89 ideas
| 1695 | Without extensive examination firm statements are hard, but studying the difficulties is profitable [Aristotle] |
| Full Idea: It is hard to make firm statements on these questions without having examined them many times, but to have gone through the various difficulties is not unprofitable. | |
| From: Aristotle (Categories [c.331 BCE], 08b23) | |
| A reaction: Suggesting that philosophy is more like drawing the map than completing the journey. |
| 16325 | Analysis rests on natural language, but its ideal is a framework which revises language [Halbach] |
| Full Idea: For me, although the enterprise of philosophical analysis is driven by natural language, its goal is not a linguistic analysis of English but rather an expressively strong framework that may at best be seen as a revision of English. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 12) | |
| A reaction: I agree, but the problem is that there are different ideals for the revision, which may be in conflict. Logicians, mathematicians, metaphysicians, scientists, moralists and aestheticians are queueing up to improve in their own way. |
| 1698 | Both sides of contraries need not exist (as health without sickness, white without black) [Aristotle] |
| Full Idea: With contraries it is not necessary if one exists for the other to exist too, for if everyone were well health would exist but not sickness, and if everything were white whiteness would exist but not black. | |
| From: Aristotle (Categories [c.331 BCE], 14a06) |
| 1697 | The contrary of good is bad, but the contrary of bad is either good or another evil [Aristotle] |
| Full Idea: What is contrary to a good thing is necessarily bad, as we see with health and sickness. But the contrary of bad is sometimes good, sometimes not, as we see with excess, opposed by both deficiency and moderation. | |
| From: Aristotle (Categories [c.331 BCE], 13b36) |
| 16292 | An explicit definition enables the elimination of what is defined [Halbach] |
| Full Idea: Explicit definitions allow for a complete elimination of the defined notion (at least in extensional contexts). | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 1) | |
| A reaction: If the context isn't extensional (concerning the things themselves) then we could define one description of it, but be unable to eliminate it under another description. Elimination is no the aim of an Aristotelian definition. Halbach refers to truth. |
| 16307 | Don't trust analogies; they are no more than a guideline [Halbach] |
| Full Idea: Arguments from analogy are to be distrusted: at best they can serve as heuristics. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 4) |
| 11034 | The differentiae of genera which are different are themselves different in kind [Aristotle] |
| Full Idea: The differentiae of genera which are different and not subordinate one to the other are themselves different in kind. | |
| From: Aristotle (Categories [c.331 BCE], 01b16) | |
| A reaction: This seems to be indicating a category mistake, as he warns us not to attribute the wrong kind of differentiae to something we are picking out. |
| 16330 | Truth-value 'gluts' allow two truth values together; 'gaps' give a partial conception of truth [Halbach] |
| Full Idea: Truth-value 'gluts' correspond to a so-called dialethic conception of truth; excluding gluts and admitting only 'gaps' leads to a conception of what is usually called 'partial' truth. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 15.2) | |
| A reaction: Talk of 'gaps' and 'gluts' seem to be the neatest way of categorising views of truth. I want a theory with no gaps or gluts. |
| 16339 | Truth axioms prove objects exist, so truth doesn't seem to be a logical notion [Halbach] |
| Full Idea: Two typed disquotation sentences, truth axioms of TB, suffice for proving that there at least two objects. Hence truth is not a logical notion if one expects logical notions to be ontologically neutral. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 21.2) |
| 16324 | Any definition of truth requires a metalanguage [Halbach] |
| Full Idea: It is plain that the distinction between object and metalanguage is required for the definability of truth. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 11) | |
| A reaction: Halbach's axiomatic approach has given up on definability, and therefore it can seek to abandon the metalanguage and examine 'type-free' theories. |
| 16293 | Traditional definitions of truth often make it more obscure, rather than less [Halbach] |
| Full Idea: A common complaint against traditional definitional theories of truth is that it is far from clear that the definiens is not more in need of clarification than the definiendum (that is, the notion of truth). | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 1) | |
| A reaction: He refers to concepts like 'correspondence', 'facts', 'coherence' or 'utility', which are said to be trickier to understand than 'true'. I suspect that philosophers like Halbach confuse 'clear' with 'precise'. Coherence is quite clear, but imprecise. |
| 16301 | If people have big doubts about truth, a definition might give it more credibility [Halbach] |
| Full Idea: If one were wondering whether truth should be considered a legitimate notion at all, a definition might be useful in dispersing doubts about its legitimacy. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 3) | |
| A reaction: Halbach is proposing to skip definitions, and try to give rules for using 'true' instead, but he doesn't rule out definitions. A definition of 'knowledge' or 'virtue' or 'democracy' might equally give those credibility. |
