Inventory of logical connectives

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Corbin Simpson

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Oct 5, 2020, 11:00:11 PM10/5/20
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coi

la gleki asks, in private communication, about several features of the inventory of logical connectives.

First, does Lojban have operators/connectives for material condition/implication? I claim no, which is contra CLL. To explain, let's take for example the family of jeks, with {ja}, {je}, {jo}, and {ju}. CLL suggests that {naja} is then the implication for this family. But is it?

Classical Boolean logic is not the only logic, and as soon as we generalize, then we have questions about implication. First, let's consider constructive Boolean logic. Imagine that we try to expand `X -> Y` to `~X \/ Y` using the above {naja} CLL rule. However, in constructive logic, `~X` "not X" is actually shorthand for `X -> _|_` "X implies falsehood", so our expansion becomes `(X -> _|_) \/ Y`. In constructive logic, implication is a primitive operation which can't be built just by satisfying a truth table. (BHK interpretations show us that constructive logic doesn't just show statements true or false, but builds structures which witness their truth/falsity.)

[omitted paragraph about linear logic; yes, it exists; no, it isn't germane]

Lojban, I think, is built on relations; this means that it has relational logic. (Lojban also has classical logic; the two logics interact through 2-categorical structure called an allegory.) In relational logic, we can reverse implications; `X -> Y` can be transformed into `Y -> X`. For example, given an implication {pa ka ce'u mlatu ce'u} which sends cats to species, we can obtain the reversal {pa ka ce'u se mlatu ce'u} which send species to cats. Recall that relational logic is many-to-many, so negation doesn't work like one might expect.

Practically, what does this mean? Well, it's bikeshed time. Personally, I recommend altering the CLL's De Morgan rules so that {naja} cannot be desugared into { ... na ... ja } with NA moving into and out of the connective, and instead freeze {naja} as "implies". I'm not 100% convinced that this is right, but it's very compatibilist: everybody still reads it as "implies" and we just disagree on what expansions are legal.

la gleki also asks about optimality of the current syntactic allocation. In particular, the quantifiers are in PA, the modal-logic operators are in CAhA, negation is in NA, NAI, and NAhE; what could be done better?

One obvious point is that all formal logics tend to get by with just one form of negation. Indeed, negation is usually tightly controlled and only mentioned in one or two axioms.

Modal logic being contained to CAhA makes it easy to be optional or to specify in terms of other grammar (thinking specifically of PU). CAhA cannot tag sumti like UI, which is unfortunate if one actually wanted to *do* modal logic.

NAhE could be extremely powerful, depending on how it's formalized.
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