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In classical logic, we have paradoxes like paradoxes of material implication. If non-classical logic like relevance logic fixes those problems, why do we still continue to use classical logic?

Shaun
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Zeus
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    Do we prefer classical logic? Well, I'll be...didn't know that. – DonAntonio Apr 02 '13 at 04:05
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    @DonAntonio: Either we do, or we don't. Therefore LEM holds and classical logic prevails once more! Huzzah! :-) – Asaf Karagila Apr 02 '13 at 07:56
  • Nearly all serious work in computer science fields like programming language design is done with intuitionistic logic, because it corresponds much more closely with what can actually be computed. – MJD Oct 16 '23 at 15:28

5 Answers5

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The "paradoxes of material implication" are not paradoxes, in the sense of contradictions, they are just non-intuitive. And any "logic", classical or not, is a human construction, which some people prefer to use to think about the real world, but this is by no means necessary; just because a logic doesn't exactly correspond how you think reality behaves is no count against its being interesting to think about, or its being useful as an approximation to or model of reality.

Zev Chonoles
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    I don't find it useful to reserve "paradox" for actual contradictions; we have a perfectly good word for that already: "contradiction"! – Zhen Lin Apr 02 '13 at 15:05
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Relevance logic enables us to avoid some prima facie issues with classical logic. But at a high price. For a start, in many systems of relevance logic we lose disjunctive syllogism, i.e. the rule that from $A \lor B$ and $\neg A$ you can infer $B$; yet intuitively that is an absolutely fundamental valid rule.

This is rather typical when choosing between formal logical systems. We start with a bunch of logical 'intuitions' we'd like a formal logic to conform to. Here's a selection: disjunctive syllogism is OK, conditional proof is OK, modus ponens is OK, entailment is unrestrictedly transitive, the indicative conditional isn't truth-functional, a contradiction doesn't imply every proposition, etc. etc. And then we find that we can't consistently satisfy all those desiderata together. Drat! What to do?

We have to look around for "best buys" that satisfy enough of the desiderata that we most care about meeting (or care about meeting in a particular context). You pay your money and you make your choice!

And experience shows that if we most care about modelling the reasoning of mathematicians doing standard textbook maths, for example, then classical logic is actually brilliant in all sorts of ways (it is a great fit AND has beautiful proof-systems AND has an elegant and intuitive semantics, etc.). Given its great positive virtues we then learn to live with the supposed failures of "relevance" (the conditional is material, a contradiction entails anything): these failures are, in the context, usually deemed a price worth paying. And that's why we (most of us) stick to classical logic (most of the time, for most purposes).

But there is no One True Logic written on tablets of stone. It's a question of costs and benefits: and your weighing of the costs and benefits can reasonably differ from the majority view.

Peter Smith
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Mathematics is all about modeling. When modeling, you are building a bridge, or rather a translation mechanism, between a problem in the real world domain, and some formal language, formal axioms, and a a logical system to be able to symbolically manipulate axioms and consequences.

The model is not trying to be a faithful representation of reality. It is trying to be a good enough approximation of reality, while providing useful and powerful ways to deduce properties of the real life problem by symbolically manipulating marks on a piece of paper using the laws of the chosen logic.

Those models that are used are precisely those that yield good enough approximations together with powerful enough proof techniques. Today, it seems that classical logic achieves good results (if not excellent results). Of course, things can change and there are reasons to consider other logical systems, and indeed people are researching nonclassical logical systems and their applications. But, before you throw away a perfectly good horse for a slightly better one, you need to think carefully.

Ittay Weiss
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  • so non-classical logic does not yet have good approximations that classical logic offers? (or not proven to be as strong as classical logic?) – Zeus Apr 02 '13 at 04:02
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    I think it is safe to say that there aren't any clear indications that developing all of mathematics using some nonclassical logic will yield better performing models for the purposes of science. It is also safe to say that there are indications pointing to the limits of classical logic for the purposes of science (e.g., quantum mechanics leading to quantum logic), warranting researching the possibilities. – Ittay Weiss Apr 02 '13 at 04:50
  • I don't know about quantum logic (maybe all the variables aren't know yet?), but, from my admittedly limited perspective, other forms of non-classical logic seem to me to be an over-reaction to inconsistencies in naive set theory. (Is $x\in x$? Maybe???) Or has the field of so-called classical logic been more or less mined out? – Dan Christensen Apr 03 '13 at 19:19
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    I suggest you read the wiki articles on intuitionism and paraconsistent logic to see just two possibilities of nonclassical logic. The field is far from being mined out. – Ittay Weiss Apr 03 '13 at 19:37
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A very common opinion of mathematics is "mathematical platonism", which holds that mathematical obejcts exist in some sense and that mathematics is the study of these objects. The rules of classical logic are closely tied to this viewpoiont.

