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Lukasiewicz's Many-Valued Logic - Arithmetic Semantic

 

Kahtan H. Alzubaidy

 

 

 

Introduction

  Procedures are presented to evaluate propositions and to check tautologies in Lukasiewicz's n-valued logic. Atomic propositions are represented

by numerical variables x,y,z,w and logical connectives by suitable functions. Evaluations and checking are done for compound propositions consisting of at most four atomic propositions.

 

 

> 

restart
> 

with(ListTools):
> 

NULL

cartesian*product*of*two*lists

 

> 

cp2:=proc(K,L)
> 

[seq(seq([x,y],y in L),x in K)];
> 

end proc;

proc (K, L) [seq(seq([x, y], `in`(y, L)), `in`(x, K))] end proc

 (1)
> 

``
> 

NULL

cartesian*product*of*three*lists

 

> 

cp3:=proc(K,L,M)
> 

local G;
> 

G:=cp2(cp2(K, L), M);
> 

[seq(Flatten(J), `in`(J, G))];
> 

end proc;

proc (K, L, M) local G; G := cp2(cp2(K, L), M); [seq(ListTools:-Flatten(J), `in`(J, G))] end proc

 (2)

cartesian*product*of*four*lists

 

> 

cp4:=proc(K,L,M,N)
> 

local G;
> 

G:=cp2(cp3(K, L, M),N);
> 

[seq(Flatten(J), `in`(J, G))];
> 

end proc;

proc (K, L, M, N) local G; G := cp2(cp3(K, L, M), N); [seq(ListTools:-Flatten(J), `in`(J, G))] end proc

 (3)
> 

NULL
> 

NULL
> 

NULL

n*truth*values

 

> 

V:=proc(n)
> 

[seq(r/(n-1),r=0..n-1)];
> 

end proc;

proc (n) [seq(r/(n-1), r = 0 .. n-1)] end proc

 (4)
> 

V(6);

[0, 1/5, 2/5, 3/5, 4/5, 1]

 (5)
> 

cp2(V(3), V(3));

[[0, 0], [0, 1/2], [0, 1], [1/2, 0], [1/2, 1/2], [1/2, 1], [1, 0], [1, 1/2], [1, 1]]

 (6)
> 

cp3(V(3), V(3), V(3));

[[0, 0, 0], [0, 0, 1/2], [0, 0, 1], [0, 1/2, 0], [0, 1/2, 1/2], [0, 1/2, 1], [0, 1, 0], [0, 1, 1/2], [0, 1, 1], [1/2, 0, 0], [1/2, 0, 1/2], [1/2, 0, 1], [1/2, 1/2, 0], [1/2, 1/2, 1/2], [1/2, 1/2, 1], [1/2, 1, 0], [1/2, 1, 1/2], [1/2, 1, 1], [1, 0, 0], [1, 0, 1/2], [1, 0, 1], [1, 1/2, 0], [1, 1/2, 1/2], [1, 1/2, 1], [1, 1, 0], [1, 1, 1/2], [1, 1, 1]]

 (7)
> 

cp4(V(2), V(2), V(2), V(2));

[[0, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0], [0, 0, 1, 1], [0, 1, 0, 0], [0, 1, 0, 1], [0, 1, 1, 0], [0, 1, 1, 1], [1, 0, 0, 0], [1, 0, 0, 1], [1, 0, 1, 0], [1, 0, 1, 1], [1, 1, 0, 0], [1, 1, 0, 1], [1, 1, 1, 0], [1, 1, 1, 1]]

 (8)

 

minimum function

 

> 

m:=proc(x,y)
> 

(x+y)/2 -abs(x-y)/2;
> 

end proc;

proc (x, y) (1/2)*x+(1/2)*y-(1/2)*abs(x-y) end proc

 (9)
> 

NULL

maximum function

 

> 

M:=proc(x,y)
> 

(x+y)/2 +abs(x-y)/2;
> 

end proc;

proc (x, y) (1/2)*x+(1/2)*y+(1/2)*abs(x-y) end proc

 (10)
> 

NULL

   logical connectives

 

negation

> 

NULL
> 

n:=proc(x)
> 

1-x;
> 

end proc;

proc (x) 1-x end proc

 (11)
> 

NULL

   conditional

 

