50 ideas
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'. |
7306 | If the only property of a name was its reference, we couldn't explain bearerless names [Miller,A] |
Full Idea: If having a reference were the only semantic property in terms of which we could explain the functioning of names, we would be in trouble with respect to names that simply have no bearer. | |
From: Alexander Miller (Philosophy of Language [1998], 2.1.1) | |
A reaction: (Miller is discussing Frege) 'Odysseus' is given as an example. Instead of switching to a bundle of descriptions, we could say that we just imagine an object which is stamped with the name. Names always try to refer. |
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). |
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. |
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). |
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. |
458 | Nothing could come out of nothing, and existence could never completely cease [Empedocles] |
Full Idea: From what in no wise exists, it is impossible for anything to come into being; for Being to perish completely is incapable of fulfilment and unthinkable. | |
From: Empedocles (fragments/reports [c.453 BCE], B012), quoted by Anon (Lyc) - On Melissus 975b1-4 |
5112 | Empedocles says things are at rest, unless love unites them, or hatred splits them [Empedocles, by Aristotle] |
Full Idea: Empedocles claims that things are alternately changing and at rest - that they are changing whenever love is creating a unity out of plurality, or hatred is creating plurality out of unity, and they are at rest in the times in between. | |
From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Physics 250b26 | |
A reaction: I suppose one must say that this an example of Ruskin's 'pathetic fallacy' - reading human emotions into the cosmos. Being constructive little creatures, we think goodness leads to construction. I'm afraid Empedocles is just wrong. |
13209 | There is no coming-to-be of anything, but only mixing and separating [Empedocles, by Aristotle] |
Full Idea: Empedocles says there is no coming-to-be of anything, but only a mingling and a divorce of what has been mingled. | |
From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 314b08 | |
A reaction: Aristotle comments that this prevents Empedocleans from distinguishing between superficial alteration and fundamental change of identity. Presumably, though, that wouldn't bother them. |
457 | Substance is not created or destroyed in mortals, but there is only mixing and exchange [Empedocles] |
Full Idea: There is no creation of substance in any one of mortal existence, nor any end in execrable death, but only mixing and exchange of what has been mixed. | |
From: Empedocles (fragments/reports [c.453 BCE], B008), quoted by Plutarch - 74: Reply to Colotes 1111f | |
A reaction: also Aristotle 314b08 |
7322 | Constitutive scepticism is about facts, and epistemological scepticism about our ability to know them [Miller,A] |
Full Idea: We should distinguish 'constitutive scepticism' (about the existence of certain sorts of facts) from the traditional 'epistemological scepticism' (which concedes that the sort of fact in question exists, but questions our right to claim knowledge of it). | |
From: Alexander Miller (Philosophy of Language [1998], 4.7) | |
A reaction: I would be inclined to call the first type 'ontological scepticism'. Miller is discussing Quine's scepticism about meaning. Atheists fall into the first group, and agnostics into the second. An important, and nicely simple, distinction. |
462 | One vision is produced by both eyes [Empedocles] |
Full Idea: One vision is produced by both eyes | |
From: Empedocles (fragments/reports [c.453 BCE], B088), quoted by Strabo - works 8.364.3 |
22765 | Wisdom and thought are shared by all things [Empedocles] |
Full Idea: Wisdom and power of thought, know thou, are shared in by all things. | |
From: Empedocles (fragments/reports [c.453 BCE]), quoted by Sextus Empiricus - Against the Logicians (two books) II.286 | |
A reaction: Sextus quotes this, saying that it is 'still more paradoxical', and that it explicitly includes plants. This may mean that Empedocles was not including inanimate matter. |
7325 | Dispositions say what we will do, not what we ought to do, so can't explain normativity [Miller,A] |
Full Idea: Dispositional facts are facts about what we will do, not about what we ought to do, and as such cannot capture the normativity of meaning. | |
From: Alexander Miller (Philosophy of Language [1998], 6.2) | |
A reaction: Miller is discussing language, but this raises a nice question for all behaviourist accounts of mental events. Perhaps there is a disposition to behave in a guilty way if you do something you think you shouldn't do. (Er, isn't 'guilt' a mental event?) |
1524 | For Empedocles thinking is almost identical to perception [Empedocles, by Theophrastus] |
Full Idea: Empedocles assumes that thinking is either identical to or very similar to sense-perception. | |
From: report of Empedocles (fragments/reports [c.453 BCE], A86) by Theophrastus - On the Senses 9 | |
A reaction: Not to be sniffed at. We can, of course, control our thinking (though we can't control the controller) and we contemplate abstractions, but that might be seen as a sort of perception. Vision is not as visual as we think. |
