2001 | Modal Logic |
7.1 | p.137 | 14970 | Normal system K has five axioms and rules |
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) |
7.1.1 | p.140 | 14971 | D is valid on every serial frame, but not where there are dead ends |
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) |
7.1.2 | p.141 | 14972 | S4 has 14 modalities, and always reduces to a maximum of three modal operators |
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. |
7.1.2 | p.141 | 14973 | In S5 all the long complex modalities reduce to just three, and their negations |
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) |
7.1.2 | p.141 | 14974 | A relation is 'Euclidean' if aRb and aRc imply bRc |
Full Idea: A relation is 'Euclidean' if aRb and aRc imply bRc. | |||
From: Max J. Cresswell (Modal Logic [2001], 7.1.2) | |||
A reaction: If a thing has a relation to two separate things, then those two things will also have that relation between them. If I am in the same family as Jim and as Jill, then Jim and Jill are in the same family. |
7.2.1 | p.149 | 14975 | A de dicto necessity is true in all worlds, but not necessarily of the same thing in each world |
Full Idea: A de dicto necessary truth says that something is φ, that this proposition is a necessary truth, i.e. that in every accessible world something (but not necessarily the same thing in each world) is φ. | |||
From: Max J. Cresswell (Modal Logic [2001], 7.2.1) | |||
A reaction: At last, a really clear and illuminating account of this term! The question is then invited of what is the truthmaker for a de dicto truth, assuming that the objects themselves are truthmakers for de re truths. |
7.2.2 | p.151 | 14976 | Reject the Barcan if quantifiers are confined to worlds, and different things exist in other worlds |
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. |