Finding, Clarifying, and Evaluating Arguments



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Finding, Clarifying, and Evaluating Arguments

Finding, Clarifying, and Evaluating Arguments

E.J. Coffman

The University of Tennessee

I. Introduction
II. Finding Arguments
III. Clarifying an Argument

A. Identify the Conclusion

B. Identify Explicit Premises

C. Add Implicit Premises

D. Apply the Charity Principle

E. Regiment the Argument


IV. Evaluating an Argument

A. Step One: Test the Argument’s Form

1. ‘Valid’ and ‘Invalid’ Defined

2. The Validity Test Explained

a. Figure out the argument’s form

b. Try to prove that the argument’s form isn’t “truth-preserving”

3. The Validity Test Illustrated

B. Step Two: Test the Argument’s Premises

1. ‘Conditional Statement’ Defined

2. The Counterexample Method Explained

a. Translate into “If P then Q” form

b. Try to imagine a counterexample

3. The Counterexample Method Illustrated

V. Summary
VI. Appendix 1: Some Common Valid and Invalid Argument Forms
VII. Appendix 2: Analyzing Concepts

________________________________________________________________________
Key Terms to Learn: argument, conclusion, premises, good argument, bad argument, argument form, deductive argument, valid, invalid, conditional statement (conditional), antecedent, consequent, counterexample, thought experiment, analysis
I. Introduction
One of this course’s main goals is to help you hone certain of your basic reasoning skills—in particular, your ability to
▪ find, clarify, and evaluate arguments presented by others,

and to


▪ present and defend arguments of your own.1
In this essay, I’ll offer you some standard guidance about becoming better at finding, clarifying, and evaluating arguments about all kinds of topics (not just those we’ll explore in this course). Along the way, I’ll introduce some special terminology that philosophers use when discussing arguments. You’ll want to learn that terminology, then use it in class meetings and writing assignments. To facilitate this process, I’ve collected—in one spot—the most important terminology introduced in this essay; see ‘Key Terms to Learn’ above.
I’ll start by addressing three questions that may have occurred to you already. First, why should you devote some of your time and energy to improving your ability to find, clarify, and evaluate arguments? Much could be said here, but most of it boils down to this: These skills are extremely important to have yet surprisingly difficult to develop.2 Devoting some time and energy to developing these skills is thus well worth the while.
Second question: Why do I say that this course aims to “hone” and “improve” your basic reasoning skills (as opposed to “teach you those skills from scratch”)? Well, I say things like “hone” and “improve” because you’re probably already a pretty decent reasoner: you probably already have some understanding of the main concepts used to find, clarify, and evaluate arguments. As a result, many of the ideas and techniques discussed below will probably seem at least somewhat familiar to you.
What will likely seem unfamiliar is the special terminology that philosophers use when discussing arguments. And that raises a third question: Why must we learn special terminology for discussing arguments?
The basic answer is this: Having a standard terminology to use when discussing arguments will be tremendously helpful as you try to become a better reasoner.

Being an excellent reasoner consists largely in being able to give clear, satisfying, illuminating assessments of the arguments you encounter. Without a standard terminology to use when discussing arguments, you may be tempted to evaluate arguments in unclear, unsatisfying ways.3 On the other hand, if you’ve learned some special terminology to use when discussing arguments, you’ll be poised to give clear, satisfying, illuminating assessments of the arguments you encounter—assessments that pinpoint exactly what’s wrong (and right) with an argument. In order to become a better reasoner, then, you’ll want to learn—and then use, in class and in writing—the special terminology for discussing arguments introduced below.


The rest of this essay offers some standard advice about how to find, clarify, and evaluate arguments. This advice should be helpful to you not only in this course, but also throughout and beyond your undergraduate career.

