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How strong is the difference between inclusive and exclusive disjunction? At least, let ∨0 be inclusive ("weak") disjunction, and ∨1 be exclusive ("strong") disjunction. Then take a fourfold disjunctive question:

A1 B1 C1 D?

Now, it seems like you can logically add in some parentheses like so:

A1 {B1 C1 D}?

Then set {B, C, D} to ~A. So we have collapsed an exclusive quadruple into an excluded middle. In general, it seems like you'd say that for an n-ary exclusive disjunction, you can code for A with the number 1, and then for the alternative with n - 1.

However, what happens here with A1 A? Suppose, that is, that you have:

A1 A1 A?

Suppose that A = ~A, i.e. A is a self-contradictory term. At any rate, perform the collapse operation:

A1 A1 A?

A1 {A1 A}?

A1 ~A?

Now, is A1 A equivalent to an empty exclusive disjunction, i.e. whereas normally when using the disjunctive form in writing, we have in mind to write down at least three glyphs (e.g. A, the connective, and some B), yet for a purely empty disjunction we "ill-form" the sentence by writing only the first A and then the connective? For its being ill-formed, then, can we say that this empty exclusion is as much as to speak of assigning 1 as the code for A and then n - 1 for ~A, so that in fact A stays at 1 but ~A goes to zero, then?

It has been argued that exclusive negation being "true" negation is at issue, such that the strong paraconsistent/dialethic theorist sees room for inclusive negation. Does the above indicate that inclusive negation and self-exclusive disjunction repeat the general duality of paraconsistent and paracomplete logics, allowing the dialethicist to not "entirely change the subject" (contra Quine, then) while keeping in view that exclusive negation, and self-inclusive disjunction, are also real logical functions, and that holding fast the LNC and the LEM (and DNE, for that matter) for them could be "correct," even if inclusive negation doesn't conform to those protocols?

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  • I don't understand much of this. "A = ~A" does not express a contradiction, unless you are using = to represent a biconditional connective, which is a bad thing to do. Writing the more conventional ⊻ for exclusive or, "A ⊻ A" is false, for any A. One could define an n-ary exclusive or operator that would be true iff an odd number of its arguments are true. In which case, the otherwise ill-formed "A ⊻" would be true iff A is true. I don't see any interesting consequences of doing so...
    – Bumble
    Commented Aug 13, 2022 at 23:58
  • I'm not sure what 'inclusive negation' means. Some dialethist logicians consider that their understanding of negation is correct and the classical one is wrong; some logical pluralists agree with Quine and hold that different logics embody a different understanding of the meaning of the logical constants; other pluralists consider that there is no single right and wrong understanding of negation and the other constants.
    – Bumble
    Commented Aug 13, 2022 at 23:58
  • @Bumble, I don't think "inclusive negation" is a current phrase, but it's meant to cover talk of "paraconsistent negation" that I've read through some of. My counterargument(?), then, is that when we ask exclusive questions, we expect exclusive answers, so if there are different forms of negation, even so, the applicability of the LNC/LEM can be universal for exclusive questions, and the dialethist can only maintain the possibility of, "A & ~A," for inclusive questions, maybe. Commented Aug 14, 2022 at 0:23

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I cannot speak to any of the concerns which led to your question. However, I can say something.

With regard to material connectivity, inclusive disjunction is linearly separable whereas exclusive disjunction is not.

Typically, this is obfuscated by practices arising with logicism. Logicism (as presented in Russell's "Principles of Mathematics") begins with an emphasis on the material conditional. Consequently, that a proposition ought to be consequent to itself becomes the ground for portraying the law of excluded middle with an inclusive disjunction. With negation being taken for granted, one has a complete set of material connectives. That is, the exclusive disjunction is merely an eliminable abbreviation.

Logicism arises relative to a folklore declaring an arithmetization of mathematics. So, too, does modern formalism. The notion of linear separability is a topological notion which is regularly deprecated in these paradigms.

While I abhor nonsensical metaphysics grounded upon mathematics, you can find discussions of linear separability and Boolean polynomials in "artificial intelligence" (a self-describing field of study). More mathematically, look for topics on switching functions with threshold functions as the linearly separable subclass.

I believe that approaches different from classical accounts arise because the principle of excluded middle is inherently vague relative to representations of material connectives. Indeed, Pavicic and Megill demonstrated that classical propositional logic is not categorical in 1999. In 2006, Eric Schecter published a "hexagon model" for classical propositional logic having a syntactic character (as opposed to the algebraic characterization of Pavicic and Megill). Interpretation of biconditionals (the negation of exclusive disjunction) is one situation Schechter uses to show that the two possible models are different.

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