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 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) 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
a+m
:3.
L
i-a
Takes the minimum of the value of pand the
value of q.
ti (p)
Takes the minimum value of
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 subsequently 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) value the value of p plus m.
10.
11.1.1
thus takes as
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.
(p, q)
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), and Tm (p) [I ~ m ~ n] as follows
V
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
Tm (p) [I ~ m ~ n] is functionally complete.
Furthermore if we define the sum of a function
~
a+l
L I. (p)
i
~
n] and
as:
i.- r
a
L tdp) = la (p)
i- a
l
a+k+l
.~ t. (p)
=
a+k
la+1+k (p) V .~ t. (p)
we can define - in terms of V and Dm.• [I
~ m ~ n,
I
~
i
~ n]
.
P= • L-1 D"+l_'., (p) •
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
D l •i (p) and Tm (p) are likewise defineahle in terms of V, and s:
D l •l (p) =
(for i
*- I)
"-1
S,,-l
n-i
Dl.i (p) =
S,,-l (
L
L
Si
(p)
i~l
"
L
Si (p) V
si
i=,,-i+2
i-I
(p))
(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) =
P Vq =
11.1.1 (p, p)
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) =
sm (p) =
p & q=
mH
rr I; (p)
,-ni
I~.l.l (p, p)
)SO(p)=p
i sA'+1 (p) = S (sA' (p))
S,,-l 1~1.l
(p, q)
ti li (p) = Im (p)
~ Um li (p) = Im+k+l (IJ)
(
~
,=m
) m+k+I
T" (p) =
m+k
and
l! I, (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)
=
Nl.1.1
NOO TpNpNOpNp
(p, q)
=
N 2 •l •l (p, q) =
NOOpNqNOpNq
8
(NOTpNN l •l •1 (P, q))
A'pq=ONpq
11.1.1 (p, q) = A'A'A'A'A' N
N 2 •l •l (8 Z (p), q) N 2,1.1 (p,
N I .I .1 (8 (p), 8(q))
8
2 •1 •l
(p, q) N 2 •l •1 (8 (p), q)
(q)) N 2 •1 •1 (p,
82
(q))
1104
for m
-=I=-
n, Tm (p) = sm T" (p)
n-l
En.l
(p) =
TI S,-l (p)
S
,~l
for
m
-=I=-
1, E".m (p)
=
S
["ICs'
Ek.m (p) =
&
(p)
Sk (T"+l-k
,JJm+ls (P)]
i
(p)
&
E".m (p))
" E,,-Hl.i (p)
p = TI
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 "successor" 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 gi (p) is defined recursi vely :
i =m
m
cp
g, (p) = gm (p) and
m+k+l
cp
i=m
'Î =m
,,-1
then GLcl (p) =
cp
i~
Si (p) takes d
1
m+k
g, (p) = 11,d,o (g""+k+l (p), cp g, (p)),
i=m
+1
on line 1 and 1 elsewhere, and
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 W l (p) = p, can be defined.
Then
sW2 Il,cl,o (W l (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).
is the function which takes the value i
the value i and otherwise takes the value c.
ba,c (p, q)
11 a, is relatively prime to n, ba • c (p, q) is an n-valued
lunction.
Theorem 6.
SHEFFER
+ a, if pand q take
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:
(i ~ 2)
Ao (p, q) = bl .n (p, sn-l (q))
Al (p, q) = bl .n (N l .n.n (p, q), T n (p, q))
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 (M h (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).
'rhe part of the proof of theorem 6 above is due to WEBB, loc. cito
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 twoarguments 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:
8)
9)
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).