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

#### [logic extending variables to predicates and relations]

33 ideas
 11033 Predications of predicates are predications of their subjects [Aristotle]
 9188 Gödel proved that first-order logic is complete, and second-order logic incomplete [Gödel, by Dummett]
 8789 Various strategies try to deal with the ontological commitments of second-order logic [Hale/Wright on Quine]
 10014 Quine rejects second-order logic, saying that predicates refer to multiple objects [Quine, by Hodes]
 10828 Quantifying over predicates is treating them as names of entities [Quine]
 13639 Quine says higher-order items are intensional, and lack a clearly defined identity relation [Quine, by Shapiro]
 10794 The nominalist is tied by standard semantics to first-order, denying higher-order abstracta [Marcus (Barcan)]
 10769 Second-order logic isn't provable, but will express set-theory and classic problems [Tharp]
 13842 Second-order completeness seems to need intensional entities and possible worlds [Hacking]
 14249 Boolos reinterprets second-order logic as plural logic [Boolos, by Oliver/Smiley]
 10830 Second-order logic metatheory is set-theoretic, and second-order validity has set-theoretic problems [Boolos]
 10225 Monadic second-order logic might be understood in terms of plural quantifiers [Boolos, by Shapiro]
 10736 Boolos showed how plural quantifiers can interpret monadic second-order logic [Boolos, by Linnebo]
 10780 Any sentence of monadic second-order logic can be translated into plural first-order logic [Boolos, by Linnebo]
 10616 Second-order arithmetic can prove new sentences of first-order [Smith,P]
 10015 Higher-order logic may be unintelligible, but it isn't set theory [Hodes]
 17791 Only second-order logic can capture mathematical structure up to isomorphism [Mayberry]
 15944 Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine]
 13629 Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro]
 13650 Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro]
 13645 In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro]
 13649 Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro]
 10298 Some say that second-order logic is mathematics, not logic [Shapiro]
 10299 If the aim of logic is to codify inferences, second-order logic is useless [Shapiro]
 10594 Henkin semantics is more plausible for plural logic than for second-order logic [Maddy]
 19296 If second-order variables range over sets, those are just objects; properties and relations aren't sets [Hale]
 11024 Semantics must precede proof in higher-order logics, since they are incomplete [Read]
 21657 Since properties can have properties, some theorists rank them in 'types' [Hofweber]
 10704 We can formalize second-order formation rules, but not inference rules [Potter]
 15926 Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine]
 10751 Second-order logic needs the sets, and its consequence has epistemological problems [Rossberg]
 10759 There are at least seven possible systems of semantics for second-order logic [Rossberg]
 10757 Henkin semantics has a second domain of predicates and relations (in upper case) [Rossberg]