The "paradoxical" formulas of material implication are verified when we interpret them as talking about truth values in a fixed model. For example, if we know that $p$ and $q$ are statements about a fixed model, we know that $p \to (q \lor \lnot q)$ will be true in the model, by reasoning by cases about the truth values of $p$ and $q$ in the model.

The completeness theorem for first-order logic says that a formula is provable in first-order logic if and only if it is true in every model. This statement has two parts:

  1. If we can prove a formula in first-order logic, the formula is true in all models. To a platonist, this means that if we already have a fixed model in mind, and we prove a formula, we know the formula will be true in that model.

  2. A formula that is true in every model is already provable in first-order logic, so we cannot extend first-order logic to a properly stronger logic (that is, one that proves more sentences) while simultaneously maintaining (1).

The completeness theorem thus says that first-order logic is the strongest logic possible (in terms of proving the most sentences) whose results are sound when applied to an arbitary model that we have already fixed. That is exactly the sort of logic that we would want, as mathematical platonists, in order to study a collection of pre-existing structures.

Relevance logic, for example, makes fewer implication formulas provable. That is of interest if we are trying to study the natural-language "implies" relationship, but it is less interesting if we are trying to study the field of real numbers and we want to generate as many formulas as possible that are true in that structure.

Carl Mummert
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  • The platonist point of view does not oppose or support classical logic. For instance, you can view a topos as the mathematical universe that interests you where all the mathematical objects platonically exist. Then the internal logic of the topos is what allows you to study these platonic objects. E.g., there is a topos in which every function $f:\mathbb R \to \mathbb R$ is continuous. In other toposes not every such function is continuous. But in any case, we study things that platonically exist. And many other forms of logic support appropriate versions of completeness theorems. – Ittay Weiss Apr 03 '13 at 19:45
  • @Ittay Weiss: one way to see that the topos position is incompatible with Platonism is that "hard-core" Platonism entails the law of the excluded middle. For example, to a hard-core Platonist a fixed real number $\alpha$ is either equal to zero to not equal to zero -- not because of any sort of formalized logic, but just from the general fact that any two objectively existing objects in the universe are either the same or different, and are thus equal or nonequal. There are topoi that don't verify this fact about real numbers; the internal logic of such topoi isn't compatible with platonism. – Carl Mummert Apr 03 '13 at 23:34
  • That's a good point. I guess it's slightly relaxed platonism then that I'm having in mind. To me, rejecting the law of excluded middle and the principle of explosion does not entail rejecting the objective existence of mathematical objects. They are still somewhere out there, but just that for an intuitionist an object may neither exist nor not exist, and paraconsistently, an object may both exist and not exist. For instance, in a naive paraconsistent approach to naive set theory, one may say that the Russell set both exists and does not exist. – Ittay Weiss Apr 04 '13 at 00:16
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As a matter of practicality:

  • I have learned classical logic
  • To the best of my knowledge, classical logic is easier to work with than non-classical logics
  • To the best of my knowledge, any non-classical logic can be expressed in mathematics developed in the language of classical logic without losing any essential features ${}^1$

Therefore, I see no merit in working with non-classical logic "from the ground up". Even in a situation where I would be inclined to work in some sort of non-classical universe, it is simply more efficient to invoke classical logic to work out the basic features rather than trying to derive them from scratch.

As an added bonus feature, this approach gives me the chance to use non-classical logic to assist in classical work -- e.g. using intuitionistic logic to reason about sheaves of sets on a topological space. (see "topos")

1: I do not consider "avoid using classical logic as a philosophical principle" an essential feature.