| 18367 | A true existence statement has its truth caused by the existence of the thing [Aristotle] |
| Full Idea: Whereas the true statement [that there is a man] is in no way the cause of the actual thing's existence, the actual thing does seem in some way the cause of the statement's being true. | |
| From: Aristotle (Categories [c.331 BCE], 14b18) | |
| A reaction: Armstrong offers this as the earliest statement of the truthmaker principle. Notice the cautious qualification 'seem in some way'. The truthmaker dependence seems even clearer in falsemaking, where the death of the man falsifies the statement. |
| 16297 | Semantic theories avoid Tarski's Theorem by sticking to a sublanguage [Halbach] |
| Full Idea: In semantic theories (e.g.Tarski's or Kripke's), a definition evades Tarski's Theorem by restricting the possible instances in the schema T[φ]↔φ to sentences of a proper sublanguage of the language formulating the equivalences. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 1) | |
| A reaction: The schema says if it's true it's affirmable, and if it's affirmable it's true. The Liar Paradox is a key reason for imposing this restriction. |
| 16337 | Disquotational truth theories are short of deductive power [Halbach] |
| Full Idea: The problem of restricted deductive power has haunted disquotational theories of truth (…because they can't prove generalisations). | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 19.5) |
| 16322 | CT proves PA consistent, which PA can't do on its own, so CT is not conservative over PA [Halbach] |
| Full Idea: Compositional Truth CT proves the consistency of Peano arithmetic, which is not provable in Peano arithmetic by Gödel's second incompleteness theorem. Hence the theory CT is not conservative over Peano arithmetic. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 8.6) |
| 16294 | Axiomatic truth doesn't presuppose a truth-definition, though it could admit it at a later stage [Halbach] |
| Full Idea: Choosing an axiomatic approach to truth might well be compatible with the view that truth is definable; the definability of truth is just not presupposed at the outset. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 1) | |
| A reaction: Is it possible that a successful axiomatisation is a successful definition? |
| 16326 | The main semantic theories of truth are Kripke's theory, and revisions semantics [Halbach] |
| Full Idea: Revision semantics is arguably the main competitor of Kripke's theory of truth among semantic truth theories. …In the former one may hope through revision to arrive at better and better models, ..sorting out unsuitable extensions of the truth predicate. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 14) | |
| A reaction: Halbach notes later that Kripke's theory (believe it or not) is considerably simpler than revision semantics. |
| 16311 | To axiomatise Tarski's truth definition, we need a binary predicate for his 'satisfaction' [Halbach] |
| Full Idea: If the clauses of Tarski's definition of truth are turned into axioms (as Davidson proposed) then a primitive binary predicate symbol for satisfaction is needed, as Tarski defined truth in terms of satisfaction. Standard language has a unary predicate. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 5.2) |
| 16318 | Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents [Halbach] |
| Full Idea: In the typed Compositional Truth theory CT, it is compositional because the truth of a sentence depends on the semantic values of the constituents of that sentence. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 8) | |
| A reaction: [axioms on p. 65 of Halbach] |
| 16299 | Gödel numbering means a theory of truth can use Peano Arithmetic as its base theory [Halbach] |
| Full Idea: Often syntactic objects are identified with their numerical codes. …Expressions of a countable formal language can be coded in the natural numbers. This allows a theory of truth to use Peano Arithmetic (with its results) as a base theory. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 2) | |
| A reaction: The numbering system is the famous device invented by Gödel for his great proof of incompleteness. This idea is a key to understanding modern analytic philosophy. It is the bridge which means philosophical theories can be treated mathematically. |
| 16340 | Truth axioms need a base theory, because that is where truth issues arise [Halbach] |
| Full Idea: Considering the truth axioms in the absence of a base theory is not very sensible because characteristically truth theoretic reasoning arises from the interplay of the truth axioms with the base theory. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 21.2) | |
| A reaction: The base theory usually seems to be either Peano arithmetic or set theory. We might say that introverted thought (e.g. in infants) has little use for truth; it is when you think about the world that truth becomes a worry. |
| 16305 | We know a complete axiomatisation of truth is not feasible [Halbach] |
| Full Idea: In the light of incompleteness phenomena, one should not expect a categorical axiomatisation of truth to be feasible, but this should not keep one from studying axiomatic theories of truth (or of arithmetic). | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 3) | |
| A reaction: This, of course, is because of Gödel's famous results. It is important to be aware in this field that there cannot be a dream of a final theory, so we are just seeing what can be learned about truth. |
| 16313 | A theory is 'conservative' if it adds no new theorems to its base theory [Halbach, by PG] |
