Topics Logic

Question What is the difference between mathematical logic and philosophical logic? Yes I know, one has more math than the other. Is Gödel's incompleteness in mathematical logic? Is modal logic in philosophical logic? Can you give other examples of different logics or questions each asks in order to distinguish the two?

Accepted:
November 10, 2008

Comments

Gödel's first incompleteness theorem says that, for any suitable formal theory which is consistent and includes enough arithmetic, there will be an arithmetical sentence -- a "Gödel sentence" -- which that theory can neither prove nor disprove. This theorem is a bit of mathematics: its proof is undoubtedly a sound mathematical proof. (That's why those obsessives who plague internet discussion groups with purported refutations of Gödel are so very annoying! -- they are refusing to follow, or are incapable of following, a relatively straightforward bit of purely mathematical reasoning.)

The question of the significance of Gödel's theorem, however, is quite another matter. To investigate that, we need to engage in philosophical reflection. Some have held, for instance, that Gödel's theorem can be used to show that minds are not machines. Let's not worry here about why that's been said: the simple point we need now is just that, to decide the merits of this interpretative view, we need not more mathematics but some philosophy.

Likewise, modal logic is a body of mathematics. It explores, proof theoretically, what can be demonstrated in various formal deductive systems. It explores, model theoretically, various mathematical structures for doing formal semantics for the languages of those systems. It proves various soundness and completeness results ("S4 is sound and complete with respect to a standard modal semantics where the accessibility relation is reflexive and transitive").

The question of the significance of this or that modal deductive system is, however, another matter. Which logic is the apt one for reflecting the intuitive notion of logical necessity? Does a modal logic need an actuality operator if it is to adequately regiment the intended range of everyday modal reasonining? How seriously should we take the metaphor of possible worlds often used in explaining modal semantics? To answer such questions we again need not more mathematics but philosophical reflection.

These remarks might suggest an obvious demarcation. We might naturally draw a line between "mathematical logic" which is involved with the mathematical investigation of various formal theories of logical interest, and "philosophical logic" which is concerned with the philosophical significance of the technical results.

Well, it would be jolly neat if that's how the terms were used in practice, but it isn't. There's a tradition of using "mathematical logic" to cover a cluster of areas which includes some topics (e.g. set theory) which we might well argue aren't strictly speaking logic but are mathematics, while we exclude other areas, like mathematical investigations of modal systems, which undoubtedly are logic. For example, a recent textbook like Peter Hinman's massive Fundamentals of Mathematical Logic doesn't mention modal logic at all.

And on the other side, "philosophical logic" is used to cover not just philosophical reflection on formal logical systems and on results about them, but also reflection on e.g. how names and definite descriptions work in ordinary language. While we'd in fact put the question of the significance of Gödel'stheorem into thephilosophy of maths course, not the philosophical logic course.

So the actual terminological situation is a bit messy and rather unprincipled. But let's not worry about that! What matters is to be clear about when we are engaged in a kind of mathematical enquiry and when we are engaged in philosophical interpretation and reflection.