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. |
| 12585 | Most people can't even define a chair [Peacocke] |
| Full Idea: Ordinary speakers are notoriously unsuccessful if asked to offer an explicit definition of the concept 'chair'. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 6.1) |
| 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) |
| 12581 | Perceptual concepts causally influence the content of our experiences [Peacocke] |
| Full Idea: Once a thinker has acquired a perceptually individuated concept, his possession of that concept can causally influence what contents his experiences possess. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 3.3) | |
| A reaction: Like having 35 different words for 'snow', I suppose. I'm never convinced by such claims. Having the concepts may well influence what you look at or listen to, but I don't see the deliverances of the senses being changed by the concepts. |
| 12579 | Perception has proto-propositions, between immediate experience and concepts [Peacocke] |
| Full Idea: Perceptual experience has a second layer of nonconceptual representational content, distinct from immediate 'scenarios' and from conceptual contents. These additional contents I call 'protopropositions', containing an individual and a property/relation. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 3.3) | |
| A reaction: When philosophers start writing this sort of thing, I want to turn to neuroscience and psychology. I suppose the philosopher's justification for this sort of speculation is epistemological, but I see no good coming of it. |
| 20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
| Full Idea: There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles. | |
| From: report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro' | |
| A reaction: I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture? |
| 12586 | Consciousness of a belief isn't a belief that one has it [Peacocke] |
| Full Idea: I dispute the view that consciousness of a belief consists in some kind of belief that one has the belief. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 6.2) | |
| A reaction: Thus if one is trying to grasp the notion of higher-order thought, it doesn't have to be just more of same but one level up. Any sensible view of the brain would suggest that one sort of activity would lead into an entirely different sort. |
| 12608 | Concepts are distinguished by roles in judgement, and are thus tied to rationality [Peacocke] |
| Full Idea: 'Concept' is a notion tied, in the classical Fregean manner, to cognitive significance. Concepts are distinct if we can judge rationally of one, without the other. Concepts are constitutively and definitionally tied to rationality in this way. | |
| From: Christopher Peacocke (Truly Understood [2008], 2.2) | |
| A reaction: It seems to a bit optimistic to say, more or less, that thinking is impossible if it isn't rational. Rational beings have been selected for. As Quine nicely observed, duffers at induction have all been weeded out - but they may have existed, briefly. |
| 18568 | Philosophy should merely give necessary and sufficient conditions for concept possession [Peacocke, by Machery] |
| Full Idea: Peacocke's 'Simple Account' says philosophers should determine the necessary and sufficient conditions for possessing a concept, and psychologists should explain how the human mind meets these conditions. | |
| From: report of Christopher Peacocke (A Study of Concepts [1992]) by Edouard Machery - Doing Without Concepts 2 | |
| A reaction: One can't restrict philosophy so easily. Psychologists could do that job themselves, and dump philosophy. Philosophy is interested in the role of concepts in meaning, experience and judgement. If psychologists can contribute to philosophy, fine. |
| 18571 | Peacocke's account of possession of a concept depends on one view of counterfactuals [Peacocke, by Machery] |
| Full Idea: Peacocke's method for discovering the possession conditions of concepts is committed to a specific account of counterfactual judgements - the Simulation Model (judgements we'd make if the antecedent were actual). | |
| From: report of Christopher Peacocke (A Study of Concepts [1992]) by Edouard Machery - Doing Without Concepts 2.3.4 | |
| A reaction: Machery concludes that the Simulation Model is incorrect. This appears to be Edgington's theory of conditionals, though Machery doesn't mention her. |
| 18572 | Peacocke's account separates psychology from philosophy, and is very sketchy [Machery on Peacocke] |
| Full Idea: Peacocke's Simple Account fails to connect the psychology and philosophy of concepts, it subordinates psychology to specific field of philosophy, it is committed to analytic/synthetic, and (most important) its method is very sketchy. | |
| From: comment on Christopher Peacocke (A Study of Concepts [1992]) by Edouard Machery - Doing Without Concepts 2.3.5 | |
| A reaction: Machery says Peacocke proposes a research programme, and he is not surprised that no one has every followed. Machery is a well-known champion of 'experimental philosophy', makes philosophy respond to the psychology. |
| 17722 | The concept 'red' is tied to what actually individuates red things [Peacocke] |
| Full Idea: The possession conditions for the concept 'red' of the colour red are tied to those very conditions which individuate the colour red. | |
| From: Christopher Peacocke (Explaining the A Priori [2000], p.267), quoted by Carrie Jenkins - Grounding Concepts 2.5 | |
| A reaction: Jenkins reports that he therefore argues that we can learn something about the word 'red' from thinking about the concept 'red', which is his new theory of the a priori. I find 'possession conditions' and 'individuation' to be very woolly concepts. |
| 11127 | If concepts just are mental representations, what of concepts we may never acquire? [Peacocke] |
| Full Idea: We might say that the concept just is the mental representation, ...but there are concepts that human beings may never acquire. ...But if concepts are individuated by their possession conditions this will not be a problem. | |
| From: Christopher Peacocke (Rationale and Maxims in Study of Concepts [2005], p.169), quoted by E Margolis/S Laurence - Concepts 1.3 | |
