47 ideas
| 22289 | Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic [Dedekind, by Potter] |
| Full Idea: Dedkind gave a rigorous proof of the principle of definition by recursion, permitting recursive definitions of addition and multiplication, and hence proofs of the familiar arithmetical laws. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 13 'Deriv' |
| 13252 | Some truths have true negations [Beall/Restall] |
| Full Idea: Dialetheism is the view that some truths have true negations. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 7.4) | |
| A reaction: The important thing to remember is that they are truths. Thus 'Are you feeling happy?' might be answered 'Yes and no'. |
| 13247 | A truthmaker is an object which entails a sentence [Beall/Restall] |
| Full Idea: The truthmaker thesis is that an object is a truthmaker for a sentence if and only if its existence entails the sentence. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 5.5.3) | |
| A reaction: The use of the word 'object' here is even odder than usual, and invites many questions. And the 'only if' seems peculiar, since all sorts of things can make a sentence true. 'There is someone in the house' for example. |
| 13249 | (∀x)(A v B) |- (∀x)A v (∃x)B) is valid in classical logic but invalid intuitionistically [Beall/Restall] |
| Full Idea: The inference of 'distribution' (∀x)(A v B) |- (∀x)A v (∃x)B) is valid in classical logic but invalid intuitionistically. It is straightforward to construct a 'stage' at which the LHS is true but the RHS is not. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 6.1.2) | |
| A reaction: This seems to parallel the iterative notion in set theory, that you must construct your hierarchy. All part of the general 'constructivist' approach to things. Is some kind of mad platonism the only alternative? |
| 13243 | Excluded middle must be true for some situation, not for all situations [Beall/Restall] |
| Full Idea: Relevant logic endorses excluded middle, ..but says instances of the law may fail. Bv¬B is true in every situation that settles the matter of B. It is necessary that there is some such situation. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 5.2) | |
| A reaction: See next idea for the unusual view of necessity on which this rests. It seems easier to assert something about all situations than just about 'some' situation. |
| 13242 | It's 'relevantly' valid if all those situations make it true [Beall/Restall] |
| Full Idea: The argument from P to A is 'relevantly' valid if and only if, for every situation in which each premise in P is true, so is A. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 5.2) | |
| A reaction: I like the idea that proper inference should have an element of relevance to it. A falsehood may allow all sorts of things, without actually implying them. 'Situations' sound promising here. |
| 13246 | Relevant logic does not abandon classical logic [Beall/Restall] |
| Full Idea: We have not abandoned classical logic in our acceptance of relevant logic. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 5.4) | |
| A reaction: It appears that classical logic is straightforwardly accepted, but there is a difference of opinion over when it is applicable. |
| 13245 | Relevant consequence says invalidity is the conclusion not being 'in' the premises [Beall/Restall] |
| Full Idea: Relevant consequence says the conclusion of a relevantly invalid argument is not 'carried in' the premises - it does not follow from the premises. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 5.3.3) | |
| A reaction: I find this appealing. It need not invalidate classical logic. It is just a tougher criterion which is introduced when you want to do 'proper' reasoning, instead of just playing games with formal systems. |
| 13254 | A doesn't imply A - that would be circular [Beall/Restall] |
| Full Idea: We could reject the inference from A to itself (on grounds of circularity). | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 8) | |
| A reaction: [Martin-Meyer System] 'It's raining today'. 'Are you implying that it is raining today?' 'No, I'm SAYING it's raining today'. Logicians don't seem to understand the word 'implication'. Logic should capture how we reason. Nice proposal. |
| 13255 | Relevant logic may reject transitivity [Beall/Restall] |
| Full Idea: Some relevant logics reject transitivity, but we defend the classical view. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 8) | |
| A reaction: [they cite Neil Tennant for this view] To reject transitivity (A?B ? B?C ? A?C) certainly seems a long way from classical logic. But in everyday inference Tennant's idea seems good. The first premise may be irrelevant to the final conclusion. |
| 13250 | Free logic terms aren't existential; classical is non-empty, with referring names [Beall/Restall] |
