting the if-clause, asking you suspiciously, “Are you sure you pushed the ignition button?” “I’m sure,” you reply, anxiously pushing it again and again. “Let me see,” he says with a hint of disdain and gets in to push the button himself. Only when it does not start for him does he say, “Well, OK, I was mistaken, but I just can’t figure out what’s wrong with it.”

The mechanic’s initial reluctance to give up the truth of the if–then statement is because he cannot imagine which secondary assumption is mistaken. And he only concedes that the if–then statement is false when he sees that evidence in favor of the if-clause—that the button has been pushed—is conclusive. (The evidence that the then-clause is false—that the car did not start—is already conclusive.) He is still ignorant of which secondary assumption to blame, but now that he is duly educated—about the truth of the if-clause and falsity of the then-clause— he can reasonably resort to the educated ignorance defense. Eventually something better than educated ignorance will be required if the car is to be driven away.

Science provides many examples of this defense. In the 18th century, for example, astronomers used the new Newtonian mechanics to accurately predict the orbits of many of the planets in our solar system. The following if–then statement describes the general shape of these predictions:

If Newtonian mechanics is true, then the orbit of planet A will be observed to be F.

(In this case, A is the name of the planet and F is a mathematical description of the predicted observed orbit of the planet around Earth.) After many successes, the astronomers did their work on the orbit of Uranus and discovered, to everyone’s surprise, that the predicted orbit did not accord with their observations. They thus found themselves with good evidence for the following three statements, not all of which could be true:

If Newtonian mechanics is true, then the orbit of Uranus will be observed to be F. Newtonian mechanics is true. (The if-clause is true.) It is not the case that the orbit of Uranus is observed to be F. (The then-clause is false).

They checked and rechecked their equipment to be sure of their evidence that the then-clause was false, but they found their surprising observations to be accurate. They reminded themselves of the mountains of other evidence in favor of the if-clause. And they checked and rechecked their calculations, in the futile hope of finding some faulty secondary assumption that would falsify the if–then statement. In the end, the only reasonable thing to do was to reject the if–then statement with a defense something like this:

This premise is probably false; the support for Newtonian theory is so strong, and the quality of this observation so good, that it is most likely that some not-yet-identified faulty secondary assumption lies behind its falsity.

Incidentally, that is where things stood until the 19th century, when the Englishman John Adams and the Frenchman Urbain Leverier, working independently, realized that the mistake had been in assuming Uranus is the outermost planet. Due to this secondary assumption, the earlier Newtonians had not factored into their calculations any gravitational pull from the other side of Uranus. They each reworked the calculations and predicted where they should be able to observe an outer planet exercising gravitational attraction on Uranus. In 1846 they independently observed this planet, later named Neptune, in the predicted location.

The strategy of saying, “There is some unidentified secondary assumption that is mistaken” should be employed with great care. Again, it works only when there is independent strong evidence in favor of the truth of the if-clause and against the truth of the then-clause. These lines are from the final letter written to his wife by one of the doomed soldiers of the German Sixth Army outside Stalingrad:

“If there is a God,” you wrote me in your last letter, “then he will bring you back to me soon and healthy.” But, dearest, if your words are weighed now you will have to make a difficult and great decision.

Her own words, quoted by her husband, committed her to the statement If God exists, then the soldier will return to his wife soon and healthy. The report of his death that she later received supported this statement: The soldier will not return to his wife soon and healthy. But by a valid denying the consequent argument, these two premises entail God does not exist. This, then, presented his wife with the difficult and great decision that the soldier foretold—she must stop believing in God, or she must go back on her own words.

Let’s set this up in the same way as we did with the auto mechanic and the Newtonians. There are three statements before her, at least one of which must be false:

If God exists, then the soldier will return to his wife soon and healthy. God exists. (The if-clause is true.) The soldier will not return to his wife soon and healthy. (The then-clause is false.)

Let’s suppose that instead of giving up her belief in God, she chose the option of going back on her words and rejecting her if–then statement. Her most reasonable defense, as we have seen, would be for her to sever the connection between the if-clause and the then-clause by identifying and rejecting the false secondary assumption. Candidates might include:

God cares about human suffering. God cares about the suffering of this particular soldier and his wife. God is able to prevent this suffering. God knows about this suffering.

But let’s further suppose that she insisted on continuing to embrace all these secondary assumptions, on the grounds that to do otherwise would be to unsuitably diminish God. Instead, she took the step that many believers in God take—the step of saying, “God’s ways are beyond the understanding of man. When I get to heaven he will reveal to me his reasons. Until then, I will continue to believe in him.” This is an attempt to use the educated ignorance defense. We give up on the if–then statement in the expectation that we will eventually discover the car’s mechanical defect, the flaw in our astronomical calculations, or the hidden mysteries of God.

Whether this is a reasonable move for the soldier’s wife depends on one condition: it is educated ignorance—and thus a reasonable defense—only if the wife has independent strong evidence that God exists (evidence for the if-clause). If she does not—if she accepts by faith alone not only God’s mysterious ways but also his very existence—then she cannot reasonably defend her rejection of the if–then statement unless she identifies and rejects the false secondary assumption.

EXERCISES Chapter 11, set (g)

In each problem there are three statements, at least one of which must be false. Provide an “educated ignorance” defense for the claim that the if–then statement is false; you’ll need to state evidence for the if-clause and against the then-clause in your defense.

Sample exercise. If the instructor is fair, then he will not give higher grades to males than to females. The instructor is fair. The instructor gave higher grades to males than females.

Sample answer. The if–then statement is probably false, although I can’t say exactly what the mistaken assumption is. Even though the record shows that in this class the males did much better than the females, he has a widespread reputation for bending over backward to treat everyone fairly. It seems likely that his reputation is deserved and that in this case there is an explanation that will eventually emerge.

• If you patch that hole, then the roof will stop leaking. You patch the hole. The roof does not stop leaking.
• If your boyfriend loves you, then he will be on time tonight. Your boyfriend loves you. He is not on time tonight.
• If you are the most talented, then you will win the talent show. You are the most talented. You do not win the talent show.
• If it is impossible to move physical objects by only thinking about it, then when Uri Geller concentrates on bending the spoon it will not bend. It is impossible to move physical objects by only thinking about it. When Uri Geller concentrates on bending the spoon it does bend.

Strategies for Evaluating the Truth of If–Then Statements

11.2.4 Secondary Assumptions and Indirect Arguments

Secondary assumptions can also play an important role in the evaluation of indirect arguments (which we have also called reductios ). Introduced in Chapter 10, such arguments, in their simplest form, exhibit the structure of denying the consequent. They begin with a statement that may seem quite innocuous and attempt to show that it is false by pointing out, in what amounts to an if–then premise, an absurd consequence that it forces on you. You accept the absurdity of the consequence by accepting a premise that says the then-clause is false. You must then conclude, by the valid form of denying the consequent, that the seemingly innocuous if-clause must be rejected. [5] An example is found in these remarks by David Wilson (no known relation to the author), adapted from a newspaper report:

Melina Mercouri, Greece’s minister of culture, swept into the staid old British Museum to examine what she called the soul of the Greek people—the Elgin Marbles. Lord Elgin took them from the Parthenon in Athens in the early 19th century. Mercouri is expected to make a formal request soon for the marbles’ return. But Dr. David Wilson, director of the British Museum, opposes the idea. “If we start dismantling our collection,” Wilson said, “it will be the beginning of the end of the museum as an international cultural institution. The logical conclusion of the forced return of the Elgin Marbles would be the utter stripping of the great museums of the world.”

Wilson’s argument can be clarified thus:

• If it is acceptable to force the British to return the Elgin Marbles to Greece, then it is acceptable to strip the great museums of the world.
• [It is not acceptable to strip the great museums of the world.]
• ∴ It is not acceptable to force the British to return the Elgin Marbles to Greece.

Melina Mercouri must avoid the conclusion without rejecting premise 2, so her only recourse is to reject the if–then premise. But when she does reject it, she is no position to respond with the educated ignorance defense; the evidence for the if-clause is exactly what is in question, so for her to simply say the if-clause is obviously true would be to beg the question. In short, her only reasonable strategy is to reject the if–then premise by identifying a faulty secondary assumption that it depends on. Here is a strong candidate for the role of faulty secondary assumption:

The only principle for returning the Elgin Marbles would be that any item, great or small, removed from its original culture, whether by consent or by force, must be returned to that culture.

This secondary assumption is clearly false. So Mercouri might defend her rejection of premise 1 as follows:

Premise 1 is almost certainly false, since it assumes that all items must be returned to their original culture; but the return of the Elgin Marbles only depends on a principle calling for the return of great national treasures that have been forcibly removed.

What Mercouri would be doing is accusing Wilson of committing the fallacy of non causa pro causa (introduced in Chapter 10). This is the fallacy of blaming the absurd consequence ( It is acceptable to strip the great museums of the world ) on what is set forth as its cause ( It is acceptable to force the British to return the Elgin Marbles to Greece ) instead of blaming the unnoticed assumption that is the real cause of the absurdity ( All items must be returned to their original culture ).

Because indirect arguments are typically offered in support of controversial conclusions, only rarely can the educated ignorance approach be used in evaluating them without begging the question. Be especially watchful for faulty secondary assumptions behind the if–then premise of indirect arguments; when there is such an assumption, the indirect argument can be criticized for committing the fallacy of non causa pro causa.

EXERCISES Chapter 11 set (h)

Clarify each of these simple indirect arguments; then evaluate only the if–then premise, on the grounds that it commits the fallacy of non causa pro causa. (Use the Elgin Marbles case as your sample.)

• If you are right in your claim that income taxes should be eliminated, then you must accept the consequence that the government will be left with no money to do even its most important business. But we would all agree that government cannot be done away with. So income taxes must remain.
• If children who misbehave are not immediately and severely punished, they will grow up with the belief that misbehavior has no negative consequences. We all agree, of course, that our children cannot be allowed to grow up with that belief. So don’t spare the rod with your children.

EXERCISES Chapter 11, set (i)

Each of the passages below indicates what could be seen as a misuse of secondary assumptions. In the Kelvin case, clarify the denying the consequent argument and identify the secondary assumption that, perhaps, Kelvin should have questioned. In the Azande case, clarify the affirming the antecedent argument that the Azande are trying to avoid, and identify the secondary assumption that they, perhaps, too readily reject to show that the if–then premise is false.

• Lord Kelvin, the leading British physicist of his day, dismissed Darwin’s work on the ground that it violated the principles of thermodynamics. The sun could be no more than 100 million years old; evolution demanded a much longer period in which to operate; therefore evolution must be rejected. Kelvin wasted no time pursuing the minutiae of the geological and palaeontological evidence on which evolution was based. Physics in the guise of thermodynamics had spoken clearly and whatever failed to fit into its scheme had to be rejected. Kelvin’s thermodynamics were later shown to be wrong. Unaware of radioactivity, he had inevitably failed to allow for its effect in his calculations.—Derek Gjertsen, Science and Philosophy
• According to the Azande, witchcraft is inherited unilinearly, from father to son, and mother to daughter. How, therefore, can I accept that my brother is a witch and yet deny that I am also infected? To prevent this absurdity arising, the Azande adopt further “elaborations of belief.” They argue, for example, that if a man is proven a witch beyond all doubt, his kin, to establish their innocence, deny that he is a member of their clan. They say he was a bastard, for among Azande a man is always of the clan of his genitor [natural father] and not of his pater [mother’s legal husband]. In this and other ways, Evans-Pritchard concluded, the Azande freed themselves from “the logical consequences of belief in biological transmission of witchcraft.”—Derek Gjertsen, Science and Philosophy

11.3 If–Then Arguments With Implicit Statements

If–then arguments, like any other sort of arguments, frequently have implicit premises or conclusions. To use a term from earlier in the book, they are frequently enthymemes. In extreme cases, only the if–then premise is explicit. Suppose, for example, that you’ve complained for the 10th time that the party across the hall is too loud, and I say to you, “Hey, if you can’t beat ‘em, join ‘em.” What I’ve actually given is an affirming the antecedent argument. I’ve explicitly provided the if–then premise; the implicit premise, obviously, is You can’t beat them; and the implicit conclusion is You should join them.

Consider the following, more sophisticated example, from a New York Review of Books review of a book of film criticism by Stanley Cavell:

When Katharine Hepburn in The Philadelphia Story brightly says, “I think men are wonderful,” Cavell hears an “allusion” to The Tempest that amounts “almost to an echo” of Miranda’s saying, “How beauteous mankind is!” If this is an echo, then Irene Dunne’s saying of her marriage, “It was pretty swell while it lasted” is a reminiscence of Gibbon’s Decline and Fall.

