16 ideas
| 10928 | Maybe we can quantify modally if the objects are intensional, but it seems unlikely [Quine] |
| Full Idea: Perhaps there is no objection to quantifying into modal contexts as long as the values of any variables thus quantified are limited to intensional objects, but they also lead to disturbing examples. | |
| From: Willard Quine (Reference and Modality [1953], §3) | |
| A reaction: [Quine goes on to give his examples] I take it that possibilities are features of actual reality, not merely objects of thought. The problem is that they are harder to know than actual objects. |
| 17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
| Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
| 17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
| Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
| 10925 | Failure of substitutivity shows that a personal name is not purely referential [Quine] |
| Full Idea: Failure of substitutivity shows that the occurrence of a personal name is not purely referential. | |
| From: Willard Quine (Reference and Modality [1953], §1) | |
| A reaction: I don't think I understand the notion of a name being 'purely' referential, as if it somehow ceased to be a word, and was completely transparent to the named object. |
| 10926 | Quantifying into referentially opaque contexts often produces nonsense [Quine] |
| Full Idea: If to a referentially opaque context of a variable we apply a quantifier, with the intention that it govern that variable from outside the referentially opaque context, then what we commonly end up with is unintended sense or nonsense. | |
| From: Willard Quine (Reference and Modality [1953], §2) |
| 17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
| Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3) |
| 17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
| Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
| A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
| 17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
| Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
| A reaction: Each expansion brings a limitation, but then you can expand again. |
| 17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
| Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |
| 9558 | All scientific tests will verify mathematics, so it is a background, not something being tested [Sober] |
| Full Idea: If mathematical statements are part of every competing hypothesis, then no matter which hypothesis comes out best in the light of observations, they will be part of the best hypothesis. They are not tested, but are a background assumption. | |
| From: Elliott Sober (Mathematics and Indispensibility [1993], 45), quoted by Charles Chihara - A Structural Account of Mathematics | |
| A reaction: This is a very nice objection to the Quine-Putnam thesis that mathematics is confirmed by the ongoing successes of science. |
| 10930 | Quantification into modal contexts requires objects to have an essence [Quine] |
| Full Idea: A reversion to Aristotelian essentialism is required if quantification into modal contexts is to be insisted on. An object must be seen as having some of its traits necessarily. | |
| From: Willard Quine (Reference and Modality [1953], §3) | |
| A reaction: This thought leads directly to Kripke's proposal of rigid designation of objects (and Lewis response of counterparts), which really gets modal logic off the ground. Quine's challenge remains - the modal logic entails a huge metaphysical commitment. |
| 14645 | To be necessarily greater than 7 is not a trait of 7, but depends on how 7 is referred to [Quine] |
| Full Idea: To be necessarily greater than 7 is not a trait of a number, but depends on the manner of referring to the number. | |
| From: Willard Quine (Reference and Modality [1953], §2) | |
| A reaction: The most concise quotation of Quine's objection to 'de re' modality. The point is whether the number might have been referred to as 'the number of planets'. So many of these problems are solved by fixing unambiguous propositions first. |
| 9201 | Whether 9 is necessarily greater than 7 depends on how '9' is described [Quine, by Fine,K] |
| Full Idea: Quine's metaphysical argument is that if 9 is 7+2 the number 9 will be necessarily greater than 7, but when 9 is described as the number of planets, the number will not be necessarily greater than 7. The necessity depends on how it is described. | |
| From: report of Willard Quine (Reference and Modality [1953]) by Kit Fine - Intro to 'Modality and Tense' p. 3 | |
| A reaction: Thus necessity would be entirely 'de dicto' and not 'de re'. It sounds like a feeble argument. If I describe the law of identity (a=a) as 'my least favourite logical principle', that won't make it contingent. Describe 9, or refer to it? See Idea 9203. |
| 10927 | Necessity only applies to objects if they are distinctively specified [Quine] |
| Full Idea: Necessity does not properly apply to the fulfilment of conditions by objects (such as the number which numbers the planets), apart from special ways of specifying them. | |
| From: Willard Quine (Reference and Modality [1953], §3) | |
| A reaction: This appears to say that the only necessity is 'de dicto', and that there is no such thing as 'de re' necessity (of the thing in itself). How can Quine deny that there might be de re necessities? His point is epistemological - how can we know them? |
| 9203 | We can't quantify in modal contexts, because the modality depends on descriptions, not objects [Quine, by Fine,K] |
| Full Idea: 'Necessarily 9>7' may be true while the sentence 'necessarily the number of planets < 7' is false, even though it is obtained by substituting a coreferential term. So quantification in these contexts is unintelligible, without a clear object. | |
| From: report of Willard Quine (Reference and Modality [1953]) by Kit Fine - Intro to 'Modality and Tense' p. 4 | |
| A reaction: This is Quine's second argument against modality. See Idea 9201 for his first. Fine attempts to refute it. The standard reply seems to be to insist that 9 must therefore be an object, which pushes materialist philosophers into reluctant platonism. |
| 10931 | We can't say 'necessarily if x is in water then x dissolves' if we can't quantify modally [Quine] |
| Full Idea: To say an object is soluble in water is to say that it would dissolve if it were in water,..which implies that 'necessarily if x is in water then x dissolves'. Yet we do not know if there is a suitable sense of 'necessarily' into which we can so quantify. | |
| From: Willard Quine (Reference and Modality [1953], §4) | |
| A reaction: This is why there has been a huge revival of scientific essentialism - because Krike seems to offer exacty the account which Quine said was missing. So can you have modal logic without rigid designation? |