If this has a factual answer maybe someone with more math and logic knowledge than myself will answer it for me. If not, move this to whatever forum is appropriate mods.
Are conditional statements and their contrapositives logically equivalent?
We are taught that they are and their truth values are obviously the same (or at least it seems obvious to me). I am bothered by the following example that I read in one of Martin Gardner’s books (Aha! Gotcha I believe).
A scientist wishes to prove the hypothesis that “all crows are black.”
The conditional for this is:
If an object is a crow then it is black.
He looks at a crow, it is black. This is a confirming instance of his hypothesis. The more crows he sees that are black without seeing any nonblack crows the more his hypothesis is confirmed.
The contrapositive for this is:
If an object is not black, then it is not a crow.
He looks on his kitchen counter, sees an object that is not black and it is not a crow (it is a cookie jar). This is a confirming instance of "all nonblack objects are not crows. Is it also a confirming instance of “all crows are black”?
If it is not, how can we consider the statements logically equivalent?
IANALogician, so I have a question. It sounds like “confirming instance” is a technical term used in logic. What’s its definition? I’m having trouble finding it used in any other context than this very one. For instance:
Hempel’s Paradox: A purple cow is a confirming instance of the hypothesis that all crows are black.
(Based on this MathWorld entry I would guess that the answer to your question is “yes”, but I can’t be sure.)
Conditional statements and their contrapositives are logically equivalent. Think about the statement “if a, then b” (e.g. if it’s a crow, it’s black). When is this true, and when is it false? Look at all four cases:
a true, b true - true (e.g. black crow)
a true, b false - false (e.g. yellow crow)
a false, b true - true (e.g. black cat)
a false, b false - true (e.g. yellow cat)
Now, think about “if not b, then not a” (e.g. if it’s not black, then it’s not a crow). The truth values are the same in all cases, so the statements are all logically equivalent.
If you’re worried about confirming evidence, then you’re worrying about probabilities (inductive reasoning), not logic (deductive reasoning), so it may be more practical to apply the original proposition than its contrapositive. If you observed all non-black things in the world and saw that none of them were crows, you would have proven the statement that all crows are black by proving the contrapositive. This isn’t a very good strategy, though. You’re better off looking at crows, or looking in places where you’re most likely to find a non-black crow.
Logically, the statements are indeed equivalent. What Knock Knock says is absolutely correct, however. You may not confuse inductive reason that establishes an empirical law with deductive reasoning that begins with established truth (that is tautologies) and infers other truths (likewise tautologies). Just because Fermat’s last theorem (or even Fermat’s little theorem that says that if p is a prime number and a is any integer, then a^p - a is divisible by p) is a tautology does not mean it is obvious.
The question raised in the OP is a serious one for philosophy of science and, AFAIK, very far from being resolved. FWIW, the statement, “All crows are black”, which is equivalent to, “All non-black things are non-crows” says more about crows than about non-black things. But that is another empirical fact, not a logical one.
They are logically equivalent in a purely truth-functional sense. This is all that matters in standard logic. Natural language, on the other hand, is never purely truth-functional.
I disagree. The assertion “Set A (crows) is a subset of B (black bodies)” is most certainly equivalent to “everything which is not in set A (crows) is not in set B (black bodies)”. That is pretty obvious to me.
I’m sure you meant “most certainly not equivalent.” Either way, he is not saying all non-crows are not black. He is saying all non-black things are not crows.
“If it is not black, it is not a crow.” This sounds like a definition, so that if you come upon a white (albino) crow, you could just say “Well that can’t be a crow”. I don’t like that.
Does contrapositive mean a statement intended for a test? rather than a definition? Ie, when you come upon an albino crow, or a featherless crow, it breaks the test, and you now can say “Not all crows are black” as a conclusion.
Actually unfledged baby crows break the conditional statement too, “If it is a crow, it is black.”
So catching up to the OP, we’re talking logic here, not definitive descriptions of crows. OK, now I get it, I think.
Thank you. I guess this is what bothers me. Two statements that are logically equivalent seem to say different things (or the same thing to different degrees).
Thank you also. I think this may be the best explanation that I have heard, although I am not sure how much of the problem is natural language and how much is the difference between scientific methods and pure logic.
If I saw a lot of crows and they were all black, I would be inclined to accept the theory that all crows are black. I am not sure how many non-black non-crow items (short of all of them) it would take to convince me of the same thing.
A little probability can help here. To simplify things, assume there are 1,000,000 crows, one of which might be non-black. If you look at 500,000 crows, you’ve got a 50 percent chance of seeing the non-black crow.
Now consider how many non-black objects there are, just on the face of the Earth. Let’s say it’s 1 trillion (10[sup]12[/sup]). You’d have to look at 500 billion non-black objects to get the same 50 percent chance of seeing that one non-black crow. So each non-white object is only one millionth as effective at showing all crows are black as each black crow.
On the other hand, if you somehow only had a thousand non-black objects and a million crows, you’d be better off looking at the non-black objects.
I think that looking at things this way, you can see some of the non-intuitiveness of logic. Okay, so if there are 1 million crows, then no matter whether 1 or 5000 or all 1 million of them are grey, the statement “All crows are black” is equally wrong in (Boolean) logic. But we would tend to think that if no crows are black, the statement is “more wrong” than if 99.9999% of crows are black.