31 ideas
| 10017 | Truth in a model is more tractable than the general notion of truth [Hodes] |
| Full Idea: Truth in a model is interesting because it provides a transparent and mathematically tractable model - in the 'ordinary' rather than formal sense of the term 'model' - of the less tractable notion of truth. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |
| A reaction: This is an important warning to those who wish to build their entire account of truth on Tarski's rigorously formal account of the term. Personally I think we should start by deciding whether 'true' can refer to the mental state of a dog. I say it can. |
| 10018 | Truth is quite different in interpreted set theory and in the skeleton of its language [Hodes] |
| Full Idea: There is an enormous difference between the truth of sentences in the interpreted language of set theory and truth in some model for the disinterpreted skeleton of that language. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.132) | |
| A reaction: This is a warning to me, because I thought truth and semantics only entered theories at the stage of 'interpretation'. I must go back and get the hang of 'skeletal' truth, which sounds rather charming. [He refers to set theory, not to logic.] |
| 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) |
| 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. |
| 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) |
| 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. |
| 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'. |
| 10015 | Higher-order logic may be unintelligible, but it isn't set theory [Hodes] |
| Full Idea: Brand higher-order logic as unintelligible if you will, but don't conflate it with set theory. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |
| A reaction: [he gives Boolos 1975 as a further reference] This is simply a corrective, because the conflation of second-order logic with set theory is an idea floating around in the literature. |
| 10011 | Identity is a level one relation with a second-order definition [Hodes] |
| Full Idea: Identity should he considered a logical notion only because it is the tip of a second-order iceberg - a level 1 relation with a pure second-order definition. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) |
| 10016 | When an 'interpretation' creates a model based on truth, this doesn't include Fregean 'sense' [Hodes] |
| Full Idea: A model is created when a language is 'interpreted', by assigning non-logical terms to objects in a set, according to a 'true-in' relation, but we must bear in mind that this 'interpretation' does not associate anything like Fregean senses with terms. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |
| A reaction: This seems like a key point (also made by Hofweber) that formal accounts of numbers, as required by logic, will not give an adequate account of the semantics of number-terms in natural languages. |
| 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). |
| 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) |
| 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) |
| 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. |
| 10027 | Mathematics is higher-order modal logic [Hodes] |
| Full Idea: I take the view that (agreeing with Aristotle) mathematics only requires the notion of a potential infinity, ...and that mathematics is higher-order modal logic. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) | |
| A reaction: Modern 'modal' accounts of mathematics I take to be heirs of 'if-thenism', which seems to have been Russell's development of Frege's original logicism. I'm beginning to think it is right. But what is the subject-matter of arithmetic? |
| 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). |
| 10026 | Arithmetic must allow for the possibility of only a finite total of objects [Hodes] |
| Full Idea: Arithmetic should be able to face boldly the dreadful chance that in the actual world there are only finitely many objects. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.148) | |
| A reaction: This seems to be a basic requirement for any account of arithmetic, but it was famously a difficulty for early logicism, evaded by making the existence of an infinity of objects into an axiom of the system. |
| 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. |
| 10021 | It is claimed that numbers are objects which essentially represent cardinality quantifiers [Hodes] |
| Full Idea: The mathematical object-theorist says a number is an object that represents a cardinality quantifier, with the representation relation as the entire essence of the nature of such objects as cardinal numbers like 4. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) | |
| A reaction: [compressed] This a classic case of a theory beginning to look dubious once you spell it our precisely. The obvious thought is to make do with the numerical quantifiers, and dispense with the objects. Do other quantifiers need objects to support them? |
| 10022 | Numerical terms can't really stand for quantifiers, because that would make them first-level [Hodes] |
| Full Idea: The dogmatic Frege is more right than wrong in denying that numerical terms can stand for numerical quantifiers, for there cannot be a language in which object-quantifiers and objects are simultaneously viewed as level zero. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.142) | |
| A reaction: Subtle. We see why Frege goes on to say that numbers are level zero (i.e. they are objects). We are free, it seems, to rewrite sentences containing number terms to suit whatever logical form appeals. Numbers are just quantifiers? |
| 10023 | Talk of mirror images is 'encoded fictions' about real facts [Hodes] |
| Full Idea: Talk about mirror images is a sort of fictional discourse. Statements 'about' such fictions are not made true or false by our whims; rather they 'encode' facts about the things reflected in mirrors. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.146) | |
| A reaction: Hodes's proposal for how we should view abstract objects (c.f. Frege and Dummett on 'the equator'). The facts involved are concrete, but Hodes is offering 'encoding fictionalism' as a linguistic account of such abstractions. He applies it to numbers. |
| 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. |