12 ideas
| 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) |
| 18935 | Semantic theory should specify when an act of naming is successful [Sawyer] |
| Full Idea: A semantic theory of names should deliver a specification of the conditions under which a name names an individual, and hence a specification of the conditions under which a name is empty. | |
| From: Sarah Sawyer (Empty Names [2012], 1) | |
| A reaction: Naming can be private, like naming my car 'Bertrand', but never tell anyone. I like Plato's remark that names are 'tools'. Do we specify conditions for successful spanner-usage? The first step must be individuation, preparatory to naming. |
| 18945 | Millians say a name just means its object [Sawyer] |
| Full Idea: The Millian view of direct reference says that the meaning of a name is the object named. | |
| From: Sarah Sawyer (Empty Names [2012], 4) | |
| A reaction: Any theory that says meaning somehow is features of the physical world strikes me as totally misguided. Napoleon is a man, so he can't be part of a sentence. He delegates that job to words (such as 'Napoleon'). |
| 18934 | Sentences with empty names can be understood, be co-referential, and even be true [Sawyer] |
| Full Idea: Some empty names sentences can be understood, so appear to be meaningful ('Pegasus was sired by Poseidon'), ...some appear to be co-referential ('Santa Claus'/'Father Christmas'), and some appear to be straightforwardly true ('Pegasus doesn't exist'). | |
| From: Sarah Sawyer (Empty Names [2012], 1) | |
| A reaction: Hang on to this, when the logicians arrive and start telling you that your talk of empty names is vacuous, because there is no object in the 'domain' to which a predicate can be attached. Meaning, reference and truth are the issues around empty names. |
| 18938 | Frege's compositional account of truth-vaues makes 'Pegasus doesn't exist' neither true nor false [Sawyer] |
| Full Idea: In Frege's account sentences such as 'Pegasus does not exist' will be neither true nor false, since the truth-value of a sentence is its referent, and the referent of a complex expression is determined by the referent of its parts. | |
| From: Sarah Sawyer (Empty Names [2012], 2) | |
| A reaction: We can keep the idea of 'sense', which is very useful for dealing with empty names, but tweak his account of truth-values to evade this problem. I'm thinking that meaning is compositional, but truth-value isn't. |
| 18947 | Definites descriptions don't solve the empty names problem, because the properties may not exist [Sawyer] |
| Full Idea: If it were possible for a definite description to be empty - not in the sense of there being no object that satisfies it, but of there being no set of properties it refers to - the problem of empty names would not have been solved. | |
| From: Sarah Sawyer (Empty Names [2012], 5) | |
| A reaction: Swoyer is thinking of properties like 'is a unicorn', which are clearly just as vulnerable to being empty as 'the unicorn' was. It seems unlikely that 'horse', 'white' and 'horn' would be empty. |
| 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) |
| 20147 | Arguments that my finger does not exist are less certain than your seeing my finger [Moore,GE] |
| Full Idea: This really is a finger ...and you all know it. ...I can safely challenge anyone to give an argument that it is not true, which does not rest upon some premise which is less certain than is the proposition which it is designed to attack. | |
| From: G.E. Moore (Some Judgements of Perception [1922], p.228), quoted by John Kekes - The Human Condition 01.3 | |
| A reaction: [In Moore's 'Philosophical Studies'] This is a particularly clear statement from Moore of his famous claim. I'm not sure what to make of an attempt to compare a sceptical argument (dreams, demons) with the sight of a finger. |