| Full Idea: A truth theory is 'conservative' if the addition of the truth predicate does not add any new theorems to the base theory. | |
| From: report of Volker Halbach (Axiomatic Theories of Truth [2011], 6 Df 6.6) by PG - Db (ideas) | |
| A reaction: Halbach presents the definition more formally, and this is my attempt at getting it into plain English. Halbach uses Peano Arithmetic as his base theory, but set theory is also sometimes used. |
| 16315 | The Tarski Biconditional theory TB is Peano Arithmetic, plus truth, plus all Tarski bi-conditionals [Halbach] |
| Full Idea: The truth theory TB (Tarski Biconditional) is all the axioms of Peano Arithmetic, including all instances of the induction schema with the truth predicate, plus all the sentences of the form T[φ] ↔ φ. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 7) | |
| A reaction: The biconditional formula is the famous 'snow is white' iff snow is white. The truth of the named sentence is equivalent to asserting the sentence. This is a typed theory of truth, and it is conservative over PA. |
| 16314 | Theories of truth are 'typed' (truth can't apply to sentences containing 'true'), or 'type-free' [Halbach] |
| Full Idea: I sort theories of truth into the large families of 'typed' and 'type-free'. Roughly, typed theories prohibit a truth predicate's application to sentences with occurrences of that predicate, and one cannot prove the truth of sentences containing 'true'. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], II Intro) | |
| A reaction: The problem sentence the typed theories are terrified of is the Liar Sentence. Typing produces a hierarchy of languages, referring down to the languages below them. |
| 16327 | Friedman-Sheard is type-free Compositional Truth, with two inference rules for truth [Halbach] |
| Full Idea: The Friedman-Sheard truth system FS is based on compositional theory CT. The axioms of FS are obtained by relaxing the type restriction on the CT-axioms, and adding rules inferring sentences from their truth, and vice versa. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 15) | |
| A reaction: The rules are called NEC and CONEC by Halbach. The system FSN is FS without the two rules. |
| 16329 | Kripke-Feferman theory KF axiomatises Kripke fixed-points, with Strong Kleene logic with gluts [Halbach] |
| Full Idea: The Kripke-Feferman theory KF is an axiomatisation of the fixed points of an operator, that is, of a Kripkean fixed-point semantics with the Strong Kleene evaluation schema with truth-value gluts. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 15.1) |
| 16332 | The KF theory is useful, but it is not a theory containing its own truth predicate [Halbach] |
| Full Idea: KF is useful for explicating Peano arithmetic, but it certainly does not come to close to being a theory that contains its own truth predicate. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 16) | |
| A reaction: Since it is a type-free theory, its main philosophical aspiration was to contain its own truth predicate, so that is bad news (for philosophers). |
| 16331 | The KF is much stronger deductively than FS, which relies on classical truth [Halbach] |
| Full Idea: The Kripke-Feferman theory is relatively deductively very strong. In particular, it is much stronger than its competitor FS, which is based on a completely classical notion of truth. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 15.3) |
| 16319 | Compositional Truth CT proves generalisations, so is preferred in discussions of deflationism [Halbach] |
| Full Idea: Compositional Truth CT and its variants has desirable generalisations among its logical consequences, so they seem to have ousted purely disquotational theories such as TB in the discussion on deflationism. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 8) |
| 16320 | Some say deflationism is axioms which are conservative over the base theory [Halbach] |
| Full Idea: Some authors have tried to understand the deflationist claim that truth is not a substantial notion as the claim that a satisfactory axiomatisation of truth should be conservative over the base theory. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 8) |
| 16338 | Deflationism says truth is a disquotation device to express generalisations, adding no new knowledge [Halbach] |
| Full Idea: There are two doctrines at the core of deflationism. The first says truth is a device of disquotation used to express generalisations, and the second says truth is a thin notion that contributes nothing to our knowledge of the world | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 21) |
| 16317 | The main problem for deflationists is they can express generalisations, but not prove them [Halbach] |
| Full Idea: The main criticism that deflationist theories based on the disquotation sentences or similar axioms have to meet was raised by Tarski: the disquotation sentences do not allow one to prove generalisations. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 7) |
| 16316 | Deflationists say truth is just for expressing infinite conjunctions or generalisations [Halbach] |
| Full Idea: Deflationists do not hold that truth is completely dispensable. They claim that truth serves the purpose of expressing infinite conjunctions or generalisations. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 7) | |
| A reaction: It is also of obvious value as a shorthand in ordinary conversation, but rigorous accounts can paraphrase that out. 'What he said is true'. 'Pick out the true sentences from p,q,r and s' seems to mean 'affirm some of them'. What does 'affirm' mean? |