This argument is an example of denying the consequent. But only one statement of the argument is explicit. The full clarification proceeds thus:

• If Hepburn’s remark “I think men are wonderful” in The Philadelphia Story is an echo of Miranda’s “How beauteous mankind is” in The Tempest, then Irene Dunne’s saying of her marriage, “It was pretty swell while it lasted” is a reminiscence of Gibbon’s Decline and Fall.
• [Irene Dunne’s saying of her marriage, “It was pretty swell while it lasted” is not a reminiscence of Gibbon’s Decline and Fall. ]
• ∴ [Hepburn’s remark “I think men are wonderful” in The Philadelphia Story is not an echo of Miranda’s “How beauteous mankind is” in The Tempest. ]

This is the reviewer’s sideways—but effective—way of saying that perhaps Cavell takes himself a bit too seriously.

11.3.1 If–Then Bridges

In the preceding examples, only the if–then premise was explicit. But in other cases, only the if–then premise is implicit. Note, for example, this episode recorded by Jean Piaget in his book, The Child’s Conception of the World :

A little girl of nine asked: “Daddy, is there really God?” The father answered that it wasn’t very certain, to which the child retorted: “There must be really, because he has a name!”

This does not look, on the face of it, like an if–then argument. But there must be an implicit premise connecting the two parts of her retort. A good clarification, it seems, is this:

• [If any name exists, then the thing it names exists.]
• God has a name.
• ∴ God exists.

Premise 1 serves as a universal if–then bridge. It is a universal if–then statement (note the term any ) and serves as a bridge of sorts between 2 and C. We might have proposed a more specific sort of bridge, as follows:

1*. [If God has a name, then God exists.]

Either bridge produces a valid argument—the first one by singular affirming the antecedent, the second one by affirming the antecedent. But the second doesn’t produce an argument that will convince us—after all, you can add a premise to any argument that says, “If the premises are true, then the conclusion is true,” and thereby say something that the arguer surely intended, without saying anything illuminating. (There is a specialized term for such an if–then statement, namely, the corresponding conditional of an argument.) When the conditional is expressed in its universal form, on the other hand, we get some idea of the general principle being assumed by the arguer.

EXERCISES Chapter 11, set (j)

Clarify each of these arguments, proposing for each a universal if–then bridge.

Sample exercise. “An idealist is one who, on noticing that a rose smells better than a cabbage, concludes that it will also make better soup.”—H. L. Mencken

• [If anything smells better than another thing, then it also tastes better.]
• A rose smells better than a cabbage.
• ∴ A rose tastes better than a cabbage.
• Twitter, as another high-tech innovation, will inevitably fragment community rather than enhance it.
• “Spirituality turns out to be central to cognitive psychology, and therefore to artificial intelligence, and therefore to computer science, and therefore to the whole history of science and technology.” —David Gelernter, The Muse in the Machine
• Most striking of their claims is that among the 7,000 people executed in this century, at least 23 people, and possibly many more, have been innocent. This alone is reason, the authors conclude, that the death penalty should be abolished.—review of Hugo Bedau and Michael Radelet, In Spite of Innocence
• “There will always be books. You can show them off on the shelf behind you during a Zoom meeting, and you still can’t do that with your iPhone.”—Robert Maxwell
• “Although these textbooks purport to be a universal guide to learning of great worth and importance, there is a single clue that points to another direction. In the six years I taught in city and country schools, no one ever stole a textbook.”—W. Ron Jones, Changing Education

11.4 Bringing It All Together

After learning a wide array of distinct skills, you now have the opportunity to use all of them together. If–then arguments provide us with our first of six groupings of arguments that can be substantial and interesting. And you are now equipped to fully clarify and evaluate them.

There is nothing new to be said, but a few things bear repeating. In your evaluation, separately evaluate the truth of the premises (considering each premise individually), the logic of the argument (naming the form if it has a name, and providing a validity counterexample if it is invalid), the soundness of the argument (which depends entirely on truth and logic), and, if necessary, the conversational relevance of the argument. Always provide a defense of your judgment, and do so as if there were a reasonable objector over your shoulder whom you were trying to persuade.

I’ll provide a sample clarification and evaluation of this brief argument found in Gilbert Harman’s The Nature of Morality :

Total pacifism might be a good principle if everyone were to follow it. But not everyone is, so it isn’t.

CLARIFICATION

• If everyone followed total pacifism, then total pacifism would be a good principle.
• Not everyone follows total pacifism.
• ∴ Total pacifism is not a good principle.

Premise 1 is probably true, since the main objection to total pacifism is that it leaves you with no defense against those who are not pacifists. But if everyone were a pacifist, that would be no problem. (This seems to be the main secondary assumption of the premise.)

Premise 2 is certainly true. We all have firsthand experience with violent people, not to mention the experience of them that we have via the media.

Invalid, fallacy of denying the antecedent. Here is a validity counterexample:

• If anything is a chihuahua, then it is a dog.
• Uga, the Georgia Bulldogs mascot, is not a chihuahua.
• ∴ Uga, the Georgia Bulldogs mascot, is not a dog

Unsound, due to invalidity.

EXERCISES Chapter 11, set (k)

Clarify and evaluate. Where appropriate, provide implicit statements in the clarification (including universal if–then bridges) and original validity counterexamples in the evaluations.

• “With the layout of the San Francisco-Oakland area, a rail line there had a better chance than most,” said rail critic Peter Gordon, a regional planner at USC. “If it doesn’t work there, and I assert it doesn’t, there is no way rail transit will work in a place like Los Angeles.” — Los Angeles Times
• “But if evolution proceeded as a lockstep, then the fossil record should display a pattern of gradual and sequential advance in organization. It does not, and I regard this failure as the most telling argument against an evolutionary ratchet.”—Stephen Gould, Panda’s Thumb (It might help you to know that Gould is a noted professor of paleontology at Harvard. By a “ratchet” and a “lockstep” he means a “gradual, uniform progression.”)
• “If each man had a definite set of rules of conduct by which he regulated his life he would be no better than a machine. But there are no such rules, so men cannot be machines.”—A. M. Turing, Mind (Turing is an expert in artificial intelligence, i.e., what it would take to make a machine think.)
• “ Q : Even so, don’t you think that the use of computers reinforces a child’s problem-solving ability? A: If that were true, then computer professionals would lead better lives than the rest of the population. We know very well that isn’t the case.”—interview with computer expert Joseph Weizenbaum, Le Nouvel Observateur (The argument is in the answer.)
• Pjeter Ivezaj, a U.S. citizen, was sentenced Wednesday to seven years in prison by a panel of judges in Yugoslavia for participating in peaceful anti-Yugoslav activities in this country. Said his brother Frano, 28, “I intend to take this matter to the court of world opinion. If U.S. citizenship has any value, which I believe it does, now is the time for the U.S. government to make a move.”— Detroit Free Press
• Scientists believe that the most likely location of our country’s next severe earthquake is a fault zone that centers on New Madrid, MO. But residents here, where the quake probably will be centered, are unimpressed. “I’m just a non-believer,” said L. H. Rector, publisher of the New Madrid Weekly Record, whose motto is “The only paper in the world that cares about New Madrid.” “It hasn’t been proved that we’re going to have one,” Rector said. “Now, out in California, they can see the fault. It’s been proved. But no one here in New Madrid has seen one fault.”— Los Angeles Times (Provide Rector’s argument with a universal if–then bridge.)
• “The church never wanted disease to be under the control of man. Timothy Dwight, president of Yale College, preached a sermon against vaccination. His idea was that if God had decreed from all eternity that a certain man should die with the smallpox, it was a frightful sin to avoid and annul that decree by the trick of vaccination. Smallpox being regarded as one of the heaviest guns in the arsenal of heaven, to spike it was the height of presumption.”—Robert Ingersoll, Ingersoll, The Immortal Infidel
• The key premise is that a human fetus is a full-fledged, actualized human life. Supposing human embryos are human beings, their innocence is beyond question. So nothing could justify our destroying them except, perhaps, the necessity of saving some other innocent human life.—Roger Wertheimer, “Understanding the Abortion Argument” (Clarify as a complex argument with an if–then bridge in the second inference.)
• “If there be righteousness in the heart, there will be beauty in the character. If there be beauty in the character, there will be harmony in the home. If there be harmony in the home, there will be order in the nation. If there be order in the nation, there will be peace in the world.”—Confucius (Suppose that this is a transitivity of implication argument with an implicit conclusion.)
• “If a man could not have done otherwise than he in fact did, then he is not responsible for his action. But if determinism is true, it is true of every action that the agent could not have done otherwise. Therefore, if determinism is true, no one is ever responsible for what he does.”—Winston Nesbitt and Stewart Candlish, Mind
• Their children, they soon saw, were being presented by EPIC, an organization to help children learn to think about values, with hypothetical situations that called for the students to make choices and decisions. On a spaceship survival trip, an EPIC question asks, “Determine what to take with you. Pretend the ship develops trouble and the load must be lightened. What could you discard?” There was, to a number of parents, something very troubling about these types of questions. The questions seemed to carry with them the presumption that the children were free to reason through to their own answers. If they could do that, it meant that there were no moral absolutes, and nothing was clearly right or wrong, good or bad. This was not the worldview of fundamentalists who believe in the literal word of the Bible. “Once you tell a child that he has to decide upon his own values system, that’s like saying that values are not real, and you can just make them up as you go along,” said Marjorie McNabb, a former Episcopalian who now attends a Baptist church. “Children would be better raised by a street gang than EPIC. At least, they’d learn two values, courage and loyalty. That’s better than no values.”— Los Angeles Times (Look for the actual argument to begin in about the middle, with the words “If they could do that. . . .”)
• In spite of the popularity of the finite-world picture, however, it is open to a devastating objection. In being finite the world must have a limiting boundary, such as Aristotle’s outermost sphere. That is impossible. This objection was put forward by the Greeks, reappeared in the scientific skepticism of the early Renaissance and probably occurs to any schoolchild who thinks about it today. On the basis of the objection, one must conclude that the universe is infinite.—J. J. Callahan, Scientific American (The stylistic variant for the if–then statement is unusual—it is “in being . . . must have . . .”)
• “There is one insuperable obstacle to a belief in ghosts. A ghost never comes naked: he appears either in a winding-sheet or ‘in his habit as he lived.’ To believe in him, then, is to believe that not only have the dead the power to make themselves visible after there is nothing left of them, but that the same power inheres in textile fabrics. Supposing the products of the loom to have this ability, what object would they have in exercising it? And why does not the apparition of a suit of clothes sometimes walk abroad without a ghost in it? These be riddles of significance. They reach away down and get a convulsive grasp on the very taproot of this flourishing faith.”—Ambrose Bierce, Devil’s Dictionary (Treat this as an indirect argument in which the only explicit premise is really the one starting “To believe in him . . .”)
• There was little doubt, then, that if the earth moved through an immovable sea of ether, there would be an ether wind, and if there were an ether wind, the Michelson-Morley apparatus would detect it. In fact, both scientists were confident that they would not only find such a wind, but that they could also determine (by rotating the slab until there was a maximum difference in the time it took light to make the two journeys) the exact direction, at any given moment, of the earth’s path through the ether. Michelson was astounded and disappointed. This time the astonishment was felt by physicists all over the world. Regardless of how Michelson and Morley turned their apparatus, they found no sign of an ether wind! Michelson never dreamed that this ‘failure’ would make the experiment one of the most successful, revolutionary experiments in the history of science. The reason Michelson and Morley were unable to detect an ether wind, Einstein said, is simple: there is no ether wind.—Martin Gardner, Relativity for the Million

11.5 Summary of Chapter Eleven

There are three common valid forms of if–then arguments: affirming the antecedent, denying the consequent, and transitivity of implication. There are two common invalid forms: the fallacy of affirming the consequent and the fallacy of denying the antecedent. When an argument takes one of these forms but has both a universal if–then premise and a conclusion about a single instance to which the universal applies, describe it in the same terms but for the addition of the phrase singular. . . .

When paraphrasing, translate variants such as P only if Q and Q assuming P into the standard constant If P then Q.

