It seems to me that there's a problem with it, and we're right back at what I've been harping on in other contexts: for one, we're ignoring the semantic content of the statements. But can we really do that when we're talking about whether evidence for one is evidence of the other? — Terrapin Station
That they're logically equivalent is that they have the same semantic content. — Michael
Logic is about form, not semantic content. You're arguing otherwise? That logical form is identical to semantics? — Terrapin Station
In logic, contraposition is an inference that says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped: the contrapositive of P → Q is thus ¬Q → ¬P. For instance, the proposition "All bats are mammals" can be restated as the conditional "If something is a bat, then it is a mammal". Now, the law says that statement is identical to the contrapositive "If something is not a mammal, then it is not a bat."
First you didn't quote that part. I asked "What part of what you quoted . . ." — Terrapin Station
Understanding is different than agreement. — Terrapin Station
I don't agree that logic deals with semantics other than formally. I had said that above. — Terrapin Station
It depends on who is assigning meaning to those statements and what meanings they're assigning, doesn't it? — Terrapin Station
Yes, I can readily accept that statement. What I'm having trouble with is finding a reason to believe the antecedent - that observation of a black raven is evidence for the proposition that all ravens are black. It is conclusive evidence for the proposition that SOME ravens are black, but I can't see why it should be any evidence at all for the ALL proposition.The thinking is that if a black raven is supporting evidence that all ravens are black (not proof, note), then by the same token a green apple is supporting evidence that all non black things are non-ravens. — unenlightened
Yes, I can readily accept that statement. What I'm having trouble with is finding a reason to believe the antecedent - that observation of a black raven is evidence for the proposition that all ravens are black. It is conclusive evidence for the proposition that SOME ravens are black, but I can't see why it should be any evidence at all for the ALL proposition. — andrewk
Yes, I can readily accept that statement. What I'm having trouble with is finding a reason to believe the antecedent - that observation of a black raven is evidence for the proposition that all ravens are black. It is conclusive evidence for the proposition that SOME ravens are black, but I can't see why it should be any evidence at all for the ALL proposition. — andrewk
There's no paradox because that isn't really how evidence works. Like I said, anyone can weave any tale to retroactively explain the facts, but a good hypothesis predicts something that we didn't know, and then we look for it and find it. That doesn't bolster the logical necessity of the theory being true or anything, it could as well just be a coincidence, but it sure as hell is damned impressive, and bolsters confidence. We then keep the hypothesis, use it in more general theories, keep trying to make new predictions, which keep bolstering confidence for as long as they keep working out, until we're all but certain of them. — Wosret
This seemed to be Popper's view (as someone else pointed out). Let us consider for a moment the proposition that singular instances provide no confirmation of a universally-quantified hypothesis or statement (e.g. occurrences of white swans do not even marginally raise the probability of the hypothesis "all swans are white") by means of a thought experiment....a single datum doesn't help us with the universal. — andrewk
After the observer inspects, say, three of the eggs and finds that they're white, can he reasonably be more confident in the truth of his hypothesis to any degree whatsoever? After all, each cup which is found to contain a white egg is one less cup which can possibly hold a non-white egg (and we've stipulated that the world consists solely of this egg carton, so there is nowhere else for a non-white egg to hide). Does each observation of a white egg therefore confirm the hypothesis (even if only incrementally)? My intuition seems to say "yes," but of course, my intuition does not constitute any sort of rigorous proof. — Arkady
Michael, you're right that this is the problem of induction. It never occurred to me before to wonder whether any statistical basis could be found for using the principle of induction, by considering it in terms of hypothesis testing. If not, that seems to lend even greater weight to Hume's insight. — andrewk
''ALL ravens are black'' is TRUE iff every raven you see is black. Observing a few ravens, so far as it's not ALL ravens, cannot PROVE this sratement.
''Everything that is not black is not a raven'' is TRUE iff every non-black thing is not a raven. Mind the word ''everything''. We must observe ALL non-black things in the universe.
One green apple will NOT suffice to prove either of these statements.
Paradox solved. — TheMadFool
Let's assume that we have 12 eggs and that they can be either white or brown. All other things being equal there's a 0.512 chance of every egg being white. We look at the first egg and see that it is white. There's now a 0.511 × 1 chance of every egg being white. Given that the second chance is greater than the first chance it then follows that our hypothesis is made more likely by the first successful observation. And assuming that evidence is anything that makes our hypothesis more likely it then follows that a single white egg is evidence that all eggs are white. — Michael
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.