What is a Proof?

“What is a Proof?

What Constitutes a Good Argument?

A proof is an argument. An argument is a set of at least two propositions, one of which is affirmed on the basis of the other(s), The proposition that is asserted on the basis of the other(s)s is called the conclusion. The proposition(s) that serves as the ground for the conclusion is the premise (premises). In order to quality as a proof, an argument should satisfy two conclusions.

  1. The form or logical structure of the argument must be valid. Mixed hypothetical syllogisms may take four possible forms, two of them valid and two of them invalid. The two valid forms have been given names and looks like this:A: Modus Ponens
    If p, then q
    Therefore, q

    B. Modus Tollens
    If p, then q
    not q
    Therefore, not p

    Any argument that has the form of either modus ponens or modus tollens is valid, which is simply to say that if the argument’s premises are true, then its conclusion must also be true. If an argument has one of these two forms, it is impossible for that argument’s premises to be true and also for its conclusion to be false.

  2. The two invalid forms of mixed hypothetical syllogism have also been given names and look like this:C. The Fallacy of Affirming the Consequent
    If p, then q
    Therefore, p

    D. The Fallacy of Denying the Antecedent
    If p, then q
    not p
    Therefore, not q

The mixed hypothetical syllogism is worth special attention here because other argument forms can be reduced to it. Take our old stand-by argument about Socrates:

P1. All men are mortal
P2. Socrates is a man
C. Therefore, Socrates is mortal

The argument is what we call a categorical syllogism and has the following form:

P1. All C is B
P2. All C is A
C. Therefore, All C is B

Another way of putting this argument-form is saying that if the two premises are true, then the conclusion must be true. If we lump the two premises together and call them p, we’re really saying that if p is true, then q (the conclusion) is true. In other words, the inference in any argument from the premises to the conclusion takes the form of a hypothetical statement in which the antecedent clause (if p) sums up all the premises and the consequent clause (then q) stands for the conclusion. One thing meant by validity, then, is this: if the premises of an argument are true, then the conclusion must be true.

But of course every student of logic knows that an argument may be considered formally valid and still contain one or more false statements. Consider the following:

P1. All dogs are reptiles.
P2. All reptiles lay eggs.
C. Therefore, all dogs lay eggs.

Even though the argument is formally valid, the conclusion and at least one of the premises is false. Because truth and validity are different properties, students of logic talk about a second test that any good argument must satisfy: it must be sound. In order for an argument to be sound, the argument must be valid, and the propositions that make up the argument must all be true.

What Constitutes a Good Proof? 

The criteria of a good (that is, a sound) argument are beyond dispute. The argument’s propositions must be true, and its inferences must be valid. Note that these characteristics are really independent of psychological or personal factors such as whether an individual’s personal history predisposes him to view the argument fovorably or unfavorably. Does the mere fact that a collection of claims and inferences constitute a good argument automatically make that argument a good proof?

The answer to this important question is complicated by the fact that the phrase good proof can be used in different ways. On the one hand, we could say that a good proof is a good (sound) argument that any reasonable person should accept. We might call this the objective notion of proof. While in many contexts, this is a perfectly legitimate notion of proof, it fails to capture the essence of what we take to be proving something in other contexts. Most of us, I imagine, have been in situations where we offered perfectly good arguments (or so we thought) that perfectly reasonable people failed to accept. Was there anything wrong with the people who were unconvinced by our argument? It would be hard to justify such a conclusion in all such cases. Or was there something wrong with our argument? Not if it really was sound.

Fortunately, there is another way of understanding the notion of a good proof. In this second sense, we can say that a good proof (in this second sense), an argument must not only satisfy certain logical criteria, it must also meet an imprtant psychological test; it must actually suceed in persuading someone to accept the conclusion. Consider the strangeness of a situation where a person responds to an argument by saying, “Although you have offered a good proof for your position, I remain unpersuaded.” In this subjective sense of proof, any argument (even a good one) that fails to persuade its targed audience falls short of being a good proof. A good proof is an argument that works.

Note  several other points about this subjective notion of proof. For one thing, proofs are person-relative. This claim actually says two things.

(1) Proofs are relative, which is simply to admit the obvious, namely, that the same argument may function as a proof for one person and result in little more than contempt from someone else

(2) Proofs are relative to individual persons. Even when an argument is directed to some large audience, the people in that audience must always respond as individuals. And their response will reflect varying features in their past and present personal history. In fact, we could take this point even further and state that proofs are relative to individual persons in particular circumstances. Had someone presented one of the more arguments for God’s existence to me when I was too young for too unprepared to appreciate it, the argument would undoubtedly have failed as a proof. Before an argument can function as a proof, any number of conditions must be satisfied. The person must understand what is being said; he must “see” that the key claims in the argument are true; he must believe that the argument is sound; and he must not have a strong emotional aversion to claims made in or implied by the conclusion.

All of this is to say that proofs must pass tests that are both logical and psychological. No argument can become a proof for some person until it persuades that person. In the real world, unfortunately, the logical and psychological requirements we have noted often get separated. While many perfectly good arguments failed to persuade large numbers of people, many perfectly bad arguments persuade people by the millions…So let us agree that no proof (that is an argument that has persuaded someone) can be a good proof unless it is also a good argument. But our analysis also forces us to admit that no good argument can also be a good proof unless it also persuades someone to accept its conclusion.

Given the person-relative nature of proofs, then, it seems highly unlikely that there is any such thing as a proof for God’s existence that will convince everyone.”

(Nash, Faith and Reason, 106-110)


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