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Single Idea 8473

[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism ]

Full Idea

The question to be posed is whether is-a-member-of should be considered a logical constant, that is, does logic include set theory. Frege, Russell and Whitehead held that it did.

Gist of Idea

The logicists held that is-a-member-of is a logical constant, making set theory part of logic

Source

Alex Orenstein (W.V. Quine [2002], Ch.5)

Book Ref

Orenstein,Alex: 'W.V. Quine' [Princeton 2002], p.99

A Reaction

This is obviously the key element in the logicist programme. The objection seems to be that while first-order logic is consistent and complete, set theory is not at all like that, and so is part of a different world.

The 14 ideas from 'W.V. Quine'

8452 Traditionally, universal sentences had existential import, but were later treated as conditional claims [Orenstein]
8454 The whole numbers are 'natural'; 'rational' numbers include fractions; the 'reals' include root-2 etc. [Orenstein]
8457 The Principle of Conservatism says we should violate the minimum number of background beliefs [Orenstein]
8458 Just individuals in Nominalism; add sets for Extensionalism; add properties, concepts etc for Intensionalism [Orenstein]
8471 Three ways for 'Socrates is human' to be true are nominalist, platonist, or Montague's way [Orenstein]
8465 Mereology has been exploited by some nominalists to achieve the effects of set theory [Orenstein]
8474 Unlike elementary logic, set theory is not complete [Orenstein]
8472 Sentential logic is consistent (no contradictions) and complete (entirely provable) [Orenstein]
8476 Axiomatization simply picks from among the true sentences a few to play a special role [Orenstein]
8473 The logicists held that is-a-member-of is a logical constant, making set theory part of logic [Orenstein]
8475 The substitution view of quantification says a sentence is true when there is a substitution instance [Orenstein]
8477 People presume meanings exist because they confuse meaning and reference [Orenstein]
8484 If two people believe the same proposition, this implies the existence of propositions [Orenstein]
8480 S4: 'poss that poss that p' implies 'poss that p'; S5: 'poss that nec that p' implies 'nec that p' [Orenstein]