13 ideas
| 19048 | Contextual definition shifted the emphasis from words to whole sentences [Quine] |
| Full Idea: Contextual definition precipitated a revolution in semantics. The primary vehicle of meaning is seen no longer as the word, but as the sentence. | |
| From: Willard Quine (Five Milestones of Empiricism [1975], p.69) | |
| A reaction: I think the idea is that the term is now supported entirely by its surrounding language, and not by its denotation of something in the world. |
| 19047 | Bentham's contextual definitions preserved terms after their denotation became doubtful [Quine] |
| Full Idea: If Bentham found some term convenient but ontologically embarrassing, contextual definition enabled him in some cases to continue to enjoy the services of the term while disclaiming its denotation. | |
| From: Willard Quine (Five Milestones of Empiricism [1975], p.68) | |
| A reaction: In Quine's terms this would be to withdraw the term from the periphery of the theory, where it has to meet the world, and make it part of the inner connections of the theory. He suggests that Bentham invented this technique. |
| 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) |
| 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) |
| 10558 | Abstract objects are actually constituted by the properties by which we conceive them [Zalta] |
| Full Idea: Where for ordinary objects one can discover the properties they exemplify, abstract objects are actually constituted or determined by the properties by which we conceive them. I use the technical term 'x encodes F' for this idea. | |
| From: Edward N. Zalta (Deriving Kripkean Claims with Abstract Objects [2006], 2 n2) | |
| A reaction: One might say that whereas concrete objects can be dubbed (in the Kripke manner), abstract objects can only be referred to by descriptions. See 10557 for more technicalities about Zalta's idea. |
| 19049 | In scientific theories sentences are too brief to be independent vehicles of empirical meaning [Quine] |
| Full Idea: We have come to recognise that in a scientific theory even a whole sentence is ordinarily too short a text to serve as an independent vehicle of empirical meaning. | |
| From: Willard Quine (Five Milestones of Empiricism [1975], p.70) |
| 19046 | Empiricism improvements: words for ideas, then sentences, then systems, then no analytic, then naturalism [Quine] |
| Full Idea: Since 1750 empiricism shows five turns for the better. First was a shift from ideas to words. Second a shift from terms to sentences. Third the shift to systems of sentences. Fourth the abandonment of analytic-synthetic dualism. Fifth was naturalism. | |
| From: Willard Quine (Five Milestones of Empiricism [1975], p.67) | |
| A reaction: [compressed] Quine must be largely credited with the last two. The first four are almost entirely linguistic in character, which is characteristic of mid-twentieth-century empiricism. I would offer the recognition of explanation as central for the sixth. |
| 10557 | Abstract objects are captured by second-order modal logic, plus 'encoding' formulas [Zalta] |
| Full Idea: My object theory is formulated in a 'syntactically second-order' modal predicate calculus modified only so as to admit a second kind of atomic formula ('xF'), which asserts that object x 'encodes' property F. | |
| From: Edward N. Zalta (Deriving Kripkean Claims with Abstract Objects [2006], p.2) | |
| A reaction: This is summarising Zalta's 1983 theory of abstract objects. See Idea 10558 for Zalta's idea in plain English. |
| 19050 | Holism in language blurs empirical synthetic and empty analytic sentences [Quine] |
| Full Idea: Holism blurs the supposed contrast beween the synthetic sentence, with its empirical content, and the analytic sentence, with its null content. | |
| From: Willard Quine (Five Milestones of Empiricism [1975], p.71) | |
| A reaction: This spells out nicely that Quine's rejection of the distinction is completely tied to his holistic view of language. The obvious phenomenon of compositionality (building sentence meaning in steps) counts against holism. |