| 16335 | In Strong Kleene logic a disjunction just needs one disjunct to be true [Halbach] |
| Full Idea: In Strong Kleene logic a disjunction of two sentences is true if at least one disjunct is true, even when the other disjunct lacks a truth value. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 18) | |
| A reaction: This sounds fine to me. 'Either I'm typing this or Homer had blue eyes' comes out true in any sensible system. |
| 16334 | In Weak Kleene logic there are 'gaps', neither true nor false if one component lacks a truth value [Halbach] |
| Full Idea: In Weak Kleene Logic, with truth-value gaps, a sentence is neither true nor false if one of its components lacks a truth value. A line of the truth table shows a gap if there is a gap anywhere in the line, and the other lines are classical. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 18) | |
| A reaction: This will presumably apply even if the connective is 'or', so a disjunction won't be true, even if one disjunct is true, when the other disjunct is unknown. 'Either 2+2=4 or Lot's wife was left-handed' sounds true to me. Odd. |
| 16309 | Every attempt at formal rigour uses some set theory [Halbach] |
| Full Idea: Almost any subject with any formal rigour employs some set theory. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 4.1) | |
| A reaction: This is partly because mathematics is often seen as founded in set theory, and formal rigour tends to be mathematical in character. |
| 16333 | The underestimated costs of giving up classical logic are found in mathematical reasoning [Halbach] |
| Full Idea: The costs of giving up classical logic are easily underestimated, …the price being paid in terms of mathematical reasoning. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 16.2) | |
| A reaction: No one cares much about such costs, until you say they are 'mathematical'. Presumably this is a message to Graham Priest and his pals. |
| 11033 | Predications of predicates are predications of their subjects [Aristotle] |
| Full Idea: Whenever one thing is predicated of another as of a subject, all things said of what is predicated will be said of the subject also. | |
| From: Aristotle (Categories [c.331 BCE], 01b10) |
| 16310 | A theory is some formulae and all of their consequences [Halbach] |
| Full Idea: A theory is a set of formulae closed under first-order logical consequence. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 5.1) |
| 16342 | You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system [Halbach] |
| Full Idea: One cannot just accept that all the theorems of Peano arithmetic are true when one accepts Peano arithmetic as the notion of truth is not available in the language of arithmetic. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1) | |
| A reaction: This is given as the reason why Kreisel and Levy (1968) introduced 'reflection principles', which allow you to assert whatever has been proved (with no mention of truth). (I think. The waters are closing over my head). |
| 16341 | Normally we only endorse a theory if we believe it to be sound [Halbach] |
| Full Idea: If one endorses a theory, so one might argue, one should also take it to be sound. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1) |
| 16344 | Soundness must involve truth; the soundness of PA certainly needs it [Halbach] |
| Full Idea: Soundness seems to be a notion essentially involving truth. At least I do not know how to fully express the soundness of Peano arithmetic without invoking a truth predicate. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1) | |
| A reaction: I suppose you could use some alternative locution such as 'assertible' or 'cuddly'. Intuitionists seem a bit vague about the truth end of things. |
| 16347 | Many new paradoxes may await us when we study interactions between frameworks [Halbach] |
| Full Idea: Paradoxes that arise from interaction of predicates such as truth, necessity, knowledge, future and past truths have receive little attention. There may be many unknown paradoxes lurking when we develop frameworks with these intensional notions. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 24.2) | |
| A reaction: Nice. This is a wonderful pointer to new research in the analytic tradition, in which formal problems will gradually iron out our metaphysical framework. |
| 16336 | The liar paradox applies truth to a negated truth (but the conditional will serve equally) [Halbach] |
| Full Idea: An essential feature of the liar paradox is the application of the truth predicate to a sentence with a negated occurrence of the truth predicate, though the negation can be avoided by using the conditional. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 19.3) |
| 11044 | One is prior to two, because its existence is implied by two [Aristotle] |
| Full Idea: One is prior to two because if there are two it follows at once that there is one, whereas if there is one there is not necessarily two. | |
| From: Aristotle (Categories [c.331 BCE], 14a29) | |
| A reaction: The axiomatic introduction of a 'successor' to a number does not seem to introduce this notion of priority, based on inclusiveness. Introducing order by '>' also does not seem to indicate any logical priority. |
| 11042 | Parts of a line join at a point, so it is continuous [Aristotle] |
| Full Idea: A line is a continuous quantity. For it is possible to find a common boundary at which its parts join together, a point. | |
| From: Aristotle (Categories [c.331 BCE], 04b33) | |
