Let A, B be propositional formulas. The equal sign ‘=’ shall be interpreted according to Kleene’s strong tables. (Kleene’s strong tables have three values: true, false, unknown, denoted as T, F, x.)

Definition: A propositional formula L is a tautology iff it evaluates as true on all combinations of T, F assignments to the propositional variables occurring in L (i.e. x is not considered.)

There are tautologies such that they evaluate as true on all combinations of T, F assignment to a proper subset of the variables occurring in L. These propositional variables are called truth-determining. The rest of the propositional variables (truth-redundant variables) could be x yet L will still evaluate as T.

Theorem 4.1 Let L(P1, P2, … Pn) be any tautology such that P1, P2, … Pn are all the variables occurring in L, and let our connectives be interpreted according to Kleene’s strong tables. Then L is equivalent to

(R1 ⋁ ~R1) & (R2 ⋁ ~R2) & … & (Rm ⋁ ~Rm) (4.1)

if and only if {Ri} is a set of truth-determining variables occurring in L. Naturally {Ri} ⊆ {Pi}.

Proof: First we prove that {Ri} contains all the truth-determining variables. So suppose that there exists a truth-determining variable G occurring in L but not in {Ri}. Suppose further that for all Ri, |Ri| = T or |Ri| = F, and |G| = x. Then the truth value of (4.1) is T but the truth value of L is x. Now suppose that {Ri} contains a redundant variable H, and |H| = x, all other variables in {Ri} are T or F. Then the truth value of (4.1) is x, but the truth value of L is T.

Truth-relevant Logic - Propositional Calculus

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