20 ideas
| 10017 | Truth in a model is more tractable than the general notion of truth [Hodes] |
| Full Idea: Truth in a model is interesting because it provides a transparent and mathematically tractable model - in the 'ordinary' rather than formal sense of the term 'model' - of the less tractable notion of truth. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |
| A reaction: This is an important warning to those who wish to build their entire account of truth on Tarski's rigorously formal account of the term. Personally I think we should start by deciding whether 'true' can refer to the mental state of a dog. I say it can. |
| 10018 | Truth is quite different in interpreted set theory and in the skeleton of its language [Hodes] |
| Full Idea: There is an enormous difference between the truth of sentences in the interpreted language of set theory and truth in some model for the disinterpreted skeleton of that language. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.132) | |
| A reaction: This is a warning to me, because I thought truth and semantics only entered theories at the stage of 'interpretation'. I must go back and get the hang of 'skeletal' truth, which sounds rather charming. [He refers to set theory, not to logic.] |
| 10397 | Abelard's mereology involves privileged and natural divisions, and principal parts [Abelard, by King,P] |
| Full Idea: Abelard's theory of substantial integral wholes is not a pure mereology in the modern sense, since he holds that there are privileged divisions; ..the division of a whole must be into its principal parts. Some wholes have a natural division. | |
| From: report of Peter Abelard (works [1135]) by Peter King - Peter Abelard 2 | |
| A reaction: This is a mereology that cuts nature at the joints, rather than Lewis's 'unrestricted composition', so I find Abelard rather appealing. |
| 10015 | Higher-order logic may be unintelligible, but it isn't set theory [Hodes] |
| Full Idea: Brand higher-order logic as unintelligible if you will, but don't conflate it with set theory. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |
| A reaction: [he gives Boolos 1975 as a further reference] This is simply a corrective, because the conflation of second-order logic with set theory is an idea floating around in the literature. |
| 10011 | Identity is a level one relation with a second-order definition [Hodes] |
| Full Idea: Identity should he considered a logical notion only because it is the tip of a second-order iceberg - a level 1 relation with a pure second-order definition. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) |
| 10016 | When an 'interpretation' creates a model based on truth, this doesn't include Fregean 'sense' [Hodes] |
| Full Idea: A model is created when a language is 'interpreted', by assigning non-logical terms to objects in a set, according to a 'true-in' relation, but we must bear in mind that this 'interpretation' does not associate anything like Fregean senses with terms. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |
| A reaction: This seems like a key point (also made by Hofweber) that formal accounts of numbers, as required by logic, will not give an adequate account of the semantics of number-terms in natural languages. |
| 10027 | Mathematics is higher-order modal logic [Hodes] |
| Full Idea: I take the view that (agreeing with Aristotle) mathematics only requires the notion of a potential infinity, ...and that mathematics is higher-order modal logic. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) | |
| A reaction: Modern 'modal' accounts of mathematics I take to be heirs of 'if-thenism', which seems to have been Russell's development of Frege's original logicism. I'm beginning to think it is right. But what is the subject-matter of arithmetic? |
| 10026 | Arithmetic must allow for the possibility of only a finite total of objects [Hodes] |
| Full Idea: Arithmetic should be able to face boldly the dreadful chance that in the actual world there are only finitely many objects. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.148) | |
| A reaction: This seems to be a basic requirement for any account of arithmetic, but it was famously a difficulty for early logicism, evaded by making the existence of an infinity of objects into an axiom of the system. |
| 10021 | It is claimed that numbers are objects which essentially represent cardinality quantifiers [Hodes] |
| Full Idea: The mathematical object-theorist says a number is an object that represents a cardinality quantifier, with the representation relation as the entire essence of the nature of such objects as cardinal numbers like 4. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) | |
| A reaction: [compressed] This a classic case of a theory beginning to look dubious once you spell it our precisely. The obvious thought is to make do with the numerical quantifiers, and dispense with the objects. Do other quantifiers need objects to support them? |
| 10022 | Numerical terms can't really stand for quantifiers, because that would make them first-level [Hodes] |
| Full Idea: The dogmatic Frege is more right than wrong in denying that numerical terms can stand for numerical quantifiers, for there cannot be a language in which object-quantifiers and objects are simultaneously viewed as level zero. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.142) | |