When judging the truth of if–then premises, concentrate chiefly on the proposed connection—whether causal or broadly logical—between the if-clause and the then-clause. Typically the if-clause is not alone presumed to be sufficient for the then-clause, but to be sufficient only in combination with secondary assumptions that are themselves not in question. The if-clause is the only one mentioned because it is presumed, in this particular context, to be the only factor in doubt. In defending your judgment that an if-then premise is true, point out the truth of the most doubtful secondary assumptions. In defending your judgment that the if–then premise is false, point out the falsity of a secondary assumption. In this way you either reinforce or sever the connection between if-clause and then-clause.

When the if-clause is clearly true and the then-clause is clearly false, you may have the opportunity to effectively show the falsity of the if–then premise without reference to secondary assumptions in two different ways. First, you may provide a truth counterexample, assuming the if–then premise is universal. And second, you may provide an “educated ignorance” defense, which requires that the evidence for the truth of the if-clause and the falsity of the then-clause is strong—much stronger than the evidence for the if–then premise itself.

If–then arguments are frequently enthymematic. When the if–then premise is the implicit statement, be especially attuned to the likely need for a universal if–then bridge.

11.6 Guidelines for Chapter Eleven

• Translate the stylistic variants for the if–then premise into the standard constant.
• When an argument has both a universal premise and a conclusion about a single thing that is encompassed by the universal premise, consider whether it is the singular version of a sentential logical form.
• Defend your judgment that an if–then statement is true by affirming the truth of the most questionable secondary assumptions. Defend your judgment that it is false by showing that a secondary assumption is false.
• Do not defend your judgment of an if–then statement by simply rewording the statement (or, if false, by rewording the denial of the statement).
• When a universal if–then statement is false, try to defend that judgment by providing a truth counterexample.
• It is reasonable to judge an if–then premise false “because some secondary assumption must be mistaken, though I don’t know which one” only if there is very powerful evidence both that the if-clause is true and that the then-clause is false.
• In indirect arguments, be alert for faulty secondary assumptions behind the if–then premise.
• Consider providing a universal if–then bridge when an explicit link between premise and conclusion has not been provided by the arguer.

11.7 Glossary for Chapter Eleven

Affirming the antecedent —valid deductive form, as follows:

Also known as modus ponens, which is Latin for “the method (or mode, from modus ) of affirming (or propounding, from ponens ).”

Antecedent —the if-clause of an if–then statement.

Consequent —the then-clause of an if–then statement.

Denying the consequent —valid deductive form, as follows:

Also known as modus tollens, which is Latin for “the method of denying.”

Educated ignorance defense —defense of your judgment that an if–then premise is false even though you cannot tell which secondary assumption is at fault (thus, it reflects ignorance); it can be a reasonable defense only if your evidence for the truth of the if-clause and for the falsity of the then-clause is especially strong (thus, the defense is educated).

Fallacy of affirming the consequent —invalid deductive form, as follows:

Fallacy of denying the antecedent —invalid deductive form, as follows:

Fallacy of singular affirming the consequent —invalid affirming the consequent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

Fallacy of singular denying the antecedent —invalid denying the antecedent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

If–then argument —one of a loosely defined group of deductive arguments that have an if–then statement as a premise. Also known as a conditional argument or hypothetical–syllogism.

If–then statement —a statement in the form of If P then Q. Also known as a conditional.

Secondary assumption —when an if–then statement is asserted, this is an assumption made, often implicit because it is not in doubt, about another factor besides the if-clause that contributes to the truth of the then-clause. Also known as auxiliary hypothesis.

Singular affirming the antecedent —valid affirming the antecedent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

Singular denying the consequent —valid denying the consequent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

Singular transitivity of implication —valid transitivity of implication in which the if–then premises are universal and the conclusion is about a single instance that is encompassed by the universal term.

Transitivity of implication —valid deductive form, as follows:

It can have any number of if–then premises. It can also have a negative conclusion, as follows:

• ∴ If not R , then not P.

Truth counterexample —strategy for defending your judgment that a universal if–then premise is false, by identifying a single instance in which the if-clause is obviously true and the then-clause is obviously false.

Universal statement —a premise with a term like all, none, anything, or nothing.

• If construction begins before July 4, then the building will be ready to be occupied before snow falls.
• If the building is not ready to be occupied before snow falls, then construction did not begin before July 4.
• The building will not be ready to be occupied before snow falls.
• ∴ Construction did not begin before July 4.
• ∴ If not R then not P.
• If you have studied formal logic, you have learned that you only need to know the truth-values of the antecedent and the consequent to know the truth-value of the if–then statement; you may have memorized truth tables in support of this. But this applies only to a specialized form of the if–then statement called the material, or truth-functional, conditional. This sort of if–then statement is not used in ordinary language. ↵
• For practical purposes, it may be helpful to think of both sorts of connections as ultimately epistemic—that is, as providing in the antecedent a reason for believing the consequent to be true, on the strictly hypothetical assumption that the antecedent is true. ↵
• In formal logic, this sort of argument must produce a logical contradiction. In common usage, however, it only needs to produce something preposterous. ↵

One of a loosely defined group of deductive arguments that have an if–then statement as a premise. Also known as a conditional argument or hypothetical–syllogism.

A statement in the form of If P then Q. Also known as a conditional.

The if-clause of an if–then statement.

The then-clause of an if–then statement.

Valid deductive form, as follows:

1. If P then Q . 2. P ∴ C . Q

1. If P then Q . 2. Not Q. ∴ C . Not P.

Invalid deductive form, as follows:

1. If P then Q . 2. Q ∴ C . P

1. If P then Q . 2. Not P ∴ C . Not Q

1. If P then Q . 2. If Q then R . ∴ C . If P then R.

∴ C . If not R , then not P.

Valid denying the consequent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

A premise with a term like all, none, anything, or nothing.

Valid affirming the antecedent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

Valid transitivity of implication in which the if–then premises are universal and the conclusion is about a single instance that is encompassed by the universal term.

Invalid affirming the consequent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

Invalid denying the antecedent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

When an if–then statement is asserted, this is an assumption made, often implicit because it is not in doubt, about another factor besides the if-clause that contributes to the truth of the then-clause. Also known as auxiliary hypothesis.

Strategy for defending your judgment that a universal if–then premise is false, by identifying a single instance in which the if-clause is obviously true and the then-clause is obviously false.

Defense of your judgment that an if–then premise is false even though you cannot tell which secondary assumption is at fault (thus, it reflects ignorance); it can be a reasonable defense only if your evidence for the truth of the if-clause and for the falsity of the then-clause is especially strong (thus, the defense is educated).

A Guide to Good Reasoning: Cultivating Intellectual Virtues Copyright © 2020 by David Carl Wilson is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License , except where otherwise noted.

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Worked examples, if...then... statements, mini-lecture., example. consider the statement "$x > 0 \rightarrow x+1>0$". is this statement true or false, example. consider the statement "if $x$ is a positive integer or a solution to $x+3>4$, then $x>0$ and $x> \frac{1}{2}$." is this statement true, example. consider the statement "$0>1 \rightarrow \sin x = 2$". is this statement true or false.

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The Craft of Writing a Strong Hypothesis

Table of Contents

Writing a hypothesis is one of the essential elements of a scientific research paper. It needs to be to the point, clearly communicating what your research is trying to accomplish. A blurry, drawn-out, or complexly-structured hypothesis can confuse your readers. Or worse, the editor and peer reviewers.

A captivating hypothesis is not too intricate. This blog will take you through the process so that, by the end of it, you have a better idea of how to convey your research paper's intent in just one sentence.

What is a Hypothesis?

The first step in your scientific endeavor, a hypothesis, is a strong, concise statement that forms the basis of your research. It is not the same as a thesis statement , which is a brief summary of your research paper .

The sole purpose of a hypothesis is to predict your paper's findings, data, and conclusion. It comes from a place of curiosity and intuition . When you write a hypothesis, you're essentially making an educated guess based on scientific prejudices and evidence, which is further proven or disproven through the scientific method.

The reason for undertaking research is to observe a specific phenomenon. A hypothesis, therefore, lays out what the said phenomenon is. And it does so through two variables, an independent and dependent variable.

The independent variable is the cause behind the observation, while the dependent variable is the effect of the cause. A good example of this is “mixing red and blue forms purple.” In this hypothesis, mixing red and blue is the independent variable as you're combining the two colors at your own will. The formation of purple is the dependent variable as, in this case, it is conditional to the independent variable.

Different Types of Hypotheses‌

Types of hypotheses

Some would stand by the notion that there are only two types of hypotheses: a Null hypothesis and an Alternative hypothesis. While that may have some truth to it, it would be better to fully distinguish the most common forms as these terms come up so often, which might leave you out of context.

Apart from Null and Alternative, there are Complex, Simple, Directional, Non-Directional, Statistical, and Associative and casual hypotheses. They don't necessarily have to be exclusive, as one hypothesis can tick many boxes, but knowing the distinctions between them will make it easier for you to construct your own.

1. Null hypothesis

A null hypothesis proposes no relationship between two variables. Denoted by H 0 , it is a negative statement like “Attending physiotherapy sessions does not affect athletes' on-field performance.” Here, the author claims physiotherapy sessions have no effect on on-field performances. Even if there is, it's only a coincidence.

2. Alternative hypothesis

Considered to be the opposite of a null hypothesis, an alternative hypothesis is donated as H1 or Ha. It explicitly states that the dependent variable affects the independent variable. A good  alternative hypothesis example is “Attending physiotherapy sessions improves athletes' on-field performance.” or “Water evaporates at 100 °C. ” The alternative hypothesis further branches into directional and non-directional.

• Directional hypothesis: A hypothesis that states the result would be either positive or negative is called directional hypothesis. It accompanies H1 with either the ‘<' or ‘>' sign.
• Non-directional hypothesis: A non-directional hypothesis only claims an effect on the dependent variable. It does not clarify whether the result would be positive or negative. The sign for a non-directional hypothesis is ‘≠.'

3. Simple hypothesis

A simple hypothesis is a statement made to reflect the relation between exactly two variables. One independent and one dependent. Consider the example, “Smoking is a prominent cause of lung cancer." The dependent variable, lung cancer, is dependent on the independent variable, smoking.

4. Complex hypothesis

In contrast to a simple hypothesis, a complex hypothesis implies the relationship between multiple independent and dependent variables. For instance, “Individuals who eat more fruits tend to have higher immunity, lesser cholesterol, and high metabolism.” The independent variable is eating more fruits, while the dependent variables are higher immunity, lesser cholesterol, and high metabolism.

5. Associative and casual hypothesis

Associative and casual hypotheses don't exhibit how many variables there will be. They define the relationship between the variables. In an associative hypothesis, changing any one variable, dependent or independent, affects others. In a casual hypothesis, the independent variable directly affects the dependent.

6. Empirical hypothesis

Also referred to as the working hypothesis, an empirical hypothesis claims a theory's validation via experiments and observation. This way, the statement appears justifiable and different from a wild guess.

Say, the hypothesis is “Women who take iron tablets face a lesser risk of anemia than those who take vitamin B12.” This is an example of an empirical hypothesis where the researcher  the statement after assessing a group of women who take iron tablets and charting the findings.

7. Statistical hypothesis

The point of a statistical hypothesis is to test an already existing hypothesis by studying a population sample. Hypothesis like “44% of the Indian population belong in the age group of 22-27.” leverage evidence to prove or disprove a particular statement.

Characteristics of a Good Hypothesis

Writing a hypothesis is essential as it can make or break your research for you. That includes your chances of getting published in a journal. So when you're designing one, keep an eye out for these pointers:

• A research hypothesis has to be simple yet clear to look justifiable enough.
• It has to be testable — your research would be rendered pointless if too far-fetched into reality or limited by technology.
• It has to be precise about the results —what you are trying to do and achieve through it should come out in your hypothesis.
• A research hypothesis should be self-explanatory, leaving no doubt in the reader's mind.
• If you are developing a relational hypothesis, you need to include the variables and establish an appropriate relationship among them.
• A hypothesis must keep and reflect the scope for further investigations and experiments.

Separating a Hypothesis from a Prediction

Outside of academia, hypothesis and prediction are often used interchangeably. In research writing, this is not only confusing but also incorrect. And although a hypothesis and prediction are guesses at their core, there are many differences between them.

A hypothesis is an educated guess or even a testable prediction validated through research. It aims to analyze the gathered evidence and facts to define a relationship between variables and put forth a logical explanation behind the nature of events.

Predictions are assumptions or expected outcomes made without any backing evidence. They are more fictionally inclined regardless of where they originate from.