7324 | Explain meaning by propositional attitudes, or vice versa, or together? [Miller,A] |
Full Idea: Grice wants to explain linguistic meaning in terms of the content of propositional attitudes, Dummett has championed the view that propositional attitudes must be explained by linguistic meaning, while Davidson says they must be explained together. | |
From: Alexander Miller (Philosophy of Language [1998], 6.1) | |
A reaction: A useful map. My intuition says propositional attitudes come first, for evolutionary reasons. We are animals first, and speakers second. Thought precedes language. A highly social animal flourishes if it can communicate. |
7323 | If truth is deflationary, sentence truth-conditions just need good declarative syntax [Miller,A] |
Full Idea: On a deflationary concept of truth, for a sentence to possess truth-conditions it is sufficient that it be disciplined by norms of correct usage, and that it possess the syntax distinctive of declarative sentences. | |
From: Alexander Miller (Philosophy of Language [1998], 5.3) | |
A reaction: Idea 6337 gives the basic deflationary claim. He mentions Boghossian as source of this point. So much the worse for the deflationary concept of truth, say I. What are the truth-conditions of "Truth rotates"? |
7315 | 'Jones is a married bachelor' does not have the logical form of a contradiction [Miller,A] |
Full Idea: The syntactic notion of contradiction (p and not-p) is well understood, but is no help in explaining analyticity, since "Jones is a married bachelor" is not of that syntactic form. | |
From: Alexander Miller (Philosophy of Language [1998], 4.2) | |
A reaction: This point is based on Quine. This means we cannot define analytic sentences as those whose denial is a contradiction, even though that seems to be true of them. Both the Kantian and the modern logical versions of analyticity are in trouble. |
7328 | The principle of charity is holistic, saying we must hold most of someone's system of beliefs to be true [Miller,A] |
Full Idea: Properly construed, the principle of charity is a holistic constraint applying, not to individual beliefs, but rather to systems of belief: we must interpret a speaker so that most of the beliefs in his system are, by our lights, true. | |
From: Alexander Miller (Philosophy of Language [1998], 8.7) | |
A reaction: This is a lot more plausible than applying the principle to individual sentences, particularly if you are in the company of habitual ironists or constitutional liars. |
7329 | Maybe we should interpret speakers as intelligible, rather than speaking truth [Miller,A] |
Full Idea: A more sophisticated version of the principle of charity holds that we interpret speakers not as necessarily having beliefs that are true by our own lights, but as having beliefs that are intelligible by our own lights. | |
From: Alexander Miller (Philosophy of Language [1998], 8.7) | |
A reaction: Consider Idea 4161 in the light of this. Presumably this means that we treat them as having a coherent set of beliefs, even if they seem to us to fail to correspond to reality. I prefer the stronger version that there has to be some proper truth in there. |
7333 | The Frege-Geach problem is that I can discuss the wrongness of murder without disapproval [Miller,A] |
Full Idea: The main problem faced by non-cognitivism is known as the Frege-Geach problem: if I say "If murder is wrong, then getting your brother to murder people is wrong", that is an unasserted context, and I don't necessarily express disapproval of murder. | |
From: Alexander Miller (Philosophy of Language [1998], 9.2) | |
A reaction: The emotivist or non-cognitivist might mount a defence by saying there is some second-order or deep-buried emotion involved. Could a robot without feelings even understand what humans meant when they said "It is morally wrong"? |
552 | Empedocles said good and evil were the basic principles [Empedocles, by Aristotle] |
Full Idea: Empedocles was the first to give evil and good as principles. | |
From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Metaphysics 985a | |
A reaction: Once you start to think that good and evil will only matter if they have causal powers, it is an easy step to the idea of a benevolent god, and a satanic anti-god. Otherwise the 'principles' could be ignored. |
589 | 'Nature' is just a word invented by people [Empedocles] |
Full Idea: Nature is but a word of human framing. | |
From: Empedocles (fragments/reports [c.453 BCE], B008), quoted by Aristotle - Metaphysics 1015a |
21823 | The principle of 'Friendship' in Empedocles is the One, and is bodiless [Empedocles, by Plotinus] |
Full Idea: In Empedocles we have a dividing principle, 'Strife', set against 'Friendship' - which is the One and is to him bodiless, while the elements represent matter. | |
From: report of Empedocles (fragments/reports [c.453 BCE]) by Plotinus - The Enneads 5.1.09 | |
A reaction: The first time I've seen the principle of Love in Empedocles identified with the One of Parmenides. Plotinus is a trustworthy reporter, I think, because he was well read, and had access to lost texts. |
2680 | Empedocles said that there are four material elements, and two further creative elements [Empedocles, by Aristotle] |
Full Idea: Empedocles holds that the corporeal elements are four, but that all the elements, including those which create motion, are six in number. | |
