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

#### [logic where variables only refer to objects]

18 ideas
 18956 Asserting first-order validity implicitly involves second-order reference to classes [Putnam]
 10767 Elementary logic is complete, but cannot capture mathematics [Tharp]
 13840 First-order logic is the strongest complete compact theory with Löwenheim-Skolem [Hacking]
 13844 A limitation of first-order logic is that it cannot handle branching quantifiers [Hacking]
 18108 First-order logic is not decidable: there is no test of whether any formula is valid [Bostock]
 18109 The completeness of first-order logic implies its compactness [Bostock]
 10478 Since first-order languages are complete, |= and |- have the same meaning [Hodges,W]
 19315 In quantified language the components of complex sentences may not be sentences [Kirkham]
 17788 First-order logic only has its main theorems because it is so weak [Mayberry]
 13624 The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro]
 13660 Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro]
 13662 First-order logic was an afterthought in the development of modern logic [Shapiro]
 13673 The notion of finitude is actually built into first-order languages [Shapiro]
 10588 First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro]
 10897 A first-order 'sentence' is a formula with no free variables [Zalabardo]
 10986 Not all validity is captured in first-order logic [Read]
 13534 In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS]
 13535 First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS]