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.

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)
  • Frankly speaking, I’m perplexed. Can you suggest some example of meaningful sentence whose denial is meaningless? – Mauro ALLEGRANZA Feb 10 at 17:30
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    Here are three examples, not sure how good they are. We can try to construct rules for when negation is allowed so that the Gödel sentence cannot be negated. We can allow self-reference in positive sentences to rule out the liar paradox, so this statement is true is okay, but this statement is not true is prohibited. Another possible idea, motivated by natural language, is prohibiting double negation ... i.e. if I have not P, then not not P is not well-formed ... which gives us the flexibility to make rules for removing and introducing negation a little bit different. – Gregory Nisbet Feb 10 at 17:39
  • My question is about the context: are you considering formal theories? If so, the issue is: if the negation sign is part of the language, the usual syntactical rules include (for what I know...): if A is a formula, then not-A is a formula. Obviously, we may have languages without negation sign; thus, my perplexity. – Mauro ALLEGRANZA Feb 11 at 7:58
  • If the context is natural language, things are very different. Natural language is full of expression without a definite truth-value: imperatives, questions, vague statement, self-referential ones. Thus, if we apply negation to a "not-meaningful" statement, what will be the resulting meaning? The answer is not so straightforward: does the imperative "Stan up!" have a truth value? No. Does it have meaning? I think so: we can understand it. If so, what happens with "Do not stand up!" Again, I'm in trouble with the truth-value, but I cannot imagine reasons to deny that it is "undertsandable". – Mauro ALLEGRANZA Feb 11 at 8:01
  • Conclusion: what is the language we are speaking of? what is the meaning of an expression? – Mauro ALLEGRANZA Feb 11 at 8:02

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.

  • In most logical systems, (independence-friendly logic is an exception) the negation is well-formed if φ is well-formed. We make this choice when inductively defining the set of well-formed formulas, L. Picking the well-formed formulas "happens before" we consider semantics or other properties. I'm wondering whether people have defended systems that are not closed under negation at a syntactic level as capturing some essential aspect of truth, the way some people have done for dialetheism. – Gregory Nisbet Feb 13 at 20:27
  • I recommend reading Jan Lukasiewicz's 'On the Principle of Contradiction in Aristotle', 1947. He was a member of famous Lvov-Warsaw logic school. In the book I've mentioned he points out several fallacies within Aristotle's reasoning. – k-wasilewski Feb 13 at 22:24

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