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How can we reason about “if P then Q” or “P only if Q” statements in propositional logic?

Are both forms of arguments correct? If not then why? To me both seem correct because If John had a gun then he shot James and if he shot James then he must had a gun.

First form:

If John shot James, then John had a gun.
John had a gun.
Therefore, John shot James.

Second form:

If John shot James, then John had a gun.
John shot James.
Therefore, John had a gun.

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marked as duplicate by Joseph Weissman Oct 30 '12 at 18:13

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

3 Answers

In the first form, The second premise doesn't necessarily mean that John shot James. He may just had a gun for some other reason. You can see this by translating this argument as:

Suppose, S = John shot James; G = John had a gun.

The first form of argument is incorrect because you cannot draw any conclusion.

1. S ⊃ G
2. G
∴ No conclusion.

The second form of argument is correct.

1. S ⊃ G
2. S
∴ G {from 1 and 2; Using modus ponens}

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Take your first form:

If John shot James, then John had a gun.
John had a gun.
Therefore, John shot James.

So write P for "John shot James" and Q for "John had a gun". Then your proposition reads:

If P, then Q.
Q.
Therefore, P.

The most important feature of propositional logic is that it doesn't look at the meaning of the sub-arguments. Therefore you can test your claim by replacing the used propositions by arbitrary others. Let's use:

  • P = "a and b are positive integers whose product is 5"
  • Q = "a and b are positive integers whose sum is 6"

Now the first line of your first form reads:

If a and b are positive integers whose product is 5, then a and b are positive integers whose sum is 6."

This is clearly a true statement, because the only positive integers whose product is 5 are 1 and 5, and the sum of those is 6.

The second line of your first form reads:

a and b are positive integers whose sum is 6.

The truth of this statement depends on the values of a and b, but it is easy to choose a and b so that this statement is true; say a=2, b=4. Note that this choice of a and b does not invalidate the first statement.

The last form of your first form reads:

Therefore, a and b are positive integers whose product is 5.

But with a=2 and b=4, the product is 8, not 5. Therefore if that form were valid, you could derive a false statement. Thus the form is not valid.

However note that the following form is valid:

If John shot James, then John had a gun.
John had no gun.
Therefore John did not shoot James.

Your second form is, of course, valid. It is a direct application of the most basic rule of reasoning, modus ponens.

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To me both seem correct because If John had a gun then he shot James and if he shot James then he must had a gun.

Beside the technically correct answer by cpx, your original question seems to be about the seemingly reasonable possibility of inferring from the fact that John shot James that John must have had a gun.

There are two kinds of inferences that you might have in mind here:

  1. A special kind of non-truth preserving inference called abduction. In fact, abduction is formally equivalent to the logical fallacy displayed in your first example, i.e. affirming the consequent.

    In order to put this kind of inference under rational constraints, it is usually formulated in a probabilistic framework. Note that the modal operator ("must") in your statement does not convey a necessary condition, but only high probability. If John shot James, it is probable that John had a gun.

    For more see http://en.wikipedia.org/wiki/Abductive_reasoning.

  2. You may wanted to express the thought that John shot James if and only if John had a gun. In this case (S → G) ∧ (G → S) holds.

    For more see http://en.wikipedia.org/wiki/Logical_biconditional.

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Whether having a gun is necessary for shooting someone depends on the definition of "having" and "shooting". You can simply define them so that "having" means "the gun is under your control" and "shooting" means "ejected a bullet from a gun into". Then there's no abduction. If John stuck a bullet in a metal pipe and heated it up with a blowtorch, and the bullet hit James, John would not have shot James in this case. I suppose my point is only that one needs to be aware of how one is using language with reasoning like this. –  Rex Kerr Oct 28 '12 at 19:08
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