| Full Idea: A logic is 'free' to the degree it refrains from existential import of its singular and general terms. Classical logic must have non-empty domain, and each name must denote in the domain. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 7.1) | |
| A reaction: My intuition is that logic should have no ontology at all, so I like the sound of 'free' logic. We can't say 'Pegasus does not exist', and then reason about Pegasus just like any other horse. |
| 10183 | An infinite set maps into its own proper subset [Dedekind, by Reck/Price] |
| Full Idea: A set is 'Dedekind-infinite' iff there exists a one-to-one function that maps a set into a proper subset of itself. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], §64) by E Reck / M Price - Structures and Structuralism in Phil of Maths n 7 | |
| A reaction: Sounds as if it is only infinite if it is contradictory, or doesn't know how big it is! |
| 22288 | We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Dedekind, by Potter] |
| Full Idea: Dedekind had an interesting proof of the Axiom of Infinity. He held that I have an a priori grasp of the idea of my self, and that every idea I can form the idea of that idea. Hence there are infinitely many objects available to me a priori. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], no. 66) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 12 'Numb' | |
| A reaction: Who said that Descartes' Cogito was of no use? Frege endorsed this, as long as the ideas are objective and not subjective. |
| 10706 | Dedekind originally thought more in terms of mereology than of sets [Dedekind, by Potter] |
| Full Idea: Dedekind plainly had fusions, not collections, in mind when he avoided the empty set and used the same symbol for membership and inclusion - two tell-tale signs of a mereological conception. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], 2-3) by Michael Potter - Set Theory and Its Philosophy 02.1 | |
| A reaction: Potter suggests that mathematicians were torn between mereology and sets, and eventually opted whole-heartedly for sets. Maybe this is only because set theory was axiomatised by Zermelo some years before Lezniewski got to mereology. |
| 13235 | Logic studies consequence; logical truths are consequences of everything, or nothing [Beall/Restall] |
| Full Idea: Nowadays we think of the consequence relation itself as the primary subject of logic, and view logical truths as degenerate instances of this relation. Logical truths follow from any set of assumptions, or from no assumptions at all. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 2.2) | |
| A reaction: This seems exactly right; the alternative is the study of necessities, but that may not involve logic. |
| 13238 | Syllogisms are only logic when they use variables, and not concrete terms [Beall/Restall] |
| Full Idea: According to the Peripatetics (Aristotelians), only syllogistic laws stated in variables belong to logic, and not their applications to concrete terms. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 2.5) | |
| A reaction: [from Lukasiewicz] Seems wrong. I take it there are logical relations between concrete things, and the variables are merely used to describe these relations. Variables lack the internal powers to drive logical necessities. Variables lack essence! |
| 13234 | The view of logic as knowing a body of truths looks out-of-date [Beall/Restall] |
| Full Idea: Through much of the 20th century the conception of logic was inherited from Frege and Russell, as knowledge of a body of logical truths, as arithmetic or geometry was a knowledge of truths. This is odd, and a historical anomaly. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 2.2) | |
| A reaction: Interesting. I have always taken this idea to be false. I presume logic has minimal subject matter and truths, and preferably none at all. |
| 13232 | Logic studies arguments, not formal languages; this involves interpretations [Beall/Restall] |
| Full Idea: Logic does not study formal languages for their own sake, which is formal grammar. Logic evaluates arguments, and primarily considers formal languages as interpreted. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 2.1) | |
| A reaction: Hodges seems to think logic just studies formal languages. The current idea strikes me as a much more sensible view. |
| 13241 | The model theory of classical predicate logic is mathematics [Beall/Restall] |
| Full Idea: The model theory of classical predicate logic is mathematics if anything is. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 4.2.1) | |
| A reaction: This is an interesting contrast to the claim of logicism, that mathematics reduces to logic. This idea explains why students of logic are surprised to find themselves involved in mathematics. |
| 13253 | There are several different consequence relations [Beall/Restall] |
