29 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] |
13520 | A 'tautology' must include connectives [Wolf,RS] |
Full Idea: 'For every number x, x = x' is not a tautology, because it includes no connectives. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.2) |
13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
Full Idea: Deduction Theorem: If T ∪ {P} |- Q, then T |- (P → Q). This is the formal justification of the method of conditional proof (CPP). Its converse holds, and is essentially modus ponens. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3) |
13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
Full Idea: Universal Specification: from ∀xP(x) we may conclude P(t), where t is an appropriate term. If something is true for all members of a domain, then it is true for some particular one that we specify. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3) |
13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
Full Idea: Universal Generalization: If we can prove P(x), only assuming what sort of object x is, we may conclude ∀xP(x) for the same x. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3) | |
A reaction: This principle needs watching closely. If you pick one person in London, with no presuppositions, and it happens to be a woman, can you conclude that all the people in London are women? Fine in logic and mathematics, suspect in life. |
13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
Full Idea: Existential Generalization (or 'proof by example'): From P(t), where t is an appropriate term, we may conclude ∃xP(x). | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3) | |
A reaction: It is amazing how often this vacuous-sounding principles finds itself being employed in discussions of ontology, but I don't quite understand why. |
13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
Full Idea: Empty Set Axiom: ∃x ∀y ¬ (y ∈ x). There is a set x which has no members (no y's). The empty set exists. There is a set with no members, and by extensionality this set is unique. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.3) | |
A reaction: A bit bewildering for novices. It says there is a box with nothing in it, or a pair of curly brackets with nothing between them. It seems to be the key idea in set theory, because it asserts the idea of a set over and above any possible members. |
13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
Full Idea: The comprehension axiom says that any collection of objects that can be clearly specified can be considered to be a set. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.2) | |
A reaction: This is virtually tautological, since I presume that 'clearly specified' means pinning down exact which items are the members, which is what a set is (by extensionality). The naïve version is, of course, not so hot. |
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) |
13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
Full Idea: One of the most appealing features of first-order logic is that the two 'turnstiles' (the syntactic single |-, and the semantic double |=), which are the two reasonable notions of logical consequence, actually coincide. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3) | |
A reaction: In the excitement about the possibility of second-order logic, plural quantification etc., it seems easy to forget the virtues of the basic system that is the target of the rebellion. The issue is how much can be 'expressed' in first-order logic. |
13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
Full Idea: The 'completeness' of first order-logic does not mean that every sentence or its negation is provable in first-order logic. We have instead the weaker result that every valid sentence is provable. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3) | |
A reaction: Peter Smith calls the stronger version 'negation completeness'. |
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) |
13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
Full Idea: Model theory helps one to understand what it takes to specify a mathematical structure uniquely. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.1) | |
A reaction: Thus it is the development of model theory which has led to the 'structuralist' view of mathematics. |
13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
Full Idea: A 'structure' in model theory has a non-empty set, the 'universe', as domain of variables, a subset for each 'relation', some 'functions', and 'constants'. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.2) |
13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
Full Idea: Model theory uses set theory to show that the theorem-proving power of the usual methods of deduction in mathematics corresponds perfectly to what must be true in actual mathematical structures. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref) | |
A reaction: That more or less says that model theory demonstrates the 'soundness' of mathematics (though normal arithmetic is famously not 'complete'). Of course, he says they 'correspond' to the truths, rather than entailing them. |
13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
Full Idea: The three foundations of first-order model theory are the Completeness theorem, the Compactness theorem, and the Löwenheim-Skolem-Tarski theorem. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3) | |
A reaction: On p.180 he notes that Compactness and LST make no mention of |- and are purely semantic, where Completeness shows the equivalence of |- and |=. All three fail for second-order logic (p.223). |
13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
Full Idea: An 'isomorphism' is a bijection between two sets that preserves all structural components. The interpretations of each constant symbol are mapped across, and functions map the relation and function symbols. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.4) |
13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
Full Idea: The Löwenheim-Skolem-Tarski theorem demonstrates a serious limitation of first-order logic, and is one of primary reasons for considering stronger logics. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.7) |
13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
Full Idea: It is valuable to know that a theory is complete, because then we know it cannot be strengthened without passing to a more powerful language. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.5) |
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) |
13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
Full Idea: Deductive logic, including first-order logic and other types of logic used in mathematics, is 'monotonic'. This means that we never retract a theorem on the basis of new givens. If T|-φ and T⊆SW, then S|-φ. Ordinary reasoning is nonmonotonic. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.7) | |
A reaction: The classic example of nonmonotonic reasoning is the induction that 'all birds can fly', which is retracted when the bird turns out to be a penguin. He says nonmonotonic logic is a rich field in computer science. |
13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
Full Idea: Less theoretically, an ordinal is an equivalence class of well-orderings. Formally, we say a set is 'transitive' if every member of it is a subset of it, and an ordinal is a transitive set, all of whose members are transitive. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.4) | |
A reaction: He glosses 'transitive' as 'every member of a member of it is a member of it'. So it's membership all the way down. This is the von Neumann rather than the Zermelo approach (which is based on singletons). |
13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
Full Idea: One of the great achievements of modern mathematics has been the unification of its many types of objects. It began with showing geometric objects numerically or algebraically, and culminated with set theory representing all the normal objects. | |
From: Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref) | |
A reaction: His use of the word 'object' begs all sorts of questions, if you are arriving from the street, where an object is something which can cause a bruise - but get used to it, because the word 'object' has been borrowed for new uses. |
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. |