Is Logic Revisable?

Is Logic Revisable?

What does this question mean? Does it mean to ask whether our current logical systems are in principle revisable? Or is it asking whether logical reality itself is revisable? Presumably not the latter: truth (reality, facts) isn’t revisable, only beliefs are. Unless we mean to be asking whether logical reality can be changed—how and by what? The question is bizarre. Can we change the laws of nature? Unlikely, to say the least. No, it must be the first thing that is meant—logical beliefs. Could our logical beliefs be false? If that is a skeptical question, the answer depends on the cogency of skepticism, and its scope. Could you be stuck in a dream in which you accept false logical propositions? That sounds possible—you might be dreaming that certain logical propositions are true but they aren’t. Or you might be logically insane. How can we rule these skeptical possibilities out? Nothing is immune to skeptical doubt, arguably. The question doesn’t seem peculiar to logic and is not particularly interesting.

Here is another interpretation: could there be another reality in which our logical laws don’t hold (though they do in our reality)? Do they hold in a fictional world, say? We can certainly imagine stories in which contradictory things happen. But this doesn’t seem very interesting, since it has no bearing our current logical knowledge. That logic doesn’t hold in logically impossible worlds is hardly news. We have to be talking about our actual world if the question is to have any bite. The simple answer to our title question is that our current systems are fallible, subject to Cartesian doubt like everything else, but that there is no sense in the idea of a revisable logical reality. If the law of non-contradiction is part of logical reality, then it isn’t revisable; but if we just mean our logical beliefs, then this belief will be open to doubt. This is just a special case of general skepticism.

However, there is one possible source of doubt that has more localized force, which I have never seen mentioned. This is that the very notion of entailment cannot be made sense of.[1] Entailment applies in any viable logical system, so if it makes no sense, then logic will totter. This would imply that no logic could be correct, because it uses an indefensible concept. Logic would be revisable in the sense that it could be revised out of existence. This wouldn’t be the claim merely that we can’t know if one proposition (or fact) entails another; it would be the stronger ontological claim that the relation in question does not and could not exist. If that were so, then even “p entails p” would fail, on account of the meaninglessness of “entails”. Perhaps surprisingly, this is not so hard to construct a case for: entailment is not easy to make sense of. What is it for one proposition to entail another (even itself)? The relation seems both internal and external at the same time: the logically connected propositions are generally distinct and yet stand in a necessary internal relation. How can that be—how can distinct things stand in necessary internal relations? Entailment seems contradictory! How can the proposition that p entail the proposition that p or q, where q is an unrelated proposition? It seems magical, contrary to reason. Similarly for all other standard entailments. How is it possible to extract one proposition from another (which doesn’t contain it)? The alleged logical relation makes no metaphysical sense. Induction has a problem because it tries to move from one proposition about the past to another about the future, but deduction has a similar problem—the problem of getting from a proposition to its logical consequences. It seems like inferring cats from dogs, or numbers from pebbles. Only propositional identity can do that. What bridges the gap, and how can one proposition be implicit in another? Why couldn’t it be that pentails q up to time time t, but then ceases to? Skeptical paradox threatens and logic begins to lose its metaphysical footing. Just as meaning can be revised out of existence, so logic could be—the whole idea of deduction falls apart.

The dialectic here is familiar and hence the range of options available. One possibility is to fall back on mysterianism: it is a mystery how entailment works, but it palpably does. It doesn’t fail of existence, but our understanding of it falls short. We must not infer non-existence from unintelligibility (to us). Logic itself doesn’t collapse, though we can’t properly understand its central concept. Logical mysterianism is then the indicated position. This seems hard to take, given the epistemic transparency of logical reasoning; but the same might be said of other fundamental aspects of reality—space and time, matter, causation. It just turns out that we are in the same epistemic position with respect to logic as we are elsewhere. We mistake the self-evidence of logical laws for their intelligibility—but the former doesn’t entail the latter. The fact is that we have no good theory of logical knowledge—no logical epistemology. The whole process of recognizing logical truth is riddled with mystery; it is one aspect of the problem of a priori knowledge, which goes back to Plato. Entailment is an inscrutable relation, begging for impatient elimination. Mysterianism allows us to resist this response. Either that or logic gets revised into oblivion by the entailment skeptic (anti-realist).[2]

[1] See my “The Problem of Deduction” and “Knowledge of Entailment”. I am applying these points to the question of the revisability of logic.

[2] I myself believe that the problem of logic (the mind-logic problem) is a lot harder than has been acknowledged historically. Even the rationalists had little positive to say about it. Logic is metaphysically baffling: what it is, how it relates to the rest of nature, how minds grasp it—all very difficult. No wonder it has had mystical associations. We love logic, but we don’t see very far into it. Logic has been around forever, but has been reluctant to yield up its secrets. Most philosophy of logic is laughably reductive. Wittgenstein was infatuated with pure logic when he was young, but hostile to it in middle age (too “sublime”). He came to think it produces intellectual monsters. I think logic is like bacteria: invisible but everywhere (and vital to life). If causation is the cement of the universe, then logic is its scaffolding (or skeleton).