Weak deduction mechanisms

The obvious strategy to solve the logical omniscience problem is to weaken epistemic logic. One denies the universal validity of the mentioned inference rules (NEC), (MON), and (CGR), or one of the essential axioms like (K). In fact, almost all attempts to solve the LOP have in common that they consider (families of) systems that are weaker than the standard modal epistemic logics in the following sense. Firstly, not all theorems of modal epistemic logic are provable in those systems. Secondly, the set of formulae known by an agent at a state is not necessarily closed under the laws of the propositional calculus: a formula $\beta$ may be provable from $\avec{\alpha}{,}{n}$ by using axioms and inference rules of the propositional calculus, but $K_i\beta$ cannot be derived from $\avec{K_i\alpha}{,}{n}$ within the epistemic logic under consideration.

To construct such weak systems we can postulate, for example, that the agent only knows some ``obvious'' logical truths, but not necessarily the ``more complicated'' ones. We can assume that the agent can draw all ``obvious'' consequences, but not any arbitrary consequence of a certain sentence. This is achieved by postulating that the deduction mechanism of the agents is not complete, that is, it is not powerful enough to allow the agents to draw all logical consequences of their knowledge ([Hin70], [Ebe74], [Ste84], [Kon86], [Wut91], [GG93]). If an agent's inference mechanism is kept very weak, then logical omniscience could be avoided.

Certain systems of modal logic are able to characterize agents whose inference mechanisms are weaker than propositional consequence. For example, non-normal systems can be used for describing an agent who does not know all logical truths, and non-monotonic modal logics can model agents whose knowledge is not closed under logical consequence^3.1. If a very weak modal logic (e.g., a classical system) is employed to model knowledge, then most versions of the LOP are solved: neither the necessitation rule (NEC), nor the monotony rule (MON), nor the axiom schema (K) is valid in a weak classical system. However, some weaker versions of the LOP still remain unsolved. All classical modal logics are closed under the congruence rule (CGR), so an agent described by such a modal system knows all logical equivalences of a sentence that he knows. Such a closure property is obviously too strong for real agents.

Another group of attempts to gain control over the LOP is to consider nonstandard logics to model agents' reasoning. The intuitive idea is as follows. Modal epistemic logic assumes that agents use classical logic (or more accurately, some extension of classical logic) in their reasoning. This causes logical omniscience because the notion of logical consequence defined by classical logic is too powerful, i.e., too much can be inferred from some base of knowledge. In particular, all tautologies are known because classical logic allows to derive them from the empty set. Hence, if the notion of logical consequence is restricted so that not all classical consequences can be drawn then certain forms of omniscience can be avoided. Such a restriction can be achieved by employing a nonstandard logic. Among the non-classical logics that have been employed for that purpose are several variants of relevance logic ([Lev84], [FHV95]) and many-valued logics ([Ho93]).

Although the approaches based on nonstandard logics solve certain forms of the LOP, they cannot eliminate the LOP completely. The agents described by those logics are not logically omniscient wrt classical logic, but they are omniscient wrt to some (nontrivial) non-classical logic. Such attempts cannot be considered satisfactory solutions to the LOP. Consequently, they are not suitable for characterizing explicit knowledge. In general, any logic that cannot model what is explicitly available to the agents but only information that must be inferred using some -- possibly incomplete -- deduction mechanism must be viewed as a logic of implicit knowledge^3.2.