| A reaction: This appears to be the essential concept of a Dedekind cut. It seems to be an open question whether a cut defines a unique number, but a boundary seems to be intrinsically unique. Aristotle wins again. |
| 16321 | The compactness theorem can prove nonstandard models of PA [Halbach] |
| Full Idea: Nonstandard models of Peano arithmetic are models of PA that are not isomorphic to the standard model. Their existence can be established with the compactness theorem or the adequacy theorem of first-order logic. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 8.3) |
| 16343 | The global reflection principle seems to express the soundness of Peano Arithmetic [Halbach] |
| Full Idea: The global reflection principle ∀x(Sent(x) ∧ Bew[PA](x) → Tx) …seems to be the full statement of the soundness claim for Peano arithmetic, as it expresses that all theorems of Peano arithmetic are true. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1) | |
| A reaction: That is, an extra principle must be introduced to express the soundness. PA is, of course, not complete. |
| 11041 | Some quantities are discrete, like number, and others continuous, like lines, time and space [Aristotle] |
| Full Idea: Of quantities, some are discrete, others continuous. ...Discrete are number and language; continuous are lines, surfaces, bodies, and also, besides these, time and place. | |
| From: Aristotle (Categories [c.331 BCE], 04b20) | |
| A reaction: This distinction seems to me to be extremely illuminating, when comparing natural numbers with real numbers, and it is the foundation of the Greek view of mathematics. |
| 9992 | The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor] |
| Full Idea: The author entirely overlooks the fact that the 'extension of a concept' in general may be quantitatively completely indeterminate. Only in certain cases is the 'extension of a concept' quantitatively determinate. | |
| From: George Cantor (Review of Frege's 'Grundlagen' [1885], 1932:440), quoted by William W. Tait - Frege versus Cantor and Dedekind | |
| A reaction: Cantor presumably has in mind various infinite sets. Tait is drawing our attention to the fact that this objection long precedes Russell's paradox, which made the objection more formal (a language Frege could understand!). |
| 16312 | To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals [Halbach] |
| Full Idea: For the reduction of Peano Arithmetic to ZF set theory, usually the set of finite von Neumann ordinals is used to represent the non-negative integers. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 6) | |
| A reaction: Halbach makes it clear that this is just one mode of reduction, relative interpretability. |
| 16308 | Set theory was liberated early from types, and recent truth-theories are exploring type-free [Halbach] |
| Full Idea: While set theory was liberated much earlier from type restrictions, interest in type-free theories of truth only developed more recently. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 4) | |
| A reaction: Tarski's theory of truth involves types (or hierarchies). |
| 11286 | Primary being must be more than mere indeterminate ultimate subject of predication [Politis on Aristotle] |
| Full Idea: He criticises his 'Categories' view, because if primary being is simply the ultimate subject of predication the primary being is, in virtue of itself, something indeterminate; it would be a necessary but not a sufficient condition for primary being. | |
| From: comment on Aristotle (Categories [c.331 BCE]) by Vassilis Politis - Aristotle and the Metaphysics 7.5 | |
| A reaction: Thus, Politis argues, primary being is essence in the later work. The words 'substance' and 'ousia' cause confusion here, and must be watched closely. Wedin argues that Aristotle merely develops his 'Categories' view, but most disagree. |
| 1700 | There are six kinds of change: generation, destruction, increase, diminution, alteration, change of place [Aristotle] |
| Full Idea: There are six kinds of change: generation, destruction, increase, diminution, alteration, change of place. A change in our affections would be an example of alteration. | |
| From: Aristotle (Categories [c.331 BCE], 15a13) |
| 16345 | That Peano arithmetic is interpretable in ZF set theory is taken by philosophers as a reduction [Halbach] |
| Full Idea: The observation that Peano arithmetic is relatively interpretable in ZF set theory is taken by many philosophers to be a reduction of numbers to sets. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 23) | |
| A reaction: Nice! Being able to express something in a different language is not the same as a reduction. Back to the drawing board. What do you really mean by a reduction? If we model something, we don't 'reduce' it to the model. |
| 18366 | Of interdependent things, the prior one causes the other's existence [Aristotle] |
| Full Idea: For of things which reciprocate as to implication of existence, that which is in some way the cause of the other's existence might reasonably by called prior by nature. | |
| From: Aristotle (Categories [c.331 BCE], 14b12) | |
| A reaction: Not so clear when you seek examples. The bus is prior to its redness, but you can't have a colourless bus, so being coloured is prior to being a bus. Aristotle's example is a man being prior to the truths about him. |
| 1699 | A thing is prior to another if it implies its existence [Aristotle] |
| Full Idea: That from which the implication of existence does not hold reciprocally is thought to be prior. | |
| From: Aristotle (Categories [c.331 BCE], 14a32) | |
| A reaction: shadows and objects |
| 3311 | The categories (substance, quality, quantity, relation, action, passion, place, time) peter out inconsequentially [Benardete,JA on Aristotle] |
