39 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'. |
| 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. |
| 11897 | A principle of individuation may pinpoint identity and distinctness, now and over time [Mackie,P] |
| Full Idea: One view of a principle of individuation is what is called a 'criterion of identity', determining answers to questions about identity and distinctness at a time and over time - a principle of distinction and persistence. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 8.2) | |
| A reaction: Since the term 'Prime Minister' might do this job, presumably there could be a de dicto as well as a de re version of individuation. The distinctness consists of chairing cabinet meetings, rather than being of a particular sex. |
| 11898 | Individuation may include counterfactual possibilities, as well as identity and persistence [Mackie,P] |
| Full Idea: A second view of the principle of individuation includes criteria of distinction and persistence, but also determines the counterfactual possibilities for a thing. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 8.5) | |
| A reaction: It would be a pretty comprehensive individuation which defined all the counterfactual truths about a thing, as well as its actual truths. This is where powers come in. We need to know a thing's powers, but not how they cash out counterfactually. |
| 11883 | A haecceity is the essential, simple, unanalysable property of being-this-thing [Mackie,P] |
| Full Idea: Socrates can be assigned a haecceity: an essential property of 'being Socrates' which (unlike the property of 'being identical with Socrates') may be regarded as what 'makes' its possessor Socrates in a non-trivial sense, but is simple and unanalysable. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 2.2) | |
| A reaction: I don't accept that there is any such property as 'being Socrates' (or even 'being identical with Socrates'), except as empty locutions or logical devices. A haecceity seems to be the 'ultimate subject of predication', with no predicates of its own. |
| 11889 | Essentialism must avoid both reduplication of essences, and multiple occupancy by essences [Mackie,P] |
| Full Idea: The argument for unshareable properties (the Reduplication Argument) suggests the danger of reduplication of Berkeley; the argument for incompatible properties (Multiple Occupancy) says Berkeley and Hume could be in the same possible object. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 2.8) | |
| A reaction: These are her arguments in favour of essential properties being necessarily incompatible between objects. Whatever the answer, it must allow essences for indistinguishables like electrons. 'Incompatible' points towards a haecceity. |
| 11877 | An individual essence is the properties the object could not exist without [Mackie,P] |
| Full Idea: By essentialism about individuals I simply mean the view that individual things have essential properties, where an essential property of an object is a property that the object could not have existed without. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 1.1) | |
| A reaction: This presumably means I could exist without a large part of my reason and consciousness, but could not exist without one of my heart valves. This seems to miss the real point of essence. I couldn't exist without oxygen - not one of my properties. |
| 11882 | No other object can possibly have the same individual essence as some object [Mackie,P] |
| Full Idea: Individual essences are essential properties that are unique to them alone. ...If a set of properties is an individual essence of A, then A has the properties essentially, and no other actual or possible object actually or possibly has them. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 2.1/2) | |
| A reaction: I'm unconvinced about this. Tigers have an essence, but individual tigers have individual essences over and above their tigerish qualities, yet the perfect identity of two tigers still seems to be possible. |
| 11886 | There are problems both with individual essences and without them [Mackie,P] |
| Full Idea: If all objects had individual essences, there would be no numerical difference without an essential difference. But if there aren't individual essences, there could be two things sharing all essential properties, differing only in accidental properties. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 2.5) | |
| A reaction: Depends how you define individual essence. Why can't two electrons have the same individual essence. To postulate a 'kind essence' which bestows the properties on each electron is to get things the wrong way round. |
| 11909 | Unlike Hesperus=Phosophorus, water=H2O needs further premisses before it is necessary [Mackie,P] |
| Full Idea: There is a disanalogy between 'necessarily water=H2O' and 'necessarily Hesperus=Phosphorus'. The second just needs the necessity of identity, but the first needs 'x is a water sample' and 'x is an H2O' sample to coincide in all possible worlds. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 10.1.) | |
| A reaction: This comment is mainly aimed at Kripke, who bases his essentialism on identities, rather than at Putnam. |
| 11899 | Why are any sortals essential, and why are only some of them essential? [Mackie,P] |
| Full Idea: Accounts of sortal essentialism do not give a satisfactory explanation of why any sortals should be essential sortals, or a satisfactory account of why some sortals should be essential while others are not. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 8.6) | |
| A reaction: A theory is not wrong, just because it cannot give a 'satisfactory explanation' of every aspect of the subject. We might, though, ask why the theory isn't doing well in this area. |
