In first order logic with equality, it is easy to define numerical quantifiers such as "there exist exactly two x such that...", or "there exist at least six x such that...". I am trying to develop a logic, more expressive than bare first order logic with equality, but not as expressive as set theory or second order logic, where there can be variable numerical quantifiers. One could say, for instance, that there are 3n+1 x such that Px, where the n is itself a variable that can be quantified over. Has anyone pursued this idea? In other words, is there any paper or book where someone has taken this idea seriously and pursued it?
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Isn't it the case that set theory is formalized in first order logic? If so how can it be that set theory is more expressive than FOL? – Trismegistos Dec 11 '13 at 09:21
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I mean, set theory is more expressive than first-order logic with equality by itself. – user107952 Dec 11 '13 at 11:12
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How theory described by FOL can be more expressive than FOL? That is contradiction. – Trismegistos Dec 11 '13 at 12:40
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Anyway, my question about variable numerical quantifiers remains unanswered. Is there a paper or text that talks about this? – user107952 Dec 11 '13 at 20:21
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See Jouko Vaananen, Models and Games (2011), Ch.10 : Generalized Quantifiers.
See page 284 :
Definition 10.1 A weak (generalized) quantifier is a mapping $Q$ which maps every non-empty set $A$ to a subset of $\mathcal P(A)$. A weak (generalized) quantifier on a domain $A$ is any subset of $\mathcal P(A)$.
See page 285 :
Example 10.2
1 . The existential quantifier ...
3 . The counting quantifier $\exists^{\ge n}$ ...
Mauro ALLEGRANZA
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