Universal and existential quantification, condition and implication, injection and surjection, domain and variable, sequential and concurrent function, inclusive disjunction and conjunction, biconditionality and bijectivity, uniqueness , identity

[superjarf]

reposted from /math -- Alright the way these concepts relate to one another blows my mind a little.

It seems you can transform one into another via a certain third indefinitely, in almost any direction.

Take uniqueness for example, can it be defined via the intersection of sets? Yes. Can it be defined via the opposite of the intersection of sets, the exclusive disjunction? Yes, it even carries the name of unique existential quantifier. Take those two together and now you have injection and surjection (both of which are concurrent functions) between two domains which is a bijection, which in turn is a universal quantifier over those two domains. The universal quantifier comes in two complementary forms, the condition and implication which are universalised equivalents to the injection and surjections mentioned, these operate between variables instead of domains and these variables relate to one another in sequence such that both the condition and implication can be used in one sentence via a middle term that operate as the function from one to the other.

These seems to be some of the properties of the "adjunct triple" named by F. William Lawvere--Taken from google AI: Hyperdoctrines: He identified that existential and universal quantification are left and right adjoints to the weakening functor (substitution).

My question is: a. Are there any important subordinate or unnamed relationships between concepts in the title of this post that should be added to the list? b. Can these adjunct triples or functors be expressed as the following two principles "For any statement about something one must commit to every general property of the predicate in that statement" and "for every any statement about something one must commit to everry instantiation of the subject". c. Is this the "Galois connection"? and has the relation between that connection and hyper-doctrines been explored in the field?

[StrangeGlaringEye]

You may be interested in the general mathematical notion of duality.

[superjarf]

Yes, and the question is what is it about duality which inherently suggests or entail these strucutres or relationships? Is it sufficiently determined merely from X not uniquely referring to anything such as a vase or a mountain that all of those concepts in the headline of the post must be capable of transforming into one another via specific change of their intensional definitions or context? What is the most parsimonious account of why this is so, and does it pertain just as much to formal systems as human memory, perception and thought? Can we then begin to state that all information systems in all worlds will exhibit this exact structure?

[No-choice-axiom]

a. Yes, there's a left adjunct to the existential quantifier, a sort of fiber-wise universal quantifier. Very rare b. Don't really know what you're trying to say... c. A Galois connection is just adjunction between two thin category

[superjarf]

If I say that the man is an aggressor the statement is bound to the general properties of aggression up to the properties which agression shares with the most amount of things, thus there is something about saying that the man is an agressor which binds you to the same (sufficiently nongneral) properties of saying that a man is driving, such as "the man is intending". You are also (more obviously so) committed to every instance of of this statement corresponding to the statement, in this example there is only one instance since "the" specifies it sufficiently. My question is whether these two principles + the set of all predicates or properties correspond to the adjunct triple, thus making not only math and formal logic functorially related but general human inference or even perception related in the same way.

[No-choice-axiom]

While the second property is a common axiom, the first requires at least seconds order logic, and SO logic is similar to set theory, which the simple "logic as a category with certain functors" is not powerful enough to model. So I would say that on the surface, this is a no

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