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?
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.
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.
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)