II. Finding Arguments
Unfortunately, there’s no foolproof method for finding arguments. Let me just state the obvious: When looking for an argument, you’re looking for a set of statements related to each other in such a way that one of those statements—the argument’s conclusion—is supposed to be rationally supported by the other statements—the argument’s premises. One of the best ways to become better at finding arguments is to actively look for them in whatever you happen to read, watch, or listen to. Here are a couple of written passages for practice:4
[1] I have a few questions for those who have raised their voices against the recent Supreme Court decision to preserve our constitutional right to engage in symbolic acts of protest, including the burning of the American flag:
Are you as outraged when our Constitution is assaulted?
Did you protest when the constitutional rights of black citizens were denied? Did you work for their rights to vote, to equal education, to fair housing?
Have you spoken out against the assault on our Constitution by the illegal maneuverings of the boys in the White House during the Iran-Contra affair?
…In short, can you honestly say that you love your flag when you have been silent in protecting all that it stands for?5
[2] Even the Fool is forced to agree that something-than-which-nothing-greater- can-be-thought exists in the mind, since he understands this when he hears it, and whatever is understood is in the mind. And surely that-than-which-a-greater- cannot-be-thought cannot exist in the mind alone. For if it exists solely in the mind, it can be thought to exist in reality also, which is greater. If then that-than- which-a-greater-cannot-be-thought exists in the mind alone, this same that-than- which-a-greater-cannot-be-thought is that-than-which-a-greater-can-be thought. But this is obviously impossible. Therefore there is absolutely no doubt that something-than-which-a-greater-cannot-be-thought exists both in the mind and in reality.6
Does passage [1] express an argument, a line of reasoning offered in support of a particular claim or thesis? How about passage [2]?
Because I want to use passage [2] to introduce the next section, I must now answer one of my own questions: passage [2] does indeed contain an argument. Moreover, a little bit of reflection on passage [2] reveals an important lesson: arguments are not always expressed as clearly as they could be! Oftentimes, before we can properly evaluate an argument we’ve found, we’ll have to do some work to clarify it—that is, to make the argument’s form (its general logical structure, its underlying logical pattern) and content (its particular premises and conclusion) as clear as possible.
III. Clarifying an Argument
Fortunately, there’s a standard method we can use to clarify an argument we’ve found. The method has five steps:
A. Identify the Argument’s Conclusion
An argument’s conclusion is its “main thesis,” the one statement that’s supposed to be rationally supported by the other statements (the argument’s premises).
B. Identify All Explicit Premises
An argument’s explicit premises are the considerations (reasons, pieces of evidence) that the author actually states on behalf of the argument’s conclusion.

C. Add Any Implicit Premises


An argument’s implicit premises are any unstated considerations (reasons, pieces of evidence) that must be added to the argument’s explicit premises so that the explicit premises are clearly connected to the argument’s conclusion.
Some authors do a good job of laying all their premises on the table. When that happens, you won’t have to spend a lot of time searching for implicit premises. Unfortunately, though, many authors—for whatever reason—fail to put all their premises in clear view. In that case, you may have to spend some time trying to find an argument’s implicit premises.
D. Apply the Charity Principle
What philosophers often call the Charity Principle says this:
▪ When clarifying an argument, make the argument as sensible as you possibly can, given what its author said when presenting it.
Why follow the Charity Principle? Much could be said here, but most of it boils down to this: Devoting precious time and energy to criticizing an argument that’s obviously doomed to failure is wasteful, and thus should be avoided at all costs. Adhering tightly to the Charity Principle when trying to clarify an argument will help ensure that we won’t waste precious time and energy considering an obviously hopeless argument.
Here are two pieces of very practical advice implied by the Charity Principle:
▪ When clarifying an argument, eliminate any unnecessary premises and language.
▪ When clarifying an argument, state and arrange the argument’s premises so as to show how they are supposed to support the argument’s conclusion. (This often involves making the argument’s wording more “uniform,” and re-ordering the argument’s premises and conclusion so that its form becomes easier to see.)
E. Regiment the Argument
To regiment an argument, first label each of its parts (its premises and its conclusion), then list each part on its own line. This exercise will not only clarify the argument’s form and content (thus helping you to follow the Charity Principle); it will also facilitate evaluation of the argument’s premises.
Let me illustrate how to regiment an argument. Consider this passage:
Our universe must have been designed by an intelligent being. After all, our universe is extremely complex. And any extremely complex thing was designed by something intelligent.
Clearly, this passage contains an argument, a line of reasoning offered in support of a particular claim or thesis. To regiment that argument, do two things: (1) Label each of its parts, then (2) List each part on its own line. Completing (1) and (2) yields something that looks like this:
P1: Every extremely complex thing was designed by an intelligent being.

