2003 | Plural Quantification Exposed |
§0 | p.71 | 10778 | Can second-order logic be ontologically first-order, with all the benefits of second-order? |
§1 | p.73 | 10779 | A comprehension axiom is 'predicative' if the formula has no bound second-order variables |
§1 | p.75 | 10781 | A 'pure logic' must be ontologically innocent, universal, and without presuppositions |
§2 | p.78 | 10782 | The modern concept of an object is rooted in quantificational logic |
§4 | p.88 | 10783 | Plural quantification depends too heavily on combinatorial and set-theoretic considerations |
2008 | Plural Quantification |
1 | p.2 | 10633 | 'Some critics admire only one another' cannot be paraphrased in singular first-order |
1.1 | p.4 | 10634 | Predicates are 'distributive' or 'non-distributive'; do individuals do what the group does? |
2 | p.5 | 10635 | Second-order quantification and plural quantification are different |
2.4 | p.8 | 10636 | Plural plurals are unnatural and need a first-level ontology |
2.4 | p.9 | 10637 | Ordinary speakers posit objects without concern for ontology |
3 | p.10 | 10638 | A pure logic is wholly general, purely formal, and directly known |
4.4 | p.14 | 10639 | Plural quantification may allow a monadic second-order theory with first-order ontology |
4.5 | p.4 | 10640 | Instead of complex objects like tables, plurally quantify over mereological atoms tablewise |
5 | p.15 | 10641 | Traditionally we eliminate plurals by quantifying over sets |
5.4 | p.20 | 10643 | We speak of a theory's 'ideological commitments' as well as its 'ontological commitments' |
2008 | Structuralism and the Notion of Dependence |
Intro | p.59 | 14083 | Structuralism is right about algebra, but wrong about sets |
1 | p.60 | 14085 | 'Deductivist' structuralism is just theories, with no commitment to objects, or modality |
I | p.60 | 14084 | Non-eliminative structuralism treats mathematical objects as positions in real abstract structures |
I | p.60 | 14086 | 'Modal' structuralism studies all possible concrete models for various mathematical theories |
I | p.61 | 14087 | 'Set-theoretic' structuralism treats mathematics as various structures realised among the sets |
II | p.65 | 14088 | An 'intrinsic' property is either found in every duplicate, or exists independent of all externals |
III | p.66 | 14089 | Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure |
III | p.68 | 14090 | In mathematical structuralism the small depends on the large, which is the opposite of physical structures |
V | p.73 | 14091 | There may be a one-way direction of dependence among sets, and among natural numbers |