14 ideas
2661 | Dialectic is speech cast in the form of logical argument [Cicero] |
Full Idea: Dialectic is speech cast in the form of logical argument. | |
From: M. Tullius Cicero (Academica [c.45 BCE], I.viii.32) |
2673 | There cannot be more than one truth [Cicero] |
Full Idea: There cannot be more than one truth. | |
From: M. Tullius Cicero (Academica [c.45 BCE], II.xlviii.147) |
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) |
2669 | Dialectic assumes that all statements are either true or false, but self-referential paradoxes are a big problem [Cicero] |
Full Idea: It is a fundamental principle of dialectic that every statement is either true or false. So is this a true proposition or a false one: "If you say that you are lying and say it truly, you lie"? | |
From: M. Tullius Cicero (Academica [c.45 BCE], II.xxix.95) |
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) |
16755 | The possible Aristotelian view that forms are real and active principles is clearly wrong [Fine,K, by Pasnau] |
Full Idea: Aristotle seems to have a possible basis for the belief [in individual forms], namely that forms are real and active principles in the world, which is denied by any right-minded modern. | |
From: report of Kit Fine (A Puzzle Concerning Matter and Form [1994], p.19) by Robert Pasnau - Metaphysical Themes 1274-1671 24.3 n8 | |
A reaction: Pasnau says this is the view of forms promoted by the scholastics, whereas Aristotle's own view should be understood as 'metaphysical'. |
2664 | If we have complete healthy senses, what more could the gods give us? [Cicero] |
Full Idea: If human nature were interrogated by some god as to whether it was content with its own senses in a sound and undamaged state or demanded something better, I cannot see what more it could ask for. | |
From: M. Tullius Cicero (Academica [c.45 BCE], II.vii.19) |
2665 | How can there be a memory of what is false? [Cicero] |
Full Idea: How can there possibly be a memory of what is false? | |
From: M. Tullius Cicero (Academica [c.45 BCE], II.vii.22) |
20800 | Every true presentation can have a false one of the same quality [Cicero] |
Full Idea: [The sceptical Academics say] what is false cannot be perceived, but every true presentation is such that there can be a false presentation of the same quality. | |
From: M. Tullius Cicero (Academica [c.45 BCE], II.40) | |
A reaction: It was the stoics who focused the discussion on 'presentations'. This claim is purely theoretical; no one has ever experienced a false presentation of talking to a family member that was as vivid as the real thing. |
2672 | Virtues must be very detached, to avoid being motivated by pleasure [Cicero] |
Full Idea: None of the virtues can exist unless they are disinterested, for virtue driven to duty by pleasure as a sort of pay is not virtue at all but a deceptive sham and pretence of virtue. | |
From: M. Tullius Cicero (Academica [c.45 BCE], II.xlvi.140) |