P2: Our universe is extremely complex.

C: Thus, our universe was designed by an intelligent being.

We’ve now regimented the argument expressed in the passage above.7 Notice that the argument’s form—its general logical structure, or underlying logical pattern—is now much easier to see. Studying the argument reveals that it has the following form:


P1: Every P is a Q.

P2: X is a P.

C: Thus, X is a Q.
In the next section, we’ll talk much more about how to figure out an argument’s form.
Finally, notice that we could now use ‘P1’ to refer to the argument’s first premise, and ‘P2’ to refer to the argument’s second premise, which would facilitate discussion of those claims.
We’ve now seen how to clarify an argument. Once you’ve clarified an argument, you’re ready to evaluate it—that is, to determine whether or not the argument’s premises really do constitute a good reason to believe its conclusion.
IV. Evaluating an Argument
Fortunately, there’s a standard method we can use to evaluate an argument we’ve found and clarified. In order to properly evaluate an argument, we need to know what it takes for an argument to qualify as good. Basically, a good argument is one that has proper form and acceptable (plausible, reasonable) premises. On the other hand, a bad argument is one that has improper form or some implausible (unacceptable, unreasonable) premises.
So, to qualify as good, an argument must pass two tests: it must have proper form, and it must have acceptable (plausible, reasonable) premises. Naturally, then, the standard method for evaluating an argument involves two steps: the first step tests the argument’s form (its general logical structure, its underlying logical pattern), while the second step tests the argument’s premises.
A. Step One: Test the Argument’s Form
When you’re testing an argument’s form (its general structure or underlying pattern), you’re trying to determine the quality of the logical connection between the argument’s premises and its conclusion. An argument that has proper logical form is such that there’s a strong logical connection between its premises and its conclusion. More precisely, an argument with proper logical form is one whose premises would be adequate rational support for its conclusion if those premises were true.
As it happens, we considered a properly formed argument in the last section. Recall this one:
P1: Every extremely complex thing was designed by an intelligent being.

P2: Our universe is extremely complex.



C: Thus, our universe was designed by an intelligent being.
Perhaps this argument won’t ultimately qualify as good, but it certainly has proper logical form or structure. For there’s a very tight logical connection between this argument’s premises and its conclusion: if this argument’s premises were true, its conclusion would have to be true too—the conclusion just “falls right out” of the premises. So, this argument definitely passes the first test involved in our standard method for evaluating arguments: it has proper logical form.
Now, if we were to continue evaluating the argument above, we would at this point proceed to the second step of argument evaluation: we would ask ourselves whether all that argument’s premises are acceptable (plausible, reasonable). And if we found all that argument’s premises to be reasonable, we would then judge the argument to be good. On the other hand, if we found the argument to involve at least one questionable premise, we would judge the argument to be bad. (By the way, what do you think: is the above argument good or bad?)
The following crucial fact about the first step of argument evaluation is now in view:
At the first step of argument evaluation, we completely ignore the question of whether the argument’s premises are actually true. Instead, we focus entirely on the argument’s form (its general structure or underlying pattern).
Repeat: At the first step of argument evaluation, we ignore the question of whether the argument’s premises are actually true, and instead focus entirely on the argument’s form.
We now know that the first step of argument evaluation is to determine whether the argument has proper logical form. Interestingly, whether or not a particular argument is properly formed depends in part on what kind of argument it is. Basically, there are two kinds of arguments: deductive arguments and inductive arguments. A deductive argument aims to prove that its conclusion is true, to establish its conclusion beyond a shadow of a doubt. A mathematical proof is a good example of a deductive argument. An inductive argument, on the other hand, attempts only to show that its conclusion is somewhat reasonable to believe. The complex case a detective slowly builds for the guilt of a particular suspect is a good example of an inductive argument.
As it happens, most of the arguments we’ll explore in this course are deductive. For the moment, then, we’ll limit ourselves to discussing how to determine whether a deductive argument you’ve found and clarified is properly formed.
1. ‘Valid’ and ‘Invalid’ Defined
When a deductive argument is properly formed, we say that the argument is valid. Repeat: In this course, a valid argument is a deductive argument that has proper logical form, a properly formed deductive argument.
To qualify as valid, a deductive argument must be such that its conclusion would have to be true if its premises were true. To put it a slightly different way, a valid argument is one whose form is such that its conclusion must be true if its premises are true. To say it one last way, a valid argument is one whose form is truth-preserving, in the sense that the form always yields a true conclusion if it is fed true premises.
Thinking about the following common valid argument forms should help you understand the concept of validity more deeply:
P1: Every P is a Q.

