28 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. |
| 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, |
| 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. |
| 13007 | Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz] |
| Full Idea: Archimedes gave a sort of definition of 'straight line' when he said it is the shortest line between two points. | |
| From: report of Archimedes (fragments/reports [c.240 BCE]) by Gottfried Leibniz - New Essays on Human Understanding 4.13 | |
| A reaction: Commentators observe that this reduces the purity of the original Euclidean axioms, because it involves distance and measurement, which are absent from the purest geometry. |
| 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) |
| 15200 | How could change consist of a conjunction of changeless facts? [McTaggart, by Le Poidevin] |
| Full Idea: McTaggart objects, to Russell 1903, that change cannot consist of a conjunction of changeless facts. | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927]) by Robin Le Poidevin - Past, Present and Future of Debate about Tense 1 (b) | |
| A reaction: I agree with McTaggart. Logicians like to model processes with domains of timeless entities, but it just won't do. |
| 14761 | Change is not just having two different qualities at different points in some series [McTaggart] |
| Full Idea: The fact that it is hot at one point in a series and cold at other points cannot give change, if neither of these facts change. If two points on a line have different properties, this doesn't give change. | |
| From: J.M.E. McTaggart (The Nature of Existence vol.2 [1927], 33.315-6), quoted by Theodore Sider - Four Dimensionalism 6.2 | |
| A reaction: [The second half compresses an example about the Meridian] This objection is aimed at Russell's view, that change is just different properties at different times. I (unlike Sider) am wholly with McTaggart on this one. Change is 'dynamic'. |
| 22628 | Substance has to exist, with no intrinsic qualities or relations [McTaggart] |
| Full Idea: Something must exist, then, and have qualities, without being itself either a quality or a relation. And this is Substance. | |
| From: J.M.E. McTaggart (The Nature of Existence vol.1 [1921], §67), quoted by R.D. Ingthorsson - A Powerful Particulars View of Causation 7.2 | |
| A reaction: Ingthorsson quotes this as 'the most extreme analytic view', which is a long way from the Aristotelian view. This is the implausible bare substrate. |
| 2608 | For McTaggart time is seen either as fixed, or as relative to events [McTaggart, by Ayer] |
| Full Idea: McTaggart says we can speak of events in time in two ways, as past, present or future, or as being before or after or simultaneous with one another. The first cannot be reduced to the second, as the second makes no provision for the passage of time. | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927], II.329-) by A.J. Ayer - The Central Questions of Philosophy 1.D |
| 22936 | A-series time positions are contradictory, and yet all events occupy all of them! [McTaggart, by Le Poidevin] |
| Full Idea: McTaggart's proof of time's unreality: A-series positions (past, present and future) are mutually incompatible, so no event can exhibit more than one of them; but since A-series events change position, all events have all A-series posititions. Absurd! | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927]) by Robin Le Poidevin - Travels in Four Dimensions 08 'McTaggart's' | |
| A reaction: I'm not convinced that this is any more contradictory than someone being married at one time and unmarried at another. No one is suggesting that an A-series event can be both past and future simultaneously. |
| 4231 | Time involves change, only the A-series explains change, but it involves contradictions, so time is unreal [McTaggart, by Lowe] |
| Full Idea: McTaggart argued that time involves change, only the A-series can explain change, the A-series involves contradictions (past, present and future), and hence time is unreal. | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927]) by E.J. Lowe - A Survey of Metaphysics p.313 | |
| A reaction: I doubt whether it is a logical contradiction to say Waterloo has been past, present and future, though it is odd. |
| 8591 | There could be no time if nothing changed [McTaggart] |
| Full Idea: It is universally admitted.... that there could be no time if nothing changed. | |
| From: J.M.E. McTaggart (The Nature of Existence vol.2 [1927], II p.11), quoted by Sydney Shoemaker - Time Without Change p.49 | |
| A reaction: This is set up alongside Aristotle (Idea 8590) to be attacked by Shoemaker. I think Shoemaker is right, and that the rejection of McTaggart's view is a key result in modern metaphysics. |
| 22935 | The B-series can be inferred from the A-series, but not the other way round [McTaggart, by Le Poidevin] |
| Full Idea: McTaggart says the A-series is more fundamental than the B-series. An objective being could not deduce the present moment of the A-series from the B-series, but the B-series can be deduced from the A-series. | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927]) by Robin Le Poidevin - Travels in Four Dimensions 08 'McTaggart's' | |
| A reaction: [summarised] This has no ontological importance for McTaggart, since he thinks time is unreal either way. But giving the A-series priority because it reveals the present moment seems to nullify the B-series as incomplete. |
| 7802 | A-series uses past, present and future; B-series uses 'before' and 'after' [McTaggart, by Girle] |
| Full Idea: The A-series puts events into past, present and future. The B-series puts events into a series based on relationships of 'before' and 'after'. McTaggart said the A-series was contradictory, and the B-series failed to cope with essential features of time. | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927]) by Rod Girle - Modal Logics and Philosophy 8.10 | |
| A reaction: The A-series is indexical. |
| 4230 | A-series expressions place things in time, and their truth varies; B-series is relative, and always true [McTaggart, by Lowe] |
| Full Idea: A-series expressions include words like 'today' and 'five weeks ago', and can be true at one time and false at another; B-series expressions are like 'simultaneously', and are always true, if true at all. | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927]) by E.J. Lowe - A Survey of Metaphysics p.308 | |
| A reaction: A-series gives time separate existence, where B-series time is purely relational. Intuition favours the A-series, but how fast do events travel against this fixed background? |
| 15199 | The B-series must depend on the A-series, because change must be explained [McTaggart, by Le Poidevin] |
| Full Idea: McTaggart's argument is 1) B-series relations are temporal relations, 2) There cannot be temporal relations unless there is change, 3) There cannot be change unless there is real A-series ordering, so there can't be a B-series unless there is an A-series. | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927], vol.ii) by Robin Le Poidevin - Past, Present and Future of Debate about Tense 1 a |