9 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) |
| 13007 | Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz] |
| Full Idea: Archimedes gave a sort of definition of 'straight line' when he said it is the shortest line between two points. | |
| From: report of Archimedes (fragments/reports [c.240 BCE]) by Gottfried Leibniz - New Essays on Human Understanding 4.13 | |
| A reaction: Commentators observe that this reduces the purity of the original Euclidean axioms, because it involves distance and measurement, which are absent from the purest geometry. |
| 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'. |
| 4304 | Descartes says there are two substance, Spinoza one, and Leibniz infinitely many [Cottingham] |
| Full Idea: Descartes was a dualist about substance, Spinoza was a monist, and Leibniz was a pluralist (an infinity of substances). | |
| From: John Cottingham (The Rationalists [1988], p.76) | |
| A reaction: Spinoza is appealing. We posit a substance, as the necessary basis for existence, but it is unclear how more than one substance can be differentiated. If mind is a separate substance, why isn't iron? Why aren't numbers? |
| 4303 | The notion of substance lies at the heart of rationalist metaphysics [Cottingham] |
| Full Idea: The notion of substance lies at the heart of rationalist metaphysics. | |
| From: John Cottingham (The Rationalists [1988], p.75) | |
| A reaction: The idea of 'substance' has had an interesting revival in modern philosophy (though not, obviously, in physics). Maybe physics and philosophy have views of reality which are not complementary, but are rivals. |
| 4306 | For rationalists, it is necessary that effects be deducible from their causes [Cottingham] |
| Full Idea: The rationalist view of causation takes it that to make effects intelligible, it must be shown that they are in principle deducible from their causes. | |
| From: John Cottingham (The Rationalists [1988], p.92) | |
| A reaction: This has intuitive appeal, but deduction is only possible with further premises, such as the laws of physics. The effects of human behaviour look a bit tricky, even if we cause them. |