44 ideas
| 9331 | How do we determine which of the sentences containing a term comprise its definition? [Horwich] |
| Full Idea: How are we to determine which of the sentences containing a term comprise its definition? | |
| From: Paul Horwich (Stipulation, Meaning and Apriority [2000], §2) | |
| A reaction: Nice question. If I say 'philosophy is the love of wisdom' and 'philosophy bores me', why should one be part of its definition and the other not? What if I stipulated that the second one is part of my definition, and the first one isn't? |
| 6334 | The function of the truth predicate? Understanding 'true'? Meaning of 'true'? The concept of truth? A theory of truth? [Horwich] |
| Full Idea: We must distinguish the function of the truth predicate, what it is to understand 'true', the meaning of 'true', grasping the concept of truth, and a theory of truth itself. | |
| From: Paul Horwich (Truth (2nd edn) [1990], Ch.2.8) | |
| A reaction: It makes you feel tired to think about it. Presumably every other philosophical analysis has to do this many jobs. Clearly Horwich wants to propose one account which will do all five jobs. Personally I don't believe these five are really distinct. |
| 6342 | Some correspondence theories concern facts; others are built up through reference and satisfaction [Horwich] |
| Full Idea: One correspondence theory (e.g. early Wittgenstein) concerns representations and facts; alternatively (Tarski, Davidson) the category of fact is eschewed, and the truth of sentences or propositions is built out of relations of reference and satisfaction. | |
| From: Paul Horwich (Truth (2nd edn) [1990], Ch.7.35) | |
| A reaction: A helpful distinction. Clearly the notion of a 'fact' is an elusive one ("how many facts are there in this room?"), so it seems quite promising to say that the parts of the sentence correspond, rather than the whole thing. |
| 6332 | The common-sense theory of correspondence has never been worked out satisfactorily [Horwich] |
| Full Idea: The common-sense notion that truth is a kind of 'correspondence with the facts' has never been worked out to anyone's satisfaction. | |
| From: Paul Horwich (Truth (2nd edn) [1990], Ch.1) | |
| A reaction: I've put this in to criticise it. Philosophy can't work by rejecting theories which can't be 'worked out', and accepting theories (like Tarski's) because they can be 'worked out'. All our theories will end up minimal, and defiant of common sense. |
| 6335 | The redundancy theory cannot explain inferences from 'what x said is true' and 'x said p', to p [Horwich] |
| Full Idea: The redundancy theory is unable to account for the inference from "Oscar's claim is true" and "Oscar's claim is that snow is white" to "the proposition 'that snow is white' is true", and hence to "snow is white". | |
| From: Paul Horwich (Truth (2nd edn) [1990], Ch.2.9) | |
| A reaction: Earlier objections appealed to the fact that the word 'true' seemed to have a use in ordinary speech, but this seems a much stronger one. In general, showing the role of a term in making inferences pins it down better than ordinary speech does. |
| 6344 | Truth is a useful concept for unarticulated propositions and generalisations about them [Horwich] |
| Full Idea: All uses of the truth predicate are explained by the hypothesis that its entire raison d'ętre is to help us say things about unarticulated propositions, and in particular to express generalisations about them. | |
| From: Paul Horwich (Truth (2nd edn) [1990], Concl) | |
| A reaction: This certain is a very deflationary notion of truth. Articulated propositions are considered to stand on their own two feet, without need of 'is true'. He makes truth sound like a language game, though. Personally I prefer to mention reality. |
| 6336 | No deflationary conception of truth does justice to the fact that we aim for truth [Horwich] |
| Full Idea: It has been suggested that no deflationary conception of truth could do justice to the fact that we aim for the truth. | |
| From: Paul Horwich (Truth (2nd edn) [1990], Ch.2.11) | |
| A reaction: (He mentions Dummett and Wright). People don't only aim for it - they become very idealistic about it, and sometimes die for it. Personally I think that any study of truth should use as its example police investigations, not philosophical analysis. |
| 23299 | Horwich's deflationary view is novel, because it relies on propositions rather than sentences [Horwich, by Davidson] |
| Full Idea: Horwich's brave and striking move is to make the primary bearers of truth propositions - not exactly a new idea in itself, but new in the context of a serious attempt to defend deflationism. | |
| From: report of Paul Horwich (Truth (2nd edn) [1990]) by Donald Davidson - The Folly of Trying to Define Truth p.30 | |
| A reaction: Davidson rejects propositions because they can't be individuated, but I totally accept propositions. I'm puzzled why this would produce a deflationist theory, since I think it points to a much more robust view. |
| 6337 | The deflationary picture says believing a theory true is a trivial step after believing the theory [Horwich] |