> 

i:=proc(x,y)
> 

 m(1,1-x+y);
> 

end proc;

proc (x, y) m(1, 1-x+y) end proc

 (12)
> 

NULL

disjunction

 

> 

o:=proc(x,y)
> 

 M(x,y);
> 

end proc;

proc (x, y) M(x, y) end proc

 (13)
> 

NULL

conjunction

 

> 

a:=proc(x,y)
> 

 m(x,y);
> 

end proc;

proc (x, y) m(x, y) end proc

 (14)
> 

NULL

  biconditional

 

> 

e:=proc(x,y)
> 

 1-abs(x-y);
> 

end proc;

proc (x, y) 1-abs(x-y) end proc

 (15)
> 

NULL

valuation*of*a*proposition*involving*one*atomic*proposition

   n is the number of truth values.

 

> 

val1:=proc(n,f)
> 

local W;
> 

W:=V(n);
> 

seq([[x],f(x)],x in W);
> 

 

> 

end proc;

proc (n, f) local W; W := V(n); seq([[x], f(x)], `in`(x, W)) end proc

 (16)
> 

``

     An¬A

 

> 

f := proc (x) options operator, arrow; o(x, n(x)) end proc;

proc (x) options operator, arrow; o(x, n(x)) end proc

 (17)
> 

val1(2, f);

[[0], 1], [[1], 1]

 (18)
> 

val1(3, f);

[[0], 1], [[1/2], 1/2], [[1], 1]

 (19)
> 

NULL

valuation*of*a*proposition*involving*two*atomic*propositions

 

> 

val2:=proc(n,f)
> 

local W,VV;
> 

W:=V(n);
> 

VV:=cp2(W,W);
> 

seq([z,f(z[1],z[2])],z in VV);
> 

 

> 

end proc;

proc (n, f) local W, VV; W := V(n); VV := cp2(W, W); seq([z, f(z[1], z[2])], `in`(z, VV)) end proc

 (20)
> 

``

   (A→ B) n (B → A)

 

> 

g := proc (x, y) options operator, arrow; o(i(x, y), i(y, x)) end proc;

proc (x, y) options operator, arrow; o(i(x, y), i(y, x)) end proc

 (21)
> 

val2(3, g);

[[0, 0], 1], [[0, 1/2], 1], [[0, 1], 1], [[1/2, 0], 1], [[1/2, 1/2], 1], [[1/2, 1], 1], [[1, 0], 1], [[1, 1/2], 1], [[1, 1], 1]

 (22)
> 

NULL

valuation*of*a*proposition*involving*three*atomic*propositions

 

> 

val3:=proc(n,f)
> 

local W,VVV,K;
> 

W:=V(n);
> 

K := cp3(W, W, W);
> 

VVV:=[seq(Flatten(J), `in`(J, K))];
> 

seq([z,f(z[1],z[2],z[3])],z in VVV);
> 

end proc;

proc (n, f) local W, VVV, K; W := V(n); K := cp3(W, W, W); VVV := [seq(ListTools:-Flatten(J), `in`(J, K))]; seq([z, f(z[1], z[2], z[3])], `in`(z, VVV)) end proc

 (23)
> 

NULL
> 

NULL

     A→(B→C) ≡ (AoB)→C

 

> 

h := proc (x, y, z) options operator, arrow; e(i(x, i(y, z)), i(a(x, y), z)) end proc;

proc (x, y, z) options operator, arrow; e(i(x, i(y, z)), i(a(x, y), z)) end proc

 (24)
> 

val3(2, h);

[[0, 0, 0], 1], [[0, 0, 1], 1], [[0, 1, 0], 1], [[0, 1, 1], 1], [[1, 0, 0], 1], [[1, 0, 1], 1], [[1, 1, 0], 1], [[1, 1, 1], 1]

 (25)
> 

val3(3, h);

