On Słupecki T-functions
Citations Over Time
Abstract
In [2] E. L. Post defined a many-valued propositional logic to be functionally complete if, for every function on the set of truth-values, there exists a formula of the logic having that function as its associated truth-value function. He proved that the logic with truth-values 1, 2, …, m and (i) a unary connective ∼ such that ∼p has truth-value i +1 (mod m ) when p has truth-value i , (ii) a binary connective ∨ such that p ∨ q has truth-value min( i, j ) when p, q have truth-values i, j respectively, is functionally complete. The many-valued logics described by Łukasiewicz and Tarski [1] are not functionally complete. These logics have truth-values 1, 2, …, m and (i) a unary connective ~ such that ~p has truth-value m −i+1 when p has truth-value i , (ii) a binary connective → such that if p, q have truth-values i, j respectively, then p → q has truth-value 1 for i ≧ j , and truth-value 1 for i ≧ j . The functional incompleteness of these logics is immediate, since there exists no formula in p having truth-value i (≠ 1 or m ) when p has truth-value 1. In [4] Słupecki showed that if a new unary connective T, such that T(p) has truth-value 2 for all truth-values assigned to p, is added to the 3-valued Łukasiewicz-Tarski logic, then the resulting logic is functionally complete. In [3] Rosser and Turquette proved this result for the m -valued ( m ≧ 3) logic.
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