For this reason, a hypothesis holds much more weight than a prediction. It sticks to the scientific method rather than pure guesswork. "Planets revolve around the Sun." is an example of a hypothesis as it is previous knowledge and observed trends. Additionally, we can test it through the scientific method.

Whereas "COVID-19 will be eradicated by 2030." is a prediction. Even though it results from past trends, we can't prove or disprove it. So, the only way this gets validated is to wait and watch if COVID-19 cases end by 2030.

Finally, How to Write a Hypothesis

Quick tips on writing a hypothesis

1.  Be clear about your research question

A hypothesis should instantly address the research question or the problem statement. To do so, you need to ask a question. Understand the constraints of your undertaken research topic and then formulate a simple and topic-centric problem. Only after that can you develop a hypothesis and further test for evidence.

2. Carry out a recce

Once you have your research's foundation laid out, it would be best to conduct preliminary research. Go through previous theories, academic papers, data, and experiments before you start curating your research hypothesis. It will give you an idea of your hypothesis's viability or originality.

Making use of references from relevant research papers helps draft a good research hypothesis. SciSpace Discover offers a repository of over 270 million research papers to browse through and gain a deeper understanding of related studies on a particular topic. Additionally, you can use SciSpace Copilot , your AI research assistant, for reading any lengthy research paper and getting a more summarized context of it. A hypothesis can be formed after evaluating many such summarized research papers. Copilot also offers explanations for theories and equations, explains paper in simplified version, allows you to highlight any text in the paper or clip math equations and tables and provides a deeper, clear understanding of what is being said. This can improve the hypothesis by helping you identify potential research gaps.

3. Create a 3-dimensional hypothesis

Variables are an essential part of any reasonable hypothesis. So, identify your independent and dependent variable(s) and form a correlation between them. The ideal way to do this is to write the hypothetical assumption in the ‘if-then' form. If you use this form, make sure that you state the predefined relationship between the variables.

In another way, you can choose to present your hypothesis as a comparison between two variables. Here, you must specify the difference you expect to observe in the results.

4. Write the first draft

Now that everything is in place, it's time to write your hypothesis. For starters, create the first draft. In this version, write what you expect to find from your research.

Clearly separate your independent and dependent variables and the link between them. Don't fixate on syntax at this stage. The goal is to ensure your hypothesis addresses the issue.

5. Proof your hypothesis

After preparing the first draft of your hypothesis, you need to inspect it thoroughly. It should tick all the boxes, like being concise, straightforward, relevant, and accurate. Your final hypothesis has to be well-structured as well.

Research projects are an exciting and crucial part of being a scholar. And once you have your research question, you need a great hypothesis to begin conducting research. Thus, knowing how to write a hypothesis is very important.

Now that you have a firmer grasp on what a good hypothesis constitutes, the different kinds there are, and what process to follow, you will find it much easier to write your hypothesis, which ultimately helps your research.

Now it's easier than ever to streamline your research workflow with SciSpace Discover . Its integrated, comprehensive end-to-end platform for research allows scholars to easily discover, write and publish their research and fosters collaboration.

It includes everything you need, including a repository of over 270 million research papers across disciplines, SEO-optimized summaries and public profiles to show your expertise and experience.

If you found these tips on writing a research hypothesis useful, head over to our blog on Statistical Hypothesis Testing to learn about the top researchers, papers, and institutions in this domain.

Frequently Asked Questions (FAQs)

1. what is the definition of hypothesis.

According to the Oxford dictionary, a hypothesis is defined as “An idea or explanation of something that is based on a few known facts, but that has not yet been proved to be true or correct”.

2. What is an example of hypothesis?

The hypothesis is a statement that proposes a relationship between two or more variables. An example: "If we increase the number of new users who join our platform by 25%, then we will see an increase in revenue."

3. What is an example of null hypothesis?

A null hypothesis is a statement that there is no relationship between two variables. The null hypothesis is written as H0. The null hypothesis states that there is no effect. For example, if you're studying whether or not a particular type of exercise increases strength, your null hypothesis will be "there is no difference in strength between people who exercise and people who don't."

4. What are the types of research?

• Fundamental research

• Applied research

• Qualitative research

• Quantitative research

• Mixed research

• Exploratory research

• Longitudinal research

• Cross-sectional research

• Field research

• Laboratory research

• Fixed research

• Flexible research

• Action research

• Policy research

• Classification research

• Comparative research

• Causal research

• Inductive research

• Deductive research

5. How to write a hypothesis?

• Your hypothesis should be able to predict the relationship and outcome.

• Avoid wordiness by keeping it simple and brief.

• Your hypothesis should contain observable and testable outcomes.

• Your hypothesis should be relevant to the research question.

6. What are the 2 types of hypothesis?

• Null hypotheses are used to test the claim that "there is no difference between two groups of data".

• Alternative hypotheses test the claim that "there is a difference between two data groups".

7. Difference between research question and research hypothesis?

A research question is a broad, open-ended question you will try to answer through your research. A hypothesis is a statement based on prior research or theory that you expect to be true due to your study. Example - Research question: What are the factors that influence the adoption of the new technology? Research hypothesis: There is a positive relationship between age, education and income level with the adoption of the new technology.

8. What is plural for hypothesis?

The plural of hypothesis is hypotheses. Here's an example of how it would be used in a statement, "Numerous well-considered hypotheses are presented in this part, and they are supported by tables and figures that are well-illustrated."

9. What is the red queen hypothesis?

The red queen hypothesis in evolutionary biology states that species must constantly evolve to avoid extinction because if they don't, they will be outcompeted by other species that are evolving. Leigh Van Valen first proposed it in 1973; since then, it has been tested and substantiated many times.

10. Who is known as the father of null hypothesis?

The father of the null hypothesis is Sir Ronald Fisher. He published a paper in 1925 that introduced the concept of null hypothesis testing, and he was also the first to use the term itself.

11. When to reject null hypothesis?

You need to find a significant difference between your two populations to reject the null hypothesis. You can determine that by running statistical tests such as an independent sample t-test or a dependent sample t-test. You should reject the null hypothesis if the p-value is less than 0.05.

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Research: Articulating Questions, Generating Hypotheses, and Choosing Study Designs

Introduction.

Articulating a clear and concise research question is fundamental to conducting a robust and useful research study. Although “getting stuck into” the data collection is the exciting part of research, this preparation stage is crucial. Clear and concise research questions are needed for a number of reasons. Initially, they are needed to enable you to search the literature effectively. They will allow you to write clear aims and generate hypotheses. They will also ensure that you can select the most appropriate research design for your study.

This paper begins by describing the process of articulating clear and concise research questions, assuming that you have minimal experience. It then describes how to choose research questions that should be answered and how to generate study aims and hypotheses from your questions. Finally, it describes briefly how your question will help you to decide on the research design and methods best suited to answering it.

TURNING CURIOSITY INTO QUESTIONS

A research question has been described as “the uncertainty that the investigator wants to resolve by performing her study” 1 or “a logical statement that progresses from what is known or believed to be true to that which is unknown and requires validation”. 2 Developing your question usually starts with having some general ideas about the areas within which you want to do your research. These might flow from your clinical work, for example. You might be interested in finding ways to improve the pharmaceutical care of patients on your wards. Alternatively, you might be interested in identifying the best antihypertensive agent for a particular subgroup of patients. Lipowski 2 described in detail how work as a practising pharmacist can be used to great advantage to generate interesting research questions and hence useful research studies. Ideas could come from questioning received wisdom within your clinical area or the rationale behind quick fixes or workarounds, or from wanting to improve the quality, safety, or efficiency of working practice.

Alternatively, your ideas could come from searching the literature to answer a query from a colleague. Perhaps you could not find a published answer to the question you were asked, and so you want to conduct some research yourself. However, just searching the literature to generate questions is not to be recommended for novices—the volume of material can feel totally overwhelming.

Use a research notebook, where you regularly write ideas for research questions as you think of them during your clinical practice or after reading other research papers. It has been said that the best way to have a great idea is to have lots of ideas and then choose the best. The same would apply to research questions!

When you first identify your area of research interest, it is likely to be either too narrow or too broad. Narrow questions (such as “How is drug X prescribed for patients with condition Y in my hospital?”) are usually of limited interest to anyone other than the researcher. Broad questions (such as “How can pharmacists provide better patient care?”) must be broken down into smaller, more manageable questions. If you are interested in how pharmacists can provide better care, for example, you might start to narrow that topic down to how pharmacists can provide better care for one condition (such as affective disorders) for a particular subgroup of patients (such as teenagers). Then you could focus it even further by considering a specific disorder (depression) and a particular type of service that pharmacists could provide (improving patient adherence). At this stage, you could write your research question as, for example, “What role, if any, can pharmacists play in improving adherence to fluoxetine used for depression in teenagers?”

TYPES OF RESEARCH QUESTIONS

Being able to consider the type of research question that you have generated is particularly useful when deciding what research methods to use. There are 3 broad categories of question: descriptive, relational, and causal.

Descriptive

One of the most basic types of question is designed to ask systematically whether a phenomenon exists. For example, we could ask “Do pharmacists ‘care’ when they deliver pharmaceutical care?” This research would initially define the key terms (i.e., describing what “pharmaceutical care” and “care” are), and then the study would set out to look for the existence of care at the same time as pharmaceutical care was being delivered.

When you know that a phenomenon exists, you can then ask description and/or classification questions. The answers to these types of questions involve describing the characteristics of the phenomenon or creating typologies of variable subtypes. In the study above, for example, you could investigate the characteristics of the “care” that pharmacists provide. Classifications usually use mutually exclusive categories, so that various subtypes of the variable will have an unambiguous category to which they can be assigned. For example, a question could be asked as to “what is a pharmacist intervention” and a definition and classification system developed for use in further research.

When seeking further detail about your phenomenon, you might ask questions about its composition. These questions necessitate deconstructing a phenomenon (such as a behaviour) into its component parts. Within hospital pharmacy practice, you might be interested in asking questions about the composition of a new behavioural intervention to improve patient adherence, for example, “What is the detailed process that the pharmacist implicitly follows during delivery of this new intervention?”

After you have described your phenomena, you may then be interested in asking questions about the relationships between several phenomena. If you work on a renal ward, for example, you may be interested in looking at the relationship between hemoglobin levels and renal function, so your question would look something like this: “Are hemoglobin levels related to level of renal function?” Alternatively, you may have a categorical variable such as grade of doctor and be interested in the differences between them with regard to prescribing errors, so your research question would be “Do junior doctors make more prescribing errors than senior doctors?” Relational questions could also be asked within qualitative research, where a detailed understanding of the nature of the relationship between, for example, the gender and career aspirations of clinical pharmacists could be sought.

Once you have described your phenomena and have identified a relationship between them, you could ask about the causes of that relationship. You may be interested to know whether an intervention or some other activity has caused a change in your variable, and your research question would be about causality. For example, you may be interested in asking, “Does captopril treatment reduce blood pressure?” Generally, however, if you ask a causality question about a medication or any other health care intervention, it ought to be rephrased as a causality–comparative question. Without comparing what happens in the presence of an intervention with what happens in the absence of the intervention, it is impossible to attribute causality to the intervention. Although a causality question would usually be answered using a comparative research design, asking a causality–comparative question makes the research design much more explicit. So the above question could be rephrased as, “Is captopril better than placebo at reducing blood pressure?”

The acronym PICO has been used to describe the components of well-crafted causality–comparative research questions. 3 The letters in this acronym stand for Population, Intervention, Comparison, and Outcome. They remind the researcher that the research question should specify the type of participant to be recruited, the type of exposure involved, the type of control group with which participants are to be compared, and the type of outcome to be measured. Using the PICO approach, the above research question could be written as “Does captopril [ intervention ] decrease rates of cardiovascular events [ outcome ] in patients with essential hypertension [ population ] compared with patients receiving no treatment [ comparison ]?”

DECIDING WHETHER TO ANSWER A RESEARCH QUESTION

Just because a question can be asked does not mean that it needs to be answered. Not all research questions deserve to have time spent on them. One useful set of criteria is to ask whether your research question is feasible, interesting, novel, ethical, and relevant. 1 The need for research to be ethical will be covered in a later paper in the series, so is not discussed here. The literature review is crucial to finding out whether the research question fulfils the remaining 4 criteria.

Conducting a comprehensive literature review will allow you to find out what is already known about the subject and any gaps that need further exploration. You may find that your research question has already been answered. However, that does not mean that you should abandon the question altogether. It may be necessary to confirm those findings using an alternative method or to translate them to another setting. If your research question has no novelty, however, and is not interesting or relevant to your peers or potential funders, you are probably better finding an alternative.

