What the Preface paradox tells us about the principle of explosion

Question

The Preface paradox (adopted on a classical logician): Imagine that a classical logician has just written a textbook (on classical logic). He has included many assertions in it and has thoroughly checked every one of them, thus he is justified in believing that all of them are true. At the same time, he knows from experience that there will be at least one mistake in there. If P means ‘every assertion the logician made in the book is true’, then this means he has reasons to rationally believe and assert both P and ¬P.

But as per the principle of explosion, shouldn't this classical logician then be doomed as a trivialist, who believes that literally everything is true? But we clearly do not regard any such textbook writer as a trivialist, not least because he probably does not believe that everything is true.

In other words, my questions:

1. Why is he not a trivialist? Is explosion not the only thing at play here? Am I missing something psychological?

2. How do we reconcile the fact that on the one hand a classical, explosion subscribing logician has contradicting beliefs, and on the other not doomed to be a trivialist?

Answer 1

I agree with Just Some Old Man's answer, but to expand on it a little...

If we think of all the statements in the textbook as propositions A₁, A₂, ..., A_n then the situation we are trying to describe is that the author of the book believes that each proposition is true, but does not believe that the conjunction of all of them is true. This cannot be represented in simple propositional calculus, because the truth of a set of propositions is closed under conjunction, i.e. if P, Q are both true then P ∧ Q is true. But probability theory provides a neat alternative. Degrees of credence can be expressed to a reasonable approximation as probabilities (we owe this insight to Richard Cox and Bruno de Finetti among others). While it is reasonable for the author to attach a high degree of credence to each proposition A₁ ... A_n it does not follow that they should attach a high degree of credence to the conjunction. The probability of a conjunction is less than or equal to the probability of each conjunct, and in typical cases, the more conjuncts there are, the less probable the conjunction will become. So there is no inconsistency in accepting that each A_i is highly probable but the conjunction of all of them is highly improbable.

This example illustrates the fact that when applying logic to real-world situations it is important to pay attention to any modalities that may be relevant, even if those modalities are not explicit.

Answer 2

If the logician somehow believes p ∧ ¬p is true, then s/he should give up logic altogether and nobody will care.

S/he may also believe p and believe not p. This is no logical problem but a psychological one since s/he doesn't need any logical principle to admit contradictory beliefs.

S/he may also have some good reasons to believe p and some good reasons to believe not p, and this is what seems to apply here. The solution is very simple, s/he should work out the exact probabilities and derive from there the probable consequences if p and the probable consequences if not p.

This is not a problem since this is essentially what happens all the time in real life. For example, when throwing a die, we believe getting a 6 has a probability of 1/6 and not getting a 6 has a probability of 5/6. The real trouble would begin if you would get a 6 and you wouldn't get a 6 at the same time, but this is a counterfactual and nobody should care until it happens.

So, either way the principle of explosion is irrelevant and there is no logical paradox.

Answer 3

I agree with the other answers, but there is another consideration at play here, which is that beliefs constitute parts of mental models of the world, and a person can use multiple mutually-contradictory mental models at different times.

This is like the way a physicist can 'believe' in Newtonian classical physics in one context, with idealisations of rigid bodies, infinite frictionless planes, point particles, and so on, and yet know perfectly well that in a relativistic quantum world those constructs are not just false but impossible. This does not lead to trivialisation because belief is model-dependent. When using a classical Newtonian model of the world, rigid bodies are possible. When using a relativistic model, they are not. There is no contradiction between the two, because they are statements in different axiomatic systems. Similarly with Euclidean geometry and spherical geometry, the statement that the angles of a triangle add up to 180 degrees is true in one and false in the other. A Euclidean geometer will be happy to say without qualification they believe the total of a triangles angles is always 180, because the axiomatic context is assumed. (i.e. We assume the axioms are true.) But if asked specifically about non-Euclidean geometries, they will be equally happy to say it is not generally true.

So when the logician says be believes all the theorems stated in the book are true, he is operating in the context of the beliefs of a prover, in which constructing an apparently valid proof of a statement should be taken to mean it is true. Those are the rules of the game. When talking about the fallibility of provers meaning that some of the statements will be false, he is using a different model, with different rules.

