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Some Analogues of the Sheffer Stroke Function in N-Valued Logic

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Some Analogues of the Sheffer Stroke Function in N-Valued Logic MA THEMA TICS SOME ANALOGUES OF THE SHEFFER STROKE FUNCTION IN n-VALUED LOGIe BY NORMAN M. MARTIN (Communicated by Prof. A. HEYTING at the meeting of June 24, 1950) 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 pro­ positional 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 func­ tionally 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) EMIL L. POST, "Introduction to a General Theory of Elementary Propositions", Am. Jl. of Math. 43, 167-168 (1921). IlOI functions and hence is functionally complete 2). In an extension to the n-valued case POST has shown that two functions, s (p), an analogue of negation, and p V q, an analogue of disjunction, will suffice 3). WEBB has proved the existence of a single function which will suffice 4). The first part of the proof of theorem 1 is equivalent to POST'S proof although it differs in details as to method. In the following, we will use the expression "n-valued SHEFFER function" to mean a two-place function which generate all the truth functions in the Logic of n truth-values 5). Theorem 1. f1.1.1 (p, q) is an n-valued SHEFFER function. The following functions (and families of functions) will be used in the pro of of the theorem: Function Truth-value properties Takes the value m when p takes the value x and q, y. Otherwise takes the value n. 2. pVq Takes the minimum of the value of pand the value of q. a+m :3. L ti (p) Takes the minimum value of i-a fa (p), fa+1 (p), ... fa+m (p). 4. Dm.i (p) Takes the value m if p takes the value i, otherwise takes n. 5. P & q Takes the maximum of the value of pand the value of q. Takes as value n + 1 minus the value of p. Takes the value m everywhere 6). 2) H. M. SHEFFER, "A Set of Five Independent Postulates for Boolean Algebras, with Application to Logical Constants", Trans. Am. Math. Soc. 14,487 -488 (1913), Properly speaking, SHEFFER did not show that the stroke function is functionally complete, but only that it can define negation and disjunction. Since POST sub­ sequently showed the latter to be functionally complete, the result follows from SHEFFER'S article. 3) POST, op. cit., p. 180-181. 4) DONALD L. WEBB, " Generation of any n-valued Logic by One Binary Operator", Proc. Nat. Acad. Sci, 21, 252-254 (1935). 5) In the remainder of this paper when a number is stated it is to be understood unIess otherwise asserted as referring to that number congruent to the stated number modulo n which is greater than zero and less or equal, to n. When truth-tables are refered to the following arrangement is presupposed (for one-place functions): The i th line of the table is that line where the argument takes the value i, (for two-place functions) if the value of the first argument is x and of the second is y then the corresponding line of the truth-table is the ith line when i = (x - 1) n + y. 8) The idea of this function is a generalisation of the T-function of SLUPECKI which together with the negation and implication of LUCASIEWICZ'S system form a functionally complete set. (Cf. note 6), "Die Logik und das Grundlagenproblem" in Gonseth, Les Entretiens de Zurich, p. 97. 70 1102 8. s (p) Takes as value the value of p plus one, I if P takes n. 9. sm (p) The m-fold application of s (p) - thus takes as value the value of p plus m. 10. 11.1.1 (p, q) Takes as value I plus the minimum of the value of pand the value of q, takes I if the value of p = the value of q = n. The family Nm.z.lJ(p,q) , m= 1,2, ... n-I; x= 1,2, ... n; y = I, 2, ... n together with the function V is functionally complete since any function (except one) can be represented by a disjunction of members of the family, where N m.z.1J (p, q) takes the value m when p takes x and q takes y: the function which takes n everywhere is the only exception and this is equivalent to Nl,l.2 (NI.l.l (p, q), N u .1 (p, q)) N mz.1J (p, q) for m> I can be defined in terms of N u .1J (p, q), - V and Tm (p) [I ~ m ~ n] as follows N m.z.1J (p, q) = T"_m+l (p) V N I.z.1I (p, q) NI.z.lJ(p, q) can in turn be defined in terms of Dl.dp) [I ~i ~n] and &: N I •rl•1J (p, q) = D I •z (p) & D I •II (q) & is to be defined by - and V by DE MORGAN'S law: p&q=pVq Accordingly, the set -, V, with the families Dl .• (p) [I ~ i ~ n] and Tm (p) [I ~ m ~ n] is functionally complete. a+l Furthermore if we define the sum of a function L I. (p) as: i.- r a L tdp) = la (p) i - a a+k+l a+k l .~ t. (p) = la+1+k (p) V .~ t. (p) we can define - in terms of V and Dm .• [I ~ m ~ n, I ~ i ~ n] .. P= L D"+l_'., (p) • • -1 With the aid of the multiple application of s (p), s' (p) (this can also be defined recursively, as follows SO (p) = p, Sk+l (p) = S (Sk (p)) ), we can define Dm .• (p) [m> I] in terms of Dl.• (p), V, and s: Dm .• (p) = sm-l (T"_m+l (p) V Dl .• (p)) ll03 Dl •i (p) and Tm (p) are likewise defineahle in terms of V, and s: "-1 Dl •l (p) = S,,-l L Si (p) i~l n-i " (for i *- I) Dl.i (p) = S,,-l ( L Si (p) V L si (p)) i-I i=,,-i+2 (for m =1= I) Tm (p) = sm-l Tl (p). Accordingly, as shown hy POST s (p) and Vare functionally complete. Both of these can he defined in terms 11.1.1 (p, q) S (p) = 11.1.1 (p, p) P Vq = S,,-l 11.1.1 (p, q) 7) We define I~.l.l (p, q) as the function which takes I if either p or q takes n and otherwise takes the maximum of the value of pand that of q, plus one. Theorem 2. I~.l,l (p, q) is an n-valued SHEFFER lunetion. This can he shown hy the following series of definitions: (E''''l (p) is the function, which takes m when p takes I, otherwise I). s (p) = I~.l.l (p, p) )SO(p)=p sm (p) = i sA'+1 (p) = S (sA' (p)) p & q = S,,-l 1~1.l (p, q) mH ( ti li (p) = Im (p) rr I; (p) ~ ,=m ,-ni ) m+k+I m+k ~ Um li (p) = Im+k+l (IJ) and l! I, (p) T" (p) = TI" Si (p) ~l 7) As in corollary to 'I'heorem 1, the LUCASIEWICZ-SLUPECKI primitives can be easily shown to be functionally complete, by the following definitions: 8 (p) = NOO TpNpNOpNp Nl.1.1 (p, q) = NOOpNqNOpNq N 2 •l •l (p, q) = 8 (NOTpNNl •l •1 (P, q)) A'pq=ONpq 11.1.1 (p, q) = A'A'A'A'A' N 2•1•l (p, q) N 2 •l •1 (8 (p), q) Z 2 N 2 •l •l (8 (p), q) N 2,1.1 (p, 8 (q)) N 2 •1•1 (p, 8 (q)) N I .I .1 (8 (p), 8(q)) 1104 for m -=I=- n, Tm (p) = sm T" (p) n-l En.l (p) = S TI S,-l (p) ,~l i for m-=I=- 1, E".m (p) = S ["ICs' (p) & ,JJm+ls (P)] Ek.m (p) = Sk (T"+l-k (p) & E".m (p)) p = TI" E,,-Hl.i (p) i - I pVq=p & q 11.1.1 (p, q) = S (pVq).
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