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Single Idea 10832
[filed under theme 5. Theory of Logic / G. Quantification / 2. Domain of Quantification
]
Full Idea
One may say that '∀x x=x' means 'everything is identical to itself', but one must realise that one's answer has a determinate sense only if the reference (range) of 'everything' is fixed.
Gist of Idea
'∀x x=x' only means 'everything is identical to itself' if the range of 'everything' is fixed
Source
George Boolos (On Second-Order Logic [1975], p.46)
Book Ref
Boolos,George: 'Logic, Logic and Logic' [Harvard 1999], p.519
A Reaction
This is the problem now discussed in the recent book 'Absolute Generality', of whether one can quantify without specifying a fixed or limited domain.
The
16 ideas
with the same theme
[specifying the objects from which quantifiers select]:
17743
|
De Morgan introduced a 'universe of discourse', to replace Boole's universe of 'all things'
[De Morgan, by Walicki]
|
9991
|
For Frege the variable ranges over all objects
[Frege, by Tait]
|
10536
|
Frege's domain for variables is all objects, but modern interpretations first fix the domain
[Dummett on Frege]
|
9871
|
Frege always, and fatally, neglected the domain of quantification
[Dummett on Frege]
|
10656
|
With 'extensive connection', boundary elements are not included in domains
[Whitehead, by Varzi]
|
10042
|
Reference to a totality need not refer to a conjunction of all its elements
[Gödel]
|
10790
|
Quantifiers are needed to refer to infinitely many objects
[Marcus (Barcan)]
|
10791
|
Substitutional semantics has no domain of objects, but place-markers for substitutions
[Marcus (Barcan)]
|
18914
|
Davidson controversially proposed to quantify over events
[Davidson, by Engelbretsen]
|
10776
|
The main quantifiers extend 'and' and 'or' to infinite domains
[Tharp]
|
13818
|
If we allow empty domains, we must allow empty names
[Bostock]
|
10832
|
'∀x x=x' only means 'everything is identical to itself' if the range of 'everything' is fixed
[Boolos]
|
17787
|
Big logic has one fixed domain, but standard logic has a domain for each interpretation
[Mayberry]
|
10007
|
Quantifiers for domains and for inference come apart if there are no entities
[Hofweber]
|
13449
|
We could have unrestricted quantification without having an all-inclusive domain
[Rayo/Uzquiano]
|
13450
|
Absolute generality is impossible, if there are indefinitely extensible concepts like sets and ordinals
[Rayo/Uzquiano]
|