The literature will also help you learn about the research designs and methods that have been used previously and hence to decide whether your potential study is feasible. As a novice researcher, it is particularly important to ask if your planned study is feasible for you to conduct. Do you or your collaborators have the necessary technical expertise? Do you have the other resources that will be needed? If you are just starting out with research, it is likely that you will have a limited budget, in terms of both time and money. Therefore, even if the question is novel, interesting, and relevant, it may not be one that is feasible for you to answer.

GENERATING AIMS AND HYPOTHESES

All research studies should have at least one research question, and they should also have at least one aim. As a rule of thumb, a small research study should not have more than 2 aims as an absolute maximum. The aim of the study is a broad statement of intention and aspiration; it is the overall goal that you intend to achieve. The wording of this broad statement of intent is derived from the research question. If it is a descriptive research question, the aim will be, for example, “to investigate” or “to explore”. If it is a relational research question, then the aim should state the phenomena being correlated, such as “to ascertain the impact of gender on career aspirations”. If it is a causal research question, then the aim should include the direction of the relationship being tested, such as “to investigate whether captopril decreases rates of cardiovascular events in patients with essential hypertension, relative to patients receiving no treatment”.

The hypothesis is a tentative prediction of the nature and direction of relationships between sets of data, phrased as a declarative statement. Therefore, hypotheses are really only required for studies that address relational or causal research questions. For the study above, the hypothesis being tested would be “Captopril decreases rates of cardiovascular events in patients with essential hypertension, relative to patients receiving no treatment”. Studies that seek to answer descriptive research questions do not test hypotheses, but they can be used for hypothesis generation. Those hypotheses would then be tested in subsequent studies.

CHOOSING THE STUDY DESIGN

The research question is paramount in deciding what research design and methods you are going to use. There are no inherently bad research designs. The rightness or wrongness of the decision about the research design is based simply on whether it is suitable for answering the research question that you have posed.

It is possible to select completely the wrong research design to answer a specific question. For example, you may want to answer one of the research questions outlined above: “Do pharmacists ‘care’ when they deliver pharmaceutical care?” Although a randomized controlled study is considered by many as a “gold standard” research design, such a study would just not be capable of generating data to answer the question posed. Similarly, if your question was, “Is captopril better than placebo at reducing blood pressure?”, conducting a series of in-depth qualitative interviews would be equally incapable of generating the necessary data. However, if these designs are swapped around, we have 2 combinations (pharmaceutical care investigated using interviews; captopril investigated using a randomized controlled study) that are more likely to produce robust answers to the questions.

The language of the research question can be helpful in deciding what research design and methods to use. Subsequent papers in this series will cover these topics in detail. For example, if the question starts with “how many” or “how often”, it is probably a descriptive question to assess the prevalence or incidence of a phenomenon. An epidemiological research design would be appropriate, perhaps using a postal survey or structured interviews to collect the data. If the question starts with “why” or “how”, then it is a descriptive question to gain an in-depth understanding of a phenomenon. A qualitative research design, using in-depth interviews or focus groups, would collect the data needed. Finally, the term “what is the impact of” suggests a causal question, which would require comparison of data collected with and without the intervention (i.e., a before–after or randomized controlled study).

CONCLUSIONS

This paper has briefly outlined how to articulate research questions, formulate your aims, and choose your research methods. It is crucial to realize that articulating a good research question involves considerable iteration through the stages described above. It is very common that the first research question generated bears little resemblance to the final question used in the study. The language is changed several times, for example, because the first question turned out not to be feasible and the second question was a descriptive question when what was really wanted was a causality question. The books listed in the “Further Reading” section provide greater detail on the material described here, as well as a wealth of other information to ensure that your first foray into conducting research is successful.

This article is the second in the CJHP Research Primer Series, an initiative of the CJHP Editorial Board and the CSHP Research Committee. The planned 2-year series is intended to appeal to relatively inexperienced researchers, with the goal of building research capacity among practising pharmacists. The articles, presenting simple but rigorous guidance to encourage and support novice researchers, are being solicited from authors with appropriate expertise.

Previous article in this series:

Bond CM. The research jigsaw: how to get started. Can J Hosp Pharm . 2014;67(1):28–30.

Competing interests: Mary Tully has received personal fees from the UK Renal Pharmacy Group to present a conference workshop on writing research questions and nonfinancial support (in the form of travel and accommodation) from the Dubai International Pharmaceuticals and Technologies Conference and Exhibition (DUPHAT) to present a workshop on conducting pharmacy practice research.

Further Reading

• Cresswell J. Research design: qualitative, quantitative and mixed methods approaches. London (UK): Sage; 2009. [ Google Scholar ]
• Haynes RB, Sackett DL, Guyatt GH, Tugwell P. Clinical epidemiology: how to do clinical practice research. 3rd ed. Philadelphia (PA): Lippincott, Williams & Wilkins; 2006. [ Google Scholar ]
• Kumar R. Research methodology: a step-by-step guide for beginners. 3rd ed. London (UK): Sage; 2010. [ Google Scholar ]
• Smith FJ. Conducting your pharmacy practice research project. London (UK): Pharmaceutical Press; 2005. [ Google Scholar ]

2.2 Compound Statements

Learning objectives.

After completing this section, you should be able to:

• Translate compound statements into symbolic form.
• Translate compound statements in symbolic form with parentheses into words.
• Apply the dominance of connectives.

Suppose your friend is trying to get a license to drive. In most places, they will need to pass some form of written test proving their knowledge of the laws and rules for driving safely. After passing the written test, your friend must also pass a road test to demonstrate that they can perform the physical task of driving safely within the rules of the law.

Consider the statement: "My friend passed the written test, but they did not pass the road test." This is an example of a compound statement , a statement formed by using a connective to join two independent clauses or logical statements. The statement, “My friend passed the written test,” is an independent clause because it is a complete thought or idea that can stand on its own. The second independent clause in this compound statement is, “My friend did not pass the road test.” The word "but" is the connective used to join these two statements together, forming a compound statement. So, did your friend acquire their driving license?.

This section introduces common logical connectives and their symbols, and allows you to practice translating compound statements between words and symbols. It also explores the order of operations, or dominance of connectives, when a single compound statement uses multiple connectives.

Common Logical Connectives

Understanding the following logical connectives, along with their properties, symbols, and names, will be key to applying the topics presented in this chapter. The chapter will discuss each connective introduced here in more detail.

The joining of two logical statements with the word "and" or "but" forms a compound statement called a conjunction . In logic, for a conjunction to be true, all the independent logical statements that make it up must be true. The symbol for a conjunction is ∧ ∧ . Consider the compound statement, “Derek Jeter played professional baseball for the New York Yankees, and he was a shortstop.” If p p represents the statement, “Derrick Jeter played professional baseball for the New York Yankees,” and if q q represents the statement, “Derrick Jeter was a short stop,” then the conjunction will be written symbolically as p ∧ q . p ∧ q .

The joining of two logical statements with the word “or” forms a compound statement called a disjunction . Unless otherwise specified, a disjunction is an inclusive or statement, which means the compound statement formed by joining two independent clauses with the word or will be true if a least one of the clauses is true. Consider the compound statement, "The office manager ordered cake for for an employee’s birthday or they ordered ice cream.” This is a disjunction because it combines the independent clause, “The office manager ordered cake for an employee’s birthday,” with the independent clause, “The office manager ordered ice cream,” using the connective, or . This disjunction is true if the office manager ordered only cake, only ice cream, or they ordered both cake and ice cream. Inclusive or means you can have one, or the other, or both!

Joining two logical statements with the word implies , or using the phrase “if first statement , then second statement,” is called a conditional or implication . The clause associated with the "if" statement is also called the hypothesis or antecedent , while the clause following the "then" statement or the word implies is called the conclusion or consequent . The conditional statement is like a one-way contract or promise. The only time the conditional statement is false, is if the hypothesis is true and the conclusion is false. Consider the following conditional statement, “If Pedro does his homework, then he can play video games.” The hypothesis/antecedent is the statement following the word if , which is “Pedro does/did his homework.” The conclusion/consequent is the statement following the word then , which is “Pedro can play his video games.”

Joining two logical statements with the connective phrase “if and only if” is called a biconditional . The connective phrase "if and only if" is represented by the symbol, ↔ . ↔ . In the biconditional statement, p ↔ q , p ↔ q , p p is called the hypothesis or antecedent and q q is called the conclusion or consequent. For a biconditional statement to be true, the truth values of p p and q q must match. They must both be true, or both be false.

The table below lists the most common connectives used in logic, along with their symbolic forms, and the common names used to describe each connective.

These connectives, along with their names, symbols, and properties, will be used throughout this chapter and should be memorized.

Example 2.8

Associate connectives with symbols and names.

For each of the following connectives, write its name and associated symbol.

• A compound statement formed with the connective word or is called a disjunction, and it is represented by the ∨ ∨ symbol.
• A compound statement formed with the connective word implies or phrase “ if …, then” is called a conditional statement or implication and is represented by the → → symbol.
• A compound statement formed with the connective words but or and is called a conjunction, and it is represented by the ∧ ∧ symbol.

Your Turn 2.8

Translating compound statements to symbolic form.

To translate a compound statement into symbolic form, we take the following steps.

• Identify and label all independent affirmative logical statements with a lower case letter, such as p p , q q , or r r .
• Identify and label any negative logical statements with a lowercase letter preceded by the negation symbol, such as ~ p ~ p , ~ q ~ q , or ~ r ~ r .
• Replace the connective words with the symbols that represent them, such as ∧ , ∨ , → , or ↔ . ∧ , ∨ , → , or ↔ .

Consider the previous example of your friend trying to get their driver’s license. Your friend passed the written test, but they did not pass the road test. Let p p represent the statement, “My friend passed the written test.” And, let ~ q ~ q represent the statement, “My friend did not pass the road test.” Because the connective but is logically equivalent to the word and , the symbol for but is the same as the symbol for and ; replace but with the symbol ∧ . ∧ . The compound statement is symbolically written as: p ∧ ~ q p ∧ ~ q . My friend passed the written test, but they did not pass the road test.

When translating English statements into symbolic form, do not assume that every connective “and” means you are dealing with a compound statement. The statement, “The stripes on the U.S. flag are red and white,” is a simple statement. The word “white” doesn’t express a complete thought, so it is not an independent clause and does not get its own symbol. This statement should be represented by a single letter, such as s s : The stripes on the U.S. flag are red and white.

Example 2.9

Translating compound statements into symbolic form.

Let p p represent the statement, “It is a warm sunny day,” and let q q represent the statement, “the family will go to the beach.” Write the symbolic form of each of the following compound statements.

• If it is a warm sunny day, then the family will go to the beach.
• The family will go to the beach, and it is a warm sunny day.
• The family will not go to the beach if and only if it is not a warm sunny day.
• The family not go to the beach, or it is a warm sunny day.
• Replace “it is a warm sunny day” with p p . Replace “the family will go to the beach.” with q q . Next. Next, because the connective is if …, then place the conditional symbol, → → , between p p and q q . The compound statement is written symbolically as: p → q . p → q .
• Replace “The family will go to the beach” with q q . Replace “it is a warm sunny day.” with p p . Next, because the connective is and , place the ∧ ∧ symbol between q q and p p . The compound statement is written symbolically as: q ∧ p . q ∧ p .
• Replace “The family will not go to the beach. with ~ q ~ q . Replace “it is not a warm sunny day” with ~ p ~ p . Next, because the connective is or , if and only if , place the biconditional symbol, ↔ ↔ between ~ q ~ q and ~ p ~ p . The compound statement is written symbolically as: ~ q ↔ ~ p . ~ q ↔ ~ p .
• Replace “The family will not go to the beach” with ~ q ~ q . Replace “it is a warm sunny day” with p p . Next, because the connective is or , place the ∨ ∨ symbol between ~ q ~ q and p p . The compound statement is written symbolically as: ~ q ∨ p . ~ q ∨ p .

Your Turn 2.9

Translating compound statements in symbolic form with parentheses into words.

When parentheses are written in a logical argument, they group a compound statement together just like when calculating numerical or algebraic expressions. Any statement in parentheses should be treated as a single component of the expression. If multiple parentheses are present, work with the inner most parentheses first.