| Full Idea: According to the deflationary picture, believing that a theory is true is a trivial step beyond believing the theory. | |
| From: Paul Horwich (Truth (2nd edn) [1990], Ch.2.17) | |
| A reaction: What has gone wrong with this picture is that you cannot (it seems to me) give a decent account of belief without mentioning truth. To believe a proposition is to hold it true. Hume's emotional account (Idea 2208) makes belief bewildering. |
| 10859 | A set is 'well-ordered' if every subset has a first element [Clegg] |
| Full Idea: For a set to be 'well-ordered' it is required that every subset of the set has a first element. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10857 | Set theory made a closer study of infinity possible [Clegg] |
| Full Idea: Set theory made a closer study of infinity possible. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10864 | Any set can always generate a larger set - its powerset, of subsets [Clegg] |
| Full Idea: The idea of the 'power set' means that it is always possible to generate a bigger one using only the elements of that set, namely the set of all its subsets. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 10872 | Extensionality: Two sets are equal if and only if they have the same elements [Clegg] |
| Full Idea: Axiom of Extension: Two sets are equal if and only if they have the same elements. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10875 | Pairing: For any two sets there exists a set to which they both belong [Clegg] |
| Full Idea: Axiom of Pairing: For any two sets there exists a set to which they both belong. So you can make a set out of two other sets. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10876 | Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg] |
| Full Idea: Axiom of Unions: For every collection of sets there exists a set that contains all the elements that belong to at least one of the sets in the collection. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10878 | Infinity: There exists a set of the empty set and the successor of each element [Clegg] |
| Full Idea: Axiom of Infinity: There exists a set containing the empty set and the successor of each of its elements. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: This is rather different from the other axioms because it contains the notion of 'successor', though that can be generated by an ordering procedure. |
| 10877 | Powers: All the subsets of a given set form their own new powerset [Clegg] |
| Full Idea: Axiom of Powers: For each set there exists a collection of sets that contains amongst its elements all the subsets of the given set. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: Obviously this must include the whole of the base set (i.e. not just 'proper' subsets), otherwise the new set would just be a duplicate of the base set. |
| 10879 | Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg] |
| Full Idea: Axiom of Choice: For every set we can provide a mechanism for choosing one member of any non-empty subset of the set. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: This axiom is unusual because it makes the bold claim that such a 'mechanism' can always be found. Cohen showed that this axiom is separate. The tricky bit is choosing from an infinite subset. |
| 10871 | Axiom of Existence: there exists at least one set [Clegg] |
| Full Idea: Axiom of Existence: there exists at least one set. This may be the empty set, but you need to start with something. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10874 | Specification: a condition applied to a set will always produce a new set [Clegg] |
| Full Idea: Axiom of Specification: For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true. So the concept 'number is even' produces a set from the integers. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: What if the condition won't apply to the set? 'Number is even' presumably won't produce a set if it is applied to a set of non-numbers. |
| 6339 | Logical form is the aspects of meaning that determine logical entailments [Horwich] |
| Full Idea: The logical forms of the sentences in a language are those aspects of their meanings that determine the relations of deductive entailment holding amongst them. | |
| From: Paul Horwich (Truth (2nd edn) [1990], Ch.6.30) | |
| A reaction: A helpful definition. Not all sentences, therefore, need to have a 'logical form'. Is the logical form the same as the underlying proposition. The two must converge, given that propositions lack the ambiguity that is often found in sentences. |
| 10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg] |
| Full Idea: Three views of mathematics: 'pure' mathematics, where it doesn't matter if it could ever have any application; 'real' mathematics, where every concept must be physically grounded; and 'applied' mathematics, using the non-real if the results are real. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.17) | |
| A reaction: Very helpful. No one can deny the activities of 'pure' mathematics, but I think it is undeniable that the origins of the subject are 'real' (rather than platonic). We do economics by pretending there are concepts like the 'average family'. |