P2: X is a P.

C: X is a Q. [Notice that the argument considered on the last page has this form.]
P1: If P then Q.

P2: P.


C: Q.
P1: If P then Q.

P2: If Q then R.

C: If P then R.
P1: If P then Q.

P2: Not Q.

C: Not P.
Each of these argument forms is valid: each is such that if it is fed true premises, it must yield a true conclusion. To put it a little differently, each is such that it simply couldn’t have all true premises and a false conclusion. In the next section, we’ll talk about how to determine whether or not a particular argument form is valid.

It’s worth emphasizing that the facts about whether or not a particular argument is valid depend solely on the argument’s form, and thus are completely independent of the facts about whether the argument’s premises are actually true. As a result, you simply do not need to know whether an argument’s premises are actually true in order to determine whether or not the argument is valid.

When a deductive argument has improper form, we say that the argument is invalid. An invalid argument is one whose form is not truth-preserving: an invalid argument form is such that even if it is fed true premises, it could still yield a false conclusion. All invalid arguments are thereby bad.
Thinking about the following common invalid argument forms should help you understand the concept of invalidity more deeply:
P1: Every P is a Q.

P2: X is a Q.

C: X is a P.
P1: If P then Q.

P2: Q.


C: P.
P1: If P then Q.

P2: Not P.

C: Not Q.
Each of these argument forms is invalid. That is, each is such that even if it is fed true premises, it could still yield a false conclusion. For example, to see that the second and third argument forms are invalid, plug George W. Bush is on Mars today into the “P” positions, and George W. Bush is on a planet today into the “Q” positions. (Can you think of an argument that shows the first form to be invalid?)

In this course, we’ll spend a fair amount of time trying to determine whether or not certain interesting and influential philosophical arguments are valid. Fortunately, there’s a good method we can use to test an argument for validity. We’ll call this method the Validity Test.


2. The Validity Test Explained
The Validity Test involves two steps: (1) Figure out the argument’s form, then (2) Try to prove that the form isn’t truth-preserving.
a. Figure out the argument’s form
To figure out an argument’s form, we replace the argument’s particular content—its particular terms, phrases, and clauses—with capital letters (P, Q, X, Y, …) that will serve as general “placeholders” for the argument’s particular content. Completing this exercise will yield a completely general argument form (structure, pattern, schema)—something that looks like this:
P1: If P then Q.

P2: P.


C: Q.
The exercise of replacing an argument’s particular content with general “placeholder” letters may initially seem difficult; but this task becomes pretty easy once you’ve seen it done a few times. Let me illustrate how to figure out an argument’s form by using an argument we considered earlier. Recall this one:
P1: Every extremely complex thing was designed by an intelligent being.

P2: Our universe is extremely complex.

C: Thus, our universe was designed by an intelligent being.
To figure out this argument’s form (which, recall, is the first step of the Validity Test), we replace its particular content—its particular terms, phrases, and clauses—with capital letters that will serve as general “placeholders” for the argument’s particular content. Completing this replacement exercise will yield something that looks like this:
P1: Every P is a Q.

