17 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) |
| 23548 | Indeterminacy is in conflict with classical logic [Fine,K] |
| Full Idea: I now believe that the existence of indeterminacy is in conflict with classical logic. | |
| From: Kit Fine (Vagueness: a global approach [2020], 3) | |
| A reaction: I think that prior to this Fine had defended classical logic. Presumably the difficulty is over Bivalence. Nietzsche spotted this problem, despite not being a logician. Logic has to simplify the world. Hence philosophy is quite different from logic. |
| 23539 | Classical semantics has referents for names, extensions for predicates, and T or F for sentences [Fine,K] |
| Full Idea: A precise language is often assigned a classical semantics, in which the semantic value of a name is its referent, the semantic value of a predicate is its extension (the objects of which it is true), and the value of a sentence is True or False. | |
| From: Kit Fine (Vagueness: a global approach [2020], 1) | |
| A reaction: Helpful to have this clear statement of how predicates are treated. This extensionalism in logic causes trouble when it creeps into philosophy, and people say that 'red' just means all the red things. No it doesn't. |
| 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) |
| 23544 | Local indeterminacy concerns a single object, and global indeterminacy covers a range [Fine,K] |
| Full Idea: Vagueness concerns 'local' indeterminacy, such as whether one man in the lineup is bald, and 'global' indeterminacy, applying to a range of cases, as when it is indeterminate how 'bald' applies to the lineup. But how do these relate? | |
| From: Kit Fine (Vagueness: a global approach [2020], 1) | |
| A reaction: This puts the focus either on objects or on predicates which are vague. |
| 23540 | Conjoining two indefinites by related sentences seems to produce a contradiction [Fine,K] |
| Full Idea: If 'P is red' and 'P is orange' are indefinite, then 'P is red and P is orange' seems false, because red and orange are exclusive. But if two conjoined indefinite sentences are false, that makes 'P is red and P is red' false, when it should be indefinite. | |
| From: Kit Fine (Vagueness: a global approach [2020], 1) | |
| A reaction: [compressed] This is the problem of 'penumbral connection', where two indefinite values are still logically related, by excluding one another. Presumably 'P is red and P is of indefinite shape' can be true? Doubtful about this argument. |
| 23546 | Standardly vagueness involves borderline cases, and a higher standpoint from which they can be seen [Fine,K] |
| Full Idea: Standard notions of vagueness all accept borderline cases, and presuppose a higher standpoint from which a judgement of being borderline F, rather than simply being F or being not F, can be made. | |
| From: Kit Fine (Vagueness: a global approach [2020], 3) | |
| A reaction: He says that the concept of borderline cases is an impediment to understanding vagueness. Proposing a third group when you are struggling to separate two other groups doesn't seem helpful, come to think of it. Limbo cases. |
| 23542 | Identifying vagueness with ignorance is the common mistake of confusing symptoms with cause [Fine,K] |
| Full Idea: We can see Epistemicism [vagueness as ignorance] as a common and misguided tendency to identify a cause with its symptoms. We are unsure how to characterise vagueness, and identify it with the resulting ignorance, instead of explaining it. | |
| From: Kit Fine (Vagueness: a global approach [2020], 1) | |
| A reaction: Love it. This echoes my repeated plea in these reactions to stop identifying features of reality with the functions which embody them or the patterns they create. We need to explain them, and must dig deeper. |
| 23541 | Supervaluation can give no answer to 'who is the last bald man' [Fine,K] |
| Full Idea: Under supervaluation there should always be someone who is the last bald man in the sequence, but there is always an acceptable way to make some other man the last bald man. | |
| From: Kit Fine (Vagueness: a global approach [2020], 1) | |
| A reaction: Fine seems to take this as a conclusive refutation of the supervaluation approach. Fine says (p.41) that supervaluation says there is a precisification for every instance. |
| 23545 | We do not have an intelligible concept of a borderline case [Fine,K] |
| Full Idea: We simply have no intelligible notion of local indeterminacy or of a borderline case. | |
| From: Kit Fine (Vagueness: a global approach [2020], 2) | |
| A reaction: He mentions cases which are near a borderline, and cases which are hard to decide, but denies that these are intrinsically borderline. If there are borderline cases between red and orange, what are the outer boundaries of the border? |
| 8886 | Being a true justified belief is not a sufficient condition for knowledge [Gettier] |
| Full Idea: The claim that someone knows a proposition if it is true, it is believed, and the person is justified in their belief is false, in that the conditions do not state a sufficient condition for the claim. | |
| From: Edmund L. Gettier (Is Justified True Belief Knowledge? [1963], p.145) | |
| A reaction: This is the beginning of the famous Gettier Problem, which has motivated most epistemology for the last forty years. Gettier implies that justification is necessary, even if it is not sufficient. He gives two counterexamples. |
| 23547 | It seems absurd that there is no identity of any kind between two objects which involve survival [Fine,K] |
| Full Idea: Pace Parfit and others, it boggles the mind that survival could be independent of any relation of identity between the currently existing object and the objects that subsequently exist. | |
| From: Kit Fine (Vagueness: a global approach [2020], 3) | |
| A reaction: Yes. If the self or mind just consists of a diachronic trail of memories such that the two ends of the trail have no connection at all, that isn't the kind of survival that any of us want. I want to live my life, not a life. |
| 23543 | We identify laws with regularities because we mistakenly identify causes with their symptoms [Fine,K] |
| Full Idea: There is a common tendency to identify a cause with its symptoms. Hence we are not sure how to characterise a law, and so we identify it with the regularities to which it gives rise. | |
| From: Kit Fine (Vagueness: a global approach [2020], 1) | |
| A reaction: A lovely clear identification of my pet hate, which is superficial accounts of things, which claim to be the last word, but actually explain nothing. |