The Four Types of The 'Accessibility Relation' in Formal Semantics
'Formal semantics' studies the meaning of statements written in symbols. The sentence, for example, is a statement about 'necessity' in 'formal semantics.' It has a meaning that can be represented by the symbol where takes the form of the 'necessity relation' described below.
So, the 'accessibility relation,' can take on at least four forms, that is, the 'accessibility relation' can be described in at least four ways.
Each type is either about 'possibility' or 'necessity' where 'possibility' and 'necessity' is defined as:
- (TS) Necessarily means that is true at every 'possible world' such that .
- Possibly means that is true at some possible world such that
The four types of will be a variation of these two general types. They will specify on what conditions a statement is true either in every possible world, or some possible. The four specific types of are:
'The Reflexive' or *Axiom (T):
This says that if every world is accessible to itself, then any world in which is true will be a world from which there is an accessible world in which is true. Notice this is a variation, more detailed description of the 'necessity' definition above.
'The Transitive' or *Axiom (4) above:
This says that is transitive, is true at a world only when is true at every world accessible from Hence, is true at a world only when is true at every world accessible from every world accessible from . Notice that this is a variation, more detailed description of the 'possibility' definition above.
'The Euclidean' or *Axiom (5) above:
This says that is euclidean. So, is true at a world if and only if is true at some world accessible from is true at a world if and only if, for every world accessible from, there is a world accessible from at which is true.
The euclidean property guarantees the truth of this. If is true at a world accessible from, then if that world is accessible from every other world accessible from, it will be true that for every world accessible from there is an accessible world in which is true. Notice that this is a variation, more detailed description of the 'necessity' definition above.
'The Symmetric' or *Axiom (B) above:
This says that is symmetric. If is true in a world, then in every world accessible from, there is a world accessible from in which is true. Since is true in, this is guaranteed to be true provided that is accessible from it, which is precisely what symmetry says.
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