So we have a language. Now we need to know when it is that we say true things in our language. In particular we need to define truth so as to respect the desired readings of the modal operators K andK. The idea is that we need to specify the conditions of interpretation for formulas inLE and
in so doing we follow the ideas of Hintikka1 by developing a possible worlds model of knowledge. The idea is to represent an agent’s uncertainty as a range of possible worlds and depict incremental learning as a kind of winnowing of possible options. The more we know about the world the fewer possibilities we entertain. In the immediate, we present only the structural model, leaving further reflections about the cogency of the idea until later.
Definition (A Kripke Structure) Let At be as above, then a possible worlds model M = <W,
∼, V >, where the following conditions hold:
• W is a non-empty set of consistent possible worlds.
• ∼ is a possibly empty relation on the set of possible worlds, i.e. ∼ ⊆ W × W, called an accessibility relation, to denote which worlds are deemed possible from the current world.2
• V is a valuation function V: W7→℘(At), so as to denote which atomic propositions are true at each world.
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2
So the idea is that an agents knowledge is represented by a model. If our agent finds that she knows φ, then there is no possible world ∼-related to w, in which φ is false because to know something to be true, is to know it couldn’t be otherwise. This is the central idea behind Hintikka’s notion of knowledge. Other kinds of principles suggest themselves. You might think that you are self aware. If you do, you presumably believe that you can reflect on what you know. As such you could insist that if you know φ, you know, you knowφ. These principles can be encoded by the type of relation ∼is, for instance if we wish to ensure that knowledge entails truth we should insist that∼is a reflexive relation, and that if Kφis true at a worldw, thenφis true at the world w since w ∼w, for all worlds by the reflexivity of the relation.
Such a setting allows to attribute certain kinds of information to any particular agent by elab- orating their knowledge-state in terms of possible worlds model with more or less restraints on the relation∼. From here on out, unless we state otherwise, we shall treat∼as an equivalence relation. We tie these notions together by defining our semantics and elaborating some consequences of this stipulation:
Definition (Truth and Satisfaction) We denote a agent’s epistemic situation as a pair (M, w), where M =<W, ∼, V >. Then,
• M, w|=φiff φ∈V(w)
• M, w|=¬φiff M, w 2φ
• M, w|=φ∨ψ iff M, w |=φor M, w|=ψ.
• M, w|= Kφiff ∀w’ (w∼w’), implies M, w’|=φ.
Note that the agent always and only considers possibilities locally, that is to say the relation
∼ is always considered with respect to a particular point of consideration i.e. w above. We defined knowledge with respect to the set of all available possibilities, however the dual notion of consistency with our knowledge, i.e. Kφcan also be defined in this setting.
Kφ iff∃w0(w∼w0)and M, w’ |=φ
We often hope that the possibilities we consider are in genuine relation to the real world. If we consider a number of possible worlds, each of which validate identical knowledge claims, we might say these worlds are indistinguishable from our point of view. How can we knowingly distinguish two possible worlds if both validate everything we know to be true? We give an example where∼
is not reflexive.
Example Consider the model M =<W,∼, V>where W ={w, u, z}, and∼={(w, w),(w, u),(u, u),(u, w)}. Furthermore, assumeAt ={p} and p6∈ V(z)
u
pw
pz
¬pNow, consider any agent whose state of information is denoted by this model. Not that p is known, since all worlds∼-related validate p. This is easy to see since no∼-related world validates anything contrary. However consider the world, z and note that M, z |= ¬p. Worse recall the semantics for K, and observe that every proposition can be said to be known, since the satisfaction clause is vacuously satisfied. In particular M, z |=K⊥. This is surely problematic?
To avoid such a situation we minimally stipulate that∼is a serial relation, then we ensure the existence of some world ∼-related toz. Our model ensures that∼ is clearly both symmetric and
also vacuously transitive. In fact, it is also reflexive if we restrict our attention to w and u we cannot distinguish the worldsw and u by the propositions they validate - they effectively collapse into one another for all intensive purposes. By definition each world validates the propositional tautologies, this last remark indicates that knowledge is closed under logical consequence. Since every world is taken to be consistent all the theorems of classical logic will be valid at each world, and so known at any world whatsoever, perhaps trivially depending on the structure of the model. This is called the problem of logical omniscience.3.