12 ideas
7791 | The simplest of the logics based on possible worlds is Lewis's S5 [Lewis,CI, by Girle] |
Full Idea: C.I.Lewis constructed five axiomatic systems of modal logic, and named them S1 to S5. It turns out that the simplest of the logics based on possible worlds is the same as Lewis's S5. | |
From: report of C.I. Lewis (works [1935]) by Rod Girle - Modal Logics and Philosophy 2.1 | |
A reaction: Nathan Salmon ('Reference and Essence' 2nd ed) claims (on p.xvii) that "the correct modal logic is weaker than S5 and weaker even than S4". Which is the greater virtue, simplicity or weakness? |
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. |
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). |
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. |
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) |
11002 | Equating necessity with informal provability is the S4 conception of necessity [Lewis,CI, by Read] |
Full Idea: C.I.Lewis's S4 system develops a sense of necessity as 'provability' in some fairly informal sense. | |
From: report of C.I. Lewis (works [1935]) by Stephen Read - Thinking About Logic Ch. 4 |
7861 | Libet says the processes initiated in the cortex can still be consciously changed [Libet, by Papineau] |
Full Idea: Libet himself points out that the conscious decisions still have the power to 'endorse' or 'cancel', so to speak, the processes initiated by the earlier cortical activity: no action will result if the action's execution is consciously countermanded. | |
From: report of Benjamin Libet (Unconscious Cerebral Initiative [1985]) by David Papineau - Thinking about Consciousness 1.4 | |
A reaction: This is why Libet's findings do not imply 'epiphenomenalism'. It seems that part of a decisive action is non-conscious, undermining the all-or-nothing view of consciousness. Searle tries to smuggle in free will at this point (Idea 3817). |
6660 | Libet found conscious choice 0.2 secs before movement, well after unconscious 'readiness potential' [Libet, by Lowe] |
Full Idea: Libet found that a subject's conscious choice to move was about a fifth of a second before movement, and thus later than the onset of the brain's so-called 'readiness potential', which seems to imply that unconscious processes initiates action. | |
From: report of Benjamin Libet (Unconscious Cerebral Initiative [1985]) by E.J. Lowe - Introduction to the Philosophy of Mind Ch.9 | |
A reaction: Of great interest to philosophers! It seems to make conscious choices epiphenomenal. The key move, I think, is to give up the idea of consciousness as being all-or-nothing. My actions are still initiated by 'me', but 'me' shades off into unconsciousness. |