Note on an independence proof of Johansson.
Ervin Nemesszeghy · Notre Dame Journal of Formal Logic · 1976
In [1], p. 124, I. Johansson proves that the propositional formula Π(ΊlαDfl) is underivable in his minimal logic.He establishes this result by the well-known matrix-method: he gives certain matrices in which all the axioms of the minimal logic are valid, the rules of the system preserve validity, but Ίl(lΊβ^α) is invalid.The matrices he uses are 5x5 matrices, i.e., of 5 rows and 5 columns for the binary connectives.The purpose of this short note is to point out that there are simpler 3x3 matrices which do the same job.The matrices for the connectives are given below.The only designated value is 1.It is easy to check that all the axioms of the minimal logic are valid in these matrices, and the rules of the system preserve validity; yet Π(ΊΊ a D a) is invalid, for if the value of 'a' is 3 then 11 (11 3^3) = ΊΊ(2 z> 3) = 113 = 2 Φ 1.