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Single Idea 13905

[filed under theme 4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀ ]

Full Idea

If all objects in a given universe had names which we knew and there were only finitely many of them, then we could always replace a universal proposition about that universe by a complex conjunction.

Clarification

A 'universe' is usually now called a 'domain'

Gist of Idea

If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers

Source

E.J. Lemmon (Beginning Logic [1965], 3.2)

Book Ref

Lemmon,E.J.: 'Beginning Logic' [Nelson 1979], p.105

The 4 ideas with the same theme [symbol showing a variable refers to 'all' objects]:

17745 For Frege, 'All A's are B's' means that the concept A implies the concept B [Frege, by Walicki]
13905 If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers [Lemmon]
13522 Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS]
13521 Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS]