# Are there practical advantages of knowing more advanced forms of logic, as opposed to just classical logic?

For example, classical logic allows one to express arguments more clearly. The LSAT's logical reasoning section tests a kind of reasoning that it considers practical, and an understanding of classical logic helps process that kind of everyday reasoning -- at the very least, in the form of modeling.

Are there similar advantages to learning more advanced forms of logic? If not, what are the advantages?

• – DBK
Mar 14, 2013 at 18:11
• In which way do you think non-classical logic is "more advanced" than classical logic? (If you didn't intend to make this claim, perhaps it would be best to rephrase you question.)
– DBK
Mar 14, 2013 at 18:14
• This sort of feels like asking "why are higher mathematics any better/more interesting than elementary mathematics?" -- at least, I'm having trouble understanding what sort of answer you might be looking for. Could you tell us a little bit more about the context and motivations of the concern? What might you have found out so far? Mar 14, 2013 at 19:53
• Trying to think about intuitionistic logic as building on classical logic is likely to confuse you; it certainly did me. It has fewer allowed rules of inference than classical, which actually changes what semantics of provability are 'intuitively' viable. (It's less about whether atomic sentences are "True" and more about whether they are "Demonstrable"; though this is the main concern of intuitionists anyhow.) Mar 15, 2013 at 11:07
• The reading I would put on the OP's question is that "classical logic" specifically means "first-order classical predicate semantics". More advanced perhaps then means something like higher order or modal logic (which can encapsulate things like relevance or intuitionistic logic through their frame or context semantics). Are there advantages to learning higher order logic? Well, yeah: Functional programming is a good one! Also, logical models of heuristics and bayesian learning ideas look like they need a higher order conception to explain how we reason probabilistically. Mar 18, 2013 at 13:15

It seems to me that the comments and answers so far have missed (or not quite made explicit) that there are two ways to read the question:

(a) Are there practical advantages to knowing richer classical logical systems (that is, systems which have some simplish classical logic as a subset)?

(b) Are there practical advantages to knowing alternative logical systems (that is, systems which in some way deviate from classical logic)?

The two answers already given have covered the practical advantages common to both pretty well, so I'll stick to the advantages specific to these senses. For (a), the advantages come in terms of ability to express and analyse more complex propositions. If you're stuck in propositional logic you can't even analyse "Socrates is a man" to anything more deep than "P", which isn't especially useful for constructing arguments. If you have a rich theory that, for instance, allows you to formulate inference rules for "K(x,p,t) -> K(x,K(x,p,t),t)" to mean "if x knows p at time t then x knows that x knows p at time t", that allows you to do lots of philosophy to a much higher level of rigour, which is a Good Thing. If that doesn't sound practical enough, think about things like rules of inference for things like modal logic, enabling us to rigorously say things like "possibly p iff not necessarily not p". I think that's certainly as practical as 'basic' logic is.

Sense (b) means dealing with stuff like quantum logic, intuitionist logic, fuzzy logic and more. Here you have some of the advantages of additional analytic power as in (a), but what's going to be more important is that seemingly viable alternative logics allow us to properly tackle questions about how fundamental logic is - do we believe in deductive logic because we just can't reason any other way, or because we're stuck in a habit, or it's a fundamental fact about the universe? Quantum theory gives rise to quantum logic, which sometimes disagrees with classical logic. So what are we supposed to do? If you accept the intuitionist thesis, a huge quantity of modern mathematics is junk. That kind of thing doesn't have practical implications in that philosophers are unlikely to get the world world to change its practices, but it is pretty significant in that it casts doubt (and light) on some of our most fundamental reasoning strategies.