[[0, 0, 0], 1], [[0, 0, 1/2], 1], [[0, 0, 1], 1], [[0, 1/2, 0], 1], [[0, 1/2, 1/2], 1], [[0, 1/2, 1], 1], [[0, 1, 0], 1], [[0, 1, 1/2], 1], [[0, 1, 1], 1], [[1/2, 0, 0], 1], [[1/2, 0, 1/2], 1], [[1/2, 0, 1], 1], [[1/2, 1/2, 0], 1/2], [[1/2, 1/2, 1/2], 1], [[1/2, 1/2, 1], 1], [[1/2, 1, 0], 1], [[1/2, 1, 1/2], 1], [[1/2, 1, 1], 1], [[1, 0, 0], 1], [[1, 0, 1/2], 1], [[1, 0, 1], 1], [[1, 1/2, 0], 1], [[1, 1/2, 1/2], 1], [[1, 1/2, 1], 1], [[1, 1, 0], 1], [[1, 1, 1/2], 1], [[1, 1, 1], 1]

 (26)
> 

NULL

valuation*of*a*proposition*involving*four*atomic*propositions

 

> 

val4:=proc(n,f)
> 

local W,VVVV,K;
> 

W:=V(n);
> 

K := cp4(W, W, W,W);
> 

VVVV:=[seq(Flatten(J), `in`(J, K))];
> 

seq([z,f(z[1],z[2],z[3],z[4])],z in VVVV);
> 

end proc;

proc (n, f) local W, VVVV, K; W := V(n); K := cp4(W, W, W, W); VVVV := [seq(ListTools:-Flatten(J), `in`(J, K))]; seq([z, f(z[1], z[2], z[3], z[4])], `in`(z, VVVV)) end proc

 (27)

 

   (AoB ) →(CnD)

  

> 

hh := proc (x, y, z, w) options operator, arrow; i(a(x, y), o(z, w)) end proc;

proc (x, y, z, w) options operator, arrow; i(a(x, y), o(z, w)) end proc

 (28)
> 

val4(2, hh);

[[0, 0, 0, 0], 1], [[0, 0, 0, 1], 1], [[0, 0, 1, 0], 1], [[0, 0, 1, 1], 1], [[0, 1, 0, 0], 1], [[0, 1, 0, 1], 1], [[0, 1, 1, 0], 1], [[0, 1, 1, 1], 1], [[1, 0, 0, 0], 1], [[1, 0, 0, 1], 1], [[1, 0, 1, 0], 1], [[1, 0, 1, 1], 1], [[1, 1, 0, 0], 0], [[1, 1, 0, 1], 1], [[1, 1, 1, 0], 1], [[1, 1, 1, 1], 1]

 (29)
> 

NULL

tautology*involving*one*proposition

 

> 

tau1:=proc(n,f)
> 

local W,L,c,x;
> 

W:=V(n);
> 

L:=[seq(f(x),x in W)];
> 

c:=0;
> 

for x in L do if x=1 then c:=c+1 fi;od;
> 

if c=nops(L) then print(tautology); else print(NO)fi;
> 

end proc;

proc (n, f) local W, L, c, x; W := V(n); L := [seq(f(x), `in`(x, W))]; c := 0; for x in L do if x = 1 then c := c+1 end if end do; if c = nops(L) then print(tautology) else print(NO) end if end proc

 (30)
> 

NULL
> 

``

     An¬A

 

> 

f := proc (x) options operator, arrow; o(x, n(x)) end proc;

proc (x) options operator, arrow; o(x, n(x)) end proc

 (31)
> 

tau1(2, f);

tautology

 (32)
> 

tau1(3, f);

NO

 (33)

    A→A

 

> 

g := proc (x) options operator, arrow; i(x, x) end proc;

proc (x) options operator, arrow; i(x, x) end proc

 (34)
> 

tau1(3, g);

tautology

 (35)
> 

``
> 

NULL

tautology*involving*two*propositions

 

> 

tau2:=proc(n,f)
> 

local W,VV,L,c,x;
> 

W:=V(n);
> 

VV:=cp2(W,W);
> 

L:=[seq(f(z[1],z[2]),z in VV)];
> 

c:=0;
> 

for x in L do if x=1 then c:=c+1 fi;od;
> 

if c=nops(L) then print(tautology); else print(NO)fi;
> 

end proc;

proc (n, f) local W, VV, L, c, x; W := V(n); VV := cp2(W, W); L := [seq(f(z[1], z[2]), `in`(z, VV))]; c := 0; for x in L do if x = 1 then c := c+1 end if end do; if c = nops(L) then print(tautology) else print(NO) end if end proc

 (36)
> 

``

     (A→ B) n (B → A)

