|Quantifiers: in the predicate logic, quantifiers are the symbol combinations (Ex) and (x) for the set of objects to which one or more properties are attributed to. A) Existence quantification (Ex)(Fx) ("At least one x"). B) Universal quantification (x)(Fx) ("Everything is F"). For other objects e.g. y, z,… are chosen. E.g. (x) (Ey) (Fx > Gy). See also quantification, generalized quantifiers._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
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Quantifiers/everyday language/Quine/Kaplan/Geach/Cresswell: not 1st order: E.g. some critics only admire each other - 2nd order: (Eφ)(Exφx u (x)(φx > x is a critics) u (x)(y)((φx u x admires y) > (x ≠ y u φy))). - That is not equivalent to any 1st order sentence - involves plural noun phrases (plural quantification).
The following is not correct: "two Fs are G".
One would have to assume that "admire" should be valid in both directions - (then x is a K u y is a K u x ≠ y ... "). - Better: "admire each other" is a predicate that is applied to pairs.
Right: "Smart and Armstrong are present" for "S. is a and A is a".
Problem: "King and Queen are a lovable couple", then "The King is an adorable ..." analog: E.g. "similar", e.g. "lessen".
Solution/Cresswell: applying predicate to quantities.
.. "admires another linguist" must be a predicate which is applied to all logicians. - This shows that quantification of higher level is required. - Problem: this leads to the fact that the possibilities to have different ranges are restricted.
Higher order quantifiers/plural quantifiers/Boolos: Thesis: these do not have to go via set theoretical entities, but can simply be interpreted as semantically primitive. ((s) basic concept). Cresswell: perhaps he is right).
Hintikka: game theory.
CresswellVsHintikka: only higher order entities. 2nd order quantification due to reference to quantities.
Branched quantifiers/Booles/Cresswell: "for every A there is a B".
(x = z ⇔ y = w) u (Ax > By)
2nd order translation: EφEψ(x)(z)((x = z ⇔ φ(x) = ψ(z)) u (Ax > Bφ(x)).
Function/unique image/assignment/logical form/Cresswell: "(x = z ⇔ φ(x) = ψ (z)" says that the function is 1: 1.
Generalization/Cresswell: If we replace W, C, A, B, and R by predicates that are true of all, and Lxyzw by Boolos ((x = z ⇔ y = w) u Ax> By) we have a proof of non-orderability of 1st order._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
M. J. Cresswell
Semantical Essays (Possible worlds and their rivals) Dordrecht Boston 1988
M. J. Cresswell
Structured Meanings Cambridge Mass. 1984