33 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'. |
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. |
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) |
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. |
16062 | A necessary relation between fact-levels seems to be a further irreducible fact [Lynch/Glasgow] |
Full Idea: It seems unavoidable that the facts about logically necessary relations between levels of facts are themselves logically distinct further facts, irreducible to the microphysical facts. | |
From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], C) | |
A reaction: I'm beginning to think that rejecting every theory of reality that is proposed by carefully exposing some infinite regress hidden in it is a rather lazy way to do philosophy. Almost as bad as rejecting anything if it can't be defined. |
16061 | If some facts 'logically supervene' on some others, they just redescribe them, adding nothing [Lynch/Glasgow] |
Full Idea: Logical supervenience, restricted to individuals, seems to imply strong reduction. It is said that where the B-facts logically supervene on the A-facts, the B-facts simply re-describe what the A-facts describe, and the B-facts come along 'for free'. | |
From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], C) | |
A reaction: This seems to be taking 'logically' to mean 'analytically'. Presumably an entailment is logically supervenient on its premisses, and may therefore be very revealing, even if some people think such things are analytic. |
16060 | Nonreductive materialism says upper 'levels' depend on lower, but don't 'reduce' [Lynch/Glasgow] |
Full Idea: The root intuition behind nonreductive materialism is that reality is composed of ontologically distinct layers or levels. …The upper levels depend on the physical without reducing to it. | |
From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], B) | |
A reaction: A nice clear statement of a view which I take to be false. This relationship is the sort of thing that drives people fishing for an account of it to use the word 'supervenience', which just says two things seem to hang out together. Fluffy materialism. |
16064 | The hallmark of physicalism is that each causal power has a base causal power under it [Lynch/Glasgow] |
Full Idea: Jessica Wilson (1999) says what makes physicalist accounts different from emergentism etc. is that each individual causal power associated with a supervenient property is numerically identical with a causal power associated with its base property. | |
From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], n 11) | |
A reaction: Hence the key thought in so-called (serious, rather than self-evident) 'emergentism' is so-called 'downward causation', which I take to be an idle daydream. |
7096 | We may still admire a person's character even if the traits are involuntary [Statman] |
Full Idea: If we focus on the evaluation of character traits, voluntariness becomes less important. We would not withdraw our admiration for a person only because we found out that his or her being such a person was not a result of voluntary choice. | |
From: Daniel Statman (Introduction to Virtue Ethics [1997], §3) | |
A reaction: The need for voluntariness does not disappear. I would not admire the only generous deed you had ever performed if it was the result of hypnotism. I might admire the hypnotist. Nevertheless, I regard this idea as a crucial truth in moral theory. |
7098 | There is a new sort of moral scepticism, about the possibility of moral theories [Statman] |
Full Idea: Since the 1980s, ethics has witnessed a new sort of moral scepticism, this time about the possibility of moral theories. | |
From: Daniel Statman (Introduction to Virtue Ethics [1997], §4) | |
A reaction: He cites McDowell, Williams, Nussbaum and Baier as the culprits. 'Particularism' (every situation is different, so there can't be rules) seems an essential part of virtue theory, but total absence of principles sounds to me like moral drift. |
7099 | With a broad concept of flourishing, it might be possible without the virtues [Statman] |
Full Idea: In a rich conception of human flourishing, both individuals and societies seem to be able to flourish without the virtues. | |
From: Daniel Statman (Introduction to Virtue Ethics [1997], §5) | |
A reaction: I can see Aristotle clutching his head in despair at this thought. It might look like flourishing, but it couldn't be the real thing. It is Aristotle's fault, though, for including external goods. Money and pleasure offer a kind of flourishing. |
7100 | Virtue theory isn't a genuine ethical theory, because it doesn't have universal application [Statman] |
Full Idea: It can be claimed that universality is a necessary property of any ethical theory and therefore virtue theory, which fails in this respect, is not a theory, and hence poses no alternative to genuine ethical theories. | |
From: Daniel Statman (Introduction to Virtue Ethics [1997], §5) | |
A reaction: Replies: a) totally universal morality is an idle dream (part of the 'Enlightenment Project' to prove everything) and we must settle for something more relative; b) virtues aren't totally universal, but they are truths about humanity. I prefer b). |
7102 | Promises create moral duties that have nothing to do with character [Statman] |
Full Idea: That duties are created irrespective of facts about character is obvious from the case of promises, which bind their makers irrespective of their motives or personality. | |
From: Daniel Statman (Introduction to Virtue Ethics [1997], §5) | |
A reaction: Just occasionally a promise can be broken, by a sensitive and wise person. I promise to give your son some money; I then discover he is a drug dealer. Promises arise out of character, and cannot be made by robots. |
7095 | Moral education is better by concrete example than abstract principle [Statman] |
Full Idea: According to virtue theory, education through moral exemplars is more effective than education focused on principles and obligations, because it is far more concrete. | |
From: Daniel Statman (Introduction to Virtue Ethics [1997], §3) | |
A reaction: Aristotle's view is that virtues must be developed from childhood, when principles don't mean much. The problem is that young people may witness highly virtuous behaviour in their exemplars, but totally fail to appreciate it without mention of principles. |
7094 | Friends express friendship even when no utility is involved [Statman] |
Full Idea: Being a good friend means acting in ways that express the friendship even when those ways do not promote overall utility. | |
From: Daniel Statman (Introduction to Virtue Ethics [1997], §3) | |
A reaction: This implies that friendship is a true virtue of character, rather than having friends just being an 'external' good. Having friends is good; being friends is a virtue. There are duties of friendship. |
7104 | The ancients recognised imperfect duties, but we have added perfect duties like justice [Statman] |
Full Idea: The advantage of modern thinkers over the ancient virtue ethicists is that in addition to imperfect duties (i.e. virtues) they also recognise the existence of perfect duties, or duties of justice, which are essential for the existence of society. | |
From: Daniel Statman (Introduction to Virtue Ethics [1997], §7) | |
A reaction: Even the Greeks had laws (e.g. Idea 422), so they understood that a society needs rules, but many laws don't seem to be moral rules (e.g. car parking), and the Greeks thought morality was about human excellence, not avoiding traffic jams. |
7093 | Behaviour may be disgusting or inhumane, but violate no duty [Statman] |
Full Idea: It is surely possible, and indeed often the case, that people who violate no duty nevertheless behave in an inhumane and a disgusting manner. | |
From: Daniel Statman (Introduction to Virtue Ethics [1997], §1) | |
A reaction: This seems right, though it is easier to be disgusting than to be inhumane if no duty is to be violated. Social duties may not require a high degree of humanity, pure Kantian duties might. |
7103 | Abortion issues focus on the mother's right over her body, and the status of the foetus [Statman] |
Full Idea: Most of the debate on abortion focuses on two issues, the mother's assumed right over her body, and the status of the foetus. | |
From: Daniel Statman (Introduction to Virtue Ethics [1997], §6) | |
A reaction: Personally I think society as a whole might have a say (if, perhaps, we are over- or under-populated, or we have a widely accepted state religion, or we are just very shocked). Mother's have virtues and duties as well as rights. |