Knowledge and belief

One of the common uses of modal logic, apart from use in the discussion of logical possibility and necessity, is to provide a logic for knowledge and a logic for belief. These logics have practical applications in artificial intelligence, especially in knowledge representation.

Logics for knowledge are epistemic logics, and logics for belief are doxastic logics. Nevertheless, the term “epistemic logic” is often taken to encompass both epistemic and doxastic logics. One of the first twentieth-century suggestions for a logic for knowledge came from Lemmon in his paper, “Is There Only One Correct System of Modal Logic?” We have already noted that Lemmon was one of the great axiomatizers of modal logic, and he presented his epistemic logic in terms of the axiomatic system S0.5 (“S nought point 5”).

The first comprehensive text in epistemic and doxastic logic, Knowledge and Belief, was published by Hintikka in 1962. This work has become a classic. Hintikka made no explicit use of possible worlds as such in the text. He used model sets instead of possible worlds. Model sets are consistent sets of sentences. He set out consistency conditions for these sets of sentences, conditions such as:

(A. ˜) If p ∈ λ and “˜ p” ∈ λ, then λ is inconsistent.

(A. &) If λ is consistent and if “p & q” ∈ λ then λ + {p, q} is consistent.