Rejecting the principle that any proposition can be meaningfully negated

Question

Is there a term for the meta-logical position that the negation of an arbitrary proposition is not a priori meaningful, or, in a stronger form, that some propositions lack negations?

I was reading this answer to this question, and noticed the claim quoted below.

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.

I'm wondering whether philosophers and logicians have doubted the meaningfulness of the negation of arbitrary propositions, and if this view has a body of work behind it.

If this view does exist, I'm curious if it's closely related to constructive or intuitionistic ideas where negation is (or can be) defined as implying a designated absurd proposition, and double negation elimination doesn't hold.

The only thing I really know of that's related to this idea is positive set theory, which structurally limits where propositions containing any negation whatsoever can appear.

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Rejecting the negation of arbitrary propositions, I think, has a strong and weak form analogous to the distinction between dialetheism and paraconsistency.

Dialetheism is the view that there are dialetheias, or true contradictions. This can be formalized in a number of different ways, but I think a fairly noncontroversial one is, for some A, the acceptance of A as true and not-A as true. Dialetheism thus rejects the principle of non-contradiction (assuming of course that the law of non-contradiction holds and there is a dialetheia is not itself a dialetheia). Dialetheism, at least my conception of it, is prior to any particular logical formalism.

I'm going to draw a distinction between dialetheism and paraconsistency, with the distinction being that paraconsistency is the ability to tolerate a contradiction as a hypothesis, without leading to the conclusion that everything is true. Merely accepting paraconsistency doesn't require accepting dialetheism.

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Is there a view that's similar in scope to dialetheism that rejects the totality of negation? I mean is there a meta-logical principle that does one of the following:

• claims that some sentences really do lack negations (analogous to dialetheism).
• rejects that the negation of an arbitrary proposition is inherently a proposition (roughly analogous to paraconsistency, i.e. we build a system that doesn't use the negation of arbitrary propositions, but doesn't commit to the existence of an unnegatable proposition)

Answer 1

It all depends on how you define logic.

In classical, propositional calculus, it is indeed true that any meaningful sentence's negation is meaningful as well.

This also holds within intuitionistic logic, within modal logic, even within fuzzy logic (of course negation means something slightly different there). Basically all common logics I know respect that notion.

Of course you can easily imagine and construct (which you were tempted to do) a logic that doesn't include the ~~p = p axiom, but for such a system to be useful and not self-contradictory, you'd have to define your truth values differently from the Aristotelian approach. My reasoning here is as follows. If you have two truth values, and they are designed to denote contradictory notions, is it even possible not to introduce the operator of negation?

Now, how is the notion of negation interpreted by meta logic, also depends on your meta logic. Let's say you define a logic that offers 10 truth values and that 3 of them are the significant values. Meta logic of such system would have to introduce some totally different apparatus to describe the notion of negation in it.

I guess the bottom line here is that your question lack in precision. It all depends on what logic and what meta logic you consider. But historically and classically: yes, every valid proposition's negation is also valid.