The determination of all Sheffer functions in $3$-valued logic, using a logical computer.
Eric Foxley · Notre Dame Journal of Formal Logic · 1962
This paper describes the method by which the results of a paper by Martin were used to enable the set of all Sheffer functions in 3-valued logic to be determined by the Nottingham University Logical Computer, and how the computer demonstrated that Martin's condition of co-substitution is superfluous.We will use the symbolism of Jan ^Lukasiewicz, and represent the functors of non-implication, joint denial, non-equivalence and incompatibility by the letters B, ], R, S respectively.The conditioned disjunction [p, q, r] will be written Dqpr, the change in the order of the variables being made as a matter of convenience.We will abbreviate any functor F written consecutively i times by F z .To solve problems in 3-valued logic on a 2-state computer, we represent each 3-valued variable by two 2-valued variables.The truth-values 1, 2, 3 of a variable p are represented by the assignments T, T; T, F; F, F respectively to the two 2-valued variables p^ and p 2The pair FT is taken as meaningless.(If we were to follow exactly a method already suggested by Rose, we would represent the truth-value 3 by both of the pairs FT and FF.If this method is adopted, any solution to the problem in which the truth-value 3 appears i times will be found in 2 1 different 2-valued forms.To avoid recording this solution 2 Z times, one of the pairs FT, FF must be chosen to represent the truth-value 3 in solutions, and the recording mechanism must be inhibited whenever any of the variables to be recorded is represented by the other pair.There is now no point in defining the latter pair to represent any particular 3-valued truth-value, since the results obtained when this pair occurs are not recorded.The pair FF was chosen to represent the truth-value 3 in solutions, since the pairs TT, TF, FF are more easily distinguished from each other in written solutions than the pairs TT, TF, FT.)For any 3-valued two-variable functor Fpq we will define r, s, t, . . ., z to be propositions taking the truth-values of Fll, F12, F13, F21, . . ., F33, where 1, 2, 3 are logical constants taking the truth-values 1, 2, 3 respectively.We will consider each of Martin's conditions in turn.