| 10861 | Beyond infinity cardinals and ordinals can come apart [Clegg] |
| Full Idea: With ordinary finite numbers ordinals and cardinals are in effect the same, but beyond infinity it is possible for two sets to have the same cardinality but different ordinals. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10860 | An ordinal number is defined by the set that comes before it [Clegg] |
| Full Idea: You can think of an ordinal number as being defined by the set that comes before it, so, in the non-negative integers, ordinal 5 is defined as {0, 1, 2, 3, 4}. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10854 | Transcendental numbers can't be fitted to finite equations [Clegg] |
| Full Idea: The 'transcendental numbers' are those irrationals that can't be fitted to a suitable finite equation, of which π is far and away the best known. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10858 | By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg] |
| Full Idea: The realisation that brought 'i' into the toolkit of physicists and engineers was that you could extend the 'number line' into a new dimension, with an imaginary number axis at right angles to it. ...We now have a 'number plane'. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.12) |
| 10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg] |
| Full Idea: It is a chicken-and-egg problem, whether the lack of zero forced forced classical mathematicians to rely mostly on a geometric approach to mathematics, or the geometric approach made 0 a meaningless concept, but the two remain strongly tied together. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10866 | Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg] |
| Full Idea: As far as Kronecker was concerned, Cantor had built a whole structure on the irrational numbers, and so that structure had no foundation at all. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10869 | The Continuum Hypothesis is independent of the axioms of set theory [Clegg] |
| Full Idea: Paul Cohen showed that the Continuum Hypothesis is independent of the axioms of set theory. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10862 | The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg] |
| Full Idea: The 'continuum hypothesis' says that aleph-one is the cardinality of the rational and irrational numbers. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 8431 | Problems with Goodman's view of counterfactuals led to a radical approach from Stalnaker and Lewis [Horwich] |
| Full Idea: In reaction to two classic difficulties in Goodman's treatment of counterfactuals - the contenability problem and the explication of law - a radically different approach was instigated by Stalnaker (1968) and has been developed by Lewis. | |
| From: Paul Horwich (Lewis's Programme [1987], p208) | |
| A reaction: [I record this for study purposes] |
| 9333 | A priori belief is not necessarily a priori justification, or a priori knowledge [Horwich] |
| Full Idea: It is one thing to believe something a priori and another for this belief to be epistemically justified. The latter is required for a priori knowledge. | |
| From: Paul Horwich (Stipulation, Meaning and Apriority [2000], §8) | |
| A reaction: Personally I would agree with this, because I don't think anything should count as knowledge if it doesn't have supporting reasons, but fans of a priori knowledge presumably think that certain basic facts are just known. They are a priori justified. |
| 9342 | Understanding needs a priori commitment [Horwich] |
| Full Idea: Understanding is itself based on a priori commitment. | |
| From: Paul Horwich (Stipulation, Meaning and Apriority [2000], §12) | |
| A reaction: This sounds plausible, but needs more justification than Horwich offers. This is the sort of New Rationalist idea I associate with Bonjour. The crucial feature of the New lot is, I take it, their fallibilism. All understanding is provisional. |
| 9332 | Meaning is generated by a priori commitment to truth, not the other way around [Horwich] |
| Full Idea: Our a priori commitment to certain sentences is not really explained by our knowledge of a word's meaning. It is the other way around. We accept a priori that the sentences are true, and thereby provide it with meaning. | |
| From: Paul Horwich (Stipulation, Meaning and Apriority [2000], §8) | |
| A reaction: This sounds like a lovely trump card, but how on earth do you decide that a sentence is true if you don't know what it means? Personally I would take it that we are committed to the truth of a proposition, before we have a sentence for it. |
| 9341 | Meanings and concepts cannot give a priori knowledge, because they may be unacceptable [Horwich] |
| Full Idea: A priori knowledge of logic and mathematics cannot derive from meanings or concepts, because someone may possess such concepts, and yet disagree with us about them. | |
| From: Paul Horwich (Stipulation, Meaning and Apriority [2000], §12) | |
