Non-Closed Logic? [closed]

Question

Our logic is usually called "closed", i.e. if there's some formula/propostion F then its negation ~F is also a formula/proposition.

Could there be also a non-closed logic, i.e. a logic where F is a formula/proposition, but ~F is not?

Answer 1

I'll address points (1) and (2); point (3) seems unrelated (and frankly, too vague to admit an answer in my opinion).

Re: (2), there is unfortunately no standard definition of "logic" (or "logical system," or "formal system," or etc.). There are various different definitions employed in different contexts. One of the standard ones - the notion of a regular logic (see e.g. the end of Ebbinghaus-Flum-Thomas) - does indeed require closure under negation, and so existential second-order logic (for example) is not a regular logic. But there are many other definitions floating around, including vastly more general ones.

Your argument in (1) that all logics have to be closed under negation seems to rest on the assumption that a logic should consist of all meaningful expressions in some context. Since every meaningful statement's negation is also meaningful (I think this is indisputable), this would indeed imply negation-closure. However, this sort of "expressive maximality" is not generally assumed. For example, we tend (to put it mildly) to consider first-order logic a logic, but it is clearly not maximally expressive since e.g. there is no first-order sentence corresponding to finiteness.

Basically, when we talk about existential second-order logic as a logic, there is no tacit assumption that negations of existential second-order sentences need not be meaningful; rather, we're merely choosing to focus on a particular set of meaningful expressions, for whatever reason - either intrinsic interest or application in some way. It may help to observe that interesting properties of logics (broadly construed) need not be "negation-symmetric." For example:

• Every satisfiable existential second-order sentence has a countable model, but there are satisfiable universal second-order sentences without countable models.

• Every finitely-satisfiable set of existential second-order sentences is satisfiable, but there are finitely-satisfiable sets of universal second-order sentences which are not satisfiable.

So existential second-order logic has the downward Lowenheim-Skolem and compactness properties, while universal second-order logic doesn't. Together with its game-theoretic analysis, this points to existential second-order logic as being "meaningfully tamer" than its dual.

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Incidentally, while I've absolutely taken a mathematical stance above, this isn't something which should be thought of as purely mathematical. For example, the standard adage "it's impossible to prove a negative" (ignoring its serious flaws) points to a fundamental difference in nature of meaningfulness between "purely existential" and "purely universal" claims. One can argue - at least in certain contexts - that some claims are "confidently affirmable but not confidently deniable." I would argue that what makes a logic "natural" is that we have some unifying idea of how its sentences are granted meaning, and while meaningfulness is preserved by negation the type of meaningfulness need not be.