8 ideas
15927 | Definition just needs negation, known variables, conjunction, disjunction, substitution and quantification [Weyl, by Lavine] |
Full Idea: For mathematics, Weyl arrived (by 1917) at a satisfactory list of definition principles: negation, identification of variables, conjunction, disjunction, substitution of constants, and existential quantification over the domain. | |
From: report of Hermann Weyl (works [1917]) by Shaughan Lavine - Understanding the Infinite V.3 | |
A reaction: Lavine summarises this as 'first-order logic with parameters'. |
21566 | 'Propositional functions' are ambiguous until the variable is given a value [Russell] |
Full Idea: By a 'propositional function' I mean something which contains a variable x, and expresses a proposition as soon as a value is assigned to x. That is to say, it differs from a proposition solely by the fact that it is ambiguous. | |
From: Bertrand Russell (The Theory of Logical Types [1910], p.216) | |
A reaction: This is Frege's notion of a 'concept', as an assertion of a predicate which still lacks a subject. |
21567 | 'All judgements made by Epimenedes are true' needs the judgements to be of the same type [Russell] |
Full Idea: Such a proposition as 'all the judgements made by Epimenedes are true' will only be prima facie capable of truth if all his judgements are of the same order. | |
From: Bertrand Russell (The Theory of Logical Types [1910], p.227) | |
A reaction: This is an attempt to use his theory of types to solve the Liar. Tarski's invocation of a meta-language is clearly in the same territory. |
23457 | Type theory cannot identify features across levels (because such predicates break the rules) [Morris,M on Russell] |
Full Idea: Russell's theory of types meant that features common to different levels of the hierarchy became uncapturable (since any attempt to capture them would involve a predicate which disobeyed the hierarchy restrictions). | |
From: comment on Bertrand Russell (The Theory of Logical Types [1910]) by Michael Morris - Guidebook to Wittgenstein's Tractatus 2H | |
A reaction: I'm not clear whether this is the main reason why type theory was abandoned. Ramsey was an important critic. |
21556 | Classes are defined by propositional functions, and functions are typed, with an axiom of reducibility [Russell, by Lackey] |
Full Idea: In Russell's mature 1910 theory of types classes are defined in terms of propositional functions, and functions themselves are regimented by a ramified theory of types mitigated by the axiom of reducibility. | |
From: report of Bertrand Russell (The Theory of Logical Types [1910]) by Douglas Lackey - Intros to Russell's 'Essays in Analysis' p.133 |
21568 | A one-variable function is only 'predicative' if it is one order above its arguments [Russell] |
Full Idea: We will define a function of one variable as 'predicative' when it is of the next order above that of its arguments, i.e. of the lowest order compatible with its having an argument. | |
From: Bertrand Russell (The Theory of Logical Types [1910], p.237) | |
A reaction: 'Predicative' just means it produces a set. This is Russell's strict restriction on which functions are predicative. |
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. |