Constructing Proofs and Refutations:
A Guide for
Novices and Undergraduates
This is a work in progress. Please email any suggestions for
improvement to Nick[DOT]Jones[AT]uah[DOT]edu.
Proofs and Refutations
We give proofs and refutations all the time. When someone accuses you of stealing his wallet, a reasonable response to the accusation is to offer a proof of why you were not in a position to steal the wallet. And, if he doesn't believe you, a reasonable response on his part is to offer a refutation of your proof and, perhaps, a competing proof for why you were in a position to do the stealing. Lawyers do this all the time, and the courtroom is perhaps the best environment to think about in trying to understand the difference between proofs and refutations.
A proof is an argument designed to convince someone else to accept a thesis as true. For instance, the scientists who recently argued that Pluto is not a planet did not merely assert this. Instead, they gave an argument to convince people who were trained to believe otherwise. They gave a proof. Generally speaking, people tend to offer proofs of their claims when those claims are not generally accepted or when it is an open question whether the claims are true. The whole point of a proof is to persuade people to believe something that they do not already believe.
A refutation, in contrast, is an argument designed to convince someone that a proof is not persuasive. For instance, if someone argued that Pluto is not a planet because it is too close to the sun, a good refutation of their argument is pointing out that there are celestial objects much closer to the sun than Pluto that, despite their proximity, are planets. Generally speaking, when people disagree with the conclusion of an argument, they tend to offer refutations of the argument. The whole point of a refutation is to persuade people that a proof of some thesis is no good. (This is not to be confused with showing that the thesis is false. Whenever one argues that something is true or false, one is giving a proof.)
A useful heuristic for understanding the difference between a proof and a refutation is to ask the following question: Is the argument designed to establish that some claim is either true or false, or is it designed to establish that some argument is not persuasive? Any argument that attempts to establish the truth or falsity of a claim is a proof. Any argument that attempts to establish the unpersuasiveness of some other argument is a refutation.
This distinction between proofs and refutations is not as straightforward as it might seem. Sometimes a refutation of an argument consists in a proof that some part of the argument is false. For instance, when a doctor's proof for why a patient has a sexually transmitted disease is that the patient's blood work came back positive for that disease, one might refute the doctor's argument by arguing that there was a mix-up in the laboratory that caused the doctor to receive blood work results for the wrong person. In this case, one's argument for why the doctor's argument is unpersuasive (i.e., one's refutation of the doctor's proof) consists in proof that the doctor's reason is false.
The important thing to keep in mind is that it is impossible to argue that some claim is true by arguing that a proof of some other claim is unpersuasive. And the reason for this is simple: truth and falsity concern facts, but persuasiveness concerns only arguments. Even if one argument for a claim is unpersuasive, there might be a different argument for the claim that is persuasive. And even if every argument for a claim is unpersuasive, the claim might still be true (in which case it would be an unprovable truth, but a truth nonetheless). For instance, even if every proof of God's existence ever given in the history of the world has a refutation, God might nonetheless exist as a being whose existence must be taken as a matter of faith rather than as a certainty. And even if every proof of God's nonexistence has a refutation, God might nonetheless not exist.
Now that we're clear on the difference between proofs and refutations, we can ask: what are some procedures for constructing good ones? The challenge in answering this question is not that proofs and refutations are entirely alien to us. The challenge is that we don't usually think about what we're doing when we give proofs and refutations, and so we don't know how to separate good procedures from bad ones. This is true of many activities we do all the time. For instance, many parents know how to swim. But some don't know any good way for teaching their children how to swim: when pressed to do so, they simply toss their children into the water and say "Sink or swim!" If the parents want do better than this, they might rent some tapes about swimming lessons, hoping that the tapes contain information about tried-and-true methods of swimming instruction. A nice thing about proofs and refutations is that there are tried-and-true methods of constructing them: following the methods greatly enhances your chances of obtaining good results, especially when you're arguing in unfamiliar waters.
How to Build a Proof
There is a five-step procedure for constructing a good proof. If you follow these five steps, you can be reasonably confident that your proof is a good proof. (There are additional steps to follow if you want your proof to be persuasive too, but first things first.)
Step 1: State the thesis to
be established by the proof.
Step 2: State a reason why someone should accept the thesis as true.
Step 3: State the connection between the reason and the thesis.
Step 4: Illustrate the connection with an example that is unrelated to the thesis.
Step 5: State the thesis again.
