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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?
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.
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.
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).
