13 ideas
| 14626 | In S5 matters of possibility and necessity are non-contingent [Williamson] |
| Full Idea: In system S5 matters of possibility and necessity are always non-contingent. | |
| From: Timothy Williamson (Modal Logic within Counterfactual Logic [2010], 3) | |
| A reaction: This will be because if something is possible in one world (because it can be seen to be true in some possible world) it will be possible for all worlds (since they can all see that world in S5). |
| 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. |
| 9935 | Mathematical truth is always compromising between ordinary language and sensible epistemology [Benacerraf] |
| Full Idea: Most accounts of the concept of mathematical truth can be identified with serving one or another of either semantic theory (matching it to ordinary language), or with epistemology (meshing with a reasonable view) - always at the expense of the other. | |
| From: Paul Benacerraf (Mathematical Truth [1973], Intro) | |
| A reaction: The gist is that language pulls you towards platonism, and epistemology pulls you towards empiricism. He argues that the semantics must give ground. He's right. |
| 17927 | Realists have semantics without epistemology, anti-realists epistemology but bad semantics [Benacerraf, by Colyvan] |
| Full Idea: Benacerraf argues that realists about mathematical objects have a nice normal semantic but no epistemology, and anti-realists have a good epistemology but an unorthodox semantics. | |
| From: report of Paul Benacerraf (Mathematical Truth [1973]) by Mark Colyvan - Introduction to the Philosophy of Mathematics 1.2 |
| 9936 | The platonist view of mathematics doesn't fit our epistemology very well [Benacerraf] |
| Full Idea: The principle defect of the standard (platonist) account of mathematical truth is that it appears to violate the requirement that our account be susceptible to integration into our over-all account of knowledge. | |
| From: Paul Benacerraf (Mathematical Truth [1973], III) | |
| A reaction: Unfortunately he goes on to defend a causal theory of justification (fashionable at that time, but implausible now). Nevertheless, his general point is well made. Your theory of what mathematics is had better make it knowable. |
| 14625 | Necessity is counterfactually implied by its negation; possibility does not counterfactually imply its negation [Williamson] |
| Full Idea: Modal thinking is logically equivalent to a type of counterfactual thinking. ...The necessary is that which is counterfactually implied by its own negation; the possible is that which does not counterfactually imply its own negation. | |
| From: Timothy Williamson (Modal Logic within Counterfactual Logic [2010], 1) | |
| A reaction: I really like this, because it builds modality on ordinary imaginative thinking. He says you just need to grasp counterfactuals, and also negation and absurdity, and you can then understand necessity and possibility. We can all do that. |
| 14623 | Strict conditionals imply counterfactual conditionals: □(A⊃B)⊃(A□→B) [Williamson] |
| Full Idea: The strict conditional implies the counterfactual conditional: □(A⊃B) ⊃ (A□→B) - suppose that A would not have held without B holding too; then if A had held, B would also have held. | |
| From: Timothy Williamson (Modal Logic within Counterfactual Logic [2010], 1) | |
| A reaction: [He then adds a reading of his formula in terms of possible worlds] This sounds rather close to modus ponens. If A implies B, and A is actually the case, what have you got? B! |
| 14624 | Counterfactual conditionals transmit possibility: (A□→B)⊃(◊A⊃◊B) [Williamson] |
| Full Idea: The counterfactual conditional transmits possibility: (A□→B) ⊃ (◊A⊃◊B). Suppose that if A had held, B would also have held; the if it is possible for A to hold, it is also possible for B to hold. | |
| From: Timothy Williamson (Modal Logic within Counterfactual Logic [2010], 1) |
| 14531 | Rather than define counterfactuals using necessity, maybe necessity is a special case of counterfactuals [Williamson, by Hale/Hoffmann,A] |
| Full Idea: Instead of regarding counterfactuals as conditionals restricted to a range of possible worlds, we can define the necessity operator by means of counterfactuals. Metaphysical necessity is a special case of ordinary counterfactual thinking. | |
| From: report of Timothy Williamson (Modal Logic within Counterfactual Logic [2010]) by Bob Hale/ Aviv Hoffmann - Introduction to 'Modality' 2 | |
| A reaction: [compressed] I very much like Williamson's approach, of basing these things on the ordinary way that ordinary people think. To me it is a welcome inclusion of psychology into metaphysics, which has been out in the cold since Frege. |
| 14628 | Imagination is important, in evaluating possibility and necessity, via counterfactuals [Williamson] |
| Full Idea: Imagination can be made to look cognitively worthless. Once we recall its fallible but vital role in evaluating counterfactual conditionals, we should be more open to the idea that it plays such a role in evaluating claims of possibility and necessity. | |
| From: Timothy Williamson (Modal Logic within Counterfactual Logic [2010], 6) | |
| A reaction: I take this to be a really important idea, because it establishes the importance of imagination within the formal framework of modern analytic philosopher (rather than in the whimsy of poets and dreamers). |
| 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). |