For a logician to believe that they are themselves a fallible prover, and thus some of the things they believe true are actually false, sounds like a version of the liar paradox. But the paradox can be avoided by keeping the contradictory beliefs in separate boxes. If you are required to keep all your beliefs in one box, then knowledge that you are fallible does indeed mean you cannot know anything. But you can define belief/truth relative to a particular fallible prover, and keep knowledge of the fallibility in an enclosing model, and get a limited form of belief.

The problem is unavoidable in sufficiently expressive systems, because of Godel's theorems. For any consistent proof system expressive enough to model arithmetic, there are statements in it that are true-but-unprovable in that system. We are only allowed to prove that these Godel statements are actually true because we are using a different proof system from the one under study, that encloses it. No sufficiently expressive proof system can prove itself consistent, (unless it is actually inconsistent!) - they must all accept the possibility of fallibility.

Answer 4

The conclusion doesn’t necessarily follow from the paradox stated as such. Just because he has justification to believe P and justification to believe !P does not mean P and !P are true, so the principle of explosion does not apply. (A real problem would be if he has justification for P and does not have justification for P.) In science for example, it is common to have justification for both of two incompatible theories.

Secondly, there’s the issue of probability which is often argued to model this situation better without contradiction because it is empirical (probability of P vs. probability of !P). Finally, a person can believe two contradictory assertions but not believe all other assertions are true, though the contradiction logically implies all other assertions are true.

The paradox is about the tension between justification for both of two contradictory assertions, not from an outright contradiction being formally proved leading to the principle of explosion, or the principle of explosion then forcing the logician to believe all assertions are true.

Putting probability aside, the paradox is important because it prods us to consider when having justification for P and having justification for !P implies it is rational to believe P and it is rational to believe !P. Then, furthermore, does the latter imply it is rational to believe P and !P (does the agglomeration principle hold here)? So falling to trivialism is contingent on a few things not necessarily entailed by the paradox.

Answer 5

The seeming contradiction leading to the undesirable principle of explosion can be readily resolved in 2 possible ways according to David Makinson's Preface paradox reference here.

This first usual way out is the orthodox probability POV:

Probabilistic perspective may restate the statements in other terms, thus resolving the paradox by making them non-contradictory. Even if the author is 99% sure each single statement in his book is true, there can still be so many statements in the book that the aggregate probability of some of them being false is very high as well. Since the principles of rational acceptance allows the author to accept a very likely statement, he may rationally chose to believe in (1). Same principles may make him rationally believe also in (2).

So assuming each statement is 99% true, if there're 200 statements in the author's book, then 0.99²⁰⁰ = 14.4%, so the probability that at least one statement is false equals 85.6% which is pretty high so one can rationally believe in (2).

A second unusual way out is from Kyburg's rejection of conjunction principle of rational beliefs:

Another way to resolve the paradox is to reject the inconsistency of both (1) and (2) being true at the same time. This is done by rejecting the conjunction principle, i.e., that belief (or rational belief) in various propositions entails a belief (or rational belief) in their conjunction. Most philosophers intuitively believe the principle to be true, but some (e.g., Kyburg) intuitively believe it to be false. This is similar to Kyburg's solution to the lottery paradox.

Answer 6

Alice told me it's sunny today. Bob told me it's cloudy today. I have no windows.

I have reason to believe it's sunny today: Alice told me.
I have reason to believe it's cloudy today: Bob told me.
I have absolutely no reason to believe it's sunny and cloudy at the same time. That would be preposterous. Either Alice or Bob was wrong.

Answer 7

There's a confusion here about what "belief" means. It can mean absolute belief, such as in propositional calculus, or probabilistic belief (or using any other form of fuzzy logic).

Suppose I believe that, with a probability of 99.5%, that each individual proposition in my book is true, which is extremely close to absolute certainty, but not quite there. Suppose there are two hundred propositions in my book. Therefore, the expected number of false propositions in my book is one, and the chance that there are no false propositions is well under one-half.

Answer 8

It is given that the logician does not believe that every statement in the book is true, but also given that the logician believes that each statement in the book is true. The resolution here is to recognize that aggregating belief is not equivalent to a belief about the aggregate. The logician does not believe that every statement in the book is true, but does believe that each statement in the book is true. A little more formally:

1. I believe that for each statement P, P is true.

vs.

2. For each statement P, I believe that P is true.

There is no contradiction to simultaneously believe that each statement is true but also believe that at least one statement is incorrect. This can be phrased as a statement about the logician's belief - the logician believes that their belief in the truth of the statements is misguided for at least one of the statements.