P2: X is a P.

C: Thus, X is a Q.
Here, I used the capital letter ‘P’ as a placeholder for ‘extremely complex thing’; ‘Q’ as a placeholder for ‘thing designed by an intelligent being’; and ‘X’ as a placeholder for ‘our universe’.8 Notice also that I left all the “logically important” terms alone—e.g., ‘every’ and ‘thus’. The result is a completely general argument form. We’ve now figured out the form exemplified by the above argument for the conclusion that our universe was designed by an intelligent being. Again, we did this by replacing the argument’s particular content with capital letters that serve as general placeholders, while leaving intact all the terms that give the argument its logical structure (e.g., ‘every’, ‘thus’).
As I suggested, those who are initially confused about how to figure out an argument’s form get the hang of it pretty quickly after seeing it done a few times. Here’s another example:
P1: If there’s horrific evil in the world, then God doesn’t exist.

P2: If Buddhism is true, then God doesn’t exist.

C: If there’s horrific evil in the world, then Buddhism is true.
To figure out this argument’s form, replace its particular content with capital letters that will serve as general placeholders, while leaving its “logically important” terms alone. Something like this will be the result:
P1: If P, then Q.

P2: If R, then Q.

C: If P, then R.
Here, I used ‘P’ as a placeholder for ‘there’s horrific evil in the world’; ‘Q’ as a placeholder for ‘God doesn’t exist’; and ‘R’ as a placeholder for ‘Buddhism is true’. Notice also that I left the “If…then…” constructions alone, since those are crucial to this argument’s logical structure. Again, the result is a completely general argument form, one that underlies the argument we’re currently considering. We’ve thus figured out the argument’s form, which is the first step we take when testing the argument for validity.
We’ve now seen how to figure out an argument’s form. But: how can we tell whether a particular argument form is valid or invalid? Here’s the answer:
b. Try to show that the argument form isn’t truth-preserving
We’ve said that a valid argument form is truth-preserving—such that if it is fed true premises, it must yield a true conclusion. To determine whether a particular argument form is valid, you try to construct an argument of that form with all true premises and a false conclusion.
If you can construct such an argument, you’ll have shown that the argument form under consideration is invalid—that is, not truth-preserving. Since all invalid arguments are bad, determining that an argument has an invalid form shows that the argument is bad.
On the other hand, if you’re unable to construct an argument of the relevant form with all true premises and a false conclusion, that’s a good sign that the form under consideration is valid. You would then proceed to the second step of argument evaluation, where you (finally!) get to test the argument’s particular premises.
3. The Validity Test Illustrated
The standard method of testing an argument for validity—the Validity Test—makes more sense once you’ve seen it done. So, let me illustrate the method by using it to test one of the arguments we encountered earlier. Recall this one:

P1: If there’s horrific evil in the world, then God doesn’t exist.

P2: If Buddhism is true, then God doesn’t exist.

C: If there’s horrific evil in the world, then Buddhism is true.


You’ll recall that the first step of the Validity Test is to figure out the argument’s form. To do that, you replace the argument’s particular content with general placeholder letters, while leaving all the “logically important” terms intact. When we replace the above argument’s particular content with general placeholder letters, we reveal an underlying argument form (structure, pattern, schema) that looks like this:
P1: If P, then Q.

P2: If R, then Q.

C: If P, then R.
We now proceed to the second step of the Validity Test, where we try to construct a different argument of the same form with all true premises and a false conclusion. Constructing such an argument would show that the argument form under consideration is invalid, and so that the argument under consideration is bad. So, what do you think: is the form currently under consideration valid or invalid?
The argument form
P1: If P, then Q.

P2: If R, then Q.

C: If P, then R.
is invalid. We can prove this by constructing an argument of that form with all true premises and a false conclusion. Such arguments can be constructed; here’s an example:
P1: If EJ is a human, then EJ is a mammal. [TRUE: All humans are mammals.]

