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Ideas for 'fragments/reports', 'Title, Unity, Authenticity of the 'Categories'' and 'Higher-Order Logic'

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3 ideas

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

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'.