Consider your friend’s struggles to get their license to drive. Let p p represent the statement, “My friend passed the written test,” let q q represent the statement, “My friend passed the road test,” and let r r represent the statement, “My friend received a driver’s license.” The statement ( p ∧ q ) → r ( p ∧ q ) → r can be translated into words as follows: the statement p ∧ q p ∧ q is grouped together to form the hypothesis of the conditional statement and r r is the conclusion. The conditional statement has the form “ if p ∧ q , p ∧ q , then r . r . ” Therefore, the written form of this statement is: “If my friend passed the written test and they passed the road test, then my friend received a driver’s license.”

Sometimes a compound statement within parentheses may need to be negated as a group. To accomplish this, add the phrase, “it is not the case that” before the translation of the phrase in parentheses. For example, using p p , q q , and r r of your friend obtaining a license, let’s translate the statement ~ ( p ∧ q ) → ~ r ~ ( p ∧ q ) → ~ r into words.

In this case, the hypothesis of the conditional statement is ~ ( p ∧ q ) ~ ( p ∧ q ) and the conclusion is ~ r . ~ r . To negate the hypothesis, add the phrase “it is not the case” before translating what is in parentheses. The translation of the hypothesis is the sentence, “It is not the case that my friend passed the written test and they passed the road test,” and the translation of the conclusion is, “My friend did not receive a driver’s license.” So, a translation of the complete conditional statement, ~ ( p ∧ q ) → ~ r ~ ( p ∧ q ) → ~ r is: “If it is not the case that my friend passed the written test and the road test, then my friend did not receive a driver’s license.”

It is acceptable to interchange proper names with pronouns and remove repeated phrases to make the written statement more readable, as long the meaning of the logical statement is not changed.

Logic Part 1B: Compound Statements, Connectives and Symbols

Example 2.10

Let p p represent the statement, “My child finished their homework,” let q q represent the statement, “My child cleaned her room,” let r r represent the statement, “My child played video games,” and let s s represent the statement, “My child streamed a movie.” Translate each of the following symbolic statements into words.

• ~ ( p ∧ q ) ~ ( p ∧ q )
• ( p ∧ q ) → ( r ∨ s ) ( p ∧ q ) → ( r ∨ s )
• ~ ( r ∨ s ) ↔ ~ ( p ∧ q ) ~ ( r ∨ s ) ↔ ~ ( p ∧ q )
• Replace ~ with “It is not the case,” and ∧ ∧ with “and.” One possible translation is: “It is not the case that my child finished their homework and cleaned their room.”
• The hypothesis of the conditional statement is, “My child finished their homework and cleaned their room.” The conclusion of the conditional statement is, “My child played video games or streamed a movie.” One possible translation of the entire statement is: “If my child finished their homework and cleaned their room, then they played video games or streamed a movie.”
• The hypothesis of the biconditional statement is ~ ( r ∨ s ) ~ ( r ∨ s ) and is written in words as: “It is not the case that my child played video games or streamed a movie.” The conclusion of the biconditional statement is ~ ( p ∧ q ) ~ ( p ∧ q ) , which translates to: “It is not the case that my child finished their homework and cleaned their room.” Because the biconditional, ↔ ↔ translates to if and only if , one possible translation of the statement is: “It is not the case that my child played video games or streamed a movie if and only if it is not the case that my child finished their homework and cleaned their room.”

Your Turn 2.10

The dominance of connectives.

The order of operations for working with algebraic and arithmetic expressions provides a set of rules that allow consistent results. For example, if you were presented with the problem 1 + 3 × 2 1 + 3 × 2 , and you were not familiar with the order of operation, you might assume that you calculate the problem from left to right. If you did so, you would add 1 and 3 to get 4, and then multiply this answer by 2 to get 8, resulting in an incorrect answer. Try inputting this expression into a scientific calculator. If you do, the calculator should return a value of 7, not 8.

Scientific Calculator

The order of operations for algebraic and arithmetic operations states that all multiplication must be applied prior to any addition. Parentheses are used to indicate which operation—addition or multiplication—should be done first. Adding parentheses can change and/or clarify the order. The parentheses in the expression 1 + ( 3 × 2 ) 1 + ( 3 × 2 ) indicate that 3 should be multiplied by 2 to get 6, and then 1 should be added to 6 to get 7: 1 + ( 3 × 2 ) = 7. 1 + ( 3 × 2 ) = 7.

As with algebraic expressions, there is a set of rules that must be applied to compound logical statements in order to evaluate them with consistent results. This set of rules is called the dominance of connectives . When evaluating compound logical statements, connectives are evaluated from least dominant to most dominant as follows:

• Parentheses are the least dominant connective. So, any expression inside parentheses must be evaluated first. Add as many parentheses as needed to any statement to specify the order to evaluate each connective.
• Next, we evaluate negations.
• Then, we evaluate conjunctions and disjunctions from left to right, because they have equal dominance.
• After evaluating all conjunctions and disjunctions, we evaluate conditionals.
• Lastly, we evaluate the most dominant connective, the biconditional. If a statement includes multiple connectives of equal dominance, then we will evaluate them from left to right.

See Figure 2.7 for a visual breakdown of the dominance of connectives.

Let’s revisit your friend’s struggles to get their driver’s license. Let p p represent the statement, “My friend passed the written test,” let q q represent the statement, “My friend passed the road test,” and let r r represent the statement, “My friend received a driver’s license.” Let's use the dominance of connectives to determine how the compound statement p ∧ ~ q → r p ∧ ~ q → r should be evaluated.

Step 1: There are no parentheses, which is least dominant of all connectives, so we can skip over that.

Step 2: Because negation is the next least dominant, we should evaluate ~ q ~ q first. We could add parentheses to the statement to make it clearer which connecting needs to be evaluated first: p ∧ ~ q → r p ∧ ~ q → r is equivalent to p ∧ ( ~ q ) → r . p ∧ ( ~ q ) → r .

Step 3: Next, we evaluate the conjunction, ∧ ∧ . p ∧ ( ~ q ) → r p ∧ ( ~ q ) → r is equivalent to ( p ∧ ( ~ q ) ) → r . ( p ∧ ( ~ q ) ) → r .

Step 4: Finally, we evaluate the conditional, → , → , as this is the most dominant connective in this compound statement.

When using spreadsheet applications, like Microsoft Excel or Google Sheets, have you noticed that core functions, such as sum, average, or rate, have the same name and syntax for use? This is not a coincidence. Various standards organizations have developed requirements that software developers must implement to meet the conditions set by governments and large customers worldwide. The OpenDocument Format OASIS Standard enables transferring data between different office productivity applications and was approved by the International Standards Organization (ISO) and IEC on May 6, 2006.

Prior to adopting these standards, a government entity, business, or individual could lose access to their own data simply because it was saved in a format no longer supported by a proprietary software product—even data they were required by law to preserve, or data they simply wished to maintain access to.

Just as rules for applying the dominance of connectives help maintain understanding and consistency when writing and interpreting compound logical statements and arguments, the standards adopted for database software maintain global integrity and accessibility across platforms and user bases.

Example 2.11

Applying the dominance of connectives.

For each of the following compound logical statements, add parentheses to indicate the order to evaluate the statement. Recall that parentheses are evaluated innermost first.

• p ∧ ~ q ∨ r p ∧ ~ q ∨ r
• q → ~ p ∧ r q → ~ p ∧ r
• ~ ( p ∨ q ) ↔ ~ p ∧ ~ q ~ ( p ∨ q ) ↔ ~ p ∧ ~ q
• Because negation is the least dominant connective, we evaluate it first: p ∧ ( ~ q ) ∨ r . p ∧ ( ~ q ) ∨ r . Because conjunction and disjunction have the same dominance, we evaluate them left to right. So, we evaluate the conjunction next, as indicated by the additional set of parentheses: ( p ∧ ( ~ q ) ) ∨ r . ( p ∧ ( ~ q ) ) ∨ r . The only remaining connective is the disjunction, so it is evaluated last, as indicated by the third set of parentheses. The complete solution is: ( ( p ∧ ( ~ q ) ) ∨ r ) . ( ( p ∧ ( ~ q ) ) ∨ r ) .
• Negation has the lowest dominance, so it is evaluated first: q → ( ~ p ) ∧ r . q → ( ~ p ) ∧ r . The remaining connectives are the conditional and the conjunction. Because conjunction has a lower precedence than the conditional, it is evaluated next, as indicated by the second set of parentheses: q → ( ( ~ p ) ∧ r ) . q → ( ( ~ p ) ∧ r ) . The last step is to evaluate the conditional, as indicated by the third set of parentheses: ( q → ( ( ~ p ) ∧ r ) ) . ( q → ( ( ~ p ) ∧ r ) ) .
• First, we evaluate the negations on the right side of the biconditional prior to the conjunction.
• Then, we evaluate the disjunction on the left side of the biconditional, followed by the negation of the disjunction on the left side.
• Lastly, after completely evaluating each side of the biconditional, we evaluate the biconditional. It does not matter which side you begin with.

Your Turn 2.11

Work it out, logic terms and properties – matching game.

Materials: For every group of four students, include 30 cards (game, trading, or playing cards), 30 individual clear plastic gaming card sleeves, and 30 card-size pieces of paper. Prepare a list of 60 questions and answers ahead of time related to definitions and problems in Statements and Quantifiers and Compound Statements . Provide each group of four students with 20 questions and their associated answers. Instruct each group to select 15 of the 20 questions, and then, for each problem selected, create one card with the question and one card with the answer. Instruct the groups to then place each problem and answer in a separate card sleeve. Once they create 15 problem cards and 15 answer cards, have students shuffle the set of cards.

To play the game, groups should exchange their card decks to ensure no team begins playing with the deck that they created. Split each four-person group into teams of two students. After shuffling the cards, one team lays cards face down on their desk in a five-by-six grid. The other team will go first.

Each player will have a turn to try matching the question with the correct answer by flipping two cards to the face up position. If a team makes a match, they get to flip another two cards; if they do not get a match, they flip the cards face down and it is the other team’s turn to flip over two cards. The game continues in this manner until teams match all question cards with their corresponding answer cards. The team with the most set of matching cards wins.

In the first module of this chapter, we evaluated the truth value of negations. In the following modules, we will discuss how to evaluate conjunctions, disjunctions, conditionals, and biconditionals, and then evaluate compound logical statements using truth tables and our knowledge of the dominance of connectives.

Check Your Understanding

Section 2.2 exercises.

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2.12: Converse, Inverse, and Contrapositive Statements

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Conditional statements drawn from an if-then statement.

Converse, Inverse, and Contrapositive

Consider the statement: If the weather is nice, then I’ll wash the car. We can rewrite this statement using letters to represent the hypothesis and conclusion.

\(p=the\: weather \:is \:nice \qquad q=I'll \:wash \:the \:car\)

Now the statement is: if \(p\), then \(q\), which can also be written as \(p\rightarrow q\).

We can also make the negations, or “nots,” of \(p\) and \(q\). The symbolic version of "not p" is \(\sim p.

\(\sim p=the \:weather \:is \:not \:nice \qquad \sim q=I \:won't \:wash \:the \:car\)

Using these “nots” and switching the order of \(p\) and \(q\), we can create three new statements.

\(Converse \qquad q\rightarrow p \qquad \underbrace{If\: I\: wash\: the\: car}_\text{q}, \underbrace{then\: the \:weather \:is \: nice}_\text{p}\).

\(Inverse \qquad \sim p\rightarrow \sim q \qquad \underbrace{If\: the\: weather\: is \:not \:nice}_\text{p}, \underbrace{\:then \:I \:won't \:wash \:the \:car}_\text{q}\).

\(Contrapositive \qquad \sim q\rightarrow \sim p \qquad \underbrace{If\: I \:don't \:wash \:the \:car}_\text{q}, \underbrace{then the weather is not nice}_\text{p}\).

If the “if-then” statement is true, then the contrapositive is also true. The contrapositive is logically equivalent to the original statement. The converse and inverse may or may not be true. When the original statement and converse are both true then the statement is a biconditional statement . In other words, if \(p\rightarrow q\) is true and \(q\rightarrow p\) is true, then \(p \leftrightarrow q\) (said “\(p\) if and only if \(q\)”).

What if you were given a conditional statement like "If I walk to school, then I will be late"? How could you rearrange and/or negate this statement to form new conditional statements?

Example \(\PageIndex{1}\)

If \(n>2\), then \(n^{2}>4\).

Find the converse, inverse, and contrapositive. Determine if each resulting statement is true or false. If it is false, find a counterexample.

The original statement is true.

