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Single Idea 9015

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

Full Idea

Universal quantification is prominent in logical practice but superfluous in theory, since (for all x)Fx obviously amounts to not(exists an x)not-Fx.

Gist of Idea

Universal quantification is widespread, but it is definable in terms of existential quantification

Source

Willard Quine (Philosophy of Logic [1970], Ch.2)

Book Ref

Quine,Willard: 'Philosophy of Logic' [Prentice-Hall 1970], p.25

A Reaction

The equivalence between these two works both ways, some you could take the universal quantifier as primitive instead, which would make general truths prior to particular ones. Is there something deep at stake here?

The 25 ideas from 'Philosophy of Logic'

10014 Quine rejects second-order logic, saying that predicates refer to multiple objects [Quine, by Hodes]
9012 Talk of 'truth' when sentences are mentioned; it reminds us that reality is the point of sentences [Quine]
9011 Truth is redundant for single sentences; we do better to simply speak the sentence [Quine]
9009 Single words are strongly synonymous if their interchange preserves truth [Quine]
9007 It makes no sense to say that two sentences express the same proposition [Quine]
9008 There is no rule for separating the information from other features of sentences [Quine]
9010 We can abandon propositions, and just talk of sentences and equivalence [Quine]
9013 We can eliminate 'or' from our basic theory, by paraphrasing 'p or q' as 'not(not-p and not-q)' [Quine]
9015 Universal quantification is widespread, but it is definable in terms of existential quantification [Quine]
9016 Names are not essential, because naming can be turned into predication [Quine]
9014 Some conditionals can be explained just by negation and conjunction: not(p and not-q) [Quine]
9017 Predicates are not names; predicates are the other parties to predication [Quine]
9018 A physical object is the four-dimensional material content of a portion of space-time [Quine]
9019 Four-d objects helps predication of what no longer exists, and quantification over items from different times [Quine]
9020 My logical grammar has sentences by predication, then negation, conjunction, and existential quantification [Quine]
9021 A good way of explaining an expression is saying what conditions make its contexts true [Quine]
10705 Putting a predicate letter in a quantifier is to make it the name of an entity [Quine]
10828 Quantifying over predicates is treating them as names of entities [Quine]
10012 Quantification theory can still be proved complete if we add identity [Quine]
9024 Excluded middle has three different definitions [Quine]
9025 You can't base quantification on substituting names for variables, if the irrationals cannot all be named [Quine]
9026 Some quantifications could be false substitutionally and true objectually, because of nameless objects [Quine]
9023 If you say that a contradiction is true, you change the meaning of 'not', and so change the subject [Quine]
9027 A sentence is logically true if all sentences with that grammatical structure are true [Quine]
9028 Maybe logical truth reflects reality, but in different ways in different languages [Quine]