Combining Philosophers

Ideas for Richard Price, Dale Jacquette and Max J. Cresswell

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7 ideas

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / a. Systems of modal logic

9456	Modal logic is multiple systems, shown in the variety of accessibility relations between worlds [Jacquette]
	Full Idea: Modal logic by its very nature is not monolithic, but fragmented into multiple systems of modal qualifications, reflected in the plurality of accessibility relations on modal model structures or logically possible worlds.
	From: Dale Jacquette (Intro to 'Philosophy of Logic' [2002], §3)
	A reaction: He implies the multiplicity is basic, and is only 'reflected' in the relations, but maybe the multiplicity is caused by incompetent logicians who can't decide whether possible worlds really are reflexive or symmetrical or transitive in their relations.

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / b. System K

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)

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / c. System D

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)

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / g. System S4

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.

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5

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)

4. Formal Logic / D. Modal Logic ML / 4. Alethic Modal Logic

7689	The modal logic of C.I.Lewis was only interpreted by Kripke and Hintikka in the 1960s [Jacquette]
	Full Idea: The modal syntax and axiom systems of C.I.Lewis (1918) were formally interpreted by Kripke and Hintikka (c.1965) who, using Z-F set theory, worked out model set-theoretical semantics for modal logics and quantified modal logics.
	From: Dale Jacquette (Ontology [2002], Ch. 2)
	A reaction: A historical note. The big question is always 'who cares?' - to which the answer seems to be 'lots of people', if they are interested in precision in discourse, in artificial intelligence, and maybe even in metaphysics. Possible worlds started here.

4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula

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.