Consider, as an illustration of this procedure, a proof that your house is on fire.
- Your house is on fire.
- Because there is heavy smoke coming from inside the house.
- And whenever something gives off heavy smoke, it probably is on fire.
- For instance, piles of leaves that emit heavy smoke usually are on fire.
- Therefore, probably your house is on fire.
In this argument, the reason that someone should accept that your house is on fire is the fact that there is heavy smoke coming from inside the house. The connection between the reason and the thesis is that heavy smoke is a good indicator of fire. And the illustration of this connection, that is unrelated to houses, is that leaves tend to give off heavy smoke when they are on fire.
To get the idea of how good proofs go, it is helpful to consider other examples from different subject areas (mathematics, biology, physics, ethics, metaphysics, theology). Read through each one slowly. Make an effort to see how each proof is an instance of the five-step procedure. But, at least for the moment, don't bother assessing whether the proofs are persuasive. The point now is to get a feel for how good proofs are structured. (This is analogous to watching people swim but not worrying about whether their technique is optimized for competitive racing.)
- 147 is not a prime number.
- Because it is divisible by a number other than 1 or itself, namely 3.
- And whenever a number is divisible by a number other than 1 or itself, it is not a prime number.
- For example, 4 is divisible by 2 and 4 is not a prime number.
- Therefore, 147 is not a prime number.
- All whales are mammals.
- Because all whales are warm-blooded.
- And whenever a living thing is warm-blooded, it is a mammal.
- For instance, humans are warm-blooded and mammals.
- Therefore, all whales are mammals.
- Gold atoms have a nucleus.
- Because when gold atoms are bombarded with particles, some of the particles pass directly through the gold but other particles bounce away from the gold at various angles.
- And the best explanation of this behavior is that gold atoms have a nucleus.
- For instance, if gold atoms were homogeneous (like pudding), then all the particles would pass through the gold.
- Therefore, gold atoms have a nucleus.
- We have a moral obligation to give aid to people who are suffering from extreme hunger.
- Because we have an abundance of resources, using those resources to aid others does not require that we sacrifice anything of moral significance, and suffering from extreme hunger is bad.
- And whenever we have the power to help those who are suffering from something bad, without our sacrificing anything of moral significance, we have a moral obligation to help.
- For example, if a child is drowning in a nearby pool and the only bad effect of a person's rescuing the child is a wetness of clothes, the person has a moral obligation to help the child.
- Therefore, we have a moral obligation to give aid to people who are suffering from extreme hunger.
- Every person has a soul.
- Because people can retain their identities despite undergoing changes.
- And the only way to explain how a person can retain its identity despite undergoing changes is to postulate that the person has a soul.
- For instance, a person can retain their identity despite total loss of memory or a complete transformation of their body, whereas a person does not retain their identity upon loss of their soul.
- Therefore, every person has a soul.
- An Intelligent Designer exists.
- Because there are some organisms in the world that are irreducibly complex.
- And the best explanation of this complexity is to postulate the existence of an Intelligent Designer.
- For instance, evolutionary theory does not explain the existence of irreducible complexity, whereas an Intelligent Designer does.
- Therefore, an Intelligent Designer exists.
If you pay careful attention to these proofs, you'll notice that there are three basic patterns. (Compare the third steps in the proofs to find the differences.) Which pattern to use is a function of the kind of reasons you have and how confident you are about those reasons. For instance, Example 1 uses the following pattern:
- Because REASON.
- And whenever REASON, then probably THESIS.
- Example in which both REASON and THESIS are true.
- Therefore, probably THESIS.
This is a good pattern to use when you think that there is some sort of correlation (perhaps a causal relation) between your reason and your thesis, but you also think there are some exceptions to this connection. For instance, heavy smoke does not always indicate fire. Sometimes it just indicates lots of friction (like when your car's radiator stops working and smoke pours out from underneath the hood), and sometimes it just indicates smoldering (like when you dump water on a fire). Putting the word 'probably' in Step 3 indicates that you're aware of such exceptions but you think they are irrelevant to the case at hand. And putting it in Step 5 indicates that, although you're confident that your thesis is more likely true than not, you're not absolutely certain that it is true (because it is possible that your thesis could be false even though your reason is true).
- Because REASON.
- And whenever REASON, then THESIS.
- Example in which both REASON and THESIS are true.
- Therefore, THESIS.
