9 ideas
| 17833 | The first-order ZF axiomatisation is highly non-categorical [Hallett,M] |
| Full Idea: The first-order Sermelo-Fraenkel axiomatisation is highly non-categorical. | |
| From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1213) |
| 17834 | Non-categoricity reveals a sort of incompleteness, with sets existing that the axioms don't reveal [Hallett,M] |
| Full Idea: The non-categoricity of the axioms which Zermelo demonstrates reveals an incompleteness of a sort, ....for this seems to show that there will always be a set (indeed, an unending sequence) that the basic axioms are incapable of revealing to be sets. | |
| From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1215) | |
| A reaction: Hallett says the incompleteness concerning Zermelo was the (transfinitely) indefinite iterability of the power set operation (which is what drives the 'iterative conception' of sets). |
| 17837 | Zermelo allows ur-elements, to enable the widespread application of set-theory [Hallett,M] |
| Full Idea: Unlike earlier writers (such as Fraenkel), Zermelo clearly allows that there might be ur-elements (that is, objects other than the empty set, which have no members). Indeed he sees in this the possibility of widespread application of set-theory. | |
| From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1217) |
| 11211 | If a sound conclusion comes from two errors that cancel out, the path of the argument must matter [Rumfitt] |
| Full Idea: If a designated conclusion follows from the premisses, but the argument involves two howlers which cancel each other out, then the moral is that the path an argument takes from premisses to conclusion does matter to its logical evaluation. | |
| From: Ian Rumfitt ("Yes" and "No" [2000], II) | |
| A reaction: The drift of this is that our view of logic should be a little closer to the reasoning of ordinary language, and we should rely a little less on purely formal accounts. |
| 11210 | Standardly 'and' and 'but' are held to have the same sense by having the same truth table [Rumfitt] |
| Full Idea: If 'and' and 'but' really are alike in sense, in what might that likeness consist? Some philosophers of classical logic will reply that they share a sense by virtue of sharing a truth table. | |
| From: Ian Rumfitt ("Yes" and "No" [2000]) | |
| A reaction: This is the standard view which Rumfitt sets out to challenge. |
| 11212 | The sense of a connective comes from primitively obvious rules of inference [Rumfitt] |
| Full Idea: A connective will possess the sense that it has by virtue of its competent users' finding certain rules of inference involving it to be primitively obvious. | |
| From: Ian Rumfitt ("Yes" and "No" [2000], III) | |
| A reaction: Rumfitt cites Peacocke as endorsing this view, which characterises the logical connectives by their rules of usage rather than by their pure semantic value. |
| 17836 | The General Continuum Hypothesis and its negation are both consistent with ZF [Hallett,M] |
| Full Idea: In 1938, Gödel showed that ZF plus the General Continuum Hypothesis is consistent if ZF is. Cohen showed that ZF and not-GCH is also consistent if ZF is, which finally shows that neither GCH nor ¬GCH can be proved from ZF itself. | |
| From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1217) |
| 15788 | Syntax and semantics are indeterminate, and modern 'semantics' is a bogus subject [Quine, by Lycan] |
| Full Idea: Quine has argued tirelessly that syntax and 'semantics' are indeterminate, and linguistic semantics of the sort that is currently in favor is a pseudoscience and a pipe dream. | |
| From: report of Willard Quine (Methodological Reflections on Current Linguistic Theory [1972]) by William Lycan - The Trouble with Possible Worlds 02 | |
| A reaction: I think the defence of such things is that they may not integrate into science very well (or even integrate at all), but semantics is intended to integrate into philosophy, and is motivated by philosophical concerns. Quine may be right! |
| 11214 | We learn 'not' along with affirmation, by learning to either affirm or deny a sentence [Rumfitt] |
| Full Idea: The standard view is that affirming not-A is more complex than affirming the atomic sentence A itself, with the latter determining its sense. But we could learn 'not' directly, by learning at once how to either affirm A or reject A. | |
| From: Ian Rumfitt ("Yes" and "No" [2000], IV) | |
| A reaction: [compressed] This seems fairly anti-Fregean in spirit, because it looks at the psychology of how we learn 'not' as a way of clarifying what we mean by it, rather than just looking at its logical behaviour (and thus giving it a secondary role). |