10 ideas
| 17879 | Axiomatising set theory makes it all relative [Skolem] |
| Full Idea: Axiomatising set theory leads to a relativity of set-theoretic notions, and this relativity is inseparably bound up with every thoroughgoing axiomatisation. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.296) |
| 9455 | Maybe proper names have the content of fixing a thing's category [Bealer] |
| Full Idea: Some say that proper names have no descriptive content, but others think that although a name does not have the right sort of descriptive content which fixes a unique referent, it has a content which fixes the sort or category to which it belongs. | |
| From: George Bealer (Propositions [1998], §7) | |
| A reaction: Presumably 'Mary', and 'Felix', and 'Rover', and 'Smallville' are cases in point. There is a well known journalist called 'Manchester', a famous man called 'Hilary', a village in Hertfordshire called 'Matching Tie'... Interesting, though. |
| 9454 | The four leading theories of definite descriptions are Frege's, Russell's, Evans's, and Prior's [Bealer] |
| Full Idea: The four leading theories of definite descriptions are Frege's, Russell's, Evans's, and Prior's, ...of which to many Frege's is the most intuitive of the four. Frege says they refer to the unique item (if it exists) which satisfies the predicate. | |
| From: George Bealer (Propositions [1998], §5) | |
| A reaction: He doesn't expound the other three, but I record this a corrective to the view that Russell has the only game in town. |
| 17878 | If a 1st-order proposition is satisfied, it is satisfied in a denumerably infinite domain [Skolem] |
| Full Idea: Löwenheim's theorem reads as follows: If a first-order proposition is satisfied in any domain at all, it is already satisfied in a denumerably infinite domain. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.293) |
| 17880 | Integers and induction are clear as foundations, but set-theory axioms certainly aren't [Skolem] |
| Full Idea: The initial foundations should be immediately clear, natural and not open to question. This is satisfied by the notion of integer and by inductive inference, by it is not satisfied by the axioms of Zermelo, or anything else of that kind. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.299) | |
| A reaction: This is a plea (endorsed by Almog) that the integers themselves should be taken as primitive and foundational. I would say that the idea of successor is more primitive than the integers. |
| 17881 | Mathematician want performable operations, not propositions about objects [Skolem] |
| Full Idea: Most mathematicians want mathematics to deal, ultimately, with performable computing operations, and not to consist of formal propositions about objects called this or that. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.300) |
| 13128 | 'Ultimate sortals' cannot explain ontological categories [Westerhoff on Wiggins] |
| Full Idea: 'Ultimate sortals' are said to be non-subordinated, disjoint from one another, and uniquely paired with each object. Because of this, the ultimate sortal cannot be a satisfactory explication of the notion of an ontological category. | |
| From: comment on David Wiggins (Identity and Spatio-Temporal Continuity [1971], p.75) by Jan Westerhoff - Ontological Categories §26 | |
| A reaction: My strong intuitions are that Wiggins is plain wrong, and Westerhoff gives the most promising reasons for my intuition. The simplest point is that objects can obviously belong to more than one category. |
| 9453 | Sentences saying the same with the same rigid designators may still express different propositions [Bealer] |
| Full Idea: The propositions behind 'Cicero is emulated more than Tully' seems to differ somehow from 'Tully is emulated more than Cicero', despite the proper names being rigid designators. | |
| From: George Bealer (Propositions [1998], §1) | |
| A reaction: Interesting, because this isn't a directly propositional attitude situation like 'believes', though it depends on such things. Bealer says this is a key modern difficulty with propositions. |
| 9452 | Propositions might be reduced to functions (worlds to truth values), or ordered sets of properties and relations [Bealer] |
| Full Idea: The reductionist view of propositions sees them as either extensional functions from possible worlds to truth values, or as ordered sets of properties, relations, and perhaps particulars. | |
| From: George Bealer (Propositions [1998], §1) | |
| A reaction: The usual problem of all functional accounts is 'what is it about x that enables it to have that function?' And if they are sets, where does the ordering come in? A proposition isn't just a list of items in some particular order. Both wrong. |
| 9451 | Modal logic and brain science have reaffirmed traditional belief in propositions [Bealer] |
| Full Idea: Philosophers have been skeptical about abstract objects, and so have been skeptical about propositions,..but with the rise of modal logic and metaphysics, and cognitive science's realism about intentional states, traditional propositions are now dominant. | |
| From: George Bealer (Propositions [1998], §1) | |
| A reaction: I personally strongly favour belief in propositions as brain states, which don't need a bizarre ontological status, but are essential to explain language, reasoning and communication. |