13 ideas
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) |
9089 | Knowledge is a quality existing subjectively in the soul [William of Ockham] |
Full Idea: Knowledge is a certain quality which exists in the soul as its subject ('existens subiective in anima'). | |
From: William of Ockham (Expositio super viii libros [1340], Prologue) | |
A reaction: One might say here that knowledge is a property, and so it might not be susceptible to further analysis. It invites the question of how you could know by introspection that you have got it, which would be an extreme internalist view. |
9091 | Sometimes 'knowledge' just concerns the conclusion, sometimes the whole demonstration [William of Ockham] |
Full Idea: Sometimes 'knowledge' means evident cognition of the conclusion alone, sometimes of the demonstration as a whole. | |
From: William of Ockham (Expositio super viii libros [1340], Prologue) | |
A reaction: 'Demonstration' will be something like Greek 'logos' - full understanding, ability to explain and give reasons. William is certainly right about normal usage. I know the answer in a quiz, without any requirement for justifications. |
9090 | Knowledge is certain cognition of something that is true [William of Ockham] |
Full Idea: Knowledge is certain cognition of something that is true. | |
From: William of Ockham (Expositio super viii libros [1340], Prologue) | |
A reaction: This view has problems. William is not facing up to the sceptical questions which can shake any degree of certainty, and also that someone who lacked self-confidence might know many things while always feeling uncertain about them. 'Cognition' must go! |
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. |