11 ideas
9184 | We can't presume that all interesting concepts can be analysed [Williamson] |
Full Idea: We have no prior reason to suppose that philosophically significant concepts have interesting analyses into necessary and sufficient conditions. | |
From: Timothy Williamson (Review of Bob Hale's 'Abstract Objects' [1988]) | |
A reaction: We might think that they are either analysable or primitive, and that failure of analysis invites us to take a concept as primitive. But maybe God can analyse it and we can't. |
10282 | Logic is the study of sound argument, or of certain artificial languages (or applying the latter to the former) [Hodges,W] |
Full Idea: A logic is a collection of closely related artificial languages, and its older meaning is the study of the rules of sound argument. The languages can be used as a framework for studying rules of argument. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.1) | |
A reaction: [Hodges then says he will stick to the languages] The suspicion is that one might confine the subject to the artificial languages simply because it is easier, and avoids the tricky philosophical questions. That approximates to computer programming. |
10283 | A formula needs an 'interpretation' of its constants, and a 'valuation' of its variables [Hodges,W] |
Full Idea: To have a truth-value, a first-order formula needs an 'interpretation' (I) of its constants, and a 'valuation' (ν) of its variables. Something in the world is attached to the constants; objects are attached to variables. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.3) |
10284 | There are three different standard presentations of semantics [Hodges,W] |
Full Idea: Semantic rules can be presented in 'Tarski style', where the interpretation-plus-valuation is reduced to the same question for simpler formulas, or the 'Henkin-Hintikka style' in terms of games, or the 'Barwise-Etchemendy style' for computers. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.3) | |
A reaction: I haven't yet got the hang of the latter two, but I note them to map the territory. |
10285 | I |= φ means that the formula φ is true in the interpretation I [Hodges,W] |
Full Idea: I |= φ means that the formula φ is true in the interpretation I. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.5) | |
A reaction: [There should be no space between the vertical and the two horizontals!] This contrasts with |-, which means 'is proved in'. That is a syntactic or proof-theoretic symbol, whereas |= is a semantic symbol (involving truth). |
10288 | Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model [Hodges,W] |
Full Idea: Downward Löwenheim-Skolem (the weakest form): If L is a first-order language with at most countably many formulas, and T is a consistent theory in L. Then T has a model with at most countably many elements. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.10) |
10289 | Up Löwenheim-Skolem: if infinite models, then arbitrarily large models [Hodges,W] |
Full Idea: Upward Löwenheim-Skolem: every first-order theory with infinite models has arbitrarily large models. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.10) |
10287 | If a first-order theory entails a sentence, there is a finite subset of the theory which entails it [Hodges,W] |
Full Idea: Compactness Theorem: suppose T is a first-order theory, ψ is a first-order sentence, and T entails ψ. Then there is a finite subset U of T such that U entails ψ. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.10) | |
A reaction: If entailment is possible, it can be done finitely. |
10286 | A 'set' is a mathematically well-behaved class [Hodges,W] |
Full Idea: A 'set' is a mathematically well-behaved class. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.6) |
9183 | Platonism claims that some true assertions have singular terms denoting abstractions, so abstractions exist [Williamson] |
Full Idea: The Fregean argument for platonism is that some true assertions contain singular terms which denote abstract objects if they denote anything; since the assertions are true, the singular terms denote. | |
From: Timothy Williamson (Review of Bob Hale's 'Abstract Objects' [1988]) | |
A reaction: I am perplexed that anyone would rest their view of reality on such an argument. The obvious comparison would be with true remarks about blatantly fictional characters, or blatantly invented concepts such as 'checkmate'. |
468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |