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Single Idea 10778
[filed under theme 5. Theory of Logic / G. Quantification / 6. Plural Quantification
]
Full Idea
According to its supporters, second-order logic allow us to pay the ontological price of a mere first-order theory and get the corresponding monadic second-order theory for free.
Clarification
'Monadic' means predicates and relations have the minimum number of places
Gist of Idea
Can second-order logic be ontologically first-order, with all the benefits of second-order?
Source
Øystein Linnebo (Plural Quantification Exposed [2003], §0)
Book Ref
-: 'Nous' [-], p.71
The
32 ideas
from Øystein Linnebo
23448
|
Mathematics is the study of all possible patterns, and is thus bound to describe the world
[Linnebo]
|
23441
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Logical truth is true in all models, so mathematical objects can't be purely logical
[Linnebo]
|
23442
|
Game Formalism has no semantics, and Term Formalism reduces the semantics
[Linnebo]
|
23443
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The axioms of group theory are not assertions, but a definition of a structure
[Linnebo]
|
23444
|
To investigate axiomatic theories, mathematics needs its own foundational axioms
[Linnebo]
|
23445
|
Naïve set theory says any formula defines a set, and coextensive sets are identical
[Linnebo]
|
23446
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You can't prove consistency using a weaker theory, but you can use a consistent theory
[Linnebo]
|
23447
|
In classical semantics singular terms refer, and quantifiers range over domains
[Linnebo]
|
10633
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'Some critics admire only one another' cannot be paraphrased in singular first-order
[Linnebo]
|
10634
|
Predicates are 'distributive' or 'non-distributive'; do individuals do what the group does?
[Linnebo]
|
10635
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Second-order quantification and plural quantification are different
[Linnebo]
|
10636
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Plural plurals are unnatural and need a first-level ontology
[Linnebo]
|
10637
|
Ordinary speakers posit objects without concern for ontology
[Linnebo]
|
10638
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A pure logic is wholly general, purely formal, and directly known
[Linnebo]
|
10639
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Plural quantification may allow a monadic second-order theory with first-order ontology
[Linnebo]
|
10640
|
Instead of complex objects like tables, plurally quantify over mereological atoms tablewise
[Linnebo]
|
10641
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Traditionally we eliminate plurals by quantifying over sets
[Linnebo]
|
10643
|
We speak of a theory's 'ideological commitments' as well as its 'ontological commitments'
[Linnebo]
|
10778
|
Can second-order logic be ontologically first-order, with all the benefits of second-order?
[Linnebo]
|
10779
|
A comprehension axiom is 'predicative' if the formula has no bound second-order variables
[Linnebo]
|
10781
|
A 'pure logic' must be ontologically innocent, universal, and without presuppositions
[Linnebo]
|
10782
|
The modern concept of an object is rooted in quantificational logic
[Linnebo]
|
10783
|
Plural quantification depends too heavily on combinatorial and set-theoretic considerations
[Linnebo]
|
14083
|
Structuralism is right about algebra, but wrong about sets
[Linnebo]
|
14085
|
'Deductivist' structuralism is just theories, with no commitment to objects, or modality
[Linnebo]
|
14084
|
Non-eliminative structuralism treats mathematical objects as positions in real abstract structures
[Linnebo]
|
14086
|
'Modal' structuralism studies all possible concrete models for various mathematical theories
[Linnebo]
|
14087
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'Set-theoretic' structuralism treats mathematics as various structures realised among the sets
[Linnebo]
|
14088
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An 'intrinsic' property is either found in every duplicate, or exists independent of all externals
[Linnebo]
|
14089
|
Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure
[Linnebo]
|
14090
|
In mathematical structuralism the small depends on the large, which is the opposite of physical structures
[Linnebo]
|
14091
|
There may be a one-way direction of dependence among sets, and among natural numbers
[Linnebo]
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