On the infinity of positive logic.
Ivo Thomas · Notre Dame Journal of Formal Logic · 1962
No direct proof seems to have been published of the theorem that no finite matrix can be adequate to the positive logic of implication, which is here proved.In the alphabet p lf p 2 , . . .,p n , {n > 1), form (p { D p f ) D p. for all i, /: 1 S i < / 5s n, and p^ D p x for all i: 1 < i <n.A n is to have all these expressions as antecedents, p x as consequent.Then A n is not a positive thesis but becomes so if any two variables are identified.Hence any n-I valued matrix that validates the positive system, validates A n » Hence no finite matrix is adequate to the positive system.Proof: That for no n is A n positive is shown by the fact that if the variables are valued by their subscripts A n has the value 1 in the infinite matrix of Dummett's LC for which cf. [l].While if any two variables are identified, there results either an antecedent equivalent in the positive system to p l9 or a pair of antecedents ρ {y p { D p l9 the consequent being always p x .