\(\underline{Converse}\): If \(n^{2}>4\), then \(n>2\).

False. If \(n^{2}=9\), \(n=−3\: or \: 3\). \((−3)^{2}=9\)

\(\underline{Inverse}\): If \(n\leq 2\), then \(n^{2}\leq 4\).

False. If \(n=−3\), then \(n^{2}=9\).

\(\underline{Contrapositive}\): If \(n^{2}\leq 4\), then \(n\leq 2\).

True. The only \(n^{2}\leq 4\) is 1 or 4. \(\sqrt{1}=\pm 1\) and\(\sqrt{4}=\pm 2\), which are both less than or equal to 2.

Example \(\PageIndex{2}\)

If I am at Disneyland, then I am in California.

\(\underline{Converse}\): If I am in California, then I am at Disneyland.

False. I could be in San Francisco.

\(\underline{Inverse}\): If I am not at Disneyland, then I am not in California.

False. Again, I could be in San Francisco.

\(\underline{Contrapositive}\): If I am not in California, then I am not at Disneyland.

True. If I am not in the state, I couldn't be at Disneyland.

Notice for the converse and inverse we can use the same counterexample.

Example \(\PageIndex{3}\)

Rewrite as a biconditional statement: Any two points are collinear.

This statement can be rewritten as:

Two points are on the same line if and only if they are collinear. Replace the “if-then” with “if and only if” in the middle of the statement.

Example \(\PageIndex{4}\)

Any two points are collinear.

First, change the statement into an “if-then” statement:

If two points are on the same line, then they are collinear.

\(\underline{Converse}\): If two points are collinear, then they are on the same line. True.

\(\underline{Inverse}\): If two points are not on the same line, then they are not collinear. True.

\(\underline{Contrapositive}\): If two points are not collinear, then they do not lie on the same line. True.

Example \(\PageIndex{5}\)

The following is a true statement:

\(m\angle ABC>90^{\circ}\) if and only if \(\angle ABC\) is an obtuse angle.

Determine the two true statements within this biconditional.

Statement 1: If \(m\angle ABC>90^{\circ}\), then \(\angle ABC\) is an obtuse angle.

Statement 2: If \(\angle ABC\) is an obtuse angle, then \(m\angle ABC>90^{\circ}\).

For questions 1-4, use the statement:

If \(AB=5\) and \(BC=5\), then \(B\) is the midpoint of \(\overline{AC}\).

• Is this a true statement? If not, what is a counterexample?
• Find the converse of this statement. Is it true?
• Find the inverse of this statement. Is it true?
• Find the contrapositive of this statement. Which statement is it the same as?

Find the converse of each true if-then statement. If the converse is true, write the biconditional statement.

• An acute angle is less than \(90^{\circ}\).
• If you are at the beach, then you are sun burnt.
• If \(x>4\), then \(x+3>7\).

For questions 8-10, determine the two true conditional statements from the given biconditional statements.

• A U.S. citizen can vote if and only if he or she is 18 or more years old.
• A whole number is prime if and only if its factors are 1 and itself.
• \(2x=18\) if and only if \(x=9\).

Review (Answers)

To see the Review answers, open this PDF file and look for section 2.4.

Additional Resources

Interactive Element

Video: Converse, Inverse and Contrapositive of a Conditional Statement Principles - Basic

Activities: Converse, Inverse, and Contrapositive Discussion Questions

Study Aids: Conditional Statements Study Guide

Practice: Converse, Inverse, and Contrapositive Statements

Real World: Converse Inverse Contrapositive

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2.1: Statements and Logical Operators

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• Ted Sundstrom
• Grand Valley State University via ScholarWorks @Grand Valley State University

PREVIEW ACTIVITY \(\PageIndex{1}\): Compound Statements

Mathematicians often develop ways to construct new mathematical objects from existing mathematical objects. It is possible to form new statements from existing statements by connecting the statements with words such as “and” and “or” or by negating the statement. A logical operator (or connective ) on mathematical statements is a word or combination of words that combines one or more mathematical statements to make a new mathematical statement. A compound statement is a statement that contains one or more operators. Because some operators are used so frequently in logic and mathematics, we give them names and use special symbols to represent them.

• The conjunction of the statements \(P\) and \(Q\) is the statement “\(P\) and \(Q\)” and its denoted by \(P \wedge Q\). The statement \(P \wedge Q\) is true only when both \(P\) and \(Q\) are true.
• The disjunction of the statements \(P\) and \(Q\) is the statement “\(P\) or \(Q\)” and its denoted by \(P \vee Q\). The statement \(P \vee Q\) is true only when at least one of \(P\) or \(Q\) is true.
• The negation ( of a statement ) of the statement \(P\) is the statement “ not \(P\) ” and is denoted by \(\urcorner P\). The negation of \(P\) is true only when \(P\) is false, and \(\urcorner P\) is false only when \(P\) is true.
• The implication or conditional is the statement “ If \(P\) then \(Q\)” and is denoted by \(P \to Q\). The statement \(P \to Q\) is often read as “\(P\) implies \(Q\), and we have seen in Section 1.1 that \(P \to Q\) is false only when \(P\) is true and \(Q\) is false.

Some comments about the disjunction. It is important to understand the use of the operator “or.” In mathematics, we use the “ inclusive or ” unless stated otherwise. This means that \(P \vee Q\) is true when both \(P\) and \(Q\) are true and also when only one of them is true. That is, \(P \vee Q\) is true when at least one of \(P\) or \(Q\) is true, or \(P \vee Q\) is false only when both \(P\) and \(Q\) are false.

A different use of the word “or” is the “ exclusive or .” For the exclusive or, the resulting statement is false when both statements are true. That is, “\(P\) exclusive or \(Q\)” is true only when exactly one of \(P\) or \(Q\) is true. In everyday life, we often use the exclusive or. When someone says, “At the intersection, turn left or go straight,” this person is using the exclusive or.

Some comments about the negation . Although the statement, \(\urcorner P\), can be read as “It is not the case that \(P\),” there are often betters ways to say or write this in English. For example, we would usually say (or write):

• The negation of the statement, “391 is prime” is “391 is not prime.”
• The negation of the statement, “\(12 < 9\)” is “\(12 \ge 9\).”

\(P\): 15 is odd \(Q\): 15 is prime write each of the following statements as English sentences and determine

whether they are true or false. (a) \(P \wedge Q\). (b) \(P \vee Q\). (c) \(P \wedge \urcorner Q\). (d) \(\urcorner P \vee \urcorner Q\).

P : 15 is odd R: 15 < 17

write each of the following statements in symbolic form using the operators\(\wedge\), \(\vee\), and \(\urcorner\)

(a) 15 \(\ge\) 17. (b) 15 is odd or 15 \(\ge\) 17. (c) 15 is even or 15 <17. (d) 15 is odd and 15 \(\ge\) 17.

PREVIEW ACTIVITY\(\PageIndex{2}\): Truth Values of Statements

We will use the following two statements for all of this Preview Activity:

• \(P\) is the statement “It is raining.”
• \(Q\) is the statement “Daisy is playing golf.”

In each of the following four parts, a truth value will be assigned to statements \(P\) and \(Q\). For example, in Question (1), we will assume that each statement is true. In Question (2), we will assume that \(P\) is true and \(Q\) is false. In each part, determine the truth value of each of the following statements:

(a) (\(P \wedge Q\)) It is raining and Daisy is playing golf.

(b) (\(P \vee Q\)) It is raining or Daisy is playing golf.

(c) (\(P \to Q\)) If it is raining, then Daisy is playing golf.

(d) (\(\urcorner P\)) It is not raining.

Which of the four statements [(a) through (d)] are true and which are false in each of the following four situations?

1. When \(P\) is true (it is raining) and \(Q\) is true (Daisy is playing golf). 2. When \(P\) is true (it is raining) and \(Q\) is false (Daisy is not playing golf). 3. When \(P\) is false (it is not raining) and \(Q\) is true (Daisy is playing golf). 4. When \(P\) is false (it is not raining) and \(Q\) is false (Daisy is not playing golf).

In the preview activities for this section, we learned about compound statements and their truth values. This information can be summarized with truth tables as is shown below.

Rather than memorizing the truth tables, for many people it is easier to remember the rules summarized in Table 2.1.

Other Forms of Conditional Statements

Conditional statements are extremely important in mathematics because almost all mathematical theorems are (or can be) stated in the form of a conditional statement in the following form:

If “certain conditions are met,” then “something happens.”

It is imperative that all students studying mathematics thoroughly understand the meaning of a conditional statement and the truth table for a conditional statement.

• If \(P\), then \(Q\).
• \(P\) implies \(Q\).
• \(P\) only if \(Q\).
• \(Q\) if \(P\).
• Whenever \(P\) is true, \(Q\) is true.
• \(Q\) is true whenever \(P\) is true.
• \(Q\) is necessary for \(P\). (This means that if \(P\) is true, then \(Q\) is necessarily true.)

In all of these cases, \(P\) is the hypothesis of the conditional statement and \(Q\) is the conclusion of the conditional statement.

Progress Check 2.1: The "Only if" statemenT

Recall that a quadrilateral is a four-sided polygon. Let \(S\) represent the following true conditional statement:

If a quadrilateral is a square, then it is a rectangle.

Write this conditional statement in English using

• the word “whenever”
• the phrase “only if”
• the phrase “is necessary for”
• the phrase “is sufficient for”

Add texts here. Do not delete this text first.

Constructing Truth Tables

Truth tables for compound statements can be constructed by using the truth tables for the basic connectives. To illustrate this, we will construct a truth table for. \((P \wedge \urcorner Q) \to R\). The first step is to determine the number of rows needed.

• For a truth table with two different simple statements, four rows are needed since there are four different combinations of truth values for the two statements. We should be consistent with how we set up the rows. The way we will do it in this text is to label the rows for the first statement with (T, T, F, F) and the rows for the second statement with (T, F, T, F). All truth tables in the text have this scheme.
• For a truth table with three different simple statements, eight rows are needed since there are eight different combinations of truth values for the three statements. Our standard scheme for this type of truth table is shown in Table 2.2 .

The next step is to determine the columns to be used. One way to do this is to work backward from the form of the given statement. For \((P \wedge \urcorner Q) \to R\), the last step is to deal with the conditional operator \((\to)\). To do this, we need to know the truth values of \((P \wedge \urcorner Q)\) and \(R\). To determine the truth values for \((P \wedge \urcorner Q)\), we need to apply the rules for the conjunction operator \((\wedge)\) and we need to know the truth values for \(P\) and \(\urcorner Q\).

Table 2.2 is a completed truth table for \((P \wedge \urcorner Q) \to R\) with the step numbers indicated at the bottom of each column. The step numbers correspond to the order in which the columns were completed.

• When completing the column for \(P \wedge \urcorner Q\), remember that the only time the conjunction is true is when both \(P\) and \(\urcorner Q\) are true.
• When completing the column for \((P \wedge \urcorner Q) \to R\), remember that the only time the conditional statement is false is when the hypothesis \((P \wedge \urcorner Q)\) is true and the conclusion, \(R\), is false.

The last column entered is the truth table for the statement \((P \wedge \urcorner Q) \to R\) using the set up in the first three columns.

Progress Check 2.2: Constructing Truth Tables

Construct a truth table for each of the following statements:

• \(P \wedge \urcorner Q\)
• \(\urcorner(P \wedge Q)\)
• \(\urcorner P \wedge \urcorner Q\)
• \(\urcorner P \vee \urcorner Q\)

Do any of these statements have the same truth table?

The Biconditional Statement

Some mathematical results are stated in the form “\(P\) if and only if \(Q\)” or “\(P\) is necessary and sufficient for \(Q\).” An example would be, “A triangle is equilateral if and only if its three interior angles are congruent.” The symbolic form for the biconditional statement “\(P\) if and only if \(Q\)” is \(P \leftrightarrow Q\). In order to determine a truth table for a biconditional statement, it is instructive to look carefully at the form of the phrase “\(P\) if and only if \(Q\).” The word “and” suggests that this statement is a conjunction. Actually it is a conjunction of the statements “\(P\) if \(Q\)” and “\(P\) only if \(Q\).” The symbolic form of this conjunction is \([(Q \to P) \wedge (P \to Q]\).

Progress Check 2.3: The Truth Table for the Biconditional Statement

Complete a truth table for \([(Q \to P) \wedge (P \to Q]\). Use the following columns: \(P\), \(Q\), \(Q \to P\), \(P \to Q\), and \([(Q \to P) \wedge (P \to Q]\). The last column of this table will be the truth for \(P \leftrightarrow Q\).

