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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.

>	cp2:=proc(K,L)

>	[seq(seq([x,y],y in L),x in K)];

end proc;

(1)

>	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)

>	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)

>	V:=proc(n)

>	[seq(r/(n-1),r=0..n-1)];

end proc;

(4)

(5)

(6)

[[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)

[[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)

function

>	m:=proc(x,y)

>	(x+y)/2 -abs(x-y)/2;

end proc;

(9)

function

>	M:=proc(x,y)

>	(x+y)/2 +abs(x-y)/2;

end proc;

(10)

logical connectives

>	n:=proc(x)

1-x;

end proc;

(11)

>	i:=proc(x,y)

>	m(1,1-x+y);

end proc;

(12)

>	o:=proc(x,y)

M(x,y);

end proc;

(13)

>	a:=proc(x,y)

m(x,y);

end proc;

(14)

biconditional

>	e:=proc(x,y)

>	1-abs(x-y);

end proc;

(15)

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;

(16)

An¬A

(17)

(18)

(19)

>	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)

(21)

[[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)

>	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)

A→(B→C) ≡ (AoB)→C

(24)

(25)

(26)

>	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)

(28)

[[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)

>	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)

An¬A

(31)

(32)

(33)

A→A

(34)

(35)

>	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)

(37)

(38)

{A→B, ¬B}~¬A

(39)

(40)

(41)

(42)

>	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)

{A→B,B→C}~A→C

(44)

(45)

>	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