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Single Idea 10299
[filed under theme 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
]
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.
Gist of Idea
If the aim of logic is to codify inferences, second-order logic is useless
Source
Stewart Shapiro (Higher-Order Logic [2001], 2.4)
Book Ref
'Blackwell Guide to Philosophical Logic', ed/tr. Goble,Lou [Blackwell 2001], p.51
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'.
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]
|