11 ideas
13913 | The four 'perfect syllogisms' are called Barbara, Celarent, Darii and Ferio [Engelbretsen/Sayward] |
Full Idea: There are four 'perfect syllogisms': Barbara (every M is P, every S is M, so every S is P); Celarent (no M is P, every S is M, so no S is P); Darii (every M is P, some S is M, so some S is P); Ferio (no M is P, some S is M, so some S is not P). | |
From: Engelbretsen,G/Sayward,C (Philosophical Logic: Intro to Advanced Topics [2011], 8) | |
A reaction: The four names are mnemonics from medieval universities. |
13914 | Syllogistic logic has one rule: what is affirmed/denied of wholes is affirmed/denied of their parts [Engelbretsen/Sayward] |
Full Idea: It has often been claimed (e.g. by Leibniz) that a single rule governs all syllogistic validity, called 'dictum de omni et null', which says that what is affirmed or denied of any whole is affirmed or denied of any part of that whole. | |
From: Engelbretsen,G/Sayward,C (Philosophical Logic: Intro to Advanced Topics [2011], 8) | |
A reaction: This seems to be the rule which is captured by Venn Diagrams. |
13915 | Syllogistic can't handle sentences with singular terms, or relational terms, or compound sentences [Engelbretsen/Sayward] |
Full Idea: Three common kinds of sentence cannot be put into syllogistic ('categorical') form: ones using singular terms ('Mars is red'), ones using relational terms ('every painter owns some brushes'), and compound sentences. | |
From: Engelbretsen,G/Sayward,C (Philosophical Logic: Intro to Advanced Topics [2011], 8) |
13916 | Term logic uses expression letters and brackets, and '-' for negative terms, and '+' for compound terms [Engelbretsen/Sayward] |
Full Idea: Term logic begins with expressions and two 'term functors'. Any simple letter is a 'term', any term prefixed by a minus ('-') is a 'negative term', and any pair of terms flanking a plus ('+') is a 'compound term'. Parenthese are used for grouping. | |
From: Engelbretsen,G/Sayward,C (Philosophical Logic: Intro to Advanced Topics [2011], 8) | |
A reaction: [see Engelbretsen and Sayward for the full formal system] |
15943 | Limitation of Size is not self-evident, and seems too strong [Lavine on Neumann] |
Full Idea: Von Neumann's Limitation of Size axiom is not self-evident, and he himself admitted that it seemed too strong. | |
From: comment on John von Neumann (An Axiomatization of Set Theory [1925]) by Shaughan Lavine - Understanding the Infinite VII.1 |
13850 | In modern logic all formal validity can be characterised syntactically [Engelbretsen/Sayward] |
Full Idea: One of the key ideas of modern formal logic is that all formally valid inferences can be specified in strictly syntactic terms. | |
From: Engelbretsen,G/Sayward,C (Philosophical Logic: Intro to Advanced Topics [2011], Ch.2) |
13849 | Classical logic rests on truth and models, where constructivist logic rests on defence and refutation [Engelbretsen/Sayward] |
Full Idea: Classical logic rests on the concepts of truth and falsity (and usually makes use of a semantic theory based on models), whereas constructivist logic accounts for inference in terms of defense and refutation. | |
From: Engelbretsen,G/Sayward,C (Philosophical Logic: Intro to Advanced Topics [2011], Intro) | |
A reaction: My instincts go with the classical view, which is that inferences do not depend on the human capacity to defend them, but sit there awaiting revelation. My view isn't platonist, because I take the inferences to be rooted in the physical world. |
13851 | Unlike most other signs, = cannot be eliminated [Engelbretsen/Sayward] |
Full Idea: Unlike ∨, →, ↔, and ∀, the sign = is not eliminable from a logic. | |
From: Engelbretsen,G/Sayward,C (Philosophical Logic: Intro to Advanced Topics [2011], Ch.3) |
13852 | Axioms are ω-incomplete if the instances are all derivable, but the universal quantification isn't [Engelbretsen/Sayward] |
Full Idea: A set of axioms is said to be ω-incomplete if, for some universal quantification, each of its instances is derivable from those axioms but the quantification is not thus derivable. | |
From: Engelbretsen,G/Sayward,C (Philosophical Logic: Intro to Advanced Topics [2011], 7) |
13007 | Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz] |
Full Idea: Archimedes gave a sort of definition of 'straight line' when he said it is the shortest line between two points. | |
From: report of Archimedes (fragments/reports [c.240 BCE]) by Gottfried Leibniz - New Essays on Human Understanding 4.13 | |
A reaction: Commentators observe that this reduces the purity of the original Euclidean axioms, because it involves distance and measurement, which are absent from the purest geometry. |
13672 | All the axioms for mathematics presuppose set theory [Neumann] |
Full Idea: There is no axiom system for mathematics, geometry, and so forth that does not presuppose set theory. | |
From: John von Neumann (An Axiomatization of Set Theory [1925]), quoted by Stewart Shapiro - Foundations without Foundationalism 8.2 | |
A reaction: Von Neumann was doubting whether set theory could have axioms, and hence the whole project is doomed, and we face relativism about such things. His ally was Skolem in this. |