| Full Idea: We are pluralists about logical consequence because we take there to be a number of different consequence relations, each reflecting different precisifications of the pre-theoretic notion of deductive logical consequence. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 8) | |
| A reaction: I don't see how you avoid the slippery slope that leads to daft logical rules like Prior's 'tonk' (from which you can infer anything you like). I say that nature imposes logical conquence on us - but don't ask me to prove it. |
| 13240 | A sentence follows from others if they always model it [Beall/Restall] |
| Full Idea: The sentence X follows logically from the sentences of the class K if and only if every model of the class K is also a model of the sentence X. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 3.2) | |
| A reaction: This why the symbol |= is often referred to as 'models'. |
| 13236 | Logical truth is much more important if mathematics rests on it, as logicism claims [Beall/Restall] |
| Full Idea: If mathematical truth reduces to logical truth then it is important what counts as logically true, …but if logicism is not a going concern, then the body of purely logical truths will be less interesting. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 2.2) | |
| A reaction: Logicism would only be one motivation for pursuing logical truths. Maybe my new 'Necessitism' will derive the Peano Axioms from broad necessary truths, rather than from logic. |
| 13237 | Preface Paradox affirms and denies the conjunction of propositions in the book [Beall/Restall] |
| Full Idea: The Paradox of the Preface is an apology, that you are committed to each proposition in the book, but admit that collectively they probably contain a mistake. There is a contradiction, of affirming and denying the conjunction of propositions. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 2.4) | |
| A reaction: This seems similar to the Lottery Paradox - its inverse perhaps. Affirm all and then deny one, or deny all and then affirm one? |
| 9823 | Numbers are free creations of the human mind, to understand differences [Dedekind] |
| Full Idea: Numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], Pref) | |
| A reaction: Does this fit real numbers and complex numbers, as well as natural numbers? Frege was concerned by the lack of objectivity in this sort of view. What sort of arithmetic might the Martians have created? Numbers register sameness too. |
| 10090 | Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman] |
| Full Idea: It was primarily Dedekind's accomplishment to define the integers, rationals and reals, taking only the system of natural numbers for granted. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by A.George / D.J.Velleman - Philosophies of Mathematics Intro |
| 17452 | Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck] |
| Full Idea: Dedekind and Cantor said the cardinals may be defined in terms of the ordinals: The cardinal number of a set S is the least ordinal onto whose predecessors the members of S can be mapped one-one. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 5 |
| 7524 | Order, not quantity, is central to defining numbers [Dedekind, by Monk] |
| Full Idea: Dedekind said that the notion of order, rather than that of quantity, is the central notion in the definition of number. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Ray Monk - Bertrand Russell: Spirit of Solitude Ch.4 | |
| A reaction: Compare Aristotle's nice question in Idea 646. My intuition is that quantity comes first, because I'm not sure HOW you could count, if you didn't think you were changing the quantity each time. Why does counting go in THAT particular order? Cf. Idea 8661. |
| 14131 | Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell] |
| Full Idea: Dedekind's ordinals are not essentially either ordinals or cardinals, but the members of any progression whatever. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - The Principles of Mathematics §243 | |
| A reaction: This is part of Russell's objection to Dedekind's structuralism. The question is always why these beautiful structures should actually be considered as numbers. I say, unlike Russell, that the connection to counting is crucial. |
| 14437 | Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell] |
| Full Idea: Dedekind set up the axiom that the gap in his 'cut' must always be filled …The method of 'postulating' what we want has many advantages; they are the same as the advantages of theft over honest toil. Let us leave them to others. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - Introduction to Mathematical Philosophy VII | |
| A reaction: This remark of Russell's is famous, and much quoted in other contexts, but I have seen the modern comment that it is grossly unfair to Dedekind. |