| 11906 | The Kripke and Putnam view of kinds makes them explanatorily basic, but has modal implications [Mackie,P] |
| Full Idea: Kripke and Putnam chose for their typical essence of kinds, sets of properties that could be thought of as explanatorily basic. ..But the modal implications of their views go well beyond this. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 10.1) | |
| A reaction: Cf. Idea 11905. The modal implications are that the explanatory essence is also necessary to the identity of the thing under discussion, such as H2O. So do basic explanations carry across into all possible worlds? |
| 11894 | Origin is not a necessity, it is just 'tenacious'; we keep it fixed in counterfactual discussions [Mackie,P] |
| Full Idea: I suggest 'tenacity of origin' rather than 'necessity of origin'. ..The most that we need is that Caesar's having something similar to his actual origin in certain respects (e.g. his actual parents) is normally kept fixed in counterfactual speculation. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 6.9) | |
| A reaction: I find necessity or essentially of origin very unconvincing, so I rather like this. Origin is just a particularly stable way to establish our reference to something. An elusive spy may have little more than date and place of birth to fix them. |
| 11887 | Transworld identity without individual essences leads to 'bare identities' [Mackie,P] |
| Full Idea: Transworld identity without individual essences leads to 'bare identities'. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 2.7) | |
| A reaction: [She gives an argument for this, based on Forbes] I certainly favour the notion of individual essences over the notion of bare identities. We must distinguish identity in reality from identity in concept. Identities are points in conceptual space. |
| 11890 | De re modality without bare identities or individual essence needs counterparts [Mackie,P] |
| Full Idea: Anyone who wishes to avoid both bare identities and individual essences, without abandoning de re modality entirely, must adopt counterpart theory. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 4.1) | |
| A reaction: This at least means that Lewis's proposal has an important place in the discussion, forcing us to think more clearly about the identities involved when we talk of possibilities. Mackie herself votes for bare indentities. |
| 11892 | Things may only be counterparts under some particular relation [Mackie,P] |
| Full Idea: A may be a counterpart of B according to one counterpart relation (similarity of origin, say), but not according to another (similarity of later history). | |
| From: Penelope Mackie (How Things Might Have Been [2006], 5.3) | |
| A reaction: Hm. Would two very diverse things have to be counterparts because they were kept in the same cupboard in different worlds? Can the counterpart relationship diverge or converge over time? Yes, I presume. |
| 11893 | Possibilities for Caesar must be based on some phase of the real Caesar [Mackie,P] |
| Full Idea: I take the 'overlap requirement' for Julius Caesar to be that, when considering how he might have been different, you have to take him as he actually was at some time in his existence, and consider possibilities consistent with that. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 6.5) | |
| A reaction: This is quite a large claim (larger than Mackie thinks?), as it seems equally applicable to properties, states of affairs and propositions, as well as to individuals. Possibility that has no contact at all with actuality is beyond our comprehension. |
| 11884 | The theory of 'haecceitism' does not need commitment to individual haecceities [Mackie,P] |
| Full Idea: The theory that things have 'haecceities' must be sharply distinguished from the theory referred to as 'haecceitism', which says there may be differences in transworld identities that do not supervene on qualitative differences. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 2.2 n7) | |
| A reaction: She says later [p,43 n] that it is possible to be a haecceitist without believing in individual haecceities, if (say) the transworld identities had no basis at all. Note that if 'thisness' is 'haecceity', then 'whatness' is 'quiddity'. |
| 11905 | Locke's kind essences are explanatory, without being necessary to the kind [Mackie,P] |
| Full Idea: One might speak of 'Lockean real essences' of a natural kind, a set of properties that is basic in the explanation of the other properties of the kind, without commitment to the essence belonging to the kind in all possible worlds. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 10.1) | |
| A reaction: I think this may be the most promising account. The essence of a tiger explains what tigers are like, but tigers may evolve into domestic pets. Questions of individuation and of explaining seem to be quite separate. |
| 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 |
| 11907 | Maybe the identity of kinds is necessary, but instances being of that kind is not [Mackie,P] |
| Full Idea: One could be an essentialist about natural kinds (of tigers, or water) while holding that every actual instance or sample of a natural kind is only accidentally an instance or a sample of that kind. | |
| From: Penelope Mackie (How Things Might Have Been [2006], 10.2) | |
| A reaction: You wonder, then, in what the necessity of the kind consists, if it is not rooted in the instances, and presumably it could only result from a stipulative definition, and hence be conventional. |
| 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. |