A Kripke frame (or simply a frame) is a pair where is a non-empty set whose elements are called worlds or possible worlds, is a binary relation on called the accessibility relation. When , we say that is accessible from . A Kripke frame is said to have property if has the property . For example, a symmetric frame is a frame whose accessibility relation is symmetric.
A Kripke model (or simply a model) for a propositional logical system (classical, intuitionistic, or modal) is a pair , where is a Kripke frame, and is a function that takes each atomic formula of to a subset of . If , we say that is true at world . We say that is a -model based on the frame if is a model for the logic .
Remark. Associated with each world , we may also define a Boolean-valued valuation on the set of all wff’s of , so that iff . In this sense, the Kripke semantics can be thought of as a generalization of the truth-value semantics for classical propositional logic. The truth-value semantics is just a Kripke model based on a frame with one world. Conversely, given a collection of valuations , we have model where iff .
Since the well-formed formulas (wff’s) of are uniquely readable, may be inductively extended so it is defined on all wff’s. The following are some examples:
in classical propositional logic PL, , where ,
in the modal propositional logic , , where , and , and
in intuitionistic propositional logic PL, , where , and .
Truth at a world can now be defined for wff’s: a wff is true at world if , and we write
if no confusion arises. If , we write . The three examples above can be now interpreted as:
means implies in PL,
means for all worlds with , we have in , and
means for all worlds with , implies in PL.
A wff is said to be valid
in a model if in true at all possible worlds in ,
in a frame if is valid in all models based on ,
in a collection of frames if is valid in all frames in .
if is valid in , or respectively.
A logic , equipped with a deductive system, is sound in if
we say that is complete in .
|Date of creation||2013-03-22 19:31:15|
|Last modified on||2013-03-22 19:31:15|
|Last modified by||CWoo (3771)|