Proof by Example

Proof by Example


B: But enough about ad hominem already.  Let’s talk about some other common logical fallacies.  Although I should mention something first: it’s not my intention to turn this into some convoluted philosophical discussion, where we wonder about existential things like “isn’t the statement that eternal truth either exists or it doesn’t in itself a question about eternal truth?”  Nor is it necessary, in my opinion, when we start talking about laws of a society, to turn the discussion into a bunch of technical legal documents.  I believe that most of what we talk about will be fairly clear from the context of previous discussions and the society in which we live.  What I mean is, I’m going to mention some assumptions when I feel it’s necessary for clarity – but I’m not necessarily going to mention every little detail, just to make sure that I’m being consistent with all possible logical dissensions.

A (laughing hysterically): Buddy, your voice is so weird!

B: Wha—?  Rude!  Wouldja FOCUS, buddy?!  Anyway, it’s no more unique than yours is.

A (laughing all the more): Is that your readymade response to everybody who makes fun of your voice?

B: Has been ever since I was a kid.  But let’s get back to logical fallacies, shall we?  I’ll start the next one with another math example (don’t worry, we’ll get away from the math soon enough): beginning with the number 1 and going up, what is the first even number one encounters?

A (finally recovering from fits of laughter): Oh dear!  My sides!…2, of course.

B (sighing impatiently): Is it a prime number?  That is, is it evenly divisible only by itself and 1?

A: ’Deed it is, son.

B: So we have this theorem: 2 is an even prime number.  What statements can we prove using this theorem?  How about the statement “it is possible that an even number is prime?”  Yes, the example of 2 being an even prime number suffices.  But how about the statement “all even numbers are prime?”  No, the example of 2 being an even prime number does not imply that all even numbers are likewise primes.

Therefore: “proof by example,” which I just used to prove the statement that there is an even number that is prime, is nevertheless, in general, a logical fallacy.  It works for some “existence” statements like the one about whether there’s a prime even number.  But just because one has an example, or perhaps several or even countless examples, for which the statement is true, does not mean that the statement is true for every single case.

A: It is interesting to note, however, that one can prove that your second statement, that of “all even numbers are prime,” is false, by the use of an example.  One need only look at the next even number on the list; that is, 4.

B: ’Deed, dude.  Please be certain, however, to note that proof by example may not be available for either the truth or falsity of a given statement.


A: So wait…aren’t you basically claiming that scientific theories such as the one about gravity are actually not true, since there’s only evidence (or “examples”) for those?

B: I’m not saying that such theories aren’t true, I’m just saying there’s no proof for those theories, since, as you parenthetically pointed out, they consist of a whole bunch of examples that are collectively referred to as “evidence.”  We’ll talk later about science and proofs.

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