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

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). On the basis of the isomorphism of s (p) in n-valued logic to the "suc cessor" function for congruence classes modulo n, we can easily obtain a considerably larger number of SHEFFER functions.

Theorem 3. 11 a and nare relatively prime, 11,1.a (p, q) and 1~.l.a (p, q) are n-valued SHEFFER lunctions. 11,1.a (p, q) is the function which takes as value the minimum of the values of pand q, plus a (n + a being considered as equal to a). I~.l.a (p, q) takes the maximum of the values, plus a. Sa (p) = 11,1.a (p, p). By EULER'S theorem S~",(")-l (p) takes as value the value of p plus one. Hence

s (p) = S~'I'<")-l (p). Thus 11,1.1 (p, q) = s,,-a+l Il.l.a (p, q).

Analogously for I~,l.a (p, q). hu (p, q) is the function which takes i + a if por q takes i and takes i + a + b, if pand q do not take any j [i ~ j < i + b ~ i-I] and p or q takes i + b (where n + c is taken as equal to c). t~.l,a (p, q) is the function which takes i + a if p or q takes i and takes i + a - b if pand q do not take any j [i + 1 ~ i - b < j ~ i] and p or q take i - b (where - c is taken as equal to n - c).

Theorem 4, 11 a and nare relatively prime, lu,a (p, q) and I~,l,a (p, q) are n-valued SHEFFER lunctions. Sa (p) = lu,a(P, p), By EULER'S theorem ~'P(n)-l (p) = S (p). Then lu,a (Si-l (p), S,-l (q)) = Il,l.i+a-l (p, q), sn+2-(a+i) 11,1.i+a-l = 11,1,1 (p, q). Analogously for IL1.a (p, q). Il.d,a (p, q) is the function which takes a + 1 if P or q takes 1; if not and por q takes d + 1, Il,d.a (p, q) takes a + d + 1; if not but either take 2 d + 1, Il,d.a (p, q) takes a + 2 d + 1, etc, until md + 1 where md is the greatest multiple of d less than n; if neither p nor q take any of these but one of them takes 2, I Ua (p, q) takes a + 2; if not and either takes d + 2, Il,d,a (p, q) takes a + d + 2 etc. until Xd + 2, where n - d < Xd + + 2 ~ n; if neither p nor q take any of these, but one of them takes 3. ll05

Il,cl,a (p, q) takes a, + 3 etc, li,cl,a (p, q) is the result of n + I-i applications of s (p) to the arguments of Il,cl,a (p, q), If in the definition of Il,cl,a (p, q) in place of starting with 1 and adding units of d with n as a limit, we start with n and subtract units of d with 1 as the limit, we obtain the function I~,cl,a (p, q), ILd,a (p, q) is the result of n - i applications of s (p) to the arguments of I:',cl,a (p, q),

Theorem 5, 11 dis not a divisor ol n and a, is relatively prime to n, li,cl,a (p, q) and I;,cl,a (p, q) are n-valued SHEFFER lunctions, s (p) can be defined as in theorem 4, Then li,cl,a (S'-l (p), Si-l (q)) = = 11,cl,i+a-l (1), q), 11,cl,o (p, q) = s,,-(i+a) 1I,'I,i +a (p, q), m+k If cp g i (p) is defined recursi vely : i =m m m+k+l m+k cp g, (p) = gm (p) and cp g, (p) = 11,d,o (g""+k+l (p), cp g, (p)), 'Î =m i = m i = m ,,-1 then GLcl (p) = cp Si (p) takes d + 1 on line 1 and 1 elsewhere, and i~ 1 K (p) = Gl,cl(Gl,cl(P)) which takes 1 on line 1 and d + 1 elsewhere, (a) Case 1. There is an x such that n = xd - 1.

Gl (p) = (sn-cl Gl,cl (p)) takes 1 on line 1 and n - d + 1 elsewhere.

Then Gk (p) = Sk-I Il,d,O (Gl (p), T,,+2-(clH) (p)) takes k on line 1 and n - d + 1 elsewhere. Define H k.m (p) as follows: Hk,l (p) = Gk (p) and Hk.m (p) = Gk (sm-l (p)) for m =1= 1. Thus Hk.m (p) takes kif P takes mand takes n - d + 1 otherwise. Any one-place function can be defined by the H-functions combined with Il,cl,o (p, q) as is obvious, since if p takes n - d + 1 and q takes j, Il,d.o (p, q) takes j and ILcl,o is symmetrical. Hence also the function Wl (p) which takes 1 on line 1, d + 1 on line 2, etc. and W2 (p) the function such that Wz Wl (p) = p, can be defined. Then

sW2 Il,cl,o (Wl (p), W l (q)) = 11.1.1 (p, q). (b) Case 2. There are numbers x and y (1 < y < d) su eh that n = xd -y. sn+ 1I -(cl+l) K (p), takes n - d + y on line 1 and y elsewhere. H",! (p) = sn-d Il.d.o (s"+tHcl+U K (p), T~ (p)) takes n on line 1 and n - d + y elsewhere. G1 (p) = S/l.cl.o (Hn,! (p), T"+1I-(cl+!) (p)) takes 1 on line 1 and n - d + y elsewhere. The remainder of the: proof is analogous to case 1 after the definition of Gl . Similarly, the proof for ILcl.a (p, q) is analogous to the proof for li,cl,a (p, q). ba,c (p, q) is the function which takes the value i + a, if pand q take the value i and otherwise takes the value c.

