10 ideas
| 8758 | We could talk of open sentences, instead of sets [Chihara, by Shapiro] |
| Full Idea: Chihara's programme is to replace talk of sets with talk of open sentences. Instead of speaking of the set of all cats, we talk about the open sentence 'x is a cat'. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Thinking About Mathematics 9.2 | |
| A reaction: As Shapiro points out, this is following up Russell's view that sets should be replaced with talk of properties. Chihara is expressing it more linguistically. I'm in favour of any attempt to get rid of sets. |
| 9390 | Logic guides thinking, but it isn't a substitute for it [Rumfitt] |
| Full Idea: Logic is part of a normative theory of thinking, not a substitute for thinking. | |
| From: Ian Rumfitt (The Logic of Boundaryless Concepts [2007], p.13) | |
| A reaction: There is some sort of logicians' dream, going back to Leibniz, of a reasoning engine, which accepts propositions and outputs inferences. I agree with this idea. People who excel at logic are often, it seems to me, modest at philosophy. |
| 10265 | Chihara's system is a variant of type theory, from which he can translate sentences [Chihara, by Shapiro] |
| Full Idea: Chihara's system is a version of type theory. Translate thus: replace variables of sets of type n with level n variables over open sentences, replace membership/predication with satisfaction, and high quantifiers with constructability quantifiers. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Philosophy of Mathematics 7.4 |
| 8759 | We can replace type theory with open sentences and a constructibility quantifier [Chihara, by Shapiro] |
| Full Idea: Chihara's system is similar to simple type theory; he replaces each type with variables over open sentences, replaces membership (or predication) with satisfaction, and replaces quantifiers over level 1+ variables with constructability quantifiers. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Thinking About Mathematics 9.2 | |
| A reaction: This is interesting for showing that type theory may not be dead. The revival of supposedly dead theories is the bread-and-butter of modern philosophy. |
| 10264 | Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ' [Chihara, by Shapiro] |
| Full Idea: Chihara has proposal a modal primitive, a 'constructability quantifier'. Syntactically it behaves like an ordinary quantifier: Φ is a formula, and x a variable. Then (Cx)Φ is a formula, read as 'it is possible to construct an x such that Φ'. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Philosophy of Mathematics 7.4 | |
| A reaction: We only think natural numbers are infinite because we see no barrier to continuing to count, i.e. to construct new numbers. We accept reals when we know how to construct them. Etc. Sounds promising to me (though not to Shapiro). |
| 9389 | Vague membership of sets is possible if the set is defined by its concept, not its members [Rumfitt] |
| Full Idea: Vagueness in respect of membership is consistency with determinacy of the set's identity, so long as a set's identity is taken to consist, not in its having such-and-such members, but in its being the extension of a concept. | |
| From: Ian Rumfitt (The Logic of Boundaryless Concepts [2007], p.5) | |
| A reaction: I find this view of sets much more appealing than the one that identifies a set with its members. The empty set is less of a problem, as well as non-existents. Logicians prefer the extensional view because it is tidy. |
| 19566 | Epistemology does not just concern knowledge; all aspects of cognitive activity are involved [Kvanvig] |
| Full Idea: Epistemology is not just knowledge. There is enquiring, reasoning, changes of view, beliefs, assumptions, presuppositions, hypotheses, true beliefs, making sense, adequacy, understanding, wisdom, responsible enquiry, and so on. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], 'What') | |
| A reaction: [abridged] Stop! I give in. His topic is whether truth is central to epistemology. Rivals seem to be knowledge-first, belief-first, and justification-first. I'm inclined to take justification as the central issue. Does it matter? |
| 19568 | Making sense of things, or finding a good theory, are non-truth-related cognitive successes [Kvanvig] |
| Full Idea: There are cognitive successes that are not obviously truth related, such as the concepts of making sense of the course of experience, and having found an empirically adequate theory. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], 'Epistemic') | |
| A reaction: He is claiming that truth is not the main aim of epistemology. He quotes Marian David for the rival view. Personally I doubt whether the concepts of 'making sense' or 'empirical adequacy' can be explicated without mentioning truth. |
| 19567 | The 'defeasibility' approach says true justified belief is knowledge if no undermining facts could be known [Kvanvig] |
| Full Idea: The 'defeasibility' approach says that having knowledge requires, in addition to justified true belief, there being no true information which, if learned, would result in the person in question no longer being justified in believing the claim. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], 'Epistemic') | |
| A reaction: I take this to be an externalist view, since it depends on information of which the cognizer may be unaware. A defeater may yet have an undiscovered counter-defeater. The only real defeater is the falsehood of the proposition. |
| 19570 | Reliabilism cannot assess the justification for propositions we don't believe [Kvanvig] |
| Full Idea: The most serious problem for reliabilism is that it cannot explain adequately the concept of propositional justification, the kind of justification one might have for a proposition one does not believe, or which one disbelieves. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], Notes 2) | |
| A reaction: I don't understand this (though I pass it on anyway). Why can't the reliabilist just offer a critique of the reliability of the justification available for the dubious proposition? |