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

[logic extending variables to predicates and relations]

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