display all the ideas for this combination of texts
7 ideas
14970 | Normal system K has five axioms and rules [Cresswell] |
Full Idea: Normal propositional modal logics derive from the minimal system K: wffs of PC are axioms; □(p⊃q)⊃(□p⊃□q); uniform substitution; modus ponens; necessitation (α→□α). | |
From: Max J. Cresswell (Modal Logic [2001], 7.1) |
14971 | D is valid on every serial frame, but not where there are dead ends [Cresswell] |
Full Idea: If a frame contains any dead end or blind world, then D is not valid on that frame, ...but D is valid on every serial frame. | |
From: Max J. Cresswell (Modal Logic [2001], 7.1.1) |
14972 | S4 has 14 modalities, and always reduces to a maximum of three modal operators [Cresswell] |
Full Idea: In S4 there are exactly 14 distinct modalities, and any modality may be reduced to one containing no more than three modal operators in sequence. | |
From: Max J. Cresswell (Modal Logic [2001], 7.1.2) | |
A reaction: The significance of this may be unclear, but it illustrates one of the rewards of using formal systems to think about modal problems. There is at least an appearance of precision, even if it is only conditional precision. |
14973 | In S5 all the long complex modalities reduce to just three, and their negations [Cresswell] |
Full Idea: S5 contains the four main reduction laws, so the first of any pair of operators may be deleted. Hence all but the last modal operator may be deleted. This leaves six modalities: p, ◊p, □p, and their negations. | |
From: Max J. Cresswell (Modal Logic [2001], 7.1.2) |
14976 | Reject the Barcan if quantifiers are confined to worlds, and different things exist in other worlds [Cresswell] |
Full Idea: If one wants the quantifiers in each world to range only over the things that exist in that world, and one doesn't believe that the same things exist in every world, one would probably not want the Barcan formula. | |
From: Max J. Cresswell (Modal Logic [2001], 7.2.2) | |
A reaction: I haven't quite got this, but it sounds to me like I should reject the Barcan formula (but Idea 9449!). I like a metaphysics to rest on the actual world (with modal properties). I assume different things could have existed, but don't. |
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) |