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4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀

[symbol showing a variable refers to 'all' objects]

4 ideas
For Frege, 'All A's are B's' means that the concept A implies the concept B [Frege, by Walicki]
     Full Idea: 'All A's are B's' meant for Frege that the concept A implies the concept B, or that to be A implies also to be B. Moreover this applies to arbitrary x which happens to be A.
     From: report of Gottlob Frege (Begriffsschrift [1879]) by Michal Walicki - Introduction to Mathematical Logic History D.2
     A reaction: This seems to hit the renate/cordate problem. If all creatures with hearts also have kidneys, does that mean that being enhearted logically implies being kidneyfied? If all chimps are hairy, is that a logical requirement? Is inclusion implication?
If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers [Lemmon]
     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.
     From: E.J. Lemmon (Beginning Logic [1965], 3.2)
Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS]
     Full Idea: Universal Specification: from ∀xP(x) we may conclude P(t), where t is an appropriate term. If something is true for all members of a domain, then it is true for some particular one that we specify.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)
Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS]
     Full Idea: Universal Generalization: If we can prove P(x), only assuming what sort of object x is, we may conclude ∀xP(x) for the same x.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)
     A reaction: This principle needs watching closely. If you pick one person in London, with no presuppositions, and it happens to be a woman, can you conclude that all the people in London are women? Fine in logic and mathematics, suspect in life.