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Followup to: The Parable of the Dagger
"All men are mortal. Socrates is a man. Therefore Socrates is mortal."
Socrates raised the glass of hemlock to his lips...
"Do you suppose," asked one of the onlookers, "that even hemlock will not be enough to kill so wise and good a man?"
"No," replied another bystander, a student of philosophy; "all men are mortal, and Socrates is a man; and if a mortal drink hemlock, surely he dies."
"Well," said the onlooker, "what if it happens that Socrates isn't mortal?"
"Nonsense," replied the student, a little sharply; "all men are mortal by definition; it is part of what we mean by the word 'man'. All men are mortal, Socrates is a man, therefore Socrates is mortal. It is not merely a guess, but a logical certainty."
"I suppose that's right..." said the onlooker. "Oh, look, Socrates already drank the hemlock while we were talking."
"Yes, he should be keeling over any minute now," said the student.
And they waited, and they waited, and they waited...
"Socrates appears not to be mortal," said the onlooker.
"Then Socrates must not be a man," replied the student. "All men are mortal, Socrates is not mortal, therefore Socrates is not a man. And that is not merely a guess, but a logical certainty."
The fundamental problem with arguing that things are true "by definition" is that you can't make reality go a different way by choosing a different definition.
You could reason, perhaps, as follows: "All things I have observed which wear clothing, speak language, and use tools, have also shared certain other properties as well, such as breathing air and pumping red blood. The last thirty 'humans' belonging to this cluster, whom I observed to drink hemlock, soon fell over and stopped moving. Socrates wears a toga, speaks fluent ancient Greek, and drank hemlock from a cup. So I predict that Socrates will keel over in the next five minutes."
But that would be mere guessing. It wouldn't be, y'know, absolutely and eternally certain. The Greek philosophers—like most prescientific philosophers—were rather fond of certainty.
Luckily the Greek philosophers have a crushing rejoinder to your questioning. You have misunderstood the meaning of "All humans are mortal," they say. It is not a mere observation. It is part of the definition of the word "human". Mortality is one of several properties that are individually necessary, and together sufficient, to determine membership in the class "human". The statement "All humans are mortal" is a logically valid truth, absolutely unquestionable. And if Socrates is human, he must be mortal: it is a logical deduction, as certain as certain can be.
But then we can never know for certain that Socrates is a "human" until after Socrates has been observed to be mortal. It does no good to observe that Socrates speaks fluent Greek, or that Socrates has red blood, or even that Socrates has human DNA. None of these characteristics are logically equivalent to mortality. You have to see him die before you can conclude that he was human.
(And even then it's not infinitely certain. What if Socrates rises from the grave a night after you see him die? Or more realistically, what if Socrates is signed up for cryonics? If mortality is defined to mean finite lifespan, then you can never really know if someone was human, until you've observed to the end of eternity—just to make sure they don't come back. Or you could think you saw Socrates keel over, but it could be an illusion projected onto your eyes with a retinal scanner. Or maybe you just hallucinated the whole thing...)
The problem with syllogisms is that they're always valid. "All humans are mortal; Socrates is human; therefore Socrates is mortal" is—if you treat it as a logical syllogism—logically valid within our own universe. It's also logically valid within neighboring Everett branches in which, due to a slightly different evolved biochemistry, hemlock is a delicious treat rather than a poison. And it's logically valid even in universes where Socrates never existed, or for that matter, where humans never existed.
The Bayesian definition of evidence favoring a hypothesis is evidence which we are more likely to see if the hypothesis is true than if it is false. Observing that a syllogism is logically valid can never be evidence favoring any empirical proposition, because the syllogism will be logically valid whether that proposition is true or false.
Syllogisms are valid in all possible worlds, and therefore, observing their validity never tells us anything about which possible world we actually live in.
This doesn't mean that logic is useless—just that logic can only tell us that which, in some sense, we already know. But we do not always believe what we know. Is the number 29384209 prime? By virtue of how I define my decimal system and my axioms of arithmetic, I have already determined my answer to this question—but I do not know what my answer is yet, and I must do some logic to find out.
Similarly, if I form the uncertain empirical generalization "Humans are vulnerable to hemlock", and the uncertain empirical guess "Socrates is human", logic can tell me that my previous guesses are predicting that Socrates will be vulnerable to hemlock.
It's been suggested that we can view logical reasoning as resolving our uncertainty about impossible possible worlds—eliminating probability mass in logically impossible worlds which we did not know to be logically impossible. In this sense, logical argument can be treated as observation.
But when you talk about an empirical prediction like "Socrates is going to keel over and stop breathing" or "Socrates is going to do fifty jumping jacks and then compete in the Olympics next year", that is a matter of possible worlds, not impossible possible worlds.
Logic can tell us which hypotheses match up to which observations, and it can tell us what these hypotheses predict for the future—it can bring old observations and previous guesses to bear on a new problem. But logic never flatly says, "Socrates will stop breathing now." Logic never dictates any empirical question; it never settles any real-world query which could, by any stretch of the imagination, go either way.
Just remember the Litany Against Logic:
Logic stays true, wherever you may go,
So logic never tells you where you live.
Part of the sequence A Human's Guide to Words
Next post: "Words as Hidden Inferences"
Previous post: "The Parable of the Dagger"