| Full Idea: The Aristotelian schedule of categories - substance, quality, quantity, relation, action, passion, place, time, and so forth - appears to peter out inconsequentially. | |
| From: comment on Aristotle (Categories [c.331 BCE]) by José A. Benardete - Metaphysics: the logical approach Ch.7 | |
| A reaction: Compare Idea 5544 for Kant's attempt to classify categories. Personally I like the way Aristotle's 'peter out'. That seems to me a more plausible character for good metaphysics. |
| 11035 | There are ten basic categories for thinking about things [Aristotle] |
| Full Idea: Of things said without any combination, each signifies either substance or quantity or qualification or a relative or where or when or being-in-a-position or having or doing or being-affected. | |
| From: Aristotle (Categories [c.331 BCE], 01b25) | |
| A reaction: This sums up the earlier of Aristotle's two metaphysical view, and each of this categories is discussed in the present text. |
| 13121 | Substance,Quantity,Quality,Relation,Place,Time,Being-in-a-position,Having,Doing,Being affected [Aristotle, by Westerhoff] |
| Full Idea: Aristotle's list of ten categories proved to be the most influential scheme found in his works: Substance, Quantity, Quality, Relation, Place, Time, Being-in-a-position, Having, Doing, Being affected. | |
| From: report of Aristotle (Categories [c.331 BCE]) by Jan Westerhoff - Ontological Categories §01 |
| 16116 | Aristotle derived categories as answers to basic questions about nature, size, quality, location etc. [Aristotle, by Gill,ML] |
| Full Idea: Aristotle seems to have worked out his list of categories by considering various questions that one might ask about a particular object, such as What is it? How big is it? How is it qualified? And Where is it? | |
| From: report of Aristotle (Categories [c.331 BCE]) by Mary Louise Gill - Aristotle on Substance | |
| A reaction: Of course, to think of his questions, Aristotle already had categories in his mind. How would he approach a proposal to recategorise reality more efficiently? |
| 21345 | Aristotle said relations are not substances, so (if they exist) they must be accidents [Aristotle, by Heil] |
| Full Idea: Aristotle categorised relations as accidents - Socrates's whiteness, the sphericity of this ball - entities dependent on substances. Relations are not substances, so they must be, if anything at all, accidents. | |
| From: report of Aristotle (Categories [c.331 BCE], §7) by John Heil - Relations 'Historical' | |
| A reaction: Heil says this thought encouraged anti-realist views of relations, which became the norm until Russell. |
| 16155 | Aristotle promoted the importance of properties and objects (rather than general and particular) [Aristotle, by Frede,M] |
| Full Idea: In 'Categories' Aristotle is taking a first step in making the distinction between objects and properties central to ontology. This plays virtually no role in Plato, and was overshadowed by the distinction between general and particular. | |
| From: report of Aristotle (Categories [c.331 BCE]) by Michael Frede - Individuals in Aristotle I | |
| A reaction: Frede says he gets in a tangle because he mixes the earlier and the new views. Because we are nowadays in a total muddle about properties, I'm thinking we should go back to the earlier view! Modern commentators make him a trope theorist. |
| 11032 | Some things said 'of' a subject are not 'in' the subject [Aristotle] |
| Full Idea: Of things there are, some are said of a subject, but are not in any subject. For example, man is said of a subject, the individual man, but is not in any subject. | |
| From: Aristotle (Categories [c.331 BCE], 01a20) | |
| A reaction: See? 'Being a man' is not a property of a man! Only the properties which are 'in' the man are properties of the man. The rest are things which are said 'of' men, usually as classifications. A classification is not a property. |
| 11038 | We call them secondary 'substances' because they reveal the primary substances [Aristotle] |
| Full Idea: It is reasonable that, after the primary substances, their species and genera should be the only other things called (secondary) substances. For only they, of things predicated, reveal the primary substance. | |
| From: Aristotle (Categories [c.331 BCE], 02b29) | |
| A reaction: This is the key passage in all of Aristotle for sortal essentialists like Wiggins, especially the word 'only'. I take it that this observation is superseded by the Metaphysics. Definition is the route to substance (which involves general terms). |
| 16739 | Four species of quality: states, capacities, affects, and forms [Aristotle, by Pasnau] |
| Full Idea: In Categories 8 there are four species of qualities: States and conditions, Natural capacities and incapacities, Affective qualities or affections, and Shape and external form. | |
| From: report of Aristotle (Categories [c.331 BCE], Ch.8) by Robert Pasnau - Metaphysical Themes 1274-1671 23.5 |
| 11037 | Colour must be in an individual body, or it is not embodied [Aristotle] |
| Full Idea: Colour is in body and therefore also in an individual body; for were it not in some individual body it would not be in body at all. | |
| From: Aristotle (Categories [c.331 BCE], 02b02) | |
| A reaction: This may be just a truism, or it may be the Aristotelian commitment to universals only existing if they are instantiated. |
| 16154 | Aristotle gave up his earlier notion of individuals, because it relied on universals [Aristotle, by Frede,M] |