P2: If EJ is a dog, then EJ is a mammal. [TRUE: All dogs are mammals.]

C: If EJ is a human, then EJ is a dog. [FALSE: No human is also a dog!]
Notice that this argument has the form currently under discussion (If P then Q; If R then Q; So, if P then R). Notice also that the argument has all true premises but a false conclusion. The existence of this argument proves that the general argument form
P1: If P, then Q.

P2: If R, then Q.

C: If P, then R.
is invalid—that is, not truth-preserving.
Finally, notice that we’ve now shown that the argument we were originally concerned with—the one above about horrific evil, God, and Buddhism—is bad. For we have shown that the argument’s underlying form (If P then Q; If R then Q; So, if P then R) is invalid; and, as you’ll recall, every argument with an invalid form is bad.
I’ve now illustrated the standard way to test an argument for validity—the Validity Test. To perform the Validity Test on an argument, (1) Figure out the argument’s form, then (2) Try to construct a different argument of the same form with all true premises and a false conclusion. If you succeed, you’ll have proved that the original argument is improperly formed, and therefore bad. On the other hand, if don’t succeed, that’s a good sign that the argument whose form you’re testing is valid. You’d then proceed to the second and final step of argument evaluation, where you (finally!) get to test the argument’s premises.

B. Test the Argument’s Premises


To test an argument’s premises, ask yourself this question:
▪ Is it reasonable for me to believe all the premises involved in this argument?
If your answer is a resounding ‘Yes’, then the argument qualifies as good: it is valid and has acceptable (plausible, reasonable) premises. On the other hand, if you have doubts about one or more of the argument’s premises, then the argument is bad (at least until something has been done to mitigate your worries about its premises).
Let me briefly highlight a thought that the previous paragraph may prompt in some readers. There’s an important—though ultimately harmless—way in which the concepts of good argument and bad argument are “subjective” or “relative”. It’s possible that different people properly arrive at different verdicts on the question whether a particular argument is good: one person may properly evaluate an argument as good, while another person properly evaluates the same argument as bad. How can this be?
What makes for this possibility is the fact that people can differ with respect to the evidence they possess. For example, professional scientists have large bodies of evidence for initially odd-sounding scientific theories, evidence that non-scientists lack. Because people can differ with respect to the evidence they possess, people can differ with respect to what’s reasonable for them to believe: what’s reasonable for you to believe is determined largely (if not completely) by your evidence.9 For example, professional physicists have evidence from various scientific experiments that makes it reasonable for them to believe that light sometimes behaves like a wave but at other times behaves like a particle. But it would be unreasonable for a typical person on the street who lacks the relevant scientific evidence to believe that light has those apparently incompatible features.
So, because people can differ with respect to the evidence they have, people can differ with respect to what’s reasonable for them to believe. And that makes for the possibility that different people properly arrive at different verdicts on the question whether a given argument is good. One way people can try to resolve such disagreement is to share with each other all the evidence they have that’s relevant to the argument under consideration.
Okay, back to our main question: How do we test an argument’s premises? As it happens, many of the arguments we’ll encounter in this course involve premises that are conditional statements (for short, conditionals). Roughly, a conditional is a claim of the form “If P then Q.” To evaluate an argument involving conditionals, you’ll need to know how to determine whether a particular conditional is reasonable for you to believe. Fortunately, there’s a good method we can use to evaluate conditionals: the Counterexample Method. I’ll describe and illustrate that method in a moment.
First, though, we need to say a little bit more about conditionals. I just said that a conditional is, roughly, a statement of the form “If P then Q.” For the purposes of this course, we’ll need a slightly more precise definition of conditional.
1. ‘Conditional Statement’ Defined
For our purposes, a conditional statement is any statement that has the same meaning as a statement of the form “If P then Q.” Interestingly, there are many different ways to say “If P then Q.” For our purposes, all these different ways of saying “If P then Q” qualify as conditional statements. Here are the main kinds of conditionals we’ll encounter in this course (repeat: all the following statements are just different ways to say “If P then Q”):
If P then Q. (“If I went to the store, then I went somewhere.”)

