Impossible LSAT Statistics Question

Here is a recent LSAT question setup (thanks to Thanatos Savehn for pointing it out):

And here are the possible answers:

Please take a moment to think and answer. Do not read below, which are my comments, until you’ve thought about it.

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My Answer. (We’ll need this later when we criticize arguments for and against “evolution.” Those who followed the Class from the beginning ought to remember all this.)

The setup makes little sense. As far as legal arguments go, it stinks. One problem: if we followed the statisticians’ prescription, it would seem impossible to form any new belief. Or even any initial belief. We would be beliefless.

But it’s only “seems”, as we’ll see.

What is a belief? It is a proposition which you hold to be at least locally true. That is, you have a list of evidence, also called premises, assumptions, and so forth, including the tacit and implicit premises of the definitions of the words and the grammar you use. The belief is deduced from this list of evidence.

If the evidence itself (all of it) is true, the belief becomes knowledge, a universal or necessary truth. If the evidence contains falsities or uncertainties, then your belief is only a local truth; i.e. true only in relation to the flawed evidence you hold. Which is why I always say you can only know what is true, but you can believe anything.

Three examples:

(1) You believe “My cat Jacques speaks French” conditional on the evidence “All cats speak French; Jacques is a cat.” Your belief is locally true; that is, the proposition follows (is deduced from) the evidence. The evidence has one true statement (“Jacques is a cat”) and one false statement (“All cats speak French”), so your belief is not universally true.

(2) You believe “Socrates” is mortal because of the old syllogism. Your belief is locally and universally true because the evidence itself is true in the same way.

(3) You believe “George wears a hat” conditional on the evidence “Most Martians wear hats; George is a Martian”. Now the proposition “George wears a hat” is only likely given the evidence, so if you do believe it, it must mean that you must have added to this list of evidence an important tacit premise. Which is something like “George is among the most.” Belief under uncertainty is an act, a decision. It is an explicit move to bend evidence toward the direction you desire or choose. The original evidence did not support your belief: you had to augment it by a pure act of will to bring you to belief. Even if you didn’t write it down or acknowledge it.

Next step in the LSAT setup was rejecting a belief “when given adequate evidence against it.” What can that mean? Let’s re-do our examples and see.

(1) You become convinced all cats can not speak French. That is, you conclude your belief (evidence) that “”All cats speak French” is false conditional on “adequate evidence”, whatever that might be. Remove that initial premise, and you have no evidence left to support your belief. But you see the twist: the evidence you use are also beliefs.

Notice that this argument is different: “If all cats speak French, and Jacques is my cat, then Jacques speaks French.” This remains true. Believing that is different than believing “All cats speak French, and Jacques is my cat, then Jacques speaks French.” That “if” makes a world of difference.

(2) Somebody gets you to accept “All men are not mortal.” That is, this becomes a new belief, conditional on whatever evidence was used. Thus your belief that “Socrates is mortal” collapses. But that new evidence, which you have come to accept, is wrong. It really is true that “All men are mortal”. So you now believe something false. You reject two truths: that “All men are mortal” and (what follows from that) “Socrates is mortal.”

There is nothing in the LSAT question to distinguish what “adequate evidence” means. You thought you had new “adequate evidence” to reject “All men are mortal.” But then “adequate evidence” can mean new false evidence. So perhaps the question was designed to imply “adequate evidence” meant only true evidence.

If that’s the case, then you can only belief true things. All your beliefs are knowledge, none are local truths, all are universal or necessary, and so you can never reject any belief. That follows from “adequate evidence” meant only true evidence. Consider that if you come to believe a new piece of “adequate” evidence, which must be true, it means there is additional evidence that this “adequate” evidence is itself true. And so on through a chain of argument anchored on propositions which are just true based on no evidence except induction. (Proving this takes us too far afield: see this lecture.)

(3) That doesn’t solve the problem of where new beliefs arise, as in this third example. Perhaps they just come from the blue, like George’s hat. It’s easy to see what “adequate evidence” is here: a demonstration that “most” does not mean “all.” That leaves us with uncertainty, and not belief. It remains only likely “George wears a hat” is true. But this is different: “It is likely George wears a hat” is itself a true proposition. Truth depends where we put the quotation marks!

Yet we are still left with many propositions in real life that don’t and can’t reach belief, but which are only uncertain given the evidence. The LSAT question gives no help what we do with this largest set of propositions we must work with in daily life. Sometimes we have to act on these propositions, and only sometimes do these acts transform the uncertainty into certainty (local truths). Other times they are just bets. For instance, the evidence of “40% chance of rain.” Do you carry an umbrella or not? Carrying one is not the same as believing it will rain.

That brings us to the bizarre end of the question setup. What in the world does survival have to do with beliefs and their number? Well, something, but not much. For instance, one can believe (based on whatever evidence) that it is the Great Turnip Gods which cause turnips to form in the ground, turnips which you eat and sustain you. This is a locally true belief, but a false universal one. Believing it has not harmed your survival, and can even be said to aid it.

And we have already seen that in order to reject false beliefs, we must necessarily increase our stock of true beliefs, which is that “adequate evidence” which is itself true.

The LSAT Answers.

(A) Doesn’t make any sense. But that’s because the setup is itself weird. It contains the implicit premise that more beliefs equals better survival, which is false, unless all beliefs are universal truths. Because most beliefs are not knowledge, and are only locally true. And anyway, you can know, and it is true, that the sun is hot, but if you’re in a spaceship hurtling out-of-control towards the Orange Orb, this knowledge will not help you live. And what could “increasing the correctness of beliefs” not “hinder one’s ability to survive” possibly mean?

Consider you hold “The sun is hot” because you believe (among other premises), Microsoft is secretly beaming heat rays into the sun. Your belief is locally true, and helps you survive. It is universally false, of course (as I hope you see). But then comes “adequate evidence” which overturns your Microsoft belief/evidence, and you come to accept the hydrogen theory. Your belief that “The sun is hot” has not changed, but it is now (or is perhaps close to) a universal truth. Your survival based on the belief has not changed, but you indeed have “increased the correctness of your beliefs” because you believe better evidence.

(B) This is true, given that new beliefs have to come from somewhere. And that new “adequate evidence” are new beliefs. At worst, when learning new “adequate evidence” the size of our collection of beliefs stays the same, but it can also increase as the “adequate evidence” in any case may contain several true beliefs.,

(C) Obviously false. See, e.g., Harvard Women’s Studies graduates.

(D) Obviously nonsensical.

(E) This is correct, as we have seen. The main setup doesn’t make the distinction between local and universal beliefs.

LSAT says the right answer is (A).

Good thing I didn’t go into law. Imagine have to blow this much time during an exam.

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