We shall consider the notion of falsehood as a primitive one which will be used as a logical operator in this formal system. Sentences with the implication will be evaluated in standard way. The notions of truth, falsehood and implication are interdependent. Falsehood and implication will be considered as basic logical connectives in FL4. Language for FL4.
Every sentential variable is wff.
The designated value is T. Note that truth-values of the intended interpretation of FL4 are similar to those of Belnap's logic. We used Belnap's symbols in our truth-value tables for initial and derivative operators and connectives. Theorem about subalgebras. Interrelations between FL4 and four-valued Belnap's logic and von Wright's truth logic. Due to Belnap, if A entails B, then the entailment preserves truth and preserves non-falsity.