Examples 2, 3, and 5 illustrate this pattern. For the most part, all mathematics proofs use this pattern. You should use this pattern whenever the connection between your reason and your thesis is a matter of definition, or whenever the connection is a consequence of some general theory (like arithmetic or biology). You also can use this pattern when, try as you might, you can think of no exceptions to the connection. (When there are exceptions, you might retreat to Pattern 1.) A good rule of thumb is to use Pattern 1 when the connection between your reason and your conclusion is based upon some causal connection in the world (because causal connections tend to have exceptions), and to use Pattern 2 when the connection does not turn on facts about the world (because, in such cases, exceptions are less likely) or when the connection is vouched for by some well-established mathematical or scientific theory (as in Examples 2 and 3).
Sometimes it is difficult to think of examples in which both your reason and your conclusion are true. (That is, sometimes it is difficult to complete Step 4.) Consider Example 4. Nuclei are not very easy to observe. So it would be very difficult to give an example of some other kind of atom that has a nucleus. Likewise with Example 6: since souls are unobservable, it is impossible to give any examples of things that have souls. Also, sometimes the only example you can give involves a case in which you take your thesis for granted. For instance, in Example 7, any example of a connection between complexity and an Intelligent Designer would have to presuppose that an Intelligent Designer exists. Whenever you encounter any of these problems, you should avoid using Patterns 1 or 2, because if people do not already accept your thesis, they won't find Step 3 of your argument convincing (and the point of giving a proof is to convince people who don't already agree with you). Instead, use the following pattern:
- Because REASON.
- And THESIS is the best (or only) explanation of REASON.
- Considerations for why alternative accounts of REASON are explanatorily inferior to THESIS.
- Therefore, probably THESIS.
In this kind of proof, the connection between your reason and your thesis is an explanatory connection rather than a causal or mathematical or definitional connection. (This is why these kinds of proofs are known as "inferences to the best explanation.") The idea behind this kind of proof is that the superior explanatory power of a thesis is good evidence that the thesis is true. One motivation for this idea is science: the theories that we currently accept as true tend to be the ones that give better explanations than any alternative theories. Another motivation is everyday reasoning. For instance, if you go to two different psychologists and they offer competing diagnoses of what your problems are, then if you can only afford to continue going to one doctor, you'll probably stay with the doctor who, in your opinion, gives the best explanation of your various psychoses. Doctors do this too: oftentimes your symptoms are compatible with a variety of ailments, and in deciding how to treat you they will opt to treat the ailment that best explains your symptoms.
There are no hard and fast criteria for deciding whether one theory is a better explanation than some other theory. (This means that there is no good algorithm for how to complete Step 4.) A good rule of thumb is to look at the simplicity, scope, and depth of the theories. Simplicity is a function, among other things, of how many things a theory postulates to exist, how many exceptions there are to the theory, and how well the theory fits in with generally accepted knowledge. The simpler a theory, the more explanatory power it tends to have. Scope is a function of how many facts a theory explains. A theory that explains everything explained by its competitors plus more besides tends to have more explanatory power than its competitors. Depth is a function of the level of detail of the explanations. Theories that give the details about why something is true tend to have more explanatory power than theories that don't. All of this means that there are three different ways to complete Step 4:
- Show why your thesis is simpler than the alternatives.
- Show why your thesis explains things that the alternatives do not.
- Show why your thesis explains things in more detail than the alternatives.
Examples 6 and 7 use the middle strategy. The general idea behind proofs that use Pattern 3 is to complete Step 2 in a way that makes completing Step 4 easy. The reason that Step 5 contains the word 'probably' is that one can never be sure that no one will ever come up with a better explanation in the future. (The history of science is littered with examples to illustrate this point.)
Regardless of which pattern you use in constructing a proof, the hardest steps are the second and fourth. The first and last steps are easy to complete once you decide what it is you want to prove and how confident you are about being able to prove it. The third step merely requires filling in one of the following schemata (and making appropriate adjustments to grammar):
- Scheme 1: Whenever [REASON, from Step 2], then probably [THESIS, from Step 5].
- Scheme 2: Whenever [REASON, from Step 2], then [THESIS, from Step 5].
- Scheme 3: The best explanation for why [REASON, from Step 2] is that [THESIS, from Step 5].
Even though Step 3 can be completed on autopilot, it is important nonetheless, because it is the way for you to indicate to other people the connection you want them to make between your reason and your thesis. No good proof should leave the audience asking why you gave that reason for your thesis.