• "If you accept the intuitionist thesis, a huge quantity of modern mathematics is junk." As I understand, in intuitionistic logic, there is no such thing as "not not junk" implies "junk". So, how would an intuitionist get to such a claim? :)
– user3164
Apr 24, 2013 at 7:47
• Take the proof that there are irrational a,b such that a^b is rational: If sqrt(2)^sqrt(2) is rational then we have what we're looking for. If sqrt(2)^sqrt(2) is irrational then consider (sqrt(2)^sqrt(2))^sqrt(2)=2 which is rational, so we have what we're looking for. Therefore there exist irrational a,b with rational a^b. Note we don't actually know what a and b are. This proof is invalid in intuitionist logic, and much of mathematics proceeds similarly through non-constructive methods. So what we think are proofs in lots of fields are actually invalid. Apr 24, 2013 at 8:02
• Although a proof may be intuitionistically invalid, that doesn't mean that there isn't a intuitionistically valid alternative. I don't see how a strict intuitionist could possibly show the latter. Catch my drift?
– user3164
Apr 24, 2013 at 8:19
• Ah, I see what you're saying. What I meant by 'mathematics' was the body of mathematical work: that is, not just a set of mathematical statements but their associated proofs too. Although a lot of classical mathematical truths can be proved intuitionistically, to an intuitionist the classical proof remains invalid (and logically not equivalent to the intuitionistic proof). Apr 24, 2013 at 8:23

It is unclear what you mean by "practical". I believe there are very strong practical advantages of knowing logic, whether practical means you just have greater clearity in structuring your arguments, or making a lot of money in a high-tech company.

1. There is a difference between syntax and semantics. This is not obvious to a lot of people, even to those with advanced education. Logic makes this viscerally clear.
2. There is a different between truth and proof in mathematics. There are things that are true. There are things that we can prove. The two are not always the same. It's amazing that such a distinction can even be made mathematically precise.
3. There is a difference between what is provable and what is algorithmically provable. This again is not obvious and being able to articulate these differences mathematically constitute some of the greatest intellectual achievements of the early 20th century.
4. Knowing about proof, antecedents, consequences and logical arguments helped me identify logical fallacies in what people would say as well as structure my own arguments and deduction better. I'm not sure what you mean by classical logic. There is a clear relationship between these two sentences which is not immediate if you've never seen quantifiers: every icecream is liked by some child, and some child likes all icecreams.
5. Temporal logics bring the benefit of logical reasoning to discussions involving time. Two of the following sentences are logically equivalent (assuming a linear model of time): "There will eventually be an earthquake in California", "it is always true that there will eventually be an earthquake in California", and "it is not the case that there will eventually be no more earthquakes in California."
6. Temporal logics are used for reasoning about computer programs and computer hardware. If you know about the algorithmic analysis of temporal logic properties of computer systems, there are specialised jobs in companies like Intel and Cadence.
7. Solvers for logics and logical theories have numerous applications in computer science and are rapidly finding use in both academic and industrial settings.

Seems quite practical to me.

I think the analogy made by Joseph Weissman ("This sort of feels like asking "why are higher mathematics any better/more interesting than elementary mathematics?"") is quite accurate. Almost everyone uses arithmetic every day, when grocery shopping, DIY, etc. The vast majority of 'mathematical problems' (in the widest sense of the word, obviously) that the general public encounters can be solved with arithmetic alone. Does that mean there is no point in studying more advanced mathematics? Of course there is, but the further you advance, the less explicit the applications become. In other words, the more advanced, the less direct, practical applications it seems to have for the general public. Also, keep in mind that discovering the usefulness of a skill (and its beauty) often comes after having learned that skill, not before learning it.

One thing formal logic can do is take ambiguous statements and make them less ambiguous, but this comes at a cost: simplification. This is a criticism I've heard before made by students who take a first course in logic: "it's much too restricted to grasp the full extent of real-life problems." I wouldn't completely disagree with that, if you keep in mind they've only studied propositional and the beginnings of predicate logic. While classic propositional logic can grasp a wide range of problems, there are also many problems that cannot be analysed by propositional logic alone. We move on to predicate logic; the addition of only two quantifiers makes predicate logic much more capable. Every new kind of logic basically does the same thing: you add new tools (operators, quantifiers, truth values) to make the analysis of more complex problems possible whilst keeping the unambiguity of formal logical analysis. In other words, studying more advanced logic - logics 'beyond' propositional and predicate logic - allows you to analyse more problems and do so more accurately (i.e. with less simplification).