The semantics for modal logic are usually given like so: First we define a frame, which consists of a non-empty set, G, whose members are generally called possible worlds, and a binary relation, R, that holds (or not) between the possible worlds of G. This binary relation is called the accessibility relation. For example, w R v means that the world v is accessible from world w. That is to say, the state of affairs known as v is a live possibility for w. This gives a pair, <G, R>.
Next, the frame is extended to a model by specifying the truth-values of all propositions at each of the worlds in G. We do so by defining a relation ⊨ between possible worlds and propositional letters. If there is a world w such that w ⊨ P, then P is true at w. A model is thus an ordered triple, <G, R, ⊨>.
Then we recursively define the truth of a formula in a model:
- w ⊨ ¬P if and only if w P
- w ⊨ (P Q) if and only if w ⊨ P and w ⊨ Q
- w ⊨ P if and only if for every element v of G, if w R v then v ⊨ P
- w ⊨ P if and only if for some element v of G, it holds that w R v and v ⊨ P
According to these semantics, a truth is necessary with respect to a possible world w if it is true at every world that is accessible to w, and possible if it is true at some world that is accessible to w. Possibility thereby depends upon the accessibility relation R, which allows us to express the relative nature of possibility. For example, we might say that given our laws of physics it is not possible for humans to travel faster than the speed of light, but that given other circumstances it could have been possible to do so. Using the accessibility relation we can translate this scenario as follows: At all of the worlds accessible to our own world, it is not the case that humans can travel faster than the speed of light, but at one of these accessible worlds there is another world accessible from those worlds but not accessible from our own at which humans can travel faster than the speed of light.
It should also be noted that the definition of makes vacuously true certain sentences, since when it speaks of "every world that is accessible to w" it takes for granted the usual mathematical interpretation of the word "every" (see vacuous truth). Hence, if a world w doesn't have any accessible worlds, any sentence beginning with is true.
The different systems of modal logic are distinguished by the properties of their corresponding accessibility relations. There are several systems that have been espoused (often called frame conditions). An accessibility relation is:
- reflexive iff w R w, for every w in G
- symmetric iff w R v implies v R w, for all w and v in G
- transitive iff w R v and v R q together imply w R q, for all w, v, q in G.
- serial iff, for each w in G there is some v in G such that w R v.
- euclidean iff, for every u,v and w, w R u and w R v implies u R v (note that it also implies: v R u)
The logics that stem from these frame conditions are:
- K := no conditions
- D := serial
- T := reflexive
- S4 := reflexive and transitive
- S5 := reflexive, symmetric, transitive and Euclidean
S5 models are reflexive transitive and euclidean. The accessibility relation R is an equivalence relation. The relation R is reflexive, symmetric and transitive. It is interesting to note how the euclidean property along with reflexivity yields symmetry and transitivity. We can prove that these frames produce the same set of valid sentences as do any frames where all worlds can see all other worlds of W(i.e., where R is a "total" relation). This gives the corresponding modal graph which is total complete (i.e., no more edges (relations) can be added).
For example, in S4:
- w ⊨ P if and only if for some element v of G, it holds that v ⊨ P and w R v.
However, in S5, we can just say that
- w ⊨ P if and only if for some element v of G, it holds that v ⊨ P.
We can drop the accessibility clause from the latter stipulation because it is trivially true of all S5 frames that w R v.
All of these logical systems can also be defined axiomatically, as is shown in the next section. For example, in S5, the axioms P → P, P → P, and P → P (corresponding to symmetry, transitivity and reflexivity, respectively) hold, whereas at least one of these axioms does not hold in each of the other, weaker logics.
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