16 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. |
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. |
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. |
19269 | 'Quus' means the same as 'plus' if the ingredients are less than 57; otherwise it just produces 5 [Kripke] |
Full Idea: I will define 'quus' by x-quus-y = x + y, if x, y < 57, and otherwise it equals 5. Who is to say that this is not the function I previously meant by '+'? | |
From: Saul A. Kripke (Wittgenstein on Rules and Private Language [1982], 2) | |
A reaction: Kripke's famous example, to illustrate the big new scepticism introduced by Wittgenstein's questions about the rationality of following a rule. I suspect that you have to delve into psychology to understand rule-following, rather than logic. |
19271 | No rule can be fully explained [Kripke] |
Full Idea: Every explanation of a rule could conceivably be misunderstood. | |
From: Saul A. Kripke (Wittgenstein on Rules and Private Language [1982], 3) | |
A reaction: This is Kripke's summary of what he takes to be Wittgenstein's scepticism about rules. |
7305 | Kripke's Wittgenstein says meaning 'vanishes into thin air' [Kripke, by Miller,A] |
Full Idea: Quine and Kripke's Wittgenstein attempt to argue that there are no facts about meaning, that the notion of meaning, as Kripke puts it, 'vanishes into thin air'. | |
From: report of Saul A. Kripke (Wittgenstein on Rules and Private Language [1982]) by Alexander Miller - Philosophy of Language Pref | |
A reaction: A tempting solution to the problem. If, though, it is possible for someone to say something that is self-evidently meaningless, or to accuse someone of speaking (deep down) without meaning, then that needs explaining. |
19270 | If you ask what is in your mind for following the addition rule, meaning just seems to vanish [Kripke] |
Full Idea: What can there be in my mind that I make use of when I follow a general rule to add in the future? It seems that the entire idea of meaning vanishes into thin air. | |
From: Saul A. Kripke (Wittgenstein on Rules and Private Language [1982], 2) | |
A reaction: Introspection probably isn't the best way to investigate the phenomenon of meaning. Indeed it seems rather old-fashioned and Cartesian. Kripke says, though, that seeking 'tacit' rules is even worse [end of note 22]. |
11076 | Community implies assertability-conditions rather than truth-conditions semantics [Kripke, by Hanna] |
Full Idea: If we take account of the fact that a speaker is in a community, then we must adopt an assertability-conditions semantics (based on what is legitimately assertible), and reject truth-conditional semantics (based on correspondence to the facts). | |
From: report of Saul A. Kripke (Wittgenstein on Rules and Private Language [1982]) by Robert Hanna - Rationality and Logic 6.1 | |
A reaction: [Part of Hanna's full summary of Kripke's argument] This sounds wrong to me. There are conditions where it is agreed that a lie should be told. Two people can be guilty of the same malapropism. |
11075 | The sceptical rule-following paradox is the basis of the private language argument [Kripke, by Hanna] |
Full Idea: Kripke argues that the 'rule-following paradox' is essential to the more controversial private language argument, and introduces a radically new form of scepticism. | |
From: report of Saul A. Kripke (Wittgenstein on Rules and Private Language [1982]) by Robert Hanna - Rationality and Logic 6.1 | |
A reaction: It certainly seems that Kripke is right to emphasise the separateness of the two, as the paradox is quite persuasive, but the private language argument seems less so. |
1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
Full Idea: Archelaus was the first person to say that the universe is boundless. | |
From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |