14 ideas
| 10476 | The idea that groups of concepts could be 'implicitly defined' was abandoned [Hodges,W] |
| Full Idea: Late nineteenth century mathematicians said that, although plus, minus and 0 could not be precisely defined, they could be partially 'implicitly defined' as a group. This nonsense was rejected by Frege and others, as expressed in Russell 1903. | |
| From: Wilfrid Hodges (Model Theory [2005], 2) | |
| A reaction: [compressed] This is helpful in understanding what is going on in Frege's 'Grundlagen'. I won't challenge Hodges's claim that such definitions are nonsense, but there is a case for understanding groups of concepts together. |
| 10478 | Since first-order languages are complete, |= and |- have the same meaning [Hodges,W] |
| Full Idea: In first-order languages the completeness theorem tells us that T |= φ holds if and only if there is a proof of φ from T (T |- φ). Since the two symbols express the same relationship, theorist often just use |- (but only for first-order!). | |
| From: Wilfrid Hodges (Model Theory [2005], 3) | |
| A reaction: [actually no spaces in the symbols] If you are going to study this kind of theory of logic, the first thing you need to do is sort out these symbols, which isn't easy! |
| 10477 | |= in model-theory means 'logical consequence' - it holds in all models [Hodges,W] |
| Full Idea: If every structure which is a model of a set of sentences T is also a model of one of its sentences φ, then this is known as the model-theoretic consequence relation, and is written T |= φ. Not to be confused with |= meaning 'satisfies'. | |
| From: Wilfrid Hodges (Model Theory [2005], 3) | |
| A reaction: See also Idea 10474, which gives the other meaning of |=, as 'satisfies'. The symbol is ALSO used in propositional logical, to mean 'tautologically implies'! Sort your act out, logicians. |
| 10474 | |= should be read as 'is a model for' or 'satisfies' [Hodges,W] |
| Full Idea: The symbol in 'I |= S' reads that if the interpretation I (about word meaning) happens to make the sentence S state something true, then I 'is a model for' S, or I 'satisfies' S. | |
| From: Wilfrid Hodges (Model Theory [2005], 1) | |
| A reaction: Unfortunately this is not the only reading of the symbol |= [no space between | and =!], so care and familiarity are needed, but this is how to read it when dealing with models. See also Idea 10477. |
| 10473 | Model theory studies formal or natural language-interpretation using set-theory [Hodges,W] |
| Full Idea: Model theory is the study of the interpretation of any language, formal or natural, by means of set-theoretic structures, with Tarski's truth definition as a paradigm. | |
| From: Wilfrid Hodges (Model Theory [2005], Intro) | |
| A reaction: My attention is caught by the fact that natural languages are included. Might we say that science is model theory for English? That sounds like Quine's persistent message. |
| 10475 | A 'structure' is an interpretation specifying objects and classes of quantification [Hodges,W] |
| Full Idea: A 'structure' in model theory is an interpretation which explains what objects some expressions refer to, and what classes some quantifiers range over. | |
| From: Wilfrid Hodges (Model Theory [2005], 1) | |
| A reaction: He cites as examples 'first-order structures' used in mathematical model theory, and 'Kripke structures' used in model theory for modal logic. A structure is also called a 'universe'. |
| 10481 | Models in model theory are structures, not sets of descriptions [Hodges,W] |
| Full Idea: The models in model-theory are structures, but there is also a common use of 'model' to mean a formal theory which describes and explains a phenomenon, or plans to build it. | |
| From: Wilfrid Hodges (Model Theory [2005], 5) | |
| A reaction: Hodges is not at all clear here, but the idea seems to be that model-theory offers a set of objects and rules, where the common usage offers a set of descriptions. Model-theory needs homomorphisms to connect models to things, |
| 10480 | First-order logic can't discriminate between one infinite cardinal and another [Hodges,W] |
| Full Idea: First-order logic is hopeless for discriminating between one infinite cardinal and another. | |
| From: Wilfrid Hodges (Model Theory [2005], 4) | |
| A reaction: This seems rather significant, since mathematics largely relies on first-order logic for its metatheory. Personally I'm tempted to Ockham's Razor out all these super-infinities, but mathematicians seem to make use of them. |
| 22392 | Morality is inescapable, in descriptive words such as 'dishonest', 'unjust' and 'uncharitable' [Foot] |
| Full Idea: There is a sense in which morality is inescapable - in moral epithets such as 'dishonest', 'unjust', 'uncharitable'; these do not cease to apply to a man because he is indifferent to the ends of morality: they may indeed apply because of his indifference. | |
| From: Philippa Foot (Morality as system of hypothetical imperatives [1972], p.172 n15) | |
| A reaction: Odysseus was admired for lying, and charity wasn't a virtue in the ancient world. They won't go away as factual descriptions, but the values attached to them vary quite a lot. |
| 23685 | Reason is not a motivator of morality [Foot, by Hacker-Wright] |
| Full Idea: In her middle period she changed her mind, and attacks moral rationalism. | |
| From: report of Philippa Foot (Morality as system of hypothetical imperatives [1972]) by John Hacker-Wright - Philippa Foot's Moral Thought Intro | |
| A reaction: That is, she doubted whether moral reasons are sufficient to motivate moral actions, which presumably therefore need desires, as the Humeans claimed. Reasons rely on merely hypothetical rules. |
| 23691 | Rejecting moral rules may be villainous, but it isn't inconsistent [Foot] |
| Full Idea: The man who rejects morality because he sees no reason to obey its rules can be convicted of villainy but not of inconsistency. | |
| From: Philippa Foot (Morality as system of hypothetical imperatives [1972], p.161) | |
| A reaction: This is 'middle period' Foot, when she decided that Hume was right about the need for a desire as moral motivator. Before and after this time, she thought there were reasons to be moral, as well as desires. |
| 16764 | The soul conserves the body, as we see by its dissolution when the soul leaves [Toletus] |
| Full Idea: Every accident of a living thing, as well as all its organs and temperaments and its dispositions are conserved by the soul. We see this from experience, since when that soul recedes, all these dissolve and become corrupted. | |
| From: Franciscus Toletus (Commentary on 'De Anima' [1572], II.1.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.5 | |
| A reaction: A nice example of observing a phenemonon, but not being able to observe the dependence relation the right way round. Compare Descartes in Idea 16763. |
| 22391 | Saying we 'ought to be moral' makes no sense, unless it relates to some other system [Foot] |
| Full Idea: 'One ought to be moral' makes no sense at all unless the 'ought' has the moral subscript, giving a tautology, or else relates morality to some other system such as prudence or etiquette. | |
| From: Philippa Foot (Morality as system of hypothetical imperatives [1972], p.169 n18) | |
| A reaction: This aims to undercut the Kantian view that morality is an absolute call to duty (filling us with wonder, like the starry heavens). Foot aims to root morality in the real world. |
| 22389 | Morality no more consists of categorical imperatives than etiquette does [Foot] |
| Full Idea: Moral judgements have no better claim to be categorical imperatives than do statements about matters of etiquette. | |
| From: Philippa Foot (Morality as system of hypothetical imperatives [1972], p.164) | |
| A reaction: Her claim is that all moral judgements are responses to situations, and so are hypothetical. This judgement of hers is the culmination of a careful discussion. |