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Annoyance comments on Guilt by Association - Less Wrong
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"If P, then Q. Q. Therefore, P."
I have taught this as a fallacy for the last two semesters in a propositional logic section of my course. However, given the prevalence of this fallacy, as well as the observed resistance of students to assimilating it, I would like to propose that this really isn't a fallacy but a disconnect between what we often naturally mean by "p implies q" and what it technically means in propositional logic.
I argue that what we tend to understand by the statement "p implies q" is that p is the set of things that result in q. This can be tested by the fact that if Q is known to have two causes P1 and P2, and someone says that P1 implies Q, it would be rather natural and logical for another person to "correct" them by adding that P2 also causes Q. But the word correct is too strong ... it is rather an augmentation.
I think that, like many statements in ordinary usage, "P implies Q" is a fundamentally Bayesian statement; a person means: "in my experience, P is the set of things that cause Q". The 'in my experience' means that it is possible (but not observed) that Q might not happen after P, and also that other things besides P might cause Q, but again this hasn't been observed or isn't recalled at the moment. A person is normally willing to be flexible on both points in the face of new evidence, but is taught in logic class to be absolute for a moment while considering the truth or falsity of the statement.
So finally, when an untrained person is told that "p implies q" is a true logical statement, in their efforts to be "absolute" they may think that both implied meanings are true: p always causes q AND only p causes q. It takes some training (deprogramming of what it has seemed to mean their whole natural lives) to understand that only the first is intended (p always causes q) when the statement is "True" in formal logic.
But since that's not what the words mean even in standard English, it's clearly a misunderstanding on the part of the students.
Doesn't it depend upon the context?
Suppose the context is some event P. Then we can talk about what things are implied by P and P implies Q has the standard/technical logical meaning. (If P implies Q1, Q2 and Q3 we naturally but not logically expect all 3 in a "true" answer: P -> Q1 ^ Q2 ^ Q3)
On the other hand, if the context is Q, and we ask "what implies Q?" then we expect a fuller answer for P; P is the set if all things that imply Q: P1 v P2 v P3 -> Q.
Perhaps, generally writing P-> Q as (P v S) <=> (Q ^ T) would more accurately capture all intended meanings (technical and natural). It would be understood that S and T are sets that complete the intended sets on each side necessary for the "if and only if" and that they could possibly be empty.
(Alas, this would make it no easier to teach. I just stress in class that P implies Q means P is one example of things that imply Q, and Q is, likewise, need only be one example of things implied by P.)
No. "P implies Q", even in regular, everyday English, does not suggest that P is the set of all possible causes for Q. Context doesn't matter.
So I would guess you don't understand why people make the mistake that "if not Q, then also not P". Do you have another hypothesis for the origin of this mistake? (Perhaps there is more than one cause, ha ha.)
Later edit: The first sentence had an obvious error. In the quotes, I meant to write, "if Q, then P" -- or, more symmetrically, "if not P, then also not Q" as the mistake that is often made from "if p then q".
I'm actually in large agreement with you about what "p implies q" means in ordinary English, but can wobble back and forth with some effort. Let me try a little harder to convince you of the interpretation I've been arguing.
Let's suppose you are told, "if P then Q". In everyday life, you can usually take this to mean that if Q then P because P would have caused Q. If Q could instead have been caused by R and R was likely, then why didn't the person say so? Why didn't the person say "if R or P then Q"?
Um... I don't think that's a mistake. Given "If P, then Q", the non-existence or falsehood of Q requires that P also not exist / be false. It leads to contradiction, otherwise.
Seriously? P→Q ⊢ ¬Q→¬P.
Perhaps people are just not good at processing asymmetrical relations. They may naturally assume, for any relation R, that aRb has the same meaning as bRa. They may not notice that conclusions they make from the mistake at this level of abstraction contradicts their correct understanding at a lower level of abstraction that includes the actual definition of implication.
Interesting, but this doesn't seem true true in general. People are pretty good at not confusing aRb and bRa when R is something like "has more status than", for example.
Good point. When the relation is obviously antisymmetric, where aRb implies not bRa, this is enough to make people realize it is not symmetric.
I wouldn't be surprised if the easiest relations for us to imagine between two variables were simply degrees of "bidirectional implication" or "mutual exclusivity".
Bing bing bing!
The real issue, of course, is why they're the easiest for us to represent.
That's coming up next.