38 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. |
| 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) |
| 16668 | Modes of things exist in some way, without being full-blown substances [Gassendi] |
| Full Idea: Modes are not nothing but something more than mere nothing; they are therefore 'res' of some kind, not substantial of course, but at least modal. | |
| From: Pierre Gassendi (Disquisitions [1644], II.3.4), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 260 | |
| A reaction: This is the great modern atomist talking pure scholastic metaphysics. He's been reading Suárez. Gassendi seems to accept more than one type of existence. |
| 16730 | If matter is entirely atoms, anything else we notice in it can only be modes [Gassendi] |
| Full Idea: Since these atoms are the whole of the corporeal matter or substance that exists in bodies, if we conceive or notice anything else to exist in these bodies, that is not a substance but only some kind of mode of the substance. | |
| From: Pierre Gassendi (Syntagma [1658], II.1.6.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 22.4 | |
| A reaction: If the atoms have a few qualities of their own, are they just modes? If they are genuine powers, then there can be emergent powers, which are rather more than mere 'modes'. |
| 20339 | Classes rarely share properties with their members - unlike universals and types [Wollheim] |
| Full Idea: Classes can share properties with their members (e.g. the class of big things is big), but this is very rare. ....In the case of both universals and types, there will be shared properties. Red things can be exhilarating, and so can redness. | |
| From: Richard Wollheim (Art and Its Objects [1968], 92) | |
| A reaction: 'Exhilarating' is an extrinsic property, so not the best illustration. This is interesting, but would need checking with a wide range of examples. (Too busy for that right now) |
| 16619 | We observe qualities, and use 'induction' to refer to the substances lying under them [Gassendi] |
| Full Idea: Nothing beyond qualities is perceived by the senses. …When we refer to the substance in which the qualities inhere, we do this through induction, by which we reason that some subject lies under the quality. | |
| From: Pierre Gassendi (Syntagma [1658], II.1.6.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 07.1 | |
| A reaction: He talks of 'induction' (in an older usage), but he seems to mean abduction, since he never makes any observations of the substances being proposed. |
| 20338 | We often treat a type as if it were a sort of token [Wollheim] |
| Full Idea: Much of the time we think and talk of a type as though it were itself a kind of token. | |
| From: Richard Wollheim (Art and Its Objects [1968], 35) | |
| A reaction: A helpful way of connecting what I call 'objectification' to the more conventional modern philosophical vocabulary. Thus I might claim that beauty is superior to truth, as if they were two tokens. |
| 3400 | Things must have parts to intermingle [Gassendi] |
| Full Idea: If you are no larger than a point, how are you joined to the whole body, which is so large? …and there can be no intermingling between things unless the parts of them can be intermingled. | |
| From: Pierre Gassendi (Objections to 'Meditations' (Fifth) [1641]), quoted by Jaegwon Kim - Philosophy of Mind p.131 | |
| A reaction: As Descartes says that mind is distinct from body because it is non-spatial, it doesn't seem quite right to describe it as a 'point', but the second half is a real problem. Being non-spatial is a real impediment to intermingling with spatial objects. |
| 20342 | Interpretation is performance for some arts, and critical for all arts [Wollheim] |
| Full Idea: Performative interpretation occurs only with certain arts, but critical intepretation pertains to all. | |
| From: Richard Wollheim (Art and Its Objects [1968], 38) | |
| A reaction: Fairly obvious, but this is the first point to make about the concept of 'interpretation'. Does the word in fact have two meanings? Or do I perform a painting when I look carefully at it? |
| 20343 | A love of nature must precede a love of art [Wollheim] |
| Full Idea: We could not have a feeling for the beauties of art unless we had been correspondingly moved in front of nature. | |
| From: Richard Wollheim (Art and Its Objects [1968], 43) | |
| A reaction: Wollheim offers this in defence of Kant's view, without necessarily agreeing. Similarly one could hardly care for fictional characters, but not for real people. So the aesthetic attitude may arise from life, rather than from art. Is art hence unimportant? |
| 20348 | A criterion of identity for works of art would be easier than a definition [Wollheim] |
| Full Idea: Maybe, rather than defining art, it would be more fruitful, and more realistic, to seek a general method of identifying works of art. | |
| From: Richard Wollheim (Art and Its Objects [1968], 60) | |
| A reaction: The whole enterprise is ruined by Marcel Duchamp! I'm more interested in identifying or defining good art. |
| 20347 | If beauty needs organisation, then totally simple things can't be beautiful [Wollheim] |
| Full Idea: It is said that beauty cannot consist in organisation because, if it did, we would not be able to predicate beauty of totally simple objects. | |
| From: Richard Wollheim (Art and Its Objects [1968], 59) | |
| A reaction: [He says this idea originates in Plotinus] I'm struggling to think of an example of something which is 'totally' simple and beautiful. Maybe a patch of colour like the breast of a bullfinch? |
| 20345 | Some say art must have verbalisable expression, and others say the opposite! [Wollheim] |
| Full Idea: The view that a work of art expresses nothing if it can't be put into other words ...is reduced by the view that a work of art has no value if what it expresses or says can be put into (other) words. | |
| From: Richard Wollheim (Art and Its Objects [1968], 49) | |
| A reaction: I prefer the second view. Poetry is what is lost in translation. Good art actually seems to evoke emotions which one virtually never feels in ordinary life. But how could that be possible? What are those emotions doing there? |
| 20331 | It is claimed that the expressive properties of artworks are non-physical [Wollheim] |
