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Ideas for 'Aristotle and Descartes on Matter', 'Higher-Order Logic' and 'Philosophy of Logic'

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5 ideas

5. Theory of Logic / A. Overview of Logic / 2. History of Logic

18954	Before the late 19th century logic was trivialised by not dealing with relations [Putnam]
	Full Idea: It was essentially the failure to develop a logic of relations that trivialised the logic studied before the end of the nineteenth century.
	From: Hilary Putnam (Philosophy of Logic [1971], Ch.3)
	A reaction: De Morgan, Peirce and Frege were, I believe, the people who put this right.

5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic

18956	Asserting first-order validity implicitly involves second-order reference to classes [Putnam]
	Full Idea: The natural understanding of first-order logic is that in writing down first-order schemata we are implicitly asserting their validity, that is, making second-order assertions. ...Thus even quantification theory involves reference to classes.
	From: Hilary Putnam (Philosophy of Logic [1971], Ch.3)
	A reaction: If, as a nominalist, you totally rejected classes, presumably you would get by in first-order logic somehow. To say 'there are no classes so there is no logical validity' sounds bonkers.

10588	First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro]
	Full Idea: Early study of first-order logic revealed a number of important features. Gödel showed that there is a complete, sound and effective deductive system. It follows that it is Compact, and there are also the downward and upward Löwenheim-Skolem Theorems.
	From: Stewart Shapiro (Higher-Order Logic [2001], 2.1)

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

10298	Some say that second-order logic is mathematics, not logic [Shapiro]
	Full Idea: Some authors argue that second-order logic (with standard semantics) is not logic at all, but is a rather obscure form of mathematics.
	From: Stewart Shapiro (Higher-Order Logic [2001], 2.4)

10299	If the aim of logic is to codify inferences, second-order logic is useless [Shapiro]
	Full Idea: If the goal of logical study is to present a canon of inference, a calculus which codifies correct inference patterns, then second-order logic is a non-starter.
	From: Stewart Shapiro (Higher-Order Logic [2001], 2.4)
	A reaction: This seems to be because it is not 'complete'. However, moves like plural quantification seem aimed at capturing ordinary language inferences, so the difficulty is only that there isn't a precise 'calculus'.