11 ideas
18491 | The idea of 'making' can be mere conceptual explanation (like 'because') [Künne] |
Full Idea: If we say 'being a child of our parent's sibling makes him your first cousin', that can be paraphrased using 'because', and this is the 'because' of conceptual explanation: the second part elucidates the sense of the first part. | |
From: Wolfgang Künne (Conceptions of Truth [2003], 3.5.2) | |
A reaction: Fans of truth-making are certainly made uncomfortable by talk of 'what makes this a good painting' or 'this made my day'. They need a bit more sharpness to the concept of 'making' a truth. |
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) |
5998 | From the necessity of the past we can infer the impossibility of what never happens [Diod.Cronus, by White,MJ] |
Full Idea: Diodorus' Master Argument inferred that since what is past (i.e. true in the past) is necessary, and the impossible cannot follow from the possible, that therefore if something neither is nor ever will be the case, then it is impossible. | |
From: report of Diodorus Cronus (fragments/reports [c.300 BCE]) by Michael J. White - Diodorus Cronus | |
A reaction: The argument is, apparently, no longer fully clear, but it seems to imply determinism, or at least a rejection of the idea that free will and determinism are compatible. (Epictetus 2.19) |
20832 | The Master Argument seems to prove that only what will happen is possible [Diod.Cronus, by Epictetus] |
Full Idea: The Master Argument: these conflict 1) what is past and true is necessary, 2) the impossible does not follow from the possible, 3) something possible neither is nor will be true. Hence only that which is or will be true is possible. | |
From: report of Diodorus Cronus (fragments/reports [c.300 BCE]) by Epictetus - The Discourses 2.19.1 | |
A reaction: [Epictetus goes on to discuss views about which of the three should be given up] It is possible there will be a sea fight tomorrow; tomorrow comes, and no sea fight; so there was necessarily no sea fight; so the impossible followed from the possible. |
14304 | Conditionals are true when the antecedent is true, and the consequent has to be true [Diod.Cronus] |
Full Idea: The connected (proposition) is true when it begins with true and neither could nor can end with false. | |
From: Diodorus Cronus (fragments/reports [c.300 BCE]), quoted by Stephen Mumford - Dispositions 03.4 | |
A reaction: [Mumford got the quote from Bochenski] This differs from the truth-functional account because it says nothing about when the antecedent is false, which fits in also with the 'supposition' view, where A is presumed. This idea adds necessity. |
6024 | Thought is unambiguous, and you should stick to what the speaker thinks they are saying [Diod.Cronus, by Gellius] |
Full Idea: No one says or thinks anything ambiguous, and nothing should be held to be being said beyond what the speaker thinks he is saying. | |
From: report of Diodorus Cronus (fragments/reports [c.300 BCE]) by Aulus Gellius - Noctes Atticae 11.12.2 | |
A reaction: A key argument in favour of propositions, implied in this remark, is that propositions are never ambiguous, though the sentences expressing them may be |