The second step is probably the one that requires the most thought. The best strategy for completing Step 2 is to ask yourself: "Why do I believe the thesis?" Make a list. These are your potential reasons. The longer, the better. If you have trouble, ask other people. Once you have a list of potential reasons, ask of each one: "Is this a reason that only I would find convincing, or would other people find it convincing too?" Cross off anything that you think other people won't buy. And, from the ones remaining, select the one that you think is the best. If you can't decide, make a different proof for each reason. It never hurts to have more than one proof of the same thesis.
Once you're figured out a reason for your thesis, complete one of the schemata for Step 3 and move on to Step 4. (If you have had previous exposure to the study of arguments, you'll notice that textbook treatments of arguments tend to omit this step. And even if you don't have prior exposure, you might be wondering what the point of this step is.) Strictly speaking, there can be good proofs that omit Step 4. But as a practical matter, completing this step decreases your chances of saying something unpersuasive without realizing it and increases the likelihood that other people will understand your proof. Thinking of examples is a good way to test the plausibility of the connection you're making between your reason and your thesis. If it turns out that it is very easy to find examples suggesting that the connection is mistaken, chances are that other people will find those examples too--in which case they won't be convinced by your argument, and probably you should think of a different reason for your thesis. For instance, if your reason for thinking that a person is dead is that their eyes are closed, you don't have a very convincing proof, because it is easy to think of cases (such as sleeping people) in which there is no connection between closed eyes and being dead. And if your reason for thinking that God exists is that the Bible says so, you don't have a very convincing proof, because there are many other books that say that God does not exist and you've given no reason for thinking that the Bible is special). Again, if your reason for thinking that eating pork is immoral is that the Koran says so, you don't have a very convincing proof, because there are religious texts that say eating pork is not immoral (perhaps the Bible) and you've given no reason to think that these other texts are mistaken.
Also, if it turns out to be incredibly difficult to complete Step 4, other people probably will not be persuaded by your proof. After all, if you can't easily come up with an example, chances are your audience can't either, in which case they'll not really understand what you're saying--and it is a fact about human nature that people tend not to have their minds changed by reasons they find unintelligible. For instance, if your reason for thinking that you will win the lottery is that the moon is eclipsing the sun, other people probably won't find your argument convincing, because they probably have no idea what celestial events have to do with the lottery. When you construct a proof, the five steps need not occur sequentially. In writing a term paper, for example, you might accomplish Step 1 in an introductory paragraph and devote a paragraph in the body of the paper to Steps 2 through 5. It is a good idea to do Steps 2 through 5 together--if they're not completed in the same paragraph, at least make sure that they are completed in adjoining paragraphs. This makes your argument much clearer and easier to follow. And it decreases the likelihood that someone else will misunderstand what your argument is or, in the worst case, not be able to figure out what your argument is. It is a good idea to make sure that you can complete Step 1 with one sentence. This helps to ensure that you end up proving what you intend to prove, because it forces you to be clear and precise about what your thesis is. It is also a good idea to spend some time explaining exactly what you mean to be saying. This helps to prevent unnecessary confusion of your audience.
Knowing how to construct proofs is useful in understanding arguments that are not your own. People often do not make their arguments fully explicit. And sometimes it is difficult to figure out what a person's argument is. The five-step procedure for constructing proofs suggests a natural procedure for figuring out what proofs other people are giving. This procedure involves asking four questions.
- Question 1: What is the thesis being argued for?
- Question 2: What is the reason given for the thesis?
- Question 3: What is the connection between the reason and the thesis supposed to be?
- Question 4: What is an example of this connection?
If you are able to answer these questions, you should have a good understanding of how the argument you are reading is supposed to go. And answering the fourth question with an example that the author does not give often produces a deeper understanding of what the author was thinking in giving the argument. Also, answering these questions makes it easier to refute a proof of an argument with which you disagree, because a necessary first step in constructing a refutation of a proof is figuring out what that proof is, and answering these four questions allows you to "reverse engineer" the author's proof.
Advanced Technique: Another Proof Pattern
Patterns 1, 2, and 3 do not exhaust the kinds of proof one can construct. Each of these patterns gives a positive reason for thinking that a thesis is true. But it is not always easy to give a positive reason. Sometimes the best that can be done is to argue that not accepting your thesis has a bad consequence. For instance, one way to prove that gay marriage should not be legalized is to argue that if gay marriage were legal then marijuana would have to be legal too. This kind of proof is called "reducing the opposition to absurdity." These arguments instantiate the following pattern:
- Because REASON.