Theorem 6. 11 a, is relatively prime to n, ba•c (p, q) is an n-valued SHEFFER lunction. 1106

s (p) = bl.1O (p, p). T 10 (p, q) = bl.1O (p, S (p)).

N u .l (p, q) = bl.1O (s1O-l bl.1O (p, q), T n (p, q)).

Then N l.x.v can be defined as Nl.".v {p, q) = Nl.l.l (sn+l-x (p), sn+l-V (q)). N;.x.1I= bI ... (Si-2 Nl,x.v(p,q),Si-lTn(p,q): We define the following functions:

Ao (p, q) = bl.n (p, sn-l (q)) Al (p, q) = bl.n (Nl.n.n (p, q), T n (p, q)) (i ~ 2) Ai (p, q) = Ao (Li-l (p, q), S (Mi-l (p, q)). where

Mo (p, q) = T n (p, q) M h (p, q) = Ah (Mh-l (p, q), Ah (Nh.n.h+l (p, q), Nh.h+l.n (p, q)) Lh (p, q) = Ah (Mh (p, q), N l.n.n (p, q).

By induction for every i, [0 < i ~ n - 1] if pand q take n then Ai takes n and if p takes the value j [0 < j ~ i] and q, n or if p takes n and q, j then Ai (p, q) takes the value j. Then the remarks made in theorem 1 with respect to the completeness of the family Ni.".!1 (p, q) together with V, hold when An-l is substituted for V. Therefore bl.n(p, q) is a

SHEFFER function 8). Then bl •c (p, q) is a SHEFFER function for any c sin ce s (p) = bl.c (p, p) and bl.n (p, q) = sn- C bl,c ((SC (p), SC (q)). If a and n are relatively prime then Sa (p) = ba.c (p, p) and by EULER'S theorem there is an X such that s (p) = s~ (p). Then sn+l-a ba.c= bl.c+n +l-a (p, q) which is a member of the family bl •c (p, q) and hen ce has been shown to be a SHEFFER function. The present position of the investigation is such that the above theorems certainly do not exhaust all the SHEFFER functions. On the contrary, the present investigator is aware of many special cases which do not fall under the theorems, SWIFT has recently discovered 90 in the 3-valued case while the above theorems give only 18 9). In the present state, the writer does not feel in a position even to conjecture as to what may be the necessary conditions (aside from the defining condition) for a function to be a SHEFFER function 10).

8) 'rhe part of the proof of theorem 6 above is due to WEBB, loc. cito 9) J. D. SWIFT, "Analogues of the SHEFFER Stroke in the 3-valued Case", Bull. Am. Math. Soc. 56, 63 (1950). 10) It is to be noted that any function which can define all functions of two arguments will also define any function of any number of arguments. For 11.11 (p, q) this can be done by constructing the appropriate analogues of the N-functions by taking the logical product of the appropriate D-functions for the number of arguments desired (e.g. the function which takes i if p, q and r take x, y respectively, and n otherwise can be defined as:

Di." (p) & Di.!I (q) & Di•• (r). Since all other functions proved to define all two-place functions also define

11•1•1 (p, q), the result follows. 1107

BIBLIOGRAPHY

1. LUCASIEWICZ, JAN, "Die Logik und das Grundlagenproblem" in Gonseth, Les EntretienIJ de Zurich BUr les FondementB et la Méthode des Sciences Mathématiques, Zurich, 82-100 (1941). 2. POST, EMIL L., "Introduction to a General Theory of Elementary Propositions", Am. Jl. of Math. 43, 163-185 (1921). 3. SHEFFER, H. M., "A Set of Five Independent Postulates for Boolean Algebras, with Application to Logical Constants", TranIJ. Am. Math. Soc. 14, 481-488 (1913). 4. SWIFT, J. D., "Analogues of the Sheffer Stroke Function in the three·valulld Case", Bul. Am. Math. Soc. 56, 63 (1950). 5. WEBB, DONALD L., "Generation of any n-valued Logic by One Binary Operator", Proc. Nat. Acad. Sei., 21, 252-254 (1935).