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) |
| 12219 | Whether a modal claim is true depends on how the object is described [Quine, by Fine,K] |
| Full Idea: Quine says if ∃x□(x>7) makes sense, then for which object x is the condition rendered true? Specify it as '9' and it is apparently rendered true, specify it as 'the number of planets' and it is apparently rendered false. | |
| From: report of Willard Quine (Three Grades of Modal Involvement [1953]) by Kit Fine - Quine on Quantifying In p.105 | |
| A reaction: This is normally characterised as Quine saying that only de dicto involvement is possible, and not de re involvement. Or that that all essences are nominal, and cannot be real. |
| 10922 | Objects are the values of variables, so a referentially opaque context cannot be quantified into [Quine] |
| Full Idea: The objects of a theory are not properly describable as the things named by the singular terms; they are the values, rather, of the variables of quantification. ..So a referentially opaque context is one that cannot properly be quantified into. | |
| From: Willard Quine (Three Grades of Modal Involvement [1953], p.174) | |
| A reaction: The point being that you cannot accurately pick out the objects in the domain |
| 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) |
| 10923 | Aristotelian essentialism says a thing has some necessary and some non-necessary properties [Quine] |
| Full Idea: What Aristotelian essentialism says is that you can have open sentences Fx and Gx, such that ∃x(nec Fx.Gx.¬nec Gx). For example, ∃x(nec(x>5). there are just x planets. ¬nec(there are just x planets)). | |
| From: Willard Quine (Three Grades of Modal Involvement [1953], p.176) | |
| A reaction: This is a denial of 'maximal essentialism', that all of a things properties might be essential. Quine is thus denying necessity, except under a description. He may be equivocating over the reference of 'there are just 9 planets'. |
| 10921 | Necessity can attach to statement-names, to statements, and to open sentences [Quine] |
| Full Idea: Three degrees necessity in logic or semantics: first and least is attaching a semantical predicate to the names of statements (as Nec '9>5'); second and more drastic attaches to statements themselves; third and gravest attaches to open sentences. | |
| From: Willard Quine (Three Grades of Modal Involvement [1953], p.158) |
| 10924 | Necessity is in the way in which we say things, and not things themselves [Quine] |
| Full Idea: Necessity resides in the way in which we say things, and not in the things we talk about. | |
| From: Willard Quine (Three Grades of Modal Involvement [1953], p.176) | |
| A reaction: This is a culminating idea of Quine's thoroughgoing empiricism, as filtered through logical positivism. I would hardly dare to accuse Quine of a use/mention confusion (his own bęte noir), but one seems to me to be lurking here. |
| 8790 | The 'doctrine of the given' is correct; some beliefs or statements are self-justifying [Chisholm] |
| Full Idea: In my opinion, the 'doctrine of the given' is correct in saying that there are some beliefs or statements which are 'self-justifying' and that among such beliefs are statements some of which concern appearances or 'ways of being appeared to'. | |
| From: Roderick Chisholm (The Myth of the Given [1964], §12) | |
| A reaction: To boldly assert that they are 'self-justifying' invites a landslide of criticisms, pointing at a regress. It might be better to say they are self-evident, or intuitively known, or primitive, or true by the natural light of reason. |