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Single Idea 19057

[filed under theme 5. Theory of Logic / G. Quantification / 1. Quantification ]

Full Idea

Classical quantification represents an infinite conjunction or disjunction, and the truth-value is determined by the infinite sum or product of the instances ....but this presupposes that all the instances already possess determinate truth-values.

Gist of Idea

Classical quantification is an infinite conjunction or disjunction - but you may not know all the instances

Source

Michael Dummett (The philosophical basis of intuitionist logic [1973], p.246)

Book Ref

Dummett,Michael: 'Truth and Other Enigmas' [Duckworth 1978], p.246

A Reaction

In the case of the universal quantifier, Dummett is doing no more than citing the classic empiricism objection to induction - that you can't make the universal claim if you don't know all the instances. The claim is still meaningful, though.

The 23 ideas with the same theme [general ideas about expressing quantities of objects]:

11149 Affirming/denying sentences are universal, particular, or indeterminate [Aristotle]
9106 The word 'every' only signifies when added to a term such as 'man', referring to all men [William of Ockham]
9950 A quantifier is a second-level predicate (which explains how it contributes to truth-conditions) [Frege, by George/Velleman]
14137 'Any' is better than 'all' where infinite classes are concerned [Russell]
9467 Wittgenstein tried unsuccessfully to reduce quantifiers to conjunctions and disjunctions [Wittgenstein, by Jacquette]
10922 Objects are the values of variables, so a referentially opaque context cannot be quantified into [Quine]
9015 Universal quantification is widespread, but it is definable in terms of existential quantification [Quine]
10926 Quantifying into referentially opaque contexts often produces nonsense [Quine]
10311 No sense can be made of quantification into opaque contexts [Quine, by Hale]
10538 Finite quantification can be eliminated in favour of disjunction and conjunction [Quine, by Dummett]
10799 Nominalists should quantify existentially at first-order, and substitutionally when higher [Marcus (Barcan)]
15891 Traditional quantifiers combine ordinary language generality and ontology assumptions [Harré]
19057 Classical quantification is an infinite conjunction or disjunction - but you may not know all the instances [Dummett]
13438 'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors [Bostock]
6042 The quantifier is overrated as an analytical tool [McGinn]
6067 Existential quantifiers just express the quantity of things, leaving existence to the predicate 'exists' [McGinn]
6890 Quantifiers turn an open sentence into one to which a truth-value can be assigned [Mautner]
18492 Not all quantification is either objectual or substitutional [Williamson]
11007 Quantifiers are second-order predicates [Read]
8452 Traditionally, universal sentences had existential import, but were later treated as conditional claims [Orenstein]
16416 The quantifier in logic is not like the ordinary English one (which has empty names, non-denoting terms etc) [Hofweber]
21643 The inferential quantifier focuses on truth; the domain quantifier focuses on reality [Hofweber]
23494 Conjunctive and disjunctive quantifiers are too specific, and are confined to the finite [Morris,M]