| 18094 | Dedekind says each cut matches a real; logicists say the cuts are the reals [Dedekind, by Bostock] |
| Full Idea: One view, favoured by Dedekind, is that the cut postulates a real number for each cut in the rationals; it does not identify real numbers with cuts. ....A view favoured by later logicists is simply to identify a real number with a cut. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by David Bostock - Philosophy of Mathematics 4.4 | |
| A reaction: Dedekind is the patriarch of structuralism about mathematics, so he has little interest in the existenc of 'objects'. |
| 9824 | In counting we see the human ability to relate, correspond and represent [Dedekind] |
| Full Idea: If we scrutinize closely what is done in counting an aggregate of things, we see the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, without which no thinking is possible. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], Pref) | |
| A reaction: I don't suppose it occurred to Dedekind that he was reasserting Hume's observation about the fundamental psychology of thought. Is the origin of our numerical ability of philosophical interest? |
| 9826 | A system S is said to be infinite when it is similar to a proper part of itself [Dedekind] |
| Full Idea: A system S is said to be infinite when it is similar to a proper part of itself. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], V.64) |
| 13508 | Dedekind gives a base number which isn't a successor, then adds successors and induction [Dedekind, by Hart,WD] |
| Full Idea: Dedekind's natural numbers: an object is in a set (0 is a number), a function sends the set one-one into itself (numbers have unique successors), the object isn't a value of the function (it isn't a successor), plus induction. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by William D. Hart - The Evolution of Logic 5 | |
| A reaction: Hart notes that since this refers to sets of individuals, it is a second-order account of numbers, what we now call 'Second-Order Peano Arithmetic'. |
| 18096 | Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Dedekind, by Bostock] |
| Full Idea: Dedekind's idea is that the set of natural numbers has zero as a member, and also has as a member the successor of each of its members, and it is the smallest set satisfying this condition. It is the intersection of all sets satisfying the condition. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by David Bostock - Philosophy of Mathematics 4.4 |
| 18841 | Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind] |
| Full Idea: It is Dedekind's categoricity result that convinces most of us that he has articulated our implicit conception of the natural numbers, since it entitles us to speak of 'the' domain (in the singular, up to isomorphism) of natural numbers. | |
| From: comment on Richard Dedekind (Nature and Meaning of Numbers [1888]) by Ian Rumfitt - The Boundary Stones of Thought 9.1 | |
| A reaction: The main rival is set theory, but that has an endlessly expanding domain. He points out that Dedekind needs second-order logic to achieve categoricity. Rumfitt says one could also add to the 1st-order version that successor is an ancestral relation. |
| 14130 | Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Dedekind, by Russell] |
| Full Idea: Dedekind proves mathematical induction, while Peano regards it as an axiom, ...and Peano's method has the advantage of simplicity, and a clearer separation between the particular and the general propositions of arithmetic. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - The Principles of Mathematics §241 |
| 8924 | Dedekind originated the structuralist conception of mathematics [Dedekind, by MacBride] |
| Full Idea: Dedekind is the philosopher-mathematician with whom the structuralist conception originates. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], §3 n13) by Fraser MacBride - Structuralism Reconsidered | |
| A reaction: Hellman says the idea grew naturally out of modern mathematics, and cites Hilbert's belief that furniture would do as mathematical objects. |
| 9153 | Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Dedekind, by Fine,K] |
| Full Idea: Dedekindian abstraction says mathematical objects are 'positions' in a model, while Cantorian abstraction says they are the result of abstracting on structurally similar objects. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Kit Fine - Cantorian Abstraction: Recon. and Defence §6 | |
| A reaction: The key debate among structuralists seems to be whether or not they are committed to 'objects'. Fine rejects the 'austere' version, which says that objects have no properties. Either version of structuralism can have abstraction as its basis. |
| 16661 | There are two sorts of category - referring to things, and to circumstances of things [Boethius] |