| Full Idea: In 'Metaphysics' Aristotle abandons the notion of an individual which he had relied on in the 'Categories', since it presupposes that there are general things, that there are universals. | |
| From: report of Aristotle (Categories [c.331 BCE]) by Michael Frede - Individuals in Aristotle Intro | |
| A reaction: Ah, very illuminating. So all the way through we have a concept of individuals, first relying on universals, and then relying on hylomorphism? I suppose a bundle theory of individuals would need universals. |
| 12351 | Genus and species are substances, because only they reveal the primary substance [Aristotle, by Wedin] |
| Full Idea: The reason Aristotle gives for calling species and genera substances is that of what is predicated only they reveal what the primary substance is. | |
| From: report of Aristotle (Categories [c.331 BCE], 02b29-37) by Michael V. Wedin - Aristotle's Theory of Substance III.6 | |
| A reaction: Thus we should not be misled into thinking that the genus and species ARE the essence. We edge our way towards the essence of an individual by subdividing its categories. |
| 1694 | Substances have no opposites, and don't come in degrees (including if the substance is a man) [Aristotle] |
| Full Idea: There is nothing contrary to substances,…. and a substance does not admit of a more and a less. If this substance is a man, it will not be more a man or less a man either than itself or than another man. | |
| From: Aristotle (Categories [c.331 BCE], 03b33) |
| 16091 | Is primary substance just an ultimate subject, or some aspect of a complex body? [Aristotle, by Gill,ML] |
| Full Idea: 'Categories' treats something's being an ultimate subject as a test for being a primary substance, but it does not treat its primary objects as complex bodies consisting of matter and form. In that case, is the composite or a feature the ultimate subject? | |
| From: report of Aristotle (Categories [c.331 BCE]) by Mary Louise Gill - Aristotle on Substance Ch.1 | |
| A reaction: Gill is trying to throw light on the difference between 'Categories' and 'Metaphysics'. Once you have hylomorphism (form-plus-matter) you have a new difficulty in explaining unity. The answer is revealed once we understand 'form'. |
| 11280 | Primary being is 'that which lies under', or 'particular substance' [Aristotle, by Politis] |
| Full Idea: In 'Categories' Aristotle argues the primary being (proté ousia) is the ultimate subject of predication (to hupokeimenon, meaning 'that which lies under'), nowadays referred to as the 'particular substance' view. | |
| From: report of Aristotle (Categories [c.331 BCE]) by Vassilis Politis - Aristotle and the Metaphysics 4.4 | |
| A reaction: Politis says that Aristotle shifts to the quite different view in 'Metaphysics', that primary being is essence, rather than mere subject of predication. |
| 11040 | A single substance can receive contrary properties [Aristotle] |
| Full Idea: It seems distinctive of substance that what is numerically one and the same is able to receive contraries. ...For example, an individual man - one and the same - becomes pale at one time and dark at another. | |
| From: Aristotle (Categories [c.331 BCE], 04a10/20) |
| 16140 | Secondary substances do have subjects, so they are not ultimate in the ontology [Aristotle, by Frede,M] |
| Full Idea: The concept of substance applies to secondary substances only with some deletions; ..it is not true that they have no subjects, and hence they are not ultimate subjects for all other elements of the ontology. | |
| From: report of Aristotle (Categories [c.331 BCE]) by Michael Frede - Title, Unity, Authenticity of the 'Categories' V | |
| A reaction: It increasingly strikes that to treat secondary substance (roughly, species) as essence is a shocking misreading of Aristotle. Frede says they are substances, because they do indeed 'underlie'. |
| 10965 | In earlier Aristotle the substances were particulars, not kinds [Aristotle, by Lawson-Tancred] |
| Full Idea: In 'Metaphysics' Aristotle changed his view, as in 'Categories' the substances, the basic realities, were particular items, notably individual men, horses, cabbages etc. | |
| From: report of Aristotle (Categories [c.331 BCE]) by Hugh Lawson-Tancred - Introductions to 'Metaphysics' p.178 | |
| A reaction: The charge is that having successfully rebelled against Plato, Aristotle gradually succumbed to his teacher's influence, and ended up with a more platonist view. For anti-platonists like myself, the 'Categories' seems to be the key text. |
| 11036 | A 'primary' substance is in each subject, with species or genera as 'secondary' substances [Aristotle] |
| Full Idea: A substance, in its most primary sense, is that which is neither said of a subject nor in a subject, e.g. the individual man or horse. The species in which things primarily called substances are, are called secondary substances, as are the genera. | |
| From: Aristotle (Categories [c.331 BCE], 02a11) | |
| A reaction: This distinction between 'primary' and 'secondary' substances is characteristic of Aristotle's earlier metaphysical view, with the later view (more unified and Platonic) in the 'Metaphysics'. |
| 8287 | Earlier Aristotle had objects as primary substances, but later he switched to substantial form [Aristotle, by Lowe] |
| Full Idea: In 'Categories' primary substances are individual concrete objects, such as a particular horse, whereas in 'Metaphysics' such things are combinations of matter and substantial form, with the latter being the primary substances. | |
| From: report of Aristotle (Categories [c.331 BCE]) by E.J. Lowe - The Possibility of Metaphysics 9.1 | |