P suffices for Q. (“My going to the store suffices for my going somewhere.”)

P is a sufficient condition for Q.

P implies Q.

P entails Q.

P only if Q.

P requires Q.

Q is required for P.

Q is necessary for P.

Q is a necessary condition for P.


Repeat: For our purposes, a conditional statement is a statement that has the same meaning as a statement of the form “If P then Q.” Since each of the statements just listed has the same meaning as a statement of the form “If P then Q,” each of those statements qualifies as a conditional.

Finally, in a statement of the form “If P then Q,” the statement in the “P” position is called the antecedent, and the statement in the “Q” position is called the consequent. For example, in the conditional


“If I went to the store, then I went somewhere”
I went to the store is the antecedent, and I went somewhere is the consequent.
We now know what a conditional is (a statement equivalent to a claim of the form “If P then Q”), and what its parts are called (antecedent, consequent). We’re finally ready to learn the standard method for evaluating conditionals: the Counterexample Method.

2. The Counterexample Method Explained


To use the Counterexample Method to evaluate a conditional statement, first
▪ translate the statement into “If P then Q” form, then
▪ try to imagine a coherent scenario in which the conditional’s antecedent holds, but its consequent doesn’t.
If you succeed in imagining a coherent scenario in which the conditional’s antecedent is true but its consequent is false, you have found a counterexample to that particular conditional, thereby showing the conditional to be false.
If the Counterexample Method seems somewhat mysterious at first, think of it this way. A conditional says that whenever its antecedent holds, its consequent will also hold: a conditional says, in effect, that there simply couldn’t be a situation in which its antecedent is true but its consequent is false. Thus, describing a coherent scenario in which the conditional’s antecedent is true but its consequent is false shows that the conditional itself is false.
Like many other things we’ve discussed, the Counterexample Method for evaluating conditionals makes more sense once you’ve seen it used a few times. So, let me illustrate the method by using it to evaluate this conditional statement:
▪ Firmly believing a claim that is in fact true suffices for your knowing the claim to be true.
To evaluate this statement with the Counterexample Method, we first translate it into “If P then Q” form, and then try to find a counterexample to it. Now, we know—perhaps by consulting the list on p.14—that the statement we want to evaluate is equivalent to the following statement:
▪ If you firmly believe a claim that is in fact true, then you know the claim to be true.

That’s the first step of the Counterexample Method: we’ve translated the statement we want to evaluate into “If P then Q” form. We can now proceed to the second step of the Counterexample Method, where we try to imagine a coherent scenario in which the statement’s antecedent holds but its consequent doesn’t.


As it happens, such scenarios exist. Consider this one:
You desperately need fifty bucks. As a result, you (somewhat foolishly) bet a friend fifty bucks that the next time she flips a coin, it will land heads. Your friend accepts the bet, and promptly flips and catches a coin. Neither of you has seen the coin yet. Due entirely to “wishful thinking,” however, you firmly believe that the coin landed heads. And as it happens, the coin did land heads. Still, you do not yet know that the coin landed heads.
This imaginary scenario is a counterexample to the conditional now under consideration. The scenario is perfectly coherent. Further, it is one in which the conditional’s antecedent (“You firmly believe a claim that is in fact true”) holds, but the conditional’s consequent (“You know the claim to be true”) doesn’t. Recall that, in the scenario, you firmly believe the true claim that the coin landed heads, but you do not yet know that the coin landed heads. The scenario thus shows the conditional under consideration to be false.

Two last things about the Counterexample Method. First, the exercise of trying to find a counterexample to a particular conditional is often called conducting a thought experiment (cool, huh?). Second, to qualify as a successful counterexample, an imagined scenario needn’t be “mundane” or “ordinary” or “familiar”. Indeed, a counterexample may involve quite “far-fetched,” “out of the ordinary” circumstances, so long as the example remains coherent. And it should here be said that a large part of what makes Philosophy so fun yet so challenging is that it’s often very hard to say whether a particular imaginary scenario is indeed coherent.

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