38 ideas
| 3123 | Science is in the business of carving nature at the joints [Segal] |
| Full Idea: Science is in the business of carving nature at the joints. | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 5) |
| 3125 | Psychology studies the way rationality links desires and beliefs to causality [Segal] |
| Full Idea: A person's desires and beliefs tend to cause what they tend to rationalise. This coordination of causality and rationalisation lies at the heart of psychology. | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 5.3) |
| 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'. |
| 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) |
| 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). |
| 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. |
| 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. |
| 3105 | Is 'Hesperus = Phosphorus' metaphysically necessary, but not logically or epistemologically necessary? [Segal] |
| Full Idea: It is metaphysically necessary that Hesperus is Phosphorus, but not logically necessary, since logical deduction could not reveal its truth, and it is not epistemologically necessary, as the ancient Greeks didn't know the identity. (Natural necessity?) | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 1.6) |
| 3106 | If claims of metaphysical necessity are based on conceivability, we should be cautious [Segal] |
| Full Idea: Since conceivability is the chief method of assessing the claims of metaphysical necessity, I think such claims are incautious. | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 1.6) |
| 3113 | The success and virtue of an explanation do not guarantee its truth [Segal] |
| Full Idea: The success and virtue of an explanation do not guarantee its truth. | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 2.2) |
| 3112 | Folk psychology is ridiculously dualist in its assumptions [Segal] |
| Full Idea: Commonsense psychology is a powerful explanatory theory, and largely correct, but it seems to be profoundly dualist, and treats minds as immaterial spirits which can transmigrate and exist disembodied. | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 2.2) | |
| A reaction: Fans of folk psychology tend to focus on central normal experience, but folk psychology also seems to range from quirky to barking mad. A 'premonition' is a widely accepted mental event. |
| 3108 | If 'water' has narrow content, it refers to both H2O and XYZ [Segal] |
| Full Idea: My view is that the concepts of both the Earth person and the Twin Earth person refer to BOTH forms of diamonds or water (H2O and XYZ). | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 1.7) | |
| A reaction: Fair enough, though that seems to imply that my current concepts may actually refer to all sorts of items of which I am currently unaware. But that may be so. |
| 3110 | Humans are made of H2O, so 'twins' aren't actually feasible [Segal] |
| Full Idea: Humans are largely made of H2O, so there could be no twin on Twin Earth, and (as Kuhn noted) nothing with a significantly different structure from H2O could be macroscopically very like water (but topaz and citrine will do). | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 2.1) | |
| A reaction: A small point, but one that appeals to essentialists like me (see under Natural Theory/Laws of Nature). We can't learn much metaphysics from impossible examples. |
| 3124 | Externalists can't assume old words refer to modern natural kinds [Segal] |
| Full Idea: The question of what a pre-scientific term extends over is extremely difficult for a Putnam-style externalist to answer. …There seems no good reason to assume that they extend over natural kinds ('whale', 'cat', 'water'). | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 5.1) | |
| A reaction: The assumption seems to be that they used to extend over descriptions, and now they extend over essences, or expert references. This can't be right. They have never changed, but now contain fewer errors. |
| 3117 | Concepts can survive a big change in extension [Segal] |
| Full Idea: We need to think of concepts as organic entities that can persist through changes of extension. | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 3.3) | |
| A reaction: This would be 'organic' in the sense of modifying and growing. This is exactly right, and the interesting problem becomes the extreme cases, where an individual stretches a concept a long way. |
| 3104 | Must we relate to some diamonds to understand them? [Segal] |
| Full Idea: Is a relationship with diamonds necessary for having a concept of diamonds? | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 1.4) | |
| A reaction: Probably not, given that I have a concept of kryptonite, and that I can invent my own concepts. Suppose I was brought up to believe that diamonds are a myth? |
| 3103 | Maybe content involves relations to a language community [Segal] |
| Full Idea: It has been argued (e.g. by Tyler Burge) that certain relations to other language users are determinants of content. | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 1.4) | |
| A reaction: Burge's idea (with Wittgenstein behind him) strikes me as plausible (more plausible than water and elms determining the content). Our concepts actually shift during conversations. |
| 3111 | Externalism can't explain concepts that have no reference [Segal] |
| Full Idea: Empty terms and concepts provide the largest problem for the externalist thesis of the world dependence of concepts. | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 2.2) | |
| A reaction: A speculative concept could then become a reality (e.g. an invention). The solution seems to be to say that there is an internal and an external component to most concepts. |
| 3109 | If content is external, so are beliefs and desires [Segal] |
| Full Idea: If we accept Putnam's externalist conclusion about the meaning of a word, it is a short step to a similar conclusion about the contents of the twins' beliefs, desires and so on. | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 2.1) | |
| A reaction: This is the key step which has launched a whole new externalist view of the nature of the mind. It is one thing to say that I don't quite know what my words mean, another that I don't know my own beliefs. |
| 3116 | Maybe experts fix content, not ordinary users [Segal] |
| Full Idea: Putnam and Burge claim that there could be two words that a misinformed subject uses to express different concepts, but that express just one concept of the experts. | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 3.2) | |
| A reaction: This pushes the concept outside the mind of the user, which leaves an ontological problem of what concepts are made of, how you individuate them, and where they are located. |
| 3121 | If content is narrow, my perfect twin shares my concepts [Segal] |
| Full Idea: To say that contents of my belief are narrow is to say that they are intrinsic to me, hence that any perfect twin of mine would have beliefs with the same contents. | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 5) | |
| A reaction: I personally find this more congenial than externalism. If my twin and I studied chemistry, we would reach identical conclusions about water, as long as we remained perfect twins. |
| 3118 | If thoughts ARE causal, we can't explain how they cause things [Segal] |
| Full Idea: If we identify a psychological property with its causal role then we lose the obvious explanation of why the event has the causal role that it has. | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 4.1) | |
| A reaction: This pinpoints very nicely one of the biggest errors in modern philosophy. There are good naturalistic reasons to reduce everything to causal role, but there is a deeper layer. Essences! |
| 3119 | Even 'mass' cannot be defined in causal terms [Segal] |
| Full Idea: We can't define mass in terms of its causal powers because massive objects do different things in different physical systems. …What an object (or concept) with a given property does depends on what it interacts with. | |
| From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 4.1) | |
| A reaction: This leaves an epistemological problem, that we believe in mass, but can only get at it within a particular gravitational or inertial system. Don't give up on ontology at this point. |