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4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / b. Terminology of ML

[definitions of the main concepts used in modal logic]

6 ideas

14684	A world is 'accessible' to another iff the first is possible according to the second [Salmon,N]
	Full Idea: A world w' is accessible to a consistent world w if and only if w' is possible in w. Being 'inaccessible to' or 'possible relative to' a consistent world is simply being possible according to that world, nothing more and nothing less.
	From: Nathan Salmon (The Logic of What Might Have Been [1989], IV)
	A reaction: More illuminating than just saying that w can 'see' w'. Accessibility is internal to worIds. It gives some connection to why we spend time examining modal logic. There is no more important metaphysical notion than what is possible according to actuality.

9734	Modern modal logic introduces 'accessibility', saying xRy means 'y is accessible from x' [Fitting/Mendelsohn]
	Full Idea: Modern modal logic takes into consideration the way the modal relates the possible worlds, called the 'accessibility' relation. .. We let R be the accessibility relation, and xRy reads as 'y is accessible from x.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.5)
	A reaction: There are various types of accessibility, and these define the various modal logics.

9736	A 'model' is a frame plus specification of propositions true at worlds, written < G,R,||- > [Fitting/Mendelsohn]
	Full Idea: A 'model' is a frame plus a specification of which propositional letters are true at which worlds. It is written as , where ||- is a relation between possible worlds and propositional letters. So Γ ||- P means P is true at world Γ.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6)

9735	A 'frame' is a set G of possible worlds, with an accessibility relation R, written < G,R > [Fitting/Mendelsohn]
	Full Idea: A 'frame' consists of a non-empty set G, whose members are generally called possible worlds, and a binary relation R, on G, generally called the accessibility relation. We say the frame is the pair so that a single object can be talked about.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6)

9741	Accessibility relations can be 'reflexive' (self-referring), 'transitive' (carries over), or 'symmetric' (mutual) [Fitting/Mendelsohn]
	Full Idea: A relation R is 'reflexive' if every world is accessible from itself; 'transitive' if the first world is related to the third world (ΓRΔ and ΔRΩ → ΓRΩ); and 'symmetric' if the accessibility relation is mutual.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.7)
	A reaction: The different systems of modal logic largely depend on how these accessibility relations are specified. There is also the 'serial' relation, which just says that any world has another world accessible to it.

13727	A 'constant' domain is the same for all worlds; 'varying' domains can be entirely separate [Fitting/Mendelsohn]
	Full Idea: In 'constant domain' semantics, the domain of each possible world is the same as every other; in 'varying domain' semantics, the domains need not coincide, or even overlap.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 4.5)