Review: Norman M. Martin, Some Analogues of the Sheffer Stroke Function in n-Valued Logic
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The interpretation of the ordinary (two-valued) propositional logic in terms of a truth-table system with values "true" and "false" or, more abstractly, the numbers "I" and "2" has become customary. This has been useful in giving an algorithm for the concept "analytic" , in giving an adequacy criterion for the definability of one function by another and it has made possible the proof of adequacy of a list of primitive terms for the definition of all truth-functions (functional completeness). If we generalize the concept of truth-function so as to allow for systems of functions of 3, 4, etc. values (preserving the "extensionality" requirement on functions examined) we obtain systems of functions of more than 2 values analogous to the truth-table interpretation of the usual propositional calculus. The problem of functional completeness (the term is due to TURQUETTE) arises in each ofthe resulting systems. Strictly speaking, this problem is notclosely connected with problems of deducibility but is rather a combinatorial question. It will be the purpose of this paper to examine the problem of functional completeness of functions in n-valued logic where by n-valued logic we mean that system of functions such that each function of the system determines, by substitution of an arbitrary numeral a for the symbol n in the definition of the n-valued function, a function in the truth table system of a values. A function is functionally complete in n-valued logic if for any natural number a, the substitution of a for n in the definition of the function will yield a functionally complete function for the system of truth tables (of the type described above) with a values. In the two-yalued propositional logic, the classical work was done with the use of the operations -, &, V, and -+ of which it was early discovered that and any of the other three will suffice to express anything desired. Formal proof of this has been provided by POST 1). SHEFFER showed that the stroke function can define the above mentioned
[1] D L Webb,et al. Generation of any N-Valued Logic by One Binary Operation. , 1935, Proceedings of the National Academy of Sciences of the United States of America.
[2] Henry M. Sheffer. A set of five independent postulates for Boolean algebras, with application to logical constants , 1913 .
[3] Emil L. Post. Introduction to a General Theory of Elementary Propositions , 1921 .