36 ideas
| 19695 | The devil was wise as an angel, and lost no knowledge when he rebelled [Whitcomb] |
| Full Idea: The devil is evil but nonetheless wise; he was a wise angel, and through no loss of knowledge, but, rather, through some sort of affective restructuring tried and failed to take over the throne. | |
| From: Dennis Whitcomb (Wisdom [2011], 'Argument') | |
| A reaction: ['affective restructuring' indeed! philosophers- don't you love 'em?] To fail at something you try to do suggests a flaw in the wisdom. And the new regime the devil wished to introduce doesn't look like a wise regime. Not convinced. |
| 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. |
| 16665 | There are entities, and then positive 'modes', modifying aspects outside the thing's essence [Suárez] |
| Full Idea: Beyond the entities there are certain real 'modes', which are positive, and in their own right act on those entities, giving them something that is outside their whole essence as individuals existing in reality. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 7.1.17), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 13.3 | |
| A reaction: Suárez is apparently the first person to formulate a proper account of properties as 'modes' of a thing, rather than as accidents which are separate, or are wholly integrated into a thing. A typical compromise proposal in philosophy. Can modes act? |
| 16666 | A mode determines the state and character of a quantity, without adding to it [Suárez] |
| Full Idea: The inherence of quantity is called its mode, because it affects that quantity, which serves to ultimately determine the state and character of its existence, but does not add to it any new proper entity, but only modifies the preexisting entity. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 7.1.17), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 13.3 | |
| A reaction: He seems to present mode as a very active thing, like someone who gives it a coat of paint, or hammers it into a new shape. I don't see how a 'mode' can have any ontological status at all. To exist, there has to be some way to exist. |
| 16667 | Substances are incomplete unless they have modes [Suárez, by Pasnau] |
| Full Idea: In the view of Suárez, substances are radically incomplete entities that cannot exist at all until determined in various ways by things of another kind, modes. …Modes are regarded as completers for their subjects. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597]) by Robert Pasnau - Metaphysical Themes 1274-1671 13.3 | |
| A reaction: This is correct. In order to be a piece of clay it needs a shape, a mass, a colour etc. Treating clay as an object independently from its shape is a misunderstanding. |
| 17007 | Forms must rule over faculties and accidents, and are the source of action and unity [Suárez] |
| Full Idea: A form is required that, as it were, rules over all those faculties and accidents, and is the source of all actions and natural motions of such a being, and in which the whole variety of accidents and powers has its root and unity. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 15.1.7), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.4 | |
| A reaction: Pasnau emphasises that this is scholastics giving a very physical and causal emphasis to forms, which made them vulnerable to doubts among the new experiment physicists. Pasnau says forms are 'metaphysical', following Leibniz. |
| 16780 | Partial forms of leaf and fruit are united in the whole form of the tree [Suárez] |
| Full Idea: In a tree the part of the form that is in the leaf is not the same character as the part that is in the fruit., but yet they are partial forms, and apt to be united ….to compose one complete form of the whole. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 15.10.30), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 26.6 | |
| A reaction: This is a common scholastic view, the main opponent of which was Aquinas, who says each thing only has one form. Do leaves have different DNA from the bark or the fruit? Presumably not (since I only have one DNA), which supports Aquinas. |
| 16758 | The best support for substantial forms is the co-ordinated unity of a natural being [Suárez] |
| Full Idea: The most powerful arguments establishing substantial forms are based on the necessity, for the perfect constitution of a natural being, that all the faculties and operations of that being are rooted in one essential principle. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 15.10.64), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.4 | |
| A reaction: Note Idea 15756, that this stability not only applies to biological entities (the usual Aristotelian examples), but also to non-living natural kinds. We might say that the drive for survival is someone united around a single entity. |
| 16743 | We can get at the essential nature of 'quantity' by knowing bulk and extension [Suárez] |
| Full Idea: We can say that the form that gives corporeal bulk [molem] or extension to things is the essential nature of quantity. To have bulk is to expel a similar bulk from the same space. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 40.4.16), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 539 | |
| A reaction: This is one step away from asking why, once we knew the bulk and extension of the thing, we would still have any interest in trying to grasp something called its 'quantity'. |
| 16742 | We only know essences through non-essential features, esp. those closest to the essence [Suárez] |
| Full Idea: We can almost never set out the essences of things, as they are in things. Instead, we work through their connection to some non-essential feature, and we seem to succeed well enough when we spell it out through the feature closest to the essence. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 40.4.16), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 23.5 | |
| A reaction: It is a common view that with geometrical figures we can actually experience the essence itself. So has science broken through, and discerned actual essences of things? |
| 22143 | Identity does not exclude possible or imagined difference [Suárez, by Boulter] |
| Full Idea: To be really the same excludes being really other, but does not exclude being other modally or mentally. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597], 7.65) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
| A reaction: So the statue and the clay are identical, but they could become separate, or be imagined as separate. |
| 22144 | Real Essential distinction: A and B are of different natural kinds [Suárez, by Boulter] |
| Full Idea: The Real Essential distinction says if A and B are not of the same natural kind, then they are essentially distinct. This is the highest degree of distinction. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
| A reaction: Boulter says Peter is essentially distinct from a cabbage, because neither has the nature of the other. |
| 22146 | Minor Real distinction: B needs A, but A doesn't need B [Suárez, by Boulter] |
| Full Idea: The Minor Real distinction is if A can exist without B, but B ceases to exist without A. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
| A reaction: This is one-way independence. Boulter's example is Peter and Peter's actual weight. |
| 22145 | Major Real distinction: A and B have independent existences [Suárez, by Boulter] |
| Full Idea: The Major Real distinction is if A can exist in the real order without B, and B can exist in the real order without A. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
| A reaction: Boulter's example is the distinction between Peter and Paul, where their identity of kind is irrelevant. This is two-way independence. |
| 22147 | Conceptual/Mental distinction: one thing can be conceived of in two different ways [Suárez, by Boulter] |
| Full Idea: The Conceptual or Mental distinction is when A and B are actually identical but we have two different ways of conceiving them. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
| A reaction: This is the Morning and Evening Star. I bet Frege never read Suarez. This seems to be Spinoza's concept of mind/body. |
| 22148 | Modal distinction: A isn't B or its property, but still needs B [Suárez, by Boulter] |
| Full Idea: The Modal distinction is when A is not B or a property of B, but still could not possibly exist without B. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
| A reaction: Duns Scotus proposed in, Ockham rejected it, but Suarez supports it. Suarez proposes that light's dependence on the Sun is distinct from the light itself, in this 'modal' way. |
| 22149 | Scholastics assess possibility by what has actually happened in reality [Suárez, by Boulter] |
| Full Idea: The scholastic view is that Actuality is our only guide to possibility in the real order. One knows that it is possible to separate A and B if one knows that A and B have actually been separated or are separate. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
| A reaction: It may be possible to separate A and B even though it has never happened, but it is hard to see how we could know that. (But if I put my pen down where it has never been before, I know I can pick it up again, even though this has not previously happened). |
| 16682 | Other things could occupy the same location as an angel [Suárez] |
| Full Idea: An angelic substance could be penetrated by other bodies in the same location. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 40.2.21), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 15.3 | |
| A reaction: So am I co-located with an angel right now? |