|Quantification: is a function within the predicate logic, in which a property is attributed to an object yet to be determined. A) Existence quantification e.g. (Ex) (Fx) "At least one object x is F". It is assumed that the object denoted by x exists. B) Universal quantification (notation (x) ...) "For all x applies ...". Both forms of quantification can be negated, covering most of the everyday cases. In addition, a subject domain must be chosen, within which the statements that result from the insertion of objects are meaningful. See also existence, non-existence, existence assumption, existence predicate, universal quantification, existence quantification, domains, opacity, intensional objects._____________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. |
John Bigelow on Quantification - Dictionary of Arguments
Names/Pronouns/Bigelow/Pargetter: we have allowed a name to be replaced by a pronoun.
Example: The stock market in Japan has collapsed.
The stock market in it has collapsed.
Quantification/Name/Pronoun/Bigelow/Pargetter: after we have replaced the name with a pronoun, we can quantify via it. Example
For each country: the stock market in it has collapsed.
Quantification of 2nd level/logic of 2nd level/Bigelow/Pargetter: then one might ask, why should this not also work for predicates, if it is possible for names?
The idea goes like this: we start with a sentence, for example:
The Wombat is sleeping tonight.
Then we replace the predicate with a pronoun:
The Wombat it today.
Instead of a pronoun, we will take something better:
The Wombat is doing whatnot today.
Then we quantify:
For a whatnot, the Wombat is doing whatnot today.
Formal: we introduce a new variable (Greek letters)ψ, ψ 1, ψ 2,... then we replace each predicate of an atomic formula to provide another atomic formula: so
Fa equi Fb
ψa ⇔ ψb.
Then we will put a quantifier of the 2nd level in front of it:
(ψ)(ψa ⇔ ψb).
Translation into everyday language: "for something, a is doing whatnot iff b is doing whatnot"._____________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. Translations: Dictionary of Arguments The note [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990