| Full Idea: The argument that works of art have properties that physical objects could not have characteristically concentrates on the expressive properties of works of art. | |
| From: Richard Wollheim (Art and Its Objects [1968], 10) | |
| A reaction: Since the idea of an object having non-physical properties strikes me as ridiculous, this gets off to a bad start. If artworks are abstract objects, then all of their properties are non-physical. |
| 20336 | Style can't be seen directly within a work, but appreciation needs a grasp of style [Wollheim] |
| Full Idea: 'Style' would seem to be a concept that cannot be applied to a work solely on the basis of what is represented and yet it is also essential to a proper understanding or appreciation of a work. | |
| From: Richard Wollheim (Art and Its Objects [1968], 32) | |
| A reaction: Sounds right. One long held musical note creates an expectation which depends on the presumed style of the piece of music. A single bar from a piece may well not exhibit its characteristic style. |
| 20337 | The traditional view is that knowledge of its genre to essential to appreciating literature [Wollheim] |
| Full Idea: From Aristotle onwards it has been a tenet of the traditional rhetoric that the proper understanding of a literary work involves the location of it in the correct genre, that is, as drama, epic or lyric. | |
| From: Richard Wollheim (Art and Its Objects [1968], 32) | |
| A reaction: Walton argues this persuasively. I've seen the climax of a Jacobean tragedy ruined by laughter from the audience. Genre dictates appropriate responses, so it is a communal concept. |
| 20333 | If artworks are not physical objects, they are either ideal entities, or collections of phenomena [Wollheim] |
| Full Idea: In denying that works of art are physical objects, one theory (the 'ideal') withdraws them altogether from experience, and a second theory ('phenomenal') pins them too it inescapably and at all points. | |
| From: Richard Wollheim (Art and Its Objects [1968], 21) | |
| A reaction: I incline towards them being transient ideals, created by human minds. As with so much, we idealise and objectify them as 'works', and abstract their image from the instance(s) we encounter. |
| 20334 | The ideal theory says art is an intuition, shaped by a particular process, and presented in public [Wollheim] |
| Full Idea: The ideal theory of Croce and Collingwood says art is first an inner intuition or expression of the artist, resulting from a particular process of organisation and unification, which can be externalised in public form. | |
| From: Richard Wollheim (Art and Its Objects [1968], 22) | |
| A reaction: [compressed] As stated this doesn't sound very controversial or 'ideal'. I take it the theory is intended to be more platonist than this expression of it suggests. I think the idea that it is an 'expression' of the artist is wrong. |
| 20335 | The ideal theory of art neglects both the audience and the medium employed [Wollheim] |
| Full Idea: Because the ideal theory makes a work of art inner or mental, the link between the artist and the audience has been severed .....and it also totally ignores the significance of the medium. | |
| From: Richard Wollheim (Art and Its Objects [1968], 23) | |
| A reaction: Emily Dickinson had virtually no audience for her poetry. The medium used to perform Bach's 'Art of Fugue' seems unimportant. For paintings of painterly painters paint matters. For some visual art many different media will suffice. |
| 20340 | A musical performance has virtually the same features as the piece of music [Wollheim] |
| Full Idea: With the usual reservations, there is nothing that can be predicated of a performance of a piece of music that could not also be predicated of that piece of music itself. | |
| From: Richard Wollheim (Art and Its Objects [1968], 37) | |
| A reaction: He offers this as evidence that it fits the performance being a token, and music (and all other art) being a type. There are quite a few 'reservations'. Music too difficult to perform. Great music always badly performed. |
| 20341 | An interpretation adds further properties to the generic piece of music [Wollheim] |
| Full Idea: Interpretation may be regarded as the production of a token that has properties in excess of those of the type. | |
| From: Richard Wollheim (Art and Its Objects [1968], 37) | |
| A reaction: I suppose so. If you play accurately everything that is written in the score, then anything else has to be an addition. If you play less than the score, you aren't quite playing that piece of music. |
| 20332 | A drawing only represents Napoleon if the artist intended it to [Wollheim] |
| Full Idea: It is necessary, if a drawing is to represent Napoleon, that the draughtsman should intend it to be Napoleon. | |
| From: Richard Wollheim (Art and Its Objects [1968], 13) | |
| A reaction: Does a perfect and intended representation of a person also count as a representation of the person's identical twin? The families of both might well order copies. |
| 16593 | Atoms are not points, but hard indivisible things, which no force in nature can divide [Gassendi] |
| Full Idea: The vulgar think atoms lack parts and are free of all magnitude, and hence nothing other than a mathematical point, but it is something solid and hard and compact, as to leave no room for division, separation and cutting. No force in nature can divide it. | |
| From: Pierre Gassendi (Syntagma [1658], II.1.3.5), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 03.2 | |
| A reaction: If you gloatingly think the atom has now been split, ask whether electrons and quarks now fit his description. Pasnau notes that though atoms are indivisible, they are not incorruptible, and could go out of existence, or be squashed. |
| 16729 | How do mere atoms produce qualities like colour, flavour and odour? [Gassendi] |
| Full Idea: If the only material principles of things are atoms, having only size, shape, and weight, or motion, then why are so many additional qualities created and existing within the things: color, heat, flavor, odor, and innumerable others? | |
| From: Pierre Gassendi (Syntagma [1658], II.1.5.7), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 22.4 | |
| A reaction: This is pretty much the 'hard question' about the mind-body relation. Bacon said that heat was just motion of matter. I would say that this problem is gradually being solved in my lifetime. |