 

> 

g := proc (x, y) options operator, arrow; o(i(x, y), i(y, x)) end proc;

proc (x, y) options operator, arrow; o(i(x, y), i(y, x)) end proc

 (37)
> 

tau2(3, g);

tautology

 (38)

   {A→B, ¬B}~¬A

 

> 

k := proc (x, y) options operator, arrow; i(a(o(x, y), n(y)), x) end proc;

proc (x, y) options operator, arrow; i(a(o(x, y), n(y)), x) end proc

 (39)
> 

tau2(2, k);

tautology

 (40)
> 

tau2(3, k);

NO

 (41)
> 

tau2(4, k);

NO

 (42)
> 

NULL
> 

NULL

tautology*involving*three*propositions

 

> 

tau3:=proc(n,f)
> 

local W,VVV,K,L,c,x;
> 

W:=V(n);
> 

K := cp3(W, W,W);VVV:=[seq(Flatten(J), `in`(J, K))];
> 

L:=[seq(f(w[1],w[2],w[3]),w in VVV)];
> 

c:=0;
> 

for x in L do if x=1 then c:=c+1 fi;od;
> 

if c=nops(L) then print(tautology); else print(NO)fi;
> 

end proc;

proc (n, f) local W, VVV, K, L, c, x; W := V(n); K := cp3(W, W, W); VVV := [seq(ListTools:-Flatten(J), `in`(J, K))]; L := [seq(f(w[1], w[2], w[3]), `in`(w, VVV))]; c := 0; for x in L do if x = 1 then c := c+1 end if end do; if c = nops(L) then print(tautology) else print(NO) end if end proc

 (43)
> 

NULL

     {A→B,B→C}~A→C

 

> 

t := proc (x, y, z) options operator, arrow; i(i(x, y), i(i(y, z), i(x, z))) end proc;

proc (x, y, z) options operator, arrow; i(i(x, y), i(i(y, z), i(x, z))) end proc

 (44)
> 

tau3(3, t);

tautology

 (45)
> 

NULL

tautology*involving*four*propositions

 

> 

tau4:=proc(n,f)
> 

local W,VVVV,K,L,c,x;
> 

W:=V(n);
> 

K := cp4(W, W,W,W);VVVV:=[seq(Flatten(J), `in`(J, K))];
> 

L:=[seq(f(w[1],w[2],w[3],w[4]),w in VVVV)];
> 

c:=0;
> 

for x in L do if x=1 then c:=c+1 fi;od;
> 

if c=nops(L) then print(tautology); else print(NO)fi;
> 

end proc;

proc (n, f) local W, VVVV, K, L, c, x; W := V(n); K := cp4(W, W, W, W); VVVV := [seq(ListTools:-Flatten(J), `in`(J, K))]; L := [seq(f(w[1], w[2], w[3], w[4]), `in`(w, VVVV))]; c := 0; for x in L do if x = 1 then c := c+1 end if end do; if c = nops(L) then print(tautology) else print(NO) end if end proc

 (46)
> 

``

   {A→B,C→D}~(AnC)→(BnD)

 

> 

NULL
> 

s := proc (x, y, z, w) options operator, arrow; i(a(i(x, y), i(z, w)), i(o(x, z), o(y, w))) end proc;

proc (x, y, z, w) options operator, arrow; i(a(i(x, y), i(z, w)), i(o(x, z), o(y, w))) end proc

 (47)
> 

tau4(2, s);

tautology

 (48)
> 

tau4(3, s);

tautology

 (49)
> 

tau4(4, s);

tautology

 (50)
> 

``
> 

u := proc (x, y) options operator, arrow; i(x, y)*x end proc;

proc (x, y) options operator, arrow; i(x, y)*x end proc

 (51)
> 

uu := proc (x, y) options operator, arrow; i(u(x, y), y) end proc;

proc (x, y) options operator, arrow; i(u(x, y), y) end proc

 (52)
> 

tau2(2, uu);

tautology

 (53)
> 

tau2(3, uu);

NO

 (54)

 

 References

 1) M. Bergmann, Many-Valued and Fuzzy Logic

     Cambridge 2008.

 2) G.Malinowski, Many-Valued Logics

     Clarendon- Oxford  1993.

 

> 

NULL