16 ideas
| 3035 | Dialectic involves conversations with short questions and brief answers [Diog. Laertius] |
| Full Idea: Dialectic is when men converse by putting short questions and giving brief answers to those who question them. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 3.1.52) |
| 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) |
| 9216 | Each area of enquiry, and its source, has its own distinctive type of necessity [Fine,K] |
| Full Idea: The three sources of necessity - the identity of things, the natural order, and the normative order - have their own peculiar forms of necessity. The three main areas of human enquiry - metaphysics, science and ethics - each has its own necessity. | |
| From: Kit Fine (The Varieties of Necessity [2002], 6) | |
| A reaction: I would treat necessity in ethics with caution, if it is not reducible to natural or metaphysical necessity. Fine's proposal is interesting, but I did not find it convincing, especially in its view that metaphysical necessity doesn't intrude into nature. |
| 1816 | Sceptics say demonstration depends on self-demonstrating things, or indemonstrable things [Diog. Laertius] |
| Full Idea: Sceptics say that every demonstration depends on things which demonstrates themselves, or on things which can't be demonstrated. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 9.Py.11) | |
| A reaction: This refers to two parts of Agrippa's Trilemma (the third being that demonstration could go on forever). He makes the first option sound very rationalist, rather than experiential. |
| 9214 | Unsupported testimony may still be believable [Fine,K] |
| Full Idea: I may have good reason to believe some testimony, for example, even though the person providing the testimony has no good reason for saying what he does. | |
| From: Kit Fine (The Varieties of Necessity [2002], 5) | |
| A reaction: Thus small children, madmen and dreamers may occasionally get things right without realising it. I take testimony to be merely one more batch of evidence which has to be assessed in building the most coherent picture possible. |
| 1819 | Scepticism has two dogmas: that nothing is definable, and every argument has an opposite argument [Diog. Laertius] |
| Full Idea: Sceptics actually assert two dogmas: that nothing should be defined, and that every argument has an opposite argument. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 9.Py.11) |
| 3064 | When sceptics say that nothing is definable, or all arguments have an opposite, they are being dogmatic [Diog. Laertius] |
| Full Idea: When sceptics say that they define nothing, and that every argument has an opposite argument, they here give a positive definition, and assert a positive dogma. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 9.11.11) |
| 3033 | Induction moves from some truths to similar ones, by contraries or consequents [Diog. Laertius] |
| Full Idea: Induction is an argument which by means of some admitted truths establishes naturally other truths which resemble them; there are two kinds, one proceeding from contraries, the other from consequents. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 3.1.23) |
| 1838 | Cyrenaic pleasure is a motion, but Epicurean pleasure is a condition [Diog. Laertius] |
| Full Idea: Cyrenaics place pleasure wholly in motion, whereas Epicurus admits it as a condition. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 10.28) | |
| A reaction: Not a distinction we meet in modern discussions. Do events within the mind count as 'motion'? If so, these two agree. If not, I'd vote for Epicurus. |
| 1769 | Cynics believe that when a man wishes for nothing he is like the gods [Diog. Laertius] |
| Full Idea: Cynics believe that when a man wishes for nothing he is like the gods. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 6.Men.3) |
| 9215 | Causation is easier to disrupt than logic, so metaphysics is part of nature, not vice versa [Fine,K] |
| Full Idea: It would be harder to break P-and-Q implying P than the connection between cause and effect. This difference in strictness means it is more plausible that natural necessities include metaphysical necessities, than vice versa. | |
| From: Kit Fine (The Varieties of Necessity [2002], 6) | |
| A reaction: I cannot see any a priori grounds for the claim that causation is more easily disrupted than logic. It seems to be based on the strategy of inferring possibilities from what can be imagined, which seems to me to lead to wild misunderstandings. |