| A reaction: I'm not sure that I understand the notion of a concept we (or any other creature) may never acquire. They no more seem to exist than buildings that were never even designed. |
| 12577 | Possessing a concept is being able to make judgements which use it [Peacocke] |
| Full Idea: Possession of any concept requires the capacity to make judgements whose content contain it. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 2.1) | |
| A reaction: Idea 12575 suggested that concept possession was an ability just to think about the concept. Why add that one must actually be able to make a judgement? Presumably to get truth in there somewhere. I may only speculate and fantasise, rather than judge. |
| 12578 | A concept is just what it is to possess that concept [Peacocke] |
| Full Idea: There can be no more to a concept than is determined by a correct account of what it is to possess that concept. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 3.2) | |
| A reaction: He calls this the Principle of Dependence. An odd idea, if you compare 'there is no more to a book than its possession conditions'. If the principle is right, I struggle with the proposal that a philosopher might demonstrate such a principle. |
| 12587 | Employing a concept isn't decided by introspection, but by making judgements using it [Peacocke] |
| Full Idea: On the account I have been developing, what makes it the case that someone is employing one concept rather than another is not constituted by his impression of whether he is, but by complex facts about explanations of his judgements. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 7.2) | |
| A reaction: I presume this brings truth into the picture, and hence establishes a link between the concept and the external world, rather than merely with other concepts. There seems to be a shadowy behaviourism lurking in the background. |
| 12605 | A sense is individuated by the conditions for reference [Peacocke] |
| Full Idea: My basic Fregean idea is that a sense is individuated by the fundamental condition for something to be its reference. | |
| From: Christopher Peacocke (Truly Understood [2008], Intro) | |
| A reaction: For something to actually be its reference (as opposed to imagined reference), truth must be involved. This needs the post-1891 Frege view of such things, and not just the view of concepts as functions which he started with. |
| 12607 | Fregean concepts have their essence fixed by reference-conditions [Peacocke] |
| Full Idea: The Fregean view is that the essence of a concept is given by the fundamental condition for something to be its reference. | |
| From: Christopher Peacocke (Truly Understood [2008], 2.1) | |
| A reaction: Peacocke is a supporter of the Fregean view. How does this work for concepts of odd creatures in a fantasy novel? Or for mistaken or confused concepts? For Burge's 'arthritis in my thigh'? I don't reject the Fregean view. |
| 12609 | Concepts have distinctive reasons and norms [Peacocke] |
| Full Idea: For each concept, there will be some reasons or norms distinctive of that concept. | |
| From: Christopher Peacocke (Truly Understood [2008], 2.3) | |
| A reaction: This is Peacocke's bold Fregean thesis (and it sounds rather Kantian to me). I dislike the word 'norms' (long story), but reasons are interesting. The trouble is the distinction between being a reason for something (its cause) and being a reason for me. |
| 12584 | An analysis of concepts must link them to something unconceptualized [Peacocke] |
| Full Idea: At some point a good account of conceptual mastery must tie the mastery to abilities and relations that do not require conceptualization by the thinker. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 5.3) | |
| A reaction: This obviously implies a physicalist commitment. Peacocke seeks, as so many do these days in philosophy of maths, to combine this commitment with some sort of Fregean "platonism without tears" (p.101). I don't buy it. |
| 12604 | Any explanation of a concept must involve reference and truth [Peacocke] |
| Full Idea: For some particular concept, we can argue that some of its distinctive features are adequately explained only by a possession-condition that involves reference and truth essentially. | |
| From: Christopher Peacocke (Truly Understood [2008], Intro) | |
| A reaction: He reached this view via the earlier assertion that it is the role in judgement which key to understanding concepts. I like any view of such things which says that truth plays a role. |
| 9335 | Concepts are constituted by their role in a group of propositions to which we are committed [Peacocke, by Greco] |
| Full Idea: Peacocke argues that it may be a condition of possessing a certain concept that one be fundamentally committed to certain propositions which contain it. A concept is constituted by playing a specific role in the cognitive economy of its possessor. | |
| From: report of Christopher Peacocke (A Study of Concepts [1992]) by John Greco - Justification is not Internal §9 | |
| A reaction: Peacocke is talking about thought and propositions rather than language. Good for him. I always have problems with this sort of view: how can something play a role if it doesn't already have intrinsic properties to make the role possible? |
| 9336 | A concept's reference is what makes true the beliefs of its possession conditions [Peacocke, by Horwich] |
| Full Idea: Peacocke has a distinctive view of reference: The reference of a concept is that which will make true the primitively compelling beliefs that provide its possession conditions. | |
| From: report of Christopher Peacocke (A Study of Concepts [1992]) by Paul Horwich - Stipulation, Meaning and Apriority §9 | |
| A reaction: The first thought is that there might occasionally be more than one referent which would do the job. It seems to be a very internal view of reference, where I take reference to be much more contextual and social. |
| 12610 | Encountering novel sentences shows conclusively that meaning must be compositional [Peacocke] |
| Full Idea: The phenomenon of understanding sentences one has never encountered before is decisive against theories of meaning which do not proceed compositionally. | |
| From: Christopher Peacocke (Truly Understood [2008], 4.3) | |
| A reaction: I agree entirely. It seems obvious, as soon as you begin to slowly construct a long and unusual sentence, and follow the mental processes of the listener. |