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Assumptions/presuppositions. I am trying to set up a logic where every connective/operator comes in at least two flavors. For example, with respect to disjunction, rather than hold the LEM rigidly over every disjunction, we confine its primary meaning/value (in the system) to binary disjunctions alone. Although it is possible to "collapse/reduce" any n-ary disjunction to a set of one disjunct vs. a set of all the other disjuncts, we say that this "misses the point" of having n-ary disjunctions overall. So in turn, we suspect that there are multiple flavors of the disjunction operator, sequenced as ∨n say, one for each general type of n-ary disjunction.

I've also "heard talk" of something going by the name "paraconsistent negation," which is presumably a negation operator that is "inclusive" in the sense of being sustainable for conjunctions of the form A&¬A. For here we might say that there are the ¬n, or we might distinguish them by correlating their order with gradients in a color palette, or whatever. Anyway, now we can confine the LNC to a specific zone, too, namely we can say that ¬0◊(A&¬0A), but this might have no effect on whether ◊(A&¬1A), say, or whatever along that line.

Question regarding the universal quantifier. First off, I might just be wrong about this, but are there any serious analysts who have argued that "any" and "all" should not be held identical before the tribunal of ∀'s own identity? My argument comes down to this: if we should expect a plurality of types for any operator scheme, still qualified by the by, by the reason we have to use such operators separately (why something like the Sheffer stroke is either pre-eminently outlandish or too sacred for ordinary use), then we should expect the quantifiers to have different "flavors" too, and we should exploit the subtle variations this makes possible. At any(!) rate, to have something like ∀0 as "any" but ∀1 as "all," would be the kind of thing we'd have in mind. {A variation on the theme: ∀V, for V the universal transet, is the universal quantifier par excellence, and the proper object of "all," wherefore there are eternally many (weirder and weirder!) degrees of universal quantification.}

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    Perhaps such multiple types are hard to justify in the straightforward logic sense of Quine's bound variable's referential integrity, but I smell "any", "every", "all" may have entirely different forces in the sense of language pragmatics in everyday life conversation under different contexts, which is outside the dominion of logic. While if you want to describe quantifiers functionally independent from some variables, there's other version such as in Independence-friendly logic... Commented Aug 23, 2022 at 5:15
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    In modeling natural languages they distinguish different quantifier types, see e.g. Contextualising generic and universal generalisations:"‘All’, ‘all the’, ‘every’ and ‘each’ are all universal quantifiers, but they differ in terms of distributivity, and on whether they allow, may resist or require QDR". You can sometimes simulate this by defining derived quantifiers w/ restricted scope from a single umbrella one.
    – Conifold
    Commented Aug 23, 2022 at 5:22
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    That we do routinely in ZFC. However, primitive quantifier types w/ distinct scopes are also used, e.g. in type theory, where they want different quantifiers for objects, predicates, etc., see Sider, p.66, or in versions of modal logic, where they distinguish quantification over actual vs merely possible objects, see Second-Order Modal Logic.
    – Conifold
    Commented Aug 23, 2022 at 5:28

2 Answers 2

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In English, the quantifiers 'all', 'any', 'every' and 'each' are somewhat different, though all of them would qualify to be called universal quantifiers. In simple cases, they are interchangeable, for example,

All mammals have hair.
Every mammal has hair.
Any mammal has hair.
Each mammal has hair.

But when they are embedded within the scope of another quantifier, or in the antecedent of a conditional, or within a modal context, they differ. For example,

Alice will be upset if any person she invited fails to show up.
Alice will be upset if every person she invited fails to show up. 

Bob has not visited every country in Europe.
Bob has not visited any country in Europe. 

It is impossible for any person here to lift this rock.
It is impossible for all the people here to lift this rock. 

Also, when we wish to distinguish what is collectively true from what is distributively true, 'all' is often used for the former, and 'each' for the latter. For example,

Bob greeted all the attendees. 
Bob greeted each attendee.