- And if THESIS were false, REASON would be false too.
- Example in which both REASON and THESIS are false.
- Therefore, THESIS. Consider, for example, the proof about gay marriage.
- Gay marriage should not be legal.
- Because marijuana should not be legal.
- And if gay marriage were legal, marijuana would be legal too.
- For example, Denmark allows both gay marriage and marijuana smoking.
- Therefore, gay marriage should not be legal.
This kind of proof is common in mathematics. The point is to show that denying your thesis requires that your audience reject something they do not want to reject. In essence, the proof appeals to the constraint that rational people should have consistent beliefs, and shows that consistency demands that one's audience accept one's thesis. The general strategy for constructing this kind of proof is to ask: "What would be wrong with saying that my thesis is false?"
How to Build a Refutation
When you disagree with the thesis of someone's proof, the rational thing to do is to construct a refutation of that proof. The first step in constructing a refutation of a proof is figuring out what that proof is. And the best way to do this is to reverse engineer the proof in order to identify the proof's reason for its thesis and the connection between reason and thesis. Once these parts of the proof are identified, you should "reconstruct" the proof to show how it completes each of the five steps discussed above. This is the second step in refuting a proof. Completing this step often stimulates ideas about what is wrong with the proof. And it ensures that you do not refute a proof that was not given and thereby fail to accomplish what you set out to do. (Doing such a thing is known as refuting a straw man. Politicians are notorious for doing this, and for sometimes doing it intentionally.)
The third step of a refutation involves examining the reason given for the thesis. (That is, examine Step 2 of the reconstructed proof.) You should ask: "Is the reason true?" If you think it is, move on to the next step. If you think it is not, then ask a further question: "Is it generally accepted that the reason is false?" If it obvious, to all parties in the debate, that the reason is false, then your refutation of the proof consists in pointing this out. (For example, you might say "The proof relies upon the claim that [REASON]. But this is false.) Chances are that your refutation of a proof will not be this easy. If it is, this is more likely an indication that you have misunderstood the argument--go back to the first step and find a reason that is not obviously false.
If the reason the proof gives for its thesis is not generally accepted as false, but if you think it is false nonetheless, then you need to construct a proof in which the thesis is the negation of the reason in the proof you are trying to refute. For instance, one way to refute the proof in Example 6 is to construct a proof for why people do not retain their identities despite undergoing changes. And one way to refute the proof in Example 7 is to construct a proof that there are not irreducibly complex organisms in the world. Your goal in constructing such refuting proofs should be to appeal to reasons that your opponent would accept as true. The cardinal rule is that you cannot appeal to the falsity of your opponent's thesis. (Doing so is called "begging the question.") This is disallowed, because the opponent has given a proof for why their thesis is not false, and it is unfair to merely assume that their proof is no good.
If you can construct a refutation of a proof's reason, then your refutation of the proof is complete. If you cannot do this, you must proceed to a fourth step in which you examine the connection between the reason and the proof's thesis. It is not unusual to have to proceed to this further step. Most proofs are strong in the reason they give for their thesis but weak in the connection they make between the reason and the thesis. Also, sometimes it is not clear how to refute a proof's reason until you have attempted to refute the connection between the reason and the proof's thesis.
When examining the connection between a proof's reason and its thesis, you should ask the question: "Is it true?" If you think it is, then you need to reassess your disagreement with the proof's thesis, because it would be irrational of you to accept the connection and the reason but reject the thesis: accepting the connection and the reason rationally commits you to accepting the thesis. It is okay to revise your initial assessment of a proof. After all, the whole point of giving proofs is to convince people who do not agree with you to change their mind, and perhaps the proof you're considering has convinced you. But if you don't want to be convinced, then you must go back to the third step in refuting a proof and try to prove that the reason given for the thesis is false.
If you think that the connection between a proof's reason and its thesis is incorrect, then what to do next depends upon the kind of connection you're considering. Recall that there are three different kinds of connection, each of which follows a different scheme:
- Scheme 1: Whenever [REASON], then probably [THESIS].
- Scheme 2: Whenever [REASON], then [THESIS].
- Scheme 3: The best explanation for why [REASON] is that [THESIS].
