13 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). |
| 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) |
| 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) |
| 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) |
| 22375 | Moral judgements need more than the relevant facts, if the same facts lead to 'x is good' and 'x is bad' [Foot] |
| Full Idea: It is suggested that anyone who has considered all the facts which could bear on his moral position has ipso facto produced a 'well founded' moral judgement, ...How 'x is good' can be well founded when 'x is bad' is equally well founded is hard to see. | |
| From: Philippa Foot (Moral Arguments [1958], p.96) | |
| A reaction: This seems to be a warning to particularists, if they hope that moral judgements just emerge from the facts. It doesn't rule out physicalist naturalism about morality, if the attitudes we bring to the facts have arisen out of further facts. |
| 22378 | We can't affirm a duty without saying why it matters if it is not performed [Foot] |
| Full Idea: I do not know what could be meant by saying it was someone's duty to do something unless there was an attempt to show why it mattered if this sort of thing was not done. | |
| From: Philippa Foot (Moral Arguments [1958], p.105) | |
| A reaction: The Kantian idea assumes that duty is an absolute, and yet each duty rests on a particular maxim which is going to be universalised. So why should that maxim be universalised, and not some other? |
| 22377 | Whether someone is rude is judged by agreed criteria, so the facts dictate the value [Foot] |
| Full Idea: Whether a man is speaking of behaviour as rude or not rude, he must use the same criteria as anyone else. ...We have here an example of a non-evaluative premise from which an evaluative conclusion can be deduced. | |
| From: Philippa Foot (Moral Arguments [1958], p.104) | |
| A reaction: We would now call 'rude' a 'thick' ethical concept (where 'good' is 'thin'). Her powerful point is, I take it, that evidence is always relevant to judgements of thick concepts, so there is no fact-value gap. 'Rude' has criteria, but 'good' may not. |
| 22376 | Facts and values are connected if we cannot choose what counts as evidence of rightness [Foot] |
| Full Idea: To show that facts and values are connected we must show that some things do and some things don't count in favour of a moral conclusion, and that no one can choose what counts as evidence for rightness or wrongness. | |
| From: Philippa Foot (Moral Arguments [1958], p.99) | |
| A reaction: But what sort of facts might do the job? I can only think of right functioning and health as facts which seem to imply value. Pleasure and misery don't quite get there. |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |