33 ideas
| 6253 | Reason is our power of finding out true propositions [Hutcheson] |
| Full Idea: Reason is our power of finding out true propositions. | |
| From: Francis Hutcheson (Treatise 4: The Moral Sense [1728], §I) | |
| A reaction: This strikes me as a very good definition. I don't see how you can define reason without mentioning truth, and you can't believe in reason if you don't believe in truth. The concept of reason entails the concept of a good reason. |
| 8368 | A correct definition is what can be substituted without loss of meaning [Ducasse] |
| Full Idea: A definition of a word is correct if the definition can be substituted for the word being defined in an assertion without in the least changing the meaning which the assertion is felt to have. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], §1) | |
| A reaction: This sounds good, but a very bland and uninformative rephrasing would fit this account, without offering anything very helpful. The word 'this' could be substituted for a lot of object words. A 'blade' is 'a thing always attached to a knife handle'. |
| 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). |
| 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. |
| 6256 | Can't the moral sense make mistakes, as the other senses do? [Hutcheson] |
| Full Idea: Can there not be a right and wrong state of our moral sense, as there is in our other senses? | |
| From: Francis Hutcheson (Treatise 4: The Moral Sense [1728], §IV) | |
| A reaction: Hutcheson replies by saying something like they are both fully reliable in normal conditions. It remains, though, a very good question for the intuitionist to face, as the moral sense is supposed to be direct and reliable, but how do you check? |
| 6252 | Happiness is a pleasant sensation, or continued state of such sensations [Hutcheson] |
| Full Idea: In the following discourse, happiness denotes pleasant sensation of any kind, or continued state of such sensations. | |
| From: Francis Hutcheson (Treatise 4: The Moral Sense [1728], Intro) | |
| A reaction: This is a very long way from Greek eudaimonia. Hutcheson seems to imply that I would be happy if I got high on drugs after my family had just burnt to death. Socrates points out that scratching an itch is a very pleasant sensation (Idea 132). |
| 6257 | You can't form moral rules without an end, which needs feelings and a moral sense [Hutcheson] |
| Full Idea: What rule of actions can be formed, without relation to some end proposed? Or what end can be proposed, without presupposing instincts, desires, affections, or a moral sense, it will not be easy to explain. | |
| From: Francis Hutcheson (Treatise 4: The Moral Sense [1728], §IV) | |
| A reaction: We have no reason to think that 'instincts, desires and affections' will give us the remotest guidance on how to behave morally well (though we would expect them to aid our survival). How could a moral sense give a reason, without spotting a rule? |
| 8367 | Causation is defined in terms of a single sequence, and constant conjunction is no part of it [Ducasse] |
| Full Idea: The correct definition of the causal relation is to be framed in terms of one single case of sequence, and constancy of conjunction is therefore no part of it. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], Intro) | |
| A reaction: This is the thesis of Ducasse's paper. I immediately warm to it. I take constant conjunction to be a consequence and symptom of causation, not its nature. There is a classic ontology/epistemology confusion to be avoided here. |
| 8372 | We see what is in common between causes to assign names to them, not to perceive them [Ducasse] |
| Full Idea: The part of a generalization concerning what is common to one individual concrete event and the causes of certain other events of the same kind is involved in the mere assigning of a name to the cause and its effect, but not in the perceiving them. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], §5) | |
| A reaction: A nice point, that we should keep distinct the recognition of a cause, and the assigning of a general name to it. Ducasse is claiming that we can directly perceive singular causation. |
| 8369 | Causes are either sufficient, or necessary, or necessitated, or contingent upon [Ducasse] |
| Full Idea: There are four causal connections: an event is sufficient for another if it is its cause; an event is necessary for another if it is a condition for it; it is necessitated by another if it is an effect; it is contingent upon another if it is a resultant. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], §2) | |
| A reaction: An event could be a condition for another without being necessary. He seems to have missed the indispensable aspect of a necessary condition. |
| 8373 | When a brick and a canary-song hit a window, we ignore the canary if we are interested in the breakage [Ducasse] |
| Full Idea: If a brick and the song of a canary strike a window, which breaks....we can truly say that the song of the canary had nothing to do with it, that is, in so far as what occurred is viewed merely as a case of breakage of window. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], §5) | |
| A reaction: This is the germ of Davidson's view, that causation is entirely dependent on the mode of description, rather than being an actual feature of reality. If one was interested in the sound of the breakage, the canary would become relevant. |
| 8370 | A cause is a change which occurs close to the effect and just before it [Ducasse] |
| Full Idea: The cause of the particular change K was such particular change C as alone occurred in the immediate environment of K immediately before. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], §3) | |
| A reaction: The obvious immediately difficulty would be overdetermination, as when it rains while I am watering my garden. The other problem would coincidence, as when I clap my hands just before a bomb goes off. |
| 8371 | Recurrence is only relevant to the meaning of law, not to the meaning of cause [Ducasse] |
| Full Idea: The supposition of recurrence is wholly irrelevant to the meaning of cause: that supposition is relevant only to the meaning of law. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], §4) | |
| A reaction: This sounds plausible, especially if our notion of laws of nature is built up from a series of caused events. But we could just have an ontology of 'similar events', out of which we build laws, and 'causation' could drop out (á la Russell). |
| 8374 | We are interested in generalising about causes and effects purely for practical purposes [Ducasse] |
| Full Idea: We are interested in causes and effects primarily for practical purposes, which needs generalizations; so the interest of concrete individual facts of causation is chiefly an indirect one, as raw material for generalizations. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], §6) | |
| A reaction: A nice explanation of why, if causation is fundamentally about single instances, people seem so interested in generalisations and laws. We want to predict, and we want to explain, and we want to intervene. |
| 6254 | We are asked to follow God's ends because he is our benefactor, but why must we do that? [Hutcheson] |
| Full Idea: The reasons assigned for actions are such as 'It is the end proposed by the Deity'. But why do we approve concurring with the divine ends? The reason is given 'He is our benefactor', but then, for what reason do we approve concurrence with a benefactor? | |
| From: Francis Hutcheson (Treatise 4: The Moral Sense [1728], §I) | |
| A reaction: Characteristic of what MacIntyre calls the 'Enlightenment Project', which is the application of Cartesian scepticism to proving the foundations of morals. Proof beyond proof is continually demanded. If you could meet God, you would obey without question. |
| 6255 | Why may God not have a superior moral sense very similar to ours? [Hutcheson] |
| Full Idea: Why may not the Deity have something of a superior kind, analogous to our moral sense, essential to him? | |
| From: Francis Hutcheson (Treatise 4: The Moral Sense [1728], §I) | |
| A reaction: This is Plato's notion of the gods, as beings who are profoundly wise, and understand all the great moral truths, but are not the actual originators of those truths. The idea that God creates morality actually serves to undermine morality. |