Combining Philosophers

Ideas for David Fair, William W. Tait and Harold Hodes

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4 ideas

5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic

10015	Higher-order logic may be unintelligible, but it isn't set theory [Hodes]
	Full Idea: Brand higher-order logic as unintelligible if you will, but don't conflate it with set theory.
	From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131)
	A reaction: [he gives Boolos 1975 as a further reference] This is simply a corrective, because the conflation of second-order logic with set theory is an idea floating around in the literature.

5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic

10011	Identity is a level one relation with a second-order definition [Hodes]
	Full Idea: Identity should he considered a logical notion only because it is the tip of a second-order iceberg - a level 1 relation with a pure second-order definition.
	From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984])

5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic

10016	When an 'interpretation' creates a model based on truth, this doesn't include Fregean 'sense' [Hodes]
	Full Idea: A model is created when a language is 'interpreted', by assigning non-logical terms to objects in a set, according to a 'true-in' relation, but we must bear in mind that this 'interpretation' does not associate anything like Fregean senses with terms.
	From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131)
	A reaction: This seems like a key point (also made by Hofweber) that formal accounts of numbers, as required by logic, will not give an adequate account of the semantics of number-terms in natural languages.

5. Theory of Logic / K. Features of Logics / 1. Axiomatisation

13416	Mathematics must be based on axioms, which are true because they are axioms, not vice versa [Tait, by Parsons,C]
	Full Idea: The axiomatic conception of mathematics is the only viable one. ...But they are true because they are axioms, in contrast to the view advanced by Frege (to Hilbert) that to be a candidate for axiomhood a statement must be true.
	From: report of William W. Tait (Intro to 'Provenance of Pure Reason' [2005], p.4) by Charles Parsons - Review of Tait 'Provenance of Pure Reason' §2
	A reaction: This looks like the classic twentieth century shift in the attitude to axioms. The Greek idea is that they must be self-evident truths, but the Tait-style view is that they are just the first steps in establishing a logical structure. I prefer the Greeks.