7 ideas
10779 | A comprehension axiom is 'predicative' if the formula has no bound second-order variables [Linnebo] |
Full Idea: If φ contains no bound second-order variables, the corresponding comprehension axiom is said to be 'predicative'; otherwise it is 'impredicative'. | |
From: Øystein Linnebo (Plural Quantification Exposed [2003], §1) | |
A reaction: ['Predicative' roughly means that a new predicate is created, and 'impredicative' means that it just uses existing predicates] |
10781 | A 'pure logic' must be ontologically innocent, universal, and without presuppositions [Linnebo] |
Full Idea: I offer these three claims as a partial analysis of 'pure logic': ontological innocence (no new entities are introduced), universal applicability (to any realm of discourse), and cognitive primacy (no extra-logical ideas are presupposed). | |
From: Øystein Linnebo (Plural Quantification Exposed [2003], §1) |
10783 | Plural quantification depends too heavily on combinatorial and set-theoretic considerations [Linnebo] |
Full Idea: If my arguments are correct, the theory of plural quantification has no right to the title 'logic'. ...The impredicative plural comprehension axioms depend too heavily on combinatorial and set-theoretic considerations. | |
From: Øystein Linnebo (Plural Quantification Exposed [2003], §4) |
10778 | Can second-order logic be ontologically first-order, with all the benefits of second-order? [Linnebo] |
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. | |
From: Øystein Linnebo (Plural Quantification Exposed [2003], §0) |
9226 | If mathematical theories conflict, it may just be that they have different subject matter [Field,H] |
Full Idea: Unlike logic, in the case of mathematics there may be no genuine conflict between alternative theories: it is natural to think that different theories, if both consistent, are simply about different subjects. | |
From: Hartry Field (Recent Debates on the A Priori [2005], 7) | |
A reaction: For this reason Field places logic at the heart of questions about a priori knowledge, rather than mathematics. My intuitions make me doubt his proposal. Given the very simple basis of, say, arithmetic, I would expect all departments to connect. |
10782 | The modern concept of an object is rooted in quantificational logic [Linnebo] |
Full Idea: Our modern general concept of an object is given content only in connection with modern quantificational logic. | |
From: Øystein Linnebo (Plural Quantification Exposed [2003], §2) | |
A reaction: [He mentions Frege, Carnap, Quine and Dummett] This is the first thing to tell beginners in modern analytical metaphysics. The word 'object' is very confusing. I think I prefer 'entity'. |
12709 | Motion is not absolute, but consists in relation [Leibniz] |
Full Idea: In reality motion is not something absolute, but consists in relation. | |
From: Gottfried Leibniz (On Motion [1677], A6.4.1968), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 3 | |
A reaction: It is often thought that motion being relative was invented by Einstein, but Leibniz wholeheartedly embraced 'Galilean relativity', and refused to even consider any absolute concept of motion. Acceleration is a bit trickier than velocity. |