27 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. |
| 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. |
| 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. |
| 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). |
| 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. |
| 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, |
| 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'. |
| 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. |
| 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. |
| 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) |
| 7096 | We may still admire a person's character even if the traits are involuntary [Statman] |
| Full Idea: If we focus on the evaluation of character traits, voluntariness becomes less important. We would not withdraw our admiration for a person only because we found out that his or her being such a person was not a result of voluntary choice. | |
| From: Daniel Statman (Introduction to Virtue Ethics [1997], §3) | |
| A reaction: The need for voluntariness does not disappear. I would not admire the only generous deed you had ever performed if it was the result of hypnotism. I might admire the hypnotist. Nevertheless, I regard this idea as a crucial truth in moral theory. |
| 7098 | There is a new sort of moral scepticism, about the possibility of moral theories [Statman] |
| Full Idea: Since the 1980s, ethics has witnessed a new sort of moral scepticism, this time about the possibility of moral theories. | |
| From: Daniel Statman (Introduction to Virtue Ethics [1997], §4) | |
| A reaction: He cites McDowell, Williams, Nussbaum and Baier as the culprits. 'Particularism' (every situation is different, so there can't be rules) seems an essential part of virtue theory, but total absence of principles sounds to me like moral drift. |
| 7099 | With a broad concept of flourishing, it might be possible without the virtues [Statman] |
| Full Idea: In a rich conception of human flourishing, both individuals and societies seem to be able to flourish without the virtues. | |
| From: Daniel Statman (Introduction to Virtue Ethics [1997], §5) | |
| A reaction: I can see Aristotle clutching his head in despair at this thought. It might look like flourishing, but it couldn't be the real thing. It is Aristotle's fault, though, for including external goods. Money and pleasure offer a kind of flourishing. |
| 7100 | Virtue theory isn't a genuine ethical theory, because it doesn't have universal application [Statman] |
| Full Idea: It can be claimed that universality is a necessary property of any ethical theory and therefore virtue theory, which fails in this respect, is not a theory, and hence poses no alternative to genuine ethical theories. | |
| From: Daniel Statman (Introduction to Virtue Ethics [1997], §5) | |
| A reaction: Replies: a) totally universal morality is an idle dream (part of the 'Enlightenment Project' to prove everything) and we must settle for something more relative; b) virtues aren't totally universal, but they are truths about humanity. I prefer b). |
| 7102 | Promises create moral duties that have nothing to do with character [Statman] |
| Full Idea: That duties are created irrespective of facts about character is obvious from the case of promises, which bind their makers irrespective of their motives or personality. | |
| From: Daniel Statman (Introduction to Virtue Ethics [1997], §5) | |
| A reaction: Just occasionally a promise can be broken, by a sensitive and wise person. I promise to give your son some money; I then discover he is a drug dealer. Promises arise out of character, and cannot be made by robots. |
| 7095 | Moral education is better by concrete example than abstract principle [Statman] |
| Full Idea: According to virtue theory, education through moral exemplars is more effective than education focused on principles and obligations, because it is far more concrete. | |
| From: Daniel Statman (Introduction to Virtue Ethics [1997], §3) | |
| A reaction: Aristotle's view is that virtues must be developed from childhood, when principles don't mean much. The problem is that young people may witness highly virtuous behaviour in their exemplars, but totally fail to appreciate it without mention of principles. |
| 7094 | Friends express friendship even when no utility is involved [Statman] |
| Full Idea: Being a good friend means acting in ways that express the friendship even when those ways do not promote overall utility. | |
| From: Daniel Statman (Introduction to Virtue Ethics [1997], §3) | |
| A reaction: This implies that friendship is a true virtue of character, rather than having friends just being an 'external' good. Having friends is good; being friends is a virtue. There are duties of friendship. |
| 7093 | Behaviour may be disgusting or inhumane, but violate no duty [Statman] |
| Full Idea: It is surely possible, and indeed often the case, that people who violate no duty nevertheless behave in an inhumane and a disgusting manner. | |
| From: Daniel Statman (Introduction to Virtue Ethics [1997], §1) | |
| A reaction: This seems right, though it is easier to be disgusting than to be inhumane if no duty is to be violated. Social duties may not require a high degree of humanity, pure Kantian duties might. |
| 7104 | The ancients recognised imperfect duties, but we have added perfect duties like justice [Statman] |
| Full Idea: The advantage of modern thinkers over the ancient virtue ethicists is that in addition to imperfect duties (i.e. virtues) they also recognise the existence of perfect duties, or duties of justice, which are essential for the existence of society. | |
| From: Daniel Statman (Introduction to Virtue Ethics [1997], §7) | |
| A reaction: Even the Greeks had laws (e.g. Idea 422), so they understood that a society needs rules, but many laws don't seem to be moral rules (e.g. car parking), and the Greeks thought morality was about human excellence, not avoiding traffic jams. |
| 7103 | Abortion issues focus on the mother's right over her body, and the status of the foetus [Statman] |
| Full Idea: Most of the debate on abortion focuses on two issues, the mother's assumed right over her body, and the status of the foetus. | |
| From: Daniel Statman (Introduction to Virtue Ethics [1997], §6) | |
| A reaction: Personally I think society as a whole might have a say (if, perhaps, we are over- or under-populated, or we have a widely accepted state religion, or we are just very shocked). Mother's have virtues and duties as well as rights. |
| 6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
| Full Idea: Hermarchus said that animal killing is justified by considerations of human safety and nourishment and by animals' inability to form contractual relations of justice with us. | |
| From: report of Hermarchus (fragments/reports [c.270 BCE]) by David A. Sedley - Hermarchus | |
| A reaction: Could the last argument be used to justify torturing animals? Or could we eat a human who was too brain-damaged to form contracts? |