Other Forms of the Biconditional Statement

As with the conditional statement, there are some common ways to express the biconditional statement, \(P \leftrightarrow Q\), in the English language.

• \(P\) is and only if \(Q\).
• \(P\) is necessary and sufficient for \(Q\).
• \(P\) implies \(Q\) and \(Q\) implies \(P\).

Tautologies and Contradictions

Definition: tautology

A tautology is a compound statement S that is true for all possible combinations of truth values of the component statements that are part of \(S\). A contradiction is a compound statement that is false for all possible combinations of truth values of the component statements that are part of \(S\).

That is, a tautology is necessarily true in all circumstances, and a contradiction is necessarily false in all circumstances.

Progress Check 2.4 (Tautologies and Contradictions)

For statements \(P\) and \(Q\):

• Use a truth table to show that \((P \vee \urcorner P)\) is a tautology.
• Use a truth table to show that \((P \wedge \urcorner P)\) is a contradiction.
• Use a truth table to determine if \(P \to (P \vee P)\) is a tautology, a contradiction, nor neither.

Exercises for Section 2.1

• Suppose that Daisy says, “If it does not rain, then I will play golf.” Later in the day you come to know that it did rain but Daisy still played golf. Was Daisy’s statement true or false? Support your conclusion.
• Suppose that \(P\) and \(Q\) are statements for which \(P \to Q\) is true and for which \(\urcorner Q\) is true. What conclusion (if any) can be made about the truth value of each of the following statements? (a) \(P\) (b) \(P \wedge Q\) (c) \(P \vee Q\)
• Suppose that \(P\) and \(Q\) are statements for which \(P \to Q\) is false. What conclusion (if any) can be made about the truth value of each of the following statements? (a) \(\urcorner P \to Q\) (b) \(Q \to P\) (c) \(P \ vee Q\)
• Suppose that \(P\) and \(Q\) are statements for which \(Q\) is false and \(\urcorner P \to Q\) is true (and it is not known if \(R\) is true or false). What conclusion (if any) can be made about the truth value of each of the following statements? (a) \(\urcorner Q \to P\) (b) \(P\) (c) \(P \wedge R\) (d) \(R \to \urcorner P\)
• Construct a truth table for each of the following statements: (a) \(P \to Q\) (b) \(Q \to P\) (c) \(\urcorner P \to \urcorner Q\) (d) \(\urcorner Q \to \urcorner P\) Do any of these statements have the same truth table?
• Construct a truth table for each of the following statements: (a) \(P \vee \urcorner Q\) (b) \(\urcorner (P \vee Q)\) (c) \(\urcorner P \vee \urcorner Q\) (d) \(\urcorner P \wedge \urcorner Q\) Do any of these statements have the same truth table?
• Construct truth table for \(P \wedge (Q \vee R)\) and \((P \wedge Q) \vee (P \wedge R)\). What do you observe.
• Laura is in the seventh grade.
• ��Laura got an A on the mathematics test or Sarah got an A on the mathematics test.
• ��If Sarah got an A on the mathematics test, then Laura is not in the seventh grade. If possible, determine the truth value of each of the following statements. Carefully explain your reasoning. (a) Laura got an A on the mathematics test. (b) Sarah got an A on the mathematics test. (c) Either Laura or Sarah did not get an A on the mathematics test.
• Let \(P\) stand for “the integer \(x\) is even,” and let \(Q\) stand for “\(x^2\) is even.” Express the conditional statement \(P \to Q\) in English using (a) The "if then" form of the conditional statement (b) The word "Implies" (c) The "only if" form of the conditional statement (d) The phrase "is necessary for" (e) The phrase "is sufficient for"
• Repeat Exercise (9) for the conditional statement \(Q \to P\).
• For statements \(P\) and \(Q\), use truth tables to determine if each of the following statements is a tautology, a contradiction, or neither. (a) \(\urcorner Q \vee (P \to Q)\). (b) \(Q \wedge (P \wedge \urcorner Q)\). (c) \((Q \wedge P) \wedge (P \to \urcorner Q)\). (d) \(\urcorner Q \to (P \wedge \urcorner P)\).
• For statements \(P\), \(Q\), and \(R\): (a) Show that \([(P \to Q) \wedge P] \to Q\) is a tautology. Note : In symbolic logic, this is an important logical argument form called modus ponens . (b) Show that \([(P \to Q) \wedge (Q \to R)] \to (P \to R)\) is atautology. Note : In symbolic logic, this is an important logical argument form called syllogism . Explorations and Activities
• Working with Truth Values of Statements. Suppose that \(P\) and \(Q\) are true statements, that \(U\) and \(V\) are false statements, and that \(W\) is a statement and it is not known if \(W\) is true or false. Which of the following statements are true, which are false, and for which statements is it not possible to determine if it is true or false? Justify your conclusions. (a) \((P \vee Q) \vee (U \wedge W)\) (f) \((\urcorner P \vee \urcorner U) \wedge (Q \vee \urcorner V)\) (b) \(P \wedge (Q \to W)\) (g) \((P \wedge \urcorner Q) \wedge (U \vee W)\) (c) \(P \wedge (W \to Q)\) (h) \((P \vee \urcorner Q) \to (U \wedge W)\) (d) \(W \to (P \wedge U)\) (i) \((P \vee W) \to (U \wedge W)\) (e) \(W \to (P \wedge \urcorner U)\) (j) \((U \wedge \urcorner V) \to (P \wedge W)\)

Conditional Statement – Definition, Truth Table, Examples, FAQs

What is a conditional statement, how to write a conditional statement, what is a biconditional statement, solved examples on conditional statements, practice problems on conditional statements, frequently asked questions about conditional statements.

A conditional statement is a statement that is written in the “If p, then q” format. Here, the statement p is called the hypothesis and q is called the conclusion. It is a fundamental concept in logic and mathematics. 

Conditional statement symbol :  p → q

A conditional statement consists of two parts.

• The “if” clause, which presents a condition or hypothesis.
• The “then” clause, which indicates the consequence or result that follows if the condition is true. 

Example : If you brush your teeth, then you won’t get cavities.

Hypothesis (Condition): If you brush your teeth

Conclusion (Consequence): then you won’t get cavities 

Conditional Statement: Definition

A conditional statement is characterized by the presence of “if” as an antecedent and “then” as a consequent. A conditional statement, also known as an “if-then” statement consists of two parts:

• The “if” clause (hypothesis): This part presents a condition, situation, or assertion. It is the initial condition that is being considered.
• The “then” clause (conclusion): This part indicates the consequence, result, or action that will occur if the condition presented in the “if” clause is true or satisfied. 

Related Worksheets

Representation of Conditional Statement

The conditional statement of the form ‘If p, then q” is represented as p → q. 

It is pronounced as “p implies q.”

Different ways to express a conditional statement are:

• p implies q
• p is sufficient for q
• q is necessary for p

Parts of a Conditional Statement

There are two parts of conditional statements, hypothesis and conclusion. The hypothesis or condition will begin with the “if” part, and the conclusion or action will begin with the “then” part. A conditional statement is also called “implication.”

Conditional Statements Examples:

Example 1: If it is Sunday, then you can go to play. 

Hypothesis: If it is Sunday

Conclusion: then you can go to play. 

Example 2: If you eat all vegetables, then you can have the dessert.

Condition: If you eat all vegetables

Conclusion: then you can have the dessert 

To form a conditional statement, follow these concise steps:

Step 1 : Identify the condition (antecedent or “if” part) and the consequence (consequent or “then” part) of the statement.

Step 2 : Use the “if… then…” structure to connect the condition and consequence.

Step 3 : Ensure the statement expresses a logical relationship where the condition leads to the consequence.

Example 1 : “If you study (condition), then you will pass the exam (consequence).” 

This conditional statement asserts that studying leads to passing the exam. If you study (condition is true), then you will pass the exam (consequence is also true).

Example 2 : If you arrange the numbers from smallest to largest, then you will have an ascending order.

Hypothesis: If you arrange the numbers from smallest to largest

Conclusion: then you will have an ascending order

Truth Table for Conditional Statement

The truth table for a conditional statement is a table used in logic to explore the relationship between the truth values of two statements. It lists all possible combinations of truth values for “p” and “q” and determines whether the conditional statement is true or false for each combination. 

The truth value of p → q is false only when p is true and q is False. 

If the condition is false, the consequence doesn’t affect the truth of the conditional; it’s always true.

In all the other cases, it is true.

The truth table is helpful in the analysis of possible combinations of truth values for hypothesis or condition and conclusion or action. It is useful to understand the presence of truth or false statements. 

Converse, Inverse, and Contrapositive

The converse, inverse, and contrapositive are three related conditional statements that are derived from an original conditional statement “p → q.” 

Consider a conditional statement: If I run, then I feel great.

• Converse: 

The converse of “p → q” is “q → p.” It reverses the order of the original statement. While the original statement says “if p, then q,” the converse says “if q, then p.” 

Converse: If I feel great, then I run.

• Inverse: 

The inverse of “p → q” is “~p → ~q,” where “” denotes negation (opposite). It negates both the antecedent (p) and the consequent (q). So, if the original statement says “if p, then q,” the inverse says “if not p, then not q.”

Inverse : If I don’t run, then I don’t feel great.

• Contrapositive: 

The contrapositive of “p → q” is “~q → ~p.” It reverses the order and also negates both the statements. So, if the original statement says “if p, then q,” the contrapositive says “if not q, then not p.”

Contrapositive: If I don’t feel great, then I don’t run.

A biconditional statement is a type of compound statement in logic that expresses a bidirectional or two-way relationship between two statements. It asserts that “p” is true if and only if “q” is true, and vice versa. In symbolic notation, a biconditional statement is represented as “p ⟺ q.”

In simpler terms, a biconditional statement means that the truth of “p” and “q” are interdependent. 

If “p” is true, then “q” must also be true, and if “q” is true, then “p” must be true. Conversely, if “p” is false, then “q” must be false, and if “q” is false, then “p” must be false. 

Biconditional statements are often used to express equality, equivalence, or conditions where two statements are mutually dependent for their truth values. 

Examples : 

• I will stop my bike if and only if the traffic light is red.  
• I will stay if and only if you play my favorite song.

Facts about Conditional Statements

• The negation of a conditional statement “p → q” is expressed as “p and not q.” It is denoted as “𝑝 ∧ ∼𝑞.” 
• The conditional statement is not logically equivalent to its converse and inverse.
• The conditional statement is logically equivalent to its contrapositive. 
• Thus, we can write p → q ∼q → ∼p

In this article, we learned about the fundamentals of conditional statements in mathematical logic, including their structure, parts, truth tables, conditional logic examples, and various related concepts. Understanding conditional statements is key to logical reasoning and problem-solving. Now, let’s solve a few examples and practice MCQs for better comprehension.

Example 1: Identify the hypothesis and conclusion. 

If you sing, then I will dance.

Solution : 

Given statement: If you sing, then I will dance.

Here, the antecedent or the hypothesis is “if you sing.”

The conclusion is “then I will dance.”

Example 2: State the converse of the statement: “If the switch is off, then the machine won’t work.” 

Here, p: The switch is off

q: The machine won’t work.

The conditional statement can be denoted as p → q.

Converse of p → q is written by reversing the order of p and q in the original statement.

Converse of  p → q is q → p.

Converse of  p → q: q → p: If the machine won’t work, then the switch is off.

Example 3: What is the truth value of the given conditional statement? 

If 2+2=5 , then pigs can fly.

Solution:  

q: Pigs can fly.

The statement p is false. Now regardless of the truth value of statement q, the overall statement will be true. 

F → F = T

Hence, the truth value of the statement is true. 

Conditional Statement - Definition, Truth Table, Examples, FAQs

Attend this quiz & Test your knowledge.

What is the antecedent in the given conditional statement? If it’s sunny, then I’ll go to the beach.

A conditional statement can be expressed as, what is the converse of “a → b”, when the antecedent is true and the consequent is false, the conditional statement is.

What is the meaning of conditional statements?

Conditional statements, also known as “if-then” statements, express a cause-and-effect or logical relationship between two propositions.

When does the truth value of a conditional statement is F?

A conditional statement is considered false when the antecedent is true and the consequent is false.

What is the contrapositive of a conditional statement?

The contrapositive reverses the order of the statements and also negates both the statements. It is equivalent in truth value to the original statement.

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