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Single Idea 11033

[filed under theme 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic ]

Full Idea

Whenever one thing is predicated of another as of a subject, all things said of what is predicated will be said of the subject also.

Gist of Idea

Predications of predicates are predications of their subjects

Source

Aristotle (Categories [c.331 BCE], 01b10)

Book Ref

Aristotle: 'Categories and De Interpretatione', ed/tr. Ackrill,J.R. [OUP 1963], p.4

The 33 ideas with the same theme [logic extending variables to predicates and relations]:

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]
10299 If the aim of logic is to codify inferences, second-order logic is useless [Shapiro]
10298 Some say that second-order logic is mathematics, not logic [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]
10757 Henkin semantics has a second domain of predicates and relations (in upper case) [Rossberg]
10759 There are at least seven possible systems of semantics for second-order logic [Rossberg]