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The only search results I got for the exact phrase "negation as a binary relation" were a cryptic essay and/or book about "Chinese opposites." Now, what I have in mind is something like A ¬ B, read as, "A negates B." This doesn't immediately look the same as something like, "A - B," since in this case what's really being negated is something about A, i.e. A is losing some of its content.

Motivation: while fiddling with my idea about the imaginary unit and the turnstile notation in logic,# I found that if I set {demi-negation = ⊣}, then I would have {proof = ⊢}, but that symbol usually figures as a binary relation, i.e. A ⊢ B. So it would seem that demi-negation would be used in the sense of, "A demi-negates B." Worse (or not), proof then is interpreted as, "A fully negates the demi-negation of B."

#For better or worse, then, so far I've ended up with no statements of the form, "A ⊢ B," being true in a normal way. Their imaginative truth value is instead -i, like all "correct"(?) statements, "A ⊣ B," having the truth value i. But this might not be such a problem: there are logics of conditionals where at least some conditionals are neither true nor false, and then in terms of substructural logic the question can be opened as to whether (A ⊢ B) → (A → B). Normally, if we have as a premise that A actually proves B, the inference of a counterpart conditional is pointless; but we might phrase the antecedent as, "If A would prove B," which seems like it might work better.

Kristian Berry
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    There is no binary negation. Negation is a unary operation. – David Gudeman May 11 '23 at 18:56
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    @DavidGudeman I would hope by now that when I go through all the trouble of citing the wide range of ideas that occur in advanced logic studies, no one would respond to my questions with such a bland denial. Obviously the classical negation operator is unary, but also I am obviously not relying on classical logic, here. – Kristian Berry May 11 '23 at 19:02
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    Incidentally, since over my earlier time here I had come to appreciate how few of this SE's contributors actually know much about advanced logic studies (and even fewer understand what they know), I thought to try asking my questions over on MathOF, where I was met with the response that my question was just "fun" and so not serious enough for them to evaluate. I know there's not much, if anything, I can do about gatekeeping besides complain, but so I guess I'll complain about it for now. – Kristian Berry May 11 '23 at 19:07
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    And I'd rather not keep all my ideas locked up in my head. If I do that, if I don't subject my thoughts to peer review, then my ambient frustration will lead me to believe that I must be "right about everything" and I'll end up like Ayn Rand or Christopher Langan (except perpetually impoverished without any apparent hope of salvation). Or worst of all, I'll end up like my mortal enemy (the founder(s) of the Internet cult I'm fighting). – Kristian Berry May 11 '23 at 19:14
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    Personally I am interested in the workings and semantics of negation, from proto-language manifestation of negation up to advanced logic. My own question about the first manifestation of negation in language in linguistics.se (years ago) was simply dismissed as little more than meaningless, if it makes you feel better. – Nikos M. May 11 '23 at 21:40
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    A theory of mine about constructive or realizable negation (that produces evidence of negative) which needs a lot of work, in fact is binary relation, like difference between two numbers is binary relation. Holding one number fixed as reference (eg zero) we have unary negation. So negation can be binary if we attach the reference we negate with respect too (eg the universe set we have as reference). Two cents. – Nikos M. May 11 '23 at 21:46
  • @KristianBerry, negation is a denial operator. To say that not P is true is to say that P is false. This is by its nature an operator that only applies to one P. There are related operators that apply to two or more propositions such as NAND and NOR, but those aren't negations. Nothing that is a binary operator can be properly be called negation. My comment was not a complaint about your speculations, but about the language you chose for it. It strikes me as an attempt to make the concept seem more interesting than it is by adding an air of paradox. – David Gudeman May 11 '23 at 21:55
  • If H is a hypothesis and P is a disproof of H, isn't contradicts(P, H) a binary relation that negates the object H? – J D May 12 '23 at 05:33
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    What you mean by "binary relation" is probably what is more commonly called binary operation as it applies to propositions rather than to domain objects (like < would, for example). Your "A negates B" sounds to me like A → ¬ B, and one can play with various versions of → and ¬ to match the specific intention of "negates". With the material implication and the standard negation this is the [Sheffer stroke NAND](https://iep.utm.edu/sheffers/), and it corresponds to the notion of [contraries](https://en.wikipedia.org/wiki/Square_of_opposition#Summary) in the traditional square of opposition. – Conifold May 12 '23 at 05:59
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    For some recent work on contrariety as a binary relation on propositions in term logic see [Humberstone](https://academic.oup.com/jigpal/advance-article-abstract/doi/10.1093/jigpal/jzac070/6761422). He suggests tightening it up from the traditional symmetric "cannot both be true, but can both be false". – Conifold May 12 '23 at 06:03
  • It seems possible to define A *negates* B. – Agent Smith May 13 '23 at 02:45
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    @KristianBerry I’m just here for moral support. The difficulty other stacks have and this one too occasionally in *fairly* answering real questions is a sign you are asking good questions. – J Kusin May 13 '23 at 15:31

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There are a few possibilities that come to mind, though none seem to be exactly what you are asking for.

One can define a kind of 'conditional negation' or 'inner negation' that has the sense that ¬A means that provided A has a truth value then ¬A says that A is false. This allows for a gappy logic that is related to connexive logic. It permits the inference from A → ¬B to ¬(A → B) for example, and for A → B to be contrary to A → ¬B . This is still a unary negation, although one could potentially extend it by developing a negation for conditionals. If one of the motivations is to be able to express the negation of "if A then B" as a binary relation on A, B then one could treat this as a kind of binary negation. For the material conditional of classical logic, the negation of A ⊃ B is A ∧ ¬B, but there is a well-known body of counterexamples and there are conditional logics in which negated conditionals work differently. For more on the concept of conditional negation, have a look at John Cantwell "The Logic of Conditional Negation" Notre Dame Journal of Formal Logic, 2008, Volume 49, Number 3, pp. 245-260.

A related option is to understand negation in the context of suppositions or contextual information. One could then have a binary operation expressing the negation of A given the supposition B, or within the context B. As above, this allows for a gappy logic where a proposition fails to have a truth value if its presuppositions are not satisfied.

One can focus on the concept of contrariety rather than negation. There are different ways in which a pair of propositions A, B may be denials of one another in a broad sense. The Stanford Encyclopedia article on negation has some material on this, particularly sections 1.5 and 1.6.

If you interested in the concept of logical subtraction, Stephen Yablo has written about it in his book Aboutness (2014) and in Stephen Yablo (2017) "If-Thenism", Australasian Philosophical Review, Volume 1, pp. 115-132.

Bumble
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  • Conditional negation seems to be the clearest route, albeit to make it "worth the effort," so far I've had to use the concept of demi-negation. Let % be demi-not and *A, B, X* be propositions. So say: "If *A* % *X* and if *B* % *X*, then if (*A&B*) then not-*X*." Then for any full negation to hold would require two prior demi-negations to hold, which sounds incorrect, although if you set demi-negative propositions to a truth value of *i*, you could picture starting with *X*, demi-negating it by multiplying its truth value by *i*, and then fully negating *X* by multiplying it by *i* again. – Kristian Berry May 18 '23 at 00:20