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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 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 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 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 '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 Second-order quantification and plural quantification are different [Linnebo]
10636 Plural plurals are unnatural and need a first-level ontology [Linnebo]
10637 Ordinary speakers posit objects without concern for ontology [Linnebo]
10638 A pure logic is wholly general, purely formal, and directly known [Linnebo]
10639 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 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 'Set-theoretic' structuralism treats mathematics as various structures realised among the sets [Linnebo]
14088 An 'intrinsic' property is either found in every duplicate, or exists independent of all externals [Linnebo]
14090 In mathematical structuralism the small depends on the large, which is the opposite of physical structures [Linnebo]
14089 Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure [Linnebo]
14091 There may be a one-way direction of dependence among sets, and among natural numbers [Linnebo]