When expressing examples of such sentences into quantifier logic, we need to be aware that sometimes 'any' works like an existential quantifier. The distinction between collective and distributive uses of quantifiers can often be handled by using plural quantification.

There are other odd differences in English. 'Every' does not work with small numbers. "Each of my parents was born in the UK" is OK, but not "Every one of my parents was born in the UK". 'All' and 'every' can be qualified, e.g. "almost all", "almost every", but not "almost each". 'Any' can be used with mass nouns, but 'each' cannot, e.g. "Any moisture that comes into contact with this surface will cause corrosion", but not, "Each moisture...". Oddities like these are why it is nice to use formal languages.

In the context of proofs over infinite domains, one might argue that 'all' and 'any' differ in another way. If F(x) is some predicate defined over the natural numbers and I say I can prove F(x) for all x, that suggests I have a proof of (∀x)F(x). But if I say I can prove F(x) for any x, that suggests rather that for any particular natural number n, I can prove F(n) and do so in a way that clearly generalizes and does not depend on that particular value. The gap between them is filled by the ω-rule.

I'm not sure if this helps you. As with many of your questions, they are very interesting but I'm not quite certain what you wanting in an answer.

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  • The main contributors here almost always have extremely useful/insightful responses, as given answers or at least as comments. Actually, a lot of the time I read people saying that they're not sure they understand what I'm saying, but they sound like they understand it better than I do half the time or something. So IDK. I would say your answer is a completely sufficient option for the priority answer to the OP question. I do want to see if anyone else is familiar with the issue, and can offer maybe a problem with my idea, that would be very interesting to have to work on. Commented Aug 24, 2022 at 3:02
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In the appendix to Blok and Pigozzi it is explained that the algebraization of first-order logic impacts the "standard" (ahem, a nonsense word for mathematics) quantifier rules associated with symbolic first-order logic.

One "hint" at what is going on can be found in Markov's "Theory of Algorithms." To avoid the reasoning of partiality characteristic of intuitionism, Markov introduces a notion of "stengthened implication" which relies on "givenness."

Although it is easy to find philosophers who shower contempt upon such a notion, this idea can be used to interpret existential quantification as a restricted quantifier using reflexive equality statements. So,

Ex Phi(x)

becomes

a=a /\ Phi(a)

while

Ax Phi(x)

becomes

a=a -> Phi(a)

Existential import associated with equality is attached to Tarski's transitivity axiom,

AxAy( x=y <-> Ez( x=z /\ z=y ))

with

Ax( ~x=x <-> Az( ~x=z /\ ~z=x ))

needed to exclude a problematic disjunction in the denied equality.

For symmetry,

AxAy( ( x=x /\ y=y ) -> ( x=y <-> y=x ) )

If you look at chapter 19 of Russell's "Principles of Mathematics," you will see that "givenness" circumvents the requirement of necessarily true reflexive equality statements.

Although this gives only a few of the needed elements to make this work, it is related to the principle of indiscernibility of non-existents from negative free logic. One needs both a discernibility relation whose denial does not carry existential import and a distinctness relation whose denial does.

An important interpretation of denied discernibility in this context would be

"is singular"

Free logics can support notions of definite description. Problematically, introducing denoting terms with defining syntaxes does not convey individuation. So, if one wishes to implement something along such lines, one needs an internal support that a term obtained through description is declared to be singular.

In October of 2015 Harvey Friedman opened a thread on free logic at the FOM mailing list. One participant, Mitchell Spector, made the simple observation which lies at the heart of these manipulations,

https://cs.nyu.edu/pipermail/fom/2015-October/019285.html

A reflexive equality can be used to manage the expressikn,

"is defined"

In a logical calculus, however, the expectation of a semantics means that defined terms denoting individuals be singular and refer to existents. So, it takes some wotk to get to a system with a calculus.

So much for how "all" and "any" can be different. As for the rest of your question I have no insight.

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