28 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.] |
| 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. |
| 1553 | No perceptible object is truly straight or curved [Protagoras] |
| Full Idea: No perceptible object is geometrically straight or curved; after all, a circle does not touch a ruler at a point, as Protagoras used to say, in arguing against the geometers. | |
| From: Protagoras (fragments/reports [c.441 BCE], B07), quoted by Aristotle - Metaphysics 998a1 |
| 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. |
| 1549 | Everything that exists consists in being perceived [Protagoras] |
| Full Idea: Everything that exists consists in being perceived. | |
| From: Protagoras (fragments/reports [c.441 BCE]), quoted by Didymus the Blind - Commentary on the Psalms (frags) | |
| A reaction: A striking anticipation of Berkeley's "esse est percipi" (to be is to be perceived). |
| 1545 | Protagoras was the first to claim that there are two contradictory arguments about everything [Protagoras, by Diog. Laertius] |
| Full Idea: Protagoras was the first to claim that there are two contradictory arguments about everything. | |
| From: report of Protagoras (fragments/reports [c.441 BCE], A01) by Diogenes Laertius - Lives of Eminent Philosophers 09.51 |
| 1547 | Man is the measure of all things - of things that are, and of things that are not [Protagoras] |
| Full Idea: He began one of his books as follows: 'Man is the measure of all things - of the things that are, that they are, and of the things that are not, that they are not'. | |
| From: Protagoras (fragments/reports [c.441 BCE], B01), quoted by Diogenes Laertius - Lives of Eminent Philosophers 09.51 |
| 3305 | There is no more purely metaphysical doctrine than Protagorean relativism [Benardete,JA on Protagoras] |
| Full Idea: No purer metaphysical doctrine can possibly be found than the Protagorean thesis that to be (anything at all) is to be relative ( to something or other). | |
| From: comment on Protagoras (fragments/reports [c.441 BCE]) by José A. Benardete - Metaphysics: the logical approach Ch.3 |
| 3313 | If my hot wind is your cold wind, then wind is neither hot nor cold, and so not as cold as itself [Benardete,JA on Protagoras] |
| Full Idea: Because the wind is cold to me but not you, Protagoras takes it to in itself neither cold nor not-cold. Accordingly, I very much doubt that he can allow the wind to be exactly as cold as itself. | |
| From: comment on Protagoras (fragments/reports [c.441 BCE]) by José A. Benardete - Metaphysics: the logical approach Ch.8 |
| 3317 | You can only state the problem of the relative warmth of an object by agreeing on the underlying object [Benardete,JA on Protagoras] |
| Full Idea: Only if the thing that is cold to me is precisely identical with the thing that is not cold to you can Protagoras launch his argument, but then it is seen to be the thing in itself that exists absolutely speaking. | |
| From: comment on Protagoras (fragments/reports [c.441 BCE]) by José A. Benardete - Metaphysics: the logical approach Ch.8 |
| 247 | God is "the measure of all things", more than any man [Plato on Protagoras] |
| Full Idea: In our view it is God who is pre-eminently the "measure of all things", much more so than any "man", as they say. | |
| From: comment on Protagoras (fragments/reports [c.441 BCE]) by Plato - The Laws 716c |
| 606 | Protagoras absurdly thought that the knowing or perceiving man is 'the measure of all things' [Aristotle on Protagoras] |
| Full Idea: When Protagoras quipped that man is the measure of all things, he had in mind, of course, the knowing or perceiving man. The grounds are that they have perception/knowledge, and these are said to be the measures of objects. Utter nonsense! | |
| From: comment on Protagoras (fragments/reports [c.441 BCE]) by Aristotle - Metaphysics 1053b |
| 612 | Relativists think if you poke your eye and see double, there must be two things [Aristotle on Protagoras] |
| Full Idea: In fact there is no difference between Protagoreanism and saying this: if you stick your finger under your eyes and make single things seem two, then they are two, just because they seem to be two. | |
| From: comment on Protagoras (fragments/reports [c.441 BCE]) by Aristotle - Metaphysics 1063a06 |
| 6016 | Early sophists thought convention improved nature; later they said nature was diminished by it [Protagoras, by Miller,FD] |
| Full Idea: Protagoras and Hippias evidently believed that convention was an improvement on nature, whereas later sophists such as Antiphon, Thrasymachus and Callicles seemed to contend that conventional morality was undermined because it was 'against nature'. | |
| From: report of Protagoras (fragments/reports [c.441 BCE]) by Fred D. Miller jr - Classical Political Thought | |
| A reaction: This gets to the heart of a much more interesting aspect of the nomos-physis (convention-nature) debate, rather than just a slanging match between relativists and the rest. The debate still goes on, over issues about the free market and intervention. |
| 1580 | For Protagoras the only bad behaviour is that which interferes with social harmony [Protagoras, by Roochnik] |
| Full Idea: For Protagoras the only constraint on human behaviour is that it not interfere with social harmony, the essential condition for human survival. | |
| From: report of Protagoras (fragments/reports [c.441 BCE]) by David Roochnik - The Tragedy of Reason p.63 |
| 205 | Protagoras contradicts himself by saying virtue is teachable, but then that it is not knowledge [Plato on Protagoras] |
| Full Idea: Protagoras claimed that virtue was teachable, but now tries to show it is not knowledge, which makes it less likely to be teachable. | |
| From: comment on Protagoras (fragments/reports [c.441 BCE]) by Plato - Protagoras 361b |
| 1659 | Protagoras seems to have made the huge move of separating punishment from revenge [Protagoras, by Vlastos] |
| Full Idea: The distinction of punishment from revenge must be regarded as one of the most momentous of the conceptual discoveries ever made by humanity in the course of its slow, tortuous, precarious, emergence from barbaric tribalism. Protagoras originated it. | |
| From: report of Protagoras (fragments/reports [c.441 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.187 |
| 532 | Successful education must go deep into the soul [Protagoras] |
| Full Idea: Education does not take root in the soul unless one goes deep. | |
| From: Protagoras (fragments/reports [c.441 BCE], B11), quoted by Plutarch - On Practice 178.25 |
| 1552 | He spent public money on education, as it benefits the individual and the state [Protagoras, by Diodorus of Sicily] |
| Full Idea: He used legislation to improve the condition of illiterate people, on the grounds that they lack one of life's great goods, and thought literacy should be a matter of public concern and expense. | |
| From: report of Protagoras (fragments/reports [c.441 BCE]) by Diodorus of Sicily - Universal History 12.13.3.3 |
| 1551 | He said he didn't know whether there are gods - but this is the same as atheism [Diogenes of Oen. on Protagoras] |
| Full Idea: He said that he did not know whether there were gods - but this is the same as saying that he knew there were no gods. | |
| From: comment on Protagoras (fragments/reports [c.441 BCE], A23) by Diogenes (Oen) - Wall inscription 11 Chil 2 |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |
| Full Idea: The Egyptians were the first to claim that the soul of a human being is immortal, and that each time the body dies the soul enters another creature just as it is being born. | |
| From: Herodotus (The Histories [c.435 BCE], 2.123.2) |