Combining Philosophers

Ideas for Hermarchus, Penelope Maddy and Brian R. Martin

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5 ideas

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

10594	Henkin semantics is more plausible for plural logic than for second-order logic [Maddy]
	Full Idea: Henkin-style semantics seem to me more plausible for plural logic than for second-order logic.
	From: Penelope Maddy (Second Philosophy [2007], III.8 n1)
	A reaction: Henkin-style semantics are presented by Shapiro as the standard semantics for second-order logic.

5. Theory of Logic / C. Ontology of Logic / 3. If-Thenism

17620	Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy]
	Full Idea: If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation.
	From: Penelope Maddy (Defending the Axioms [2011], 3.3)
	A reaction: I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics.

5. Theory of Logic / E. Structures of Logic / 1. Logical Form

18168	'Propositional functions' are propositions with a variable as subject or predicate [Maddy]
	Full Idea: A 'propositional function' is generated when one of the terms of the proposition is replaced by a variable, as in 'x is wise' or 'Socrates'.
	From: Penelope Maddy (Naturalism in Mathematics [1997], I.1)
	A reaction: This implies that you can only have a propositional function if it is derived from a complete proposition. Note that the variable can be in either subject or in predicate position. It extends Frege's account of a concept as 'x is F'.

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

17605	Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy]
	Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers.
	From: Penelope Maddy (Defending the Axioms [2011], 1.3)

17625	If two mathematical themes coincide, that suggest a single deep truth [Maddy]
	Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth.
	From: Penelope Maddy (Defending the Axioms [2011], 5.3ii)