The method to follow in constructing a refutation of a proof is highly sensitive to which of these schemes the proof invokes. If the proof you are trying to refute uses Scheme 1, then you need to argue that the situations in which both the proof's reason and its thesis are true are exceptional situations. That is, you need to construct a proof for the thesis "The situations in which both REASON and THESIS are true are exceptional." One way to do this is to try to list all the situations in which THESIS is true, and then point out that many of these situations are ones in which THESIS is false. So your refuting proof should have the following pattern:
- The connection between REASON and THESIS is mistaken.
- Because many situations in which REASON is true are ones in which THESIS is false.
- And if many situations in which REASON is true are ones in which THESIS is false, probably the connection between REASON and THESIS is mistaken.
- Instead of giving an example that illustrates Step 3, give the examples that support Step 2.
- Therefore, the connection between REASON and THESIS probably is mistaken.
For example, consider the proof that Pluto probably is not a planet because it is too close to the sun. As noted, a good refutation of their argument is pointing out that there are celestial objects much closer to the sun than Pluto that, despite their proximity, are planets. This refutation attacks the connection between solar proximity and planetary status. It fits the following pattern:
- The connection between solar proximity and planetary status is mistaken.
- Because there are many planet-like objects closer to the sun than Pluto, and such objects are planets.
- And if many planet-like objects closer to the sun than Pluto are planets, probably the connection between solar proximity and planetary status is mistaken.
- For example, consider Mercury, Venus Earth, Mars, and Saturn.
- Therefore, the connection between solar proximity and planetary status is mistaken.
The general idea in trying to refute Scheme 1 connections is to show that a proof's reason does not make its thesis more likely than not.
If the proof you are trying to refute uses Scheme 2, then you need to argue that there is at least one case in which the proof's reason is true but its thesis is false. This does not require the construction of a proof. It merely requires giving an example. This is why this method of refutation is known as "refutation by counterexample." For instance, to refute the connection in Example 3, you need only give one instance of something that is warm-blooded but not a mammal. An important fact to note here is that it is much easier to refute proofs that use Scheme 2 than it is to refute proofs that use Scheme 1. Even though the theses of proofs that use Scheme 1 are more certain than the theses of proofs that use Scheme 2, the Scheme 1 proofs are more fragile because they are easier to refute. (For this reason, if you make a proof that uses Scheme 1 and someone refutes it, a good fallback strategy is to revise your original proof into one that uses Scheme 2.)
If the proof you are trying to refute uses Scheme 3, then you need to argue that there is a theory that is a better explanation of the proof's thesis. There are many ways to do this, and no simple way to explain how to do this. You might invent a better explanation or bring into consideration an explanation that the proof omits. (For this reason, if you make a proof that uses Scheme 3, it is important to consider as many possible alternative explanations as possible.) You might argue that, even though an alternative explanation does not explain what the thesis explains, the alternative is a superior explanation in other respects (such as simplicity or scope). You might even argue that the proof's thesis does not explain what it purports to. The general idea is to show that the proof's thesis is not among the best available explanations. To summarize, there are four steps in constructing a refutation of a proof.
To summarize, there are four steps in constructing a refutation of a proof.
Step 1: Figure out the
reason the proof gives for its thesis and the connection between this
reason and the thesis.
Step 2: Reconstruct the proof.
Step 3: Examine the reason the proof gives for its thesis.
- If you disagree with the reason and the reason is obviously false, probably your completion of Step 1 is incorrect. Go back and try again.
- If you disagree with the reason and the reason is not obviously false, construct a proof for the negation of the reason. Do not assume the falsity of the thesis whose proof you are trying to refute.
- If you agree with the reason, proceed to the next step.
- If you agree with the connection, either reassess your opposition to the proof's thesis or return to the previous step.
- If you disagree with the connection, then show that the connection is mistaken. Adjust your method for doing this to the kind of connection the proof uses.
Making Proofs Persuasive
The obvious purpose of constructing a refutation of a proof is to refute proofs of theses with which you disagree. But constructing refutations has another, less obvious purpose: it can make your proofs better. Your goal in constructing a proof of some thesis should not merely be coming up with any proof whatsoever. Your goal should be to construct a proof that other people will find persuasive. And the best way to figure out whether you've succeeded in constructing a persuasive proof is to ask yourself: "How well does my proof stand up against refutations?" If it is easy to show that the reason you give for your thesis is false, or that the connection between your reason and your thesis is mistaken, then your proof is not very persuasive (because it easily refutable) and you should try again. The harder it is to come up with a refutation of your proof, the more persuasive the proof is.
