12 ideas
| 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) |
| 8386 | Events are picked out by descriptions, and facts by whole sentences [Crane] |
| Full Idea: Events are picked out using descriptions ('The death of Caesar'), while facts are picked out using whole sentences ('Caesar died'). | |
| From: Tim Crane (Causation [1995], 1.4.2) | |
| A reaction: Useful, and interesting. He mentions that Kim's usage doesn't agree with this. For analysis purposes, this means that an event is a more minimal item than a fact, and many facts will contain events as components. |
| 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? |
| 8387 | A cause has its effects in virtue of its properties [Crane] |
| Full Idea: Some philosophers say that a cause has its effects in virtue of its properties. | |
| From: Tim Crane (Causation [1995], 1.4.2) | |
| A reaction: The trouble with this approach, I think, is that it encourages us to invent dubious properties, because every explanation of an effect will require one. Dormative properties, for example, are ascribed to sleeping pills. |
| 8384 | The regularity theory explains a causal event by other items than the two that are involved [Crane] |
| Full Idea: An unsatisfactory aspect of the regularity thesis is that it explains why this A caused this B in terms of facts about things other that this A and this B. But we want to know what it is about this A and this B that makes one the cause of the other? | |
| From: Tim Crane (Causation [1995], 1.3) | |
| A reaction: Well said. This is the failing of any attempt to define things by their relationships (e.g. functional definitions). Hume, of course, was only relying on regularity because when he focused on the actual A and B, they had no helpful experiences to offer. |