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3 ideas
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) |
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'. |