Assessing the susceptibility of your proof to refutation requires skill and impartiality. If you make a proof in good faith, chances are that you think that the proof is not at all susceptible to refutation. (Otherwise, what would be the point of offering the proof in the first place? You'd just be wasting everyone else's time.) So when you ask how easy it is to refute your proof, you need to be impartial: you have to pretend that you don't already agree with your thesis. And doing this requires some skill. Probably the best strategy to practice is to ask: "What would an opponent probably say in trying to refute my argument?" If you can anticipate these kinds of objections, you can take them into account in strengthening your proof. For instance, you might modify your reason or the connection between your reason and your thesis. Or you might construct an additional proof for why a certain kind of refutation is no good. (Think of this as a sort of pre-emptive strike designed to shield your proof from the obvious strategies people might adopt for refuting it.) For example, if you can anticipate how someone might try to prove that your reason is false, you can sketch how that proof might go and then refute it. Being able to do this is a sign of philosophical sophistication. And it has the psychological effect of making your audience take your proof more seriously (perhaps because they pick up on the fact that you take your proof seriously).
In assessing the persuasiveness of your proof, keep in mind that the weak point of a proof tends to be the connection between its reason and its thesis (Step 3 in constructing a proof). This holds true of your proofs just as much as it does of other people's proofs. It is relatively easy to state what the connection is supposed to be: just fill in one of the schemata for Step 3. But it is much harder to construct a proof that has a hard-to-refute connection. So even though the hardest part of constructing a good proof is thinking of a reason for your thesis, the hardest part of constructing a persuasive proof is thinking of a reason that is persuasively connected to your thesis.
One good strategy to follow in constructing a persuasive proof is to devote a lot of mental energy to trying to refute the connection between your reason and your thesis. The more effort you put in to trying to refute your proof, the more effort other people will have to put in. And the more effort you devote to thinking about the connection between your reason and your thesis, the more you will have to say to your audience about why they should not reject the connection. Anticipating refutations is a necessary first step in pre-empting those refutations: once you figure out the obvious objections, you can say something about why those objections are not persuasive. (This is why it usually is wise to devote a separate paragraph to Step 3 of your proof, in which you anticipate the obvious objections and show why your audience should not find them persuasive.) Also, sometimes looking for refutations of your connection suggests a more persuasive proof for the same thesis, based upon a more plausible connection between your reason and your thesis.
Another good strategy to follow in constructing a persuasive proof is to use a connection that is vouched for by some well-established theory, such as arithmetic, physics, psychology, or some other generally accepted theory. The idea in basing your connection on a well-established theory is that if someone wants to refute your proof, then the only way they can refute the connection you make between your reason and your thesis is to argue against that well-established theory--and this is hard to do, analogous to swimming against a strong current. Of course, this is only effective if your audience already accepts the theory to which you appeal. If they don't, then you'll have to construct a proof for why that theory is true. For instance, traditional Christian theology can provide powerful reasons and connections for theses about ethical behavior. But it only does this for those who already accept that theology. Audiences that reject the theology will have pre-set refutations of any proofs relying upon the theology, and such proofs will fail to be persuasive if they are not backed up with proofs of the traditional theology and persuasive refutations of standard objections.
Purpose of this Guide
When one consults philosophy or logic textbooks for advice about argumentation, one tends to find discussions of the difference between deductive and inductive arguments, catalogs of valid and invalid inference forms, and lists of informal fallacies. But one tends to find very little discussion of how to go about constructing an argument of one's own beyond "Make sure your argument avoids all the fallacies and conforms to one of the valid inference forms." This is good advice, so far as it goes. But, unless one is prepared to master memorize the different inference forms and master informal fallacies, it conveys no appreciation of what makes good arguments good and bad arguments bad and thereby conveys no sense of how to go about making good arguments and criticizing bad ones. This is unfortunate, because being able to construct good proofs and refutations does not require the tools usually given to undergraduates. The purpose of this Guide is to remedy this deficiency by giving newcomers to the art of reasoning methods for constructing good proofs of their ideas and good refutations of ideas with which they disagree. (Scholars of Indian philosophy will notice that the procedure for constructing proofs given in this Guide adopts and extends the rules of argument set forth by the Nyaya philosopers.)