| Full Idea: Is it not now clear what the difference is between items in the categories? Some serve to refer to a thing, whereas others serve to refer to the circumstances of a thing. | |
| From: Boethius (Concerning the Trinity [c.518], Ch. 4), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 12.5 |
| 9825 | A thing is completely determined by all that can be thought concerning it [Dedekind] |
| Full Idea: A thing (an object of our thought) is completely determined by all that can be affirmed or thought concerning it. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], I.1) | |
| A reaction: How could you justify this as an observation? Why can't there be unthinkable things (even by God)? Presumably Dedekind is offering a stipulative definition, but we may then be confusing epistemology with ontology. |
| 13244 | Relevant necessity is always true for some situation (not all situations) [Beall/Restall] |
| Full Idea: In relevant logic, the necessary truths are not those which are true in every situation; rather, they are those for which it is necessary that there is a situation making them true. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 5.2) | |
| A reaction: This seems to rest on the truthmaker view of such things, which I find quite attractive (despite Merricks's assault). Always ask what is making some truth necessary. This leads you to essences. |
| 13239 | Judgement is always predicating a property of a subject [Beall/Restall] |
| Full Idea: All judgement, for Kant, is essentially the predication of some property to some subject. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 2.5) | |
| A reaction: Presumably the denial of a predicate could be a judgement, or the affirmation of ambiguous predicates? |
| 9189 | Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dedekind, by Dummett] |
| Full Idea: By applying the operation of abstraction to a system of objects isomorphic to the natural numbers, Dedekind believed that we obtained the abstract system of natural numbers, each member having only properties consequent upon its position. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Michael Dummett - The Philosophy of Mathematics | |
| A reaction: Dummett is scornful of the abstractionism. He cites Benacerraf as a modern non-abstractionist follower of Dedekind's view. There seems to be a suspicion of circularity in it. How many objects will you abstract from to get seven? |
| 9827 | We derive the natural numbers, by neglecting everything of a system except distinctness and order [Dedekind] |
| Full Idea: If in an infinite system, set in order, we neglect the special character of the elements, simply retaining their distinguishability and their order-relations to one another, then the elements are the natural numbers, created by the human mind. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], VI.73) | |
| A reaction: [compressed] This is the classic abstractionist view of the origin of number, but with the added feature that the order is first imposed, so that ordinals remain after the abstraction. This, of course, sounds a bit circular, as well as subjective. |
| 9979 | Dedekind has a conception of abstraction which is not psychologistic [Dedekind, by Tait] |
| Full Idea: Dedekind's conception is psychologistic only if that is the only way to understand the abstraction that is involved, which it is not. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by William W. Tait - Frege versus Cantor and Dedekind IV | |
| A reaction: This is a very important suggestion, implying that we can retain some notion of abstractionism, while jettisoning the hated subjective character of private psychologism, which seems to undermine truth and logic. |
| 13248 | We can rest truth-conditions on situations, rather than on possible worlds [Beall/Restall] |
| Full Idea: Situation semantics is a variation of the truth-conditional approach, taking the salient unit of analysis not to be the possible world, or some complete consistent index, but rather the more modest 'situation'. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 5.5.4) | |
| A reaction: When I read Davidson (and implicitly Frege) this is what I always assumed was meant. The idea that worlds are meant has crept in to give truth conditions for modal statements. Hence situation semantics must cover modality. |
| 13233 | Propositions commit to content, and not to any way of spelling it out [Beall/Restall] |
| Full Idea: Our talk of propositions expresses commitment to the general notion of content, without a commitment to any particular way of spelling this out. | |
| From: JC Beall / G Restall (Logical Pluralism [2006], 2.1) | |
| A reaction: As a fan of propositions I like this. It leaves open the question of whether the content belongs to the mind or the language. Animals entertain propositions, say I. |