| A reaction: A good argument. The thing to focus on is not whether such ideas are a priori, but whether they are knowledge. I think we should employ the word 'intuition' for a priori candidates for knowledge, and demand further justification for actual knowledge. |
| 9334 | If we stipulate the meaning of 'number' to make Hume's Principle true, we first need Hume's Principle [Horwich] |
| Full Idea: If we stipulate the meaning of 'the number of x's' so that it makes Hume's Principle true, we must accept Hume's Principle. But a precondition for this stipulation is that Hume's Principle be accepted a priori. | |
| From: Paul Horwich (Stipulation, Meaning and Apriority [2000], §9) | |
| A reaction: Yet another modern Quinean argument that all attempts at defining things are circular. I am beginning to think that the only a priori knowledge we have is of when a group of ideas is coherent. Calling it 'intuition' might be more accurate. |
| 9339 | A priori knowledge (e.g. classical logic) may derive from the innate structure of our minds [Horwich] |
| Full Idea: One potential source of a priori knowledge is the innate structure of our minds. We might, for example, have an a priori commitment to classical logic. | |
| From: Paul Horwich (Stipulation, Meaning and Apriority [2000], §11) | |
| A reaction: Horwich points out that to be knowledge it must also say that we ought to believe it. I'm wondering whether if we divided the whole territory of the a priori up into intuitions and then coherent justifications, the whole problem would go away. |
| 2799 | Bayes' theorem explains why very surprising predictions have a higher value as evidence [Horwich] |
| Full Idea: Bayesianism can explain the fact that in science surprising predictions have greater evidential value, as the equation produces a higher degree of confirmation. | |
| From: Paul Horwich (Bayesianism [1992], p.42) |
| 2798 | Probability of H, given evidence E, is prob(H) x prob(E given H) / prob(E) [Horwich] |
| Full Idea: Bayesianism says ideally rational people should have degrees of belief (not all-or-nothing beliefs), corresponding with probability theory. Probability of H, given evidence E, is prob(H) X prob(E given H) / prob(E). | |
| From: Paul Horwich (Bayesianism [1992], p.41) |
| 6338 | We could know the truth-conditions of a foreign sentence without knowing its meaning [Horwich] |
| Full Idea: Someone who does not understand German and is told 'Schnee ist weiss' is true if frozen H2O is white, does not understand the German sentence, even though he knows the truth-conditions. | |
| From: Paul Horwich (Truth (2nd edn) [1990], Ch.5.22 n1) | |
| A reaction: This sounds like a powerful objection to Davidson's well-known claim that meaning is truth-conditions. Horwich likes the idea that meaning is use, but I think a similar objection arises - you can use a sentence well without knowing its meaning. |
| 6340 | There are Fregean de dicto propositions, and Russellian de re propositions, or a mixture [Horwich] |
| Full Idea: There are pure, Fregean, abstract, de dicto propositions, in which a compositional structure is filled only with senses; there are pure, Russellian, concrete, de re propositions, which are filled with referents; and there are mixed propositions. | |
| From: Paul Horwich (Truth (2nd edn) [1990], Ch.6.31) | |
| A reaction: Once Frege has distinguished sense from reference, this distinction of propositions is likely to follow. The current debate over the internalist and externalist accounts of concepts seems to continue the debate. A mixed strategy sounds good. |
| 6341 | Right translation is a mapping of languages which preserves basic patterns of usage [Horwich] |
| Full Idea: The right translation between words of two languages is the mapping that preserves basic patterns of usage - where usage is characterised non-semantically, in terms of circumstances of application, assertibility conditions and inferential role. | |
| From: Paul Horwich (Truth (2nd edn) [1990], Ch.6.32) | |
| A reaction: It still strikes me that if you ask why a piece of language is used in a certain way, you find yourself facing something deeper about meaning than mere usage. Horwich cites Wittgenstein and Quine in his support. Could a machine pass his test? |
| 6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
| Full Idea: Hermarchus said that animal killing is justified by considerations of human safety and nourishment and by animals' inability to form contractual relations of justice with us. | |
| From: report of Hermarchus (fragments/reports [c.270 BCE]) by David A. Sedley - Hermarchus | |
| A reaction: Could the last argument be used to justify torturing animals? Or could we eat a human who was too brain-damaged to form contracts? |
| 8432 | Analyse counterfactuals using causation, not the other way around [Horwich] |
| Full Idea: In my view, counterfactual conditionals are analysed in terms of causation. | |
| From: Paul Horwich (Lewis's Programme [1987], p.208) | |
| A reaction: This immediately sounds more plausible to me. Counterfactual claims are rather human, whereas causation (if we accept it) seems a feature of nature. The key question is whether some sort of 'dependency' is a feature of counterfactuals. |