We use a language L in which we have the propositional connectives A (and), V (or), and -i (not) as primitive. Alternatively we could take some as primitive and define others via the usual definitions, which work even in Kleene's three-valued logic. We also allow quantifiers V and 3, taking both as primitive. Quantifiers will not be used in the actual writing of programs, only in their analysis.
We systematically use statement for formula with no free variables. For a formula P, and a substitution 0, P6 is a closed instance of P if it is a statement. We sometimes write P{x) to indicate that x is the only free variable of P, and P{t) for the closed instance resulting from substituting t for x.
The idea behind Kleene's logic is simple. Say P is a statement with the informal meaning /(3) = 5, where / is a function we have a Turing machine to compute. Similarly say Q means g(4) = 7, where g is another Turing calculable function. By running the Turing machines we may determine the truth or falsity of P and Q, or we may never learn anything if the machines fail to halt. So, use a logic with truth values t (true), f (false), and u (undefined or undetermined) to mirror the situation.
In this logic, how should values be assigned to P A Q, say? Certainly in cases not involving u, truth values should behave classically. Now say P has value f but Q is u. Still Q " has" a truth value; we just don't know what it is. Since P isi, P A Q will be f no matter whether Q turns out to be t or f. Consequently P A Q is given the value f in this case. On the other hauid, say P has value t but Q is u. If we eventually discover that Q i&t, P A Q will turn out to have value t. But if we discover Q is f, P A Q will be f. Given no further information then, we must say the value of P A Q is u.
Then a table for the connective A is as follows: