19 ideas
| 21489 | Super-ordinate disciplines give laws or principles; subordinate disciplines give concrete cases [Peirce, by Atkin] |
| Full Idea: In Peirce's system, a super-ordinate discipline provides general laws or principles for subordinate disciplines, which in turn provide concrete examples of those general laws. | |
| From: report of Charles Sanders Peirce (works [1892]) by Albert Atkin - Peirce 1 'System' | |
| A reaction: Does he really mean that subordinate disciplines have no principles or laws? That can't be right. |
| 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. |
| 19095 | Pragmatic 'truth' is a term to cover the many varied aims of enquiry [Peirce, by Misak] |
| Full Idea: In Peirce's naturalist view of truth, it is a catch-all for the particular local aims of enquiry - empirical adequacy, predictive power, coherence, simplicity, elegance, explanatory power, a reliable guide to action, fruitfulness, great understanding. | |
| From: report of Charles Sanders Peirce (works [1892]) by Cheryl Misak - Pragmatism and Deflationism 1 | |
| A reaction: The aims I cited in my thesis on explanation. One given, for me, is that truth is an ideal, which may or may not be attainable, to varying degrees. It is just what thinking aims at. I suspect, though, that these listed items have one thing in common. |
| 19097 | Peirce did not think a belief was true if it was useful [Peirce, by Misak] |
| Full Idea: Peirce was not in the slightest bit tempted by the thought that a belief is true if it is useful. | |
| From: report of Charles Sanders Peirce (works [1892]) by Cheryl Misak - Pragmatism and Deflationism 2 | |
| A reaction: All students of the pragmatic theory of truth should start with this idea, because it rejects the caricature view of pragmatic truth, a view which is easily rebutted. James seems to have been guilty of this sin. |
| 21494 | If truth is the end of enquiry, what if it never ends, or ends prematurely? [Atkin on Peirce] |
| Full Idea: Two related worries about Peirce's account of truth are (from Royce) what are we to make of truth if enquiry never reaches an end, and (from Russell) what are we to make of truth if enquiry ends prematurely? | |
| From: comment on Charles Sanders Peirce (works [1892]) by Albert Atkin - Peirce 3 'issues' | |
| A reaction: The defence of Peirce must be that the theory is not holistic - referring to the whole Truth about absolutely everything. The discovery of the periodic table seems to me to support Peirce. In many areas basic enquiry has reached an end. |
| 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. |
| 21493 | Pure mathematics deals only with hypotheses, of which the reality does not matter [Peirce] |
| Full Idea: The pure mathematician deals exclusively with hypotheses. Whether or not there is any corresponding real thing, he does not care. | |
| From: Charles Sanders Peirce (works [1892], CP5.567), quoted by Albert Atkin - Peirce 3 'separation' | |
| A reaction: [Dated 1902] Maybe we should identify a huge branch of human learning as Hyptheticals. Professor of Hypotheticals at Cambridge University. The trouble is it would have to include computer games. So why does maths matter more than games? |
| 19102 | Bivalence is a regulative assumption of enquiry - not a law of logic [Peirce, by Misak] |
| Full Idea: Peirce takes bivalence not to be a law of logic, but a regulative assumption of enquiry. | |
| From: report of Charles Sanders Peirce (works [1892]) by Cheryl Misak - Pragmatism and Deflationism 2 n10 | |
| A reaction: I like this. For most enquiries it's either true or not true, it's either there or it's not there. When you aren't faced with these simple dichotomies (in history, or quantum mechanics) you can relax, and allow truth value gaps etc. |
| 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'. |
| 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. |
| 10352 | The real is the idea in which the community ultimately settles down [Peirce] |
| Full Idea: The real is the idea in which the community ultimately settles down. | |
| From: Charles Sanders Peirce (works [1892]), quoted by Martin Kusch - Knowledge by Agreement Ch.16 | |
| A reaction: If this is anti-realism, then I don't like it. If it is realist, then it is probably a bit on the optimistic side (if you think about cultures that are into witchcraft and voodoo). |
| 13498 | Peirce and others began the mapping out of relations [Peirce, by Hart,WD] |
| Full Idea: It was Peirce and Schröder in the nineteenth century who began a systematic taxonomy of relations. | |
| From: report of Charles Sanders Peirce (works [1892], 4) by William D. Hart - The Evolution of Logic 4 |
| 21491 | Peirce's later realism about possibilities and generalities went beyond logical positivism [Peirce, by Atkin] |
| Full Idea: The realism about possibilities, generalities, tendencies and habits that we find in Peirce's later maxim is something that the logical positivists would have been uncomfortable with. | |
| From: report of Charles Sanders Peirce (works [1892]) by Albert Atkin - Peirce 2 'Concl' | |
| A reaction: Atkin examines the various later statements of the earlier maxim, given here in Idea 21490. Ryle and Quine express the empiricist and logical positivist approach to dispositions. |
| 16376 | The possible can only be general, and the force of actuality is needed to produce a particular [Peirce] |
| Full Idea: The possible is necessarily general ..It is only actuality, the force of existence, which bursts the fluidity of the general and produces a discrete unit. | |
| From: Charles Sanders Peirce (works [1892]), quoted by François Recanati - Mental Files 13.1 | |
| A reaction: [Papers 4 1967:147] This was quoted by Prior, and is often cited. Recanati is interested in the notion of a singular thought being tied to actuality, by generating a mental file. |
| 19107 | Inquiry is not standing on bedrock facts, but standing in hope on a shifting bog [Peirce] |
| Full Idea: Inquiry is not standing upon a bedrock of fact. It is walking up a bog, and can only say, this ground seems to hold for the present. Here I will stay until it begins to give way. | |
| From: Charles Sanders Peirce (works [1892], CP 5.589), quoted by Gottfried Leibniz - Letter to Newton 4 | |
| A reaction: [I don't know which article this lovely quote comes from] |