| A reaction: Subtle. We see why Frege goes on to say that numbers are level zero (i.e. they are objects). We are free, it seems, to rewrite sentences containing number terms to suit whatever logical form appeals. Numbers are just quantifiers? |
| 10023 | Talk of mirror images is 'encoded fictions' about real facts [Hodes] |
| Full Idea: Talk about mirror images is a sort of fictional discourse. Statements 'about' such fictions are not made true or false by our whims; rather they 'encode' facts about the things reflected in mirrors. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.146) | |
| A reaction: Hodes's proposal for how we should view abstract objects (c.f. Frege and Dummett on 'the equator'). The facts involved are concrete, but Hodes is offering 'encoding fictionalism' as a linguistic account of such abstractions. He applies it to numbers. |
| 10396 | If 'animal' is wholly present in Socrates and an ass, then 'animal' is rational and irrational [Abelard, by King,P] |
| Full Idea: Abelard argued that if the universal 'animal' were completely present in both Socrates and an ass, making each wholly an animal, then the same thing, animal, will be simultaneously rational and irrational, with contraries present in the whole thing. | |
| From: report of Peter Abelard (works [1135]) by Peter King - Peter Abelard 2 | |
| A reaction: If we have universals for rationality and irrationality, they can distinguish the two. But we must also say that rationality is not an aspect of animal, which seems to mean that mind isn't either. What is the essence of an animal? Not reason? |
| 10395 | Abelard was an irrealist about virtually everything apart from concrete individuals [Abelard, by King,P] |
| Full Idea: Abelard was an irrealist about universals, but also about propositions, events, times other than the present, natural kinds, relations, wholes, absolute space, hylomorphic composites, and the like. The concrete individual is enough to populate the world. | |
| From: report of Peter Abelard (works [1135]) by Peter King - Peter Abelard 2 | |
| A reaction: If a Nominalist claims that 'only particulars exist', this makes him an extreme nominalist, and remarkably materialistic for his time (though he accepted the soul, as well as God). |
| 15384 | Only words can be 'predicated of many'; the universality is just in its mode of signifying [Abelard, by Panaccio] |
| Full Idea: Abelard concluded that only words can be 'predicated of many'. A universal is nothing but a general linguistic predicate, and its universality depends not on its mode of being, but on its mode of signifying. | |
| From: report of Peter Abelard (works [1135]) by Claude Panaccio - Medieval Problem of Universals 'Peter' | |
| A reaction: Abelard seems to be the originator of what is now called Predicate Nominalism, with Nelson Goodman as his modern representative. If it is just words, is there no fact of two things having the 'same' property? |
| 8481 | The de dicto-de re modality distinction dates back to Abelard [Abelard, by Orenstein] |
| Full Idea: The de dicto-de re modality distinction dates back to Abelard. | |
| From: report of Peter Abelard (works [1135]) by Alex Orenstein - W.V. Quine Ch.7 | |
| A reaction: Most modern philosophers couldn't (apparently) care less where a concept originated, but one of the principles of this database is that such things do matter. I'm not sure why, but if we want the whole picture, we need the historical picture. |
| 15385 | Abelard's problem is the purely singular aspects of things won't account for abstraction [Panaccio on Abelard] |
| Full Idea: Abelard's problem is that it is not clear how singular forms could do the job they are supposed to do - to account for abstraction, namely - if they were purely singular aspects. | |
| From: comment on Peter Abelard (works [1135]) by Claude Panaccio - Medieval Problem of Universals 'Peter' | |
| A reaction: A very nice question! If we say that abstracta are just acquired by ignoring all but that feature in some objects, how do we identify 'that' feature in order to select it? The instances must share something in common to be abstracted. |
| 15383 | Nothing external can truly be predicated of an object [Abelard, by Panaccio] |
| Full Idea: Abelard argued from the commonly accepted definition of a universal as 'what can be predicated of man', that no external thing can ever be predicated of anything. | |
| From: report of Peter Abelard (works [1135]) by Claude Panaccio - Medieval Problem of Universals 'Peter' | |
| A reaction: It sounds to me as if Abelard is confusing predicates with properties! Maybe no external can be a property of anything, but I take predicates to just be part of what you can say about anything, and that had better included external facts. |
| 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 |
| 10398 | Natural kinds are not special; they are just well-defined resemblance collections [Abelard, by King,P] |
| Full Idea: In Abelard's view a natural kind is a well-defined collection of things that have the same features, so that natural kinds have no special status, being no more than discrete integral wholes whose principle of membership is similarity. | |
| From: report of Peter Abelard (works [1135]) by Peter King - Peter Abelard 2 | |
| A reaction: I take a natural kind to be a completely stable and invariant class of things. Presumably this invariance has an underlying explanation, but Abelard seems to take the Humean line that we cannot penetrate beyond the experienced surface. |
| 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. |