| A reaction: Lowe claims there is no real difference. Aristotle came to think that matter was not part of primary substance, so the shift seems to be that substance was concrete, but then he decided it was abstract. Physicists will prefer 'Metaphysics'. |
| 12350 | Things are called 'substances' because they are subjects for everything else [Aristotle] |
| Full Idea: It is because the primary substances are subjects for everything else that they are called substances [ousiai] most strictly. | |
| From: Aristotle (Categories [c.331 BCE], 03a04) | |
| A reaction: This points to a rather minimal account of substance, as possibly the 'bare particular' which has no other role than to have properties. This expands in 'Metaphysics' to be matter which has form, making properties possible. |
| 11039 | A primary substance reveals a 'this', which is an individual unit [Aristotle] |
| Full Idea: Every substance seems to signify a certain 'this'. As regards the primary substances, it is indisputably true that each of them signifies a certain 'this'; for the thing revealed is individual and numerically one. | |
| From: Aristotle (Categories [c.331 BCE], 03b10) | |
| A reaction: The notion of 'primary' substance is confined to this earlier metaphysics of Aristotle. |
| 12361 | Primary substances are ontological in 'Categories', and explanatory in 'Metaphysics' [Aristotle, by Wedin] |
| Full Idea: The primacy of 'Categories' primary substances is a kind of ontological primacy, whereas the primacy of form is a kind of structural or explanatory primacy. | |
| From: report of Aristotle (Categories [c.331 BCE]) by Michael V. Wedin - Aristotle's Theory of Substance X.9 | |
| A reaction: 'Structural' and 'explanatory' sound very different, since the former sounds ontological and the latter epistemological (and more subjective). |
| 3315 | Aristotle denigrates the category of relation, but for modern absolutists self-relation is basic [Benardete,JA on Aristotle] |
| Full Idea: Aristotle denigrates the whole category of relations, but modern logical absolutists single out self-relation (in the mode of identity) as metaphysically privileged. | |
| From: comment on Aristotle (Categories [c.331 BCE]) by José A. Benardete - Metaphysics: the logical approach Ch.8 | |
| A reaction: I think this refers to Plantinga and Merrihew Adams, who make identity-with-itself the basic component of individual existences. |
| 16346 | Maybe necessity is a predicate, not the usual operator, to make it more like truth [Halbach] |
| Full Idea: Should necessity be treated as a predicate rather than (as in modal logic) as a sentential operator? It is odd to assign different status to necessity and truth, hampering their interaction. That all necessities are true can't be expressed by an operator. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 24.2) | |
| A reaction: [compressed] Halbach and Horsten consistently treat truth as a predicate, but maybe truth is an operator. Making necessity a predicate and not an operator would be a huge upheaval in the world of modal logic. Nice move! |
| 12349 | Only what can be said of many things is a predicable [Aristotle, by Wedin] |
| Full Idea: Aristotle reminds us that nothing is to count as predicable that cannot be said-of many things. | |
| From: report of Aristotle (Categories [c.331 BCE]) by Michael V. Wedin - Aristotle's Theory of Substance III.1 | |
| A reaction: Thus there wouldn't be any predicates if there were not universals. Could we have proper names for individual qualities (tropes), in the way that we have them for individual objects? |
| 11837 | Some predicates signify qualification of a substance, others the substance itself [Aristotle] |
| Full Idea: 'White' signifies nothing but a qualification, whereas the species ('man') and the genus ('animal') mark off the qualification of substance - they signify substance of a certain qualification. | |
| From: Aristotle (Categories [c.331 BCE], 03b18) | |
| A reaction: This is making a fundamental distinction between two different types of predication. I would describe them as one attributing a real property, and the other attributing a category (as a result of the properties). I don't think 'substance' helps here. |
| 16298 | We need propositions to ascribe the same beliefs to people with different languages [Halbach] |
| Full Idea: Being able to ascribe the same proposition as a belief to persons who do not have a common language seems to be one of the main reasons to employ propositions. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 2) | |
| A reaction: Propositions concern beliefs, as well as sentence meanings. I would want to say that a dog and I could believe the same thing, and that is a non-linguistic reason to believe in propositions. Maybe 'translation' cuts out the proposition middleman? |
| 11043 | It is not possible for fire to be cold or snow black [Aristotle] |
| Full Idea: It is not possible for fire to be cold or snow black. | |
| From: Aristotle (Categories [c.331 BCE], 12b01) |
| 1696 | Change goes from possession to loss (as in baldness), but not the other way round [Aristotle] |
| Full Idea: Change occurs from possession to privation, but from privation to possession is impossible; one who has gone blind does not recover sight nor does a bald man regain his hair nor does a toothless man grow new ones. | |
| From: Aristotle (Categories [c.331 BCE], 13a35) | |
| A reaction: Although this seems like an insight into entropy, it isn't an accurate observation, since trees lose their leaves, and then regain them in spring. Maybe somewhere men regrow their hair each spring. |