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| 10633 | 'Some critics admire only one another' cannot be paraphrased in singular first-order |
| Full Idea: The Geach-Kaplan sentence 'Some critics admire only one another' provably has no singular first-order paraphrase using only its predicates. | |||
| From: Øystein Linnebo (Plural Quantification [2008], 1) | |||
| A reaction: There seems to be a choice of either going second-order (picking out a property), or going plural (collectively quantifying), or maybe both. |
| 10638 | A pure logic is wholly general, purely formal, and directly known |
| Full Idea: The defining features of a pure logic are its absolute generality (the objects of discourse are irrelevant), and its formality (logical truths depend on form, not matter), and its cognitive primacy (no extra-logical understanding is needed to grasp it). | |||
| From: Øystein Linnebo (Plural Quantification [2008], 3) | |||
| A reaction: [compressed] This strikes me as very important. The above description seems to contain no ontological commitment at all, either to the existence of something, or to two things, or to numbers, or to a property. Pure logic seems to be 'if-thenism'. |
| 10635 | Second-order quantification and plural quantification are different |
| Full Idea: Second-order quantification and plural quantification are generally regarded as different forms of quantification. | |||
| From: Øystein Linnebo (Plural Quantification [2008], 2) |
| 10640 | Instead of complex objects like tables, plurally quantify over mereological atoms tablewise |
| Full Idea: Plural quantification can be used to eliminate the commitment of science and common sense to complex objects. We can use plural quantification over mereological atoms arranged tablewise or chairwise. | |||
| From: Øystein Linnebo (Plural Quantification [2008], 4.5) | |||
| A reaction: [He cites Hossack and van Ingwagen] |
| 10641 | Traditionally we eliminate plurals by quantifying over sets |
| Full Idea: The traditional view in analytic philosophy has been that all plural locutions should be paraphrased away by quantifying over sets, though Boolos and other objected that this is unnatural and unnecessary. | |||
| From: Øystein Linnebo (Plural Quantification [2008], 5) |
| 10636 | Plural plurals are unnatural and need a first-level ontology |
| Full Idea: Higher-order plural quantification (plural plurals) is often rejected because plural quantification is supposedly ontological innocent, with no plural things to be plural, and because it is not found in ordinary English. | |||
| From: Øystein Linnebo (Plural Quantification [2008], 2.4) | |||
| A reaction: [Summary; he cites Boolos as a notable rejector] Linnebo observes that Icelandic contains a word 'tvennir' which means 'two pairs of'. |
| 10639 | Plural quantification may allow a monadic second-order theory with first-order ontology |
| Full Idea: Plural quantification seems to offer ontological economy. We can pay the price of a mere first-order theory and then use plural quantification to get for free the corresponding monadic second-order theory, which would be an ontological bargain. | |||
| From: Øystein Linnebo (Plural Quantification [2008], 4.4) | |||
| A reaction: [He mentions Hellman's modal structuralism in mathematics] |
| 10643 | We speak of a theory's 'ideological commitments' as well as its 'ontological commitments' |
| Full Idea: Some philosophers speak about a theory's 'ideological commitments' and not just about its 'ontological commitments'. | |||
| From: Øystein Linnebo (Plural Quantification [2008], 5.4) | |||
| A reaction: This is a third strategy for possibly evading one's ontological duty, along with fiddling with the words 'exist' or 'object'. An ideological commitment to something to which one is not actually ontologically committed conjures up stupidity and dogma. |
| 10637 | Ordinary speakers posit objects without concern for ontology |
| Full Idea: Maybe ordinary speakers aren't very concerned about their ontological commitments, and sometimes find it convenient to posit objects. | |||
| From: Øystein Linnebo (Plural Quantification [2008], 2.4) | |||
| A reaction: I think this is the whole truth about the ontological commitment of ordinary language. We bring abstraction under control by pretending it is a world of physical objects. The 'left wing' in politics, 'dark deeds', a 'huge difference'. |
| 10634 | Predicates are 'distributive' or 'non-distributive'; do individuals do what the group does? |
| Full Idea: The predicate 'is on the table' is 'distributive', since some things are on the table if each one is, whereas the predicate 'form a circle' is 'non-distributive', since it is not analytic that when some things form a circle, each one forms a circle. | |||
| From: Øystein Linnebo (Plural Quantification [2008], 1.1) | |||
| A reaction: The first predicate can have singular or plural subjects, but the second requires a plural subject? Hm. 'The rope forms a circle'. The second is example is not true, as well as not analytic. |