From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 314a16 |
6002 | Empedocles says bone is water, fire and earth in ratio 2:4:2 [Empedocles, by Inwood] |
Full Idea: Empedocles used numerical ratios to explain different kinds of matter; for example, bone is two parts water, four parts fire, two parts earth; and blood is an equal blend of all four elements. | |
From: report of Empedocles (fragments/reports [c.453 BCE]) by Brad Inwood - Empedocles | |
A reaction: Why isn't the ration 1:2:1? This presumably shows the influence of Pythagoras (who had also been based in Italy, like Empedocles), as well as that of the earlier naturalistic philosophers. It was a very good theory, though wrong. |
13207 | Fire, Water, Air and Earth are elements, being simple as well as homoeomerous [Empedocles, by Aristotle] |
Full Idea: Empedocles says that Fire, Water, Air and Earth are four elements, and are thus 'simple' rather than flesh, bone and bodies which, like these, are 'homoeomeries'. | |
From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 314a26 | |
A reaction: The translation is not quite clear. I take it that flesh and bone may look simple, because they are homoeomerous, but they are not really - but what is his evidence for that? Compare Idea 13208. |
459 | All change is unity through love or division through hate [Empedocles] |
Full Idea: These elements never cease their continuous exchange, sometimes uniting under the influence of Love, so that all become One, at other times again moving apart through the hostile force of Hate. | |
From: Empedocles (fragments/reports [c.453 BCE], B017), quoted by Simplicius - On Aristotle's 'Physics' 158.1- |
13218 | The elements combine in coming-to-be, but how do the elements themselves come-to-be? [Aristotle on Empedocles] |
Full Idea: Empedocles says it is evident that all the other bodies down to the 'elements' have their coming-to-be and their passing-away: but it is not clear how the 'elements' themselves, severally in their aggregated masses, come-to-be and pass-away. | |
From: comment on Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 325b20 | |
A reaction: Presumably the elements are like axioms - and are just given. How do electrons and quarks come-to-be? |
13225 | Love and Strife only explain movement if their effects are distinctive [Aristotle on Empedocles] |
Full Idea: It is not an adequate explanation to say that 'Love and Strife set things moving', unless the very nature of Love is a movement of this kind and the very nature of Strife a movement of that kind. | |
From: comment on Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 333b23 | |
A reaction: I take this to be of interest for showing Aristotle's quest for explanations, and his unwillingness to be fobbed off with anything superficial. I take a task of philosophy to be to push explanations further than others wish to go. |
460 | If the one Being ever diminishes it would no longer exist, and what could ever increase it? [Empedocles] |
Full Idea: Besides these elements, nothing else comes into being, nor does anything cease. For if they had been perishing continuously, they would Be no more; and what could increase the Whole? And whence could it have come? | |
From: Empedocles (fragments/reports [c.453 BCE], B017), quoted by Simplicius - On Aristotle's 'Physics' 158.1- |
5090 | Maybe bodies are designed by accident, and the creatures that don't work are destroyed [Empedocles, by Aristotle] |
Full Idea: Is it just an accident that teeth and other parts of the body seem to have some purpose, and creatures survive because they happen to be put together in a useful way? Everything else has been destroyed, as Empedocles says of his 'cow with human head'. | |
From: report of Empedocles (fragments/reports [c.453 BCE], 61) by Aristotle - Physics 198b29 | |
A reaction: Good grief! Has no one ever noticed that Empedocles proposed the theory of evolution? It isn't quite natural selection, because we aren't told what does the 'destroying', but it is a little flash of genius that was quietly forgotten. |
461 | God is a pure, solitary, and eternal sphere [Empedocles] |
Full Idea: God is equal in all directions to himself and altogether eternal, a rounded Sphere enjoying a circular solitude. | |
From: Empedocles (fragments/reports [c.453 BCE], B028), quoted by John Stobaeus - Anthology 1.15.2 |
466 | God is pure mind permeating the universe [Empedocles] |
Full Idea: God is mind, holy and ineffable, and only mind, which darts through the whole cosmos with its swift thought. | |
From: Empedocles (fragments/reports [c.453 BCE], B134), quoted by Ammonius - On 'De Interpretatione' 4.5.249.6 |
1719 | In Empedocles' theory God is ignorant because, unlike humans, he doesn't know one of the elements (strife) [Aristotle on Empedocles] |
Full Idea: It is a consequence of Empedocles' view that God is the most unintelligent thing, for he alone is ignorant of one of the elements, namely strife, whereas mortal creatures are familiar with them all. | |
From: comment on Empedocles (fragments/reports [c.453 BCE]) by Aristotle - De Anima 410b08 |
1522 | It is wretched not to want to think clearly about the gods [Empedocles] |
Full Idea: Wretched is he who cares not for clear thinking about the gods. | |
From: Empedocles (fragments/reports [c.453 BCE], B132), quoted by Clement - Miscellanies 5.140.5.1 |