Ternary logic

From Rosetta Code
Jump to: navigation, search
Task
Ternary logic
You are encouraged to solve this task according to the task description, using any language you may know.
This page uses content from Wikipedia. The original article was at Ternary logic. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)

In logic, a three-valued logic (also trivalent, ternary, or trinary logic, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false and some indeterminate third value. This is contrasted with the more commonly known bivalent logics (such as classical sentential or boolean logic) which provide only for true and false. Conceptual form and basic ideas were initially created by Łukasiewicz, Lewis and Sulski. These were then re-formulated by Grigore Moisil in an axiomatic algebraic form, and also extended to n-valued logics in 1945.

Example Ternary Logic Operators in Truth Tables:
not a
¬
True False
Maybe Maybe
False True
a and b
True Maybe False
True True Maybe False
Maybe Maybe Maybe False
False False False False
a or b
True Maybe False
True True True True
Maybe True Maybe Maybe
False True Maybe False
if a then b
True Maybe False
True True Maybe False
Maybe True Maybe Maybe
False True True True
a is equivalent to b
True Maybe False
True True Maybe False
Maybe Maybe Maybe Maybe
False False Maybe True

Task:

  • Define a new type that emulates ternary logic by storing data trits.
  • Given all the binary logic operators of the original programming language, reimplement these operators for the new Ternary logic type trit.
  • Generate a sampling of results using trit variables.
  • Kudos for actually thinking up a test case algorithm where ternary logic is intrinsically useful, optimises the test case algorithm and is preferable to binary logic.

Note: Setun (Сетунь) was a balanced ternary computer developed in 1958 at Moscow State University. The device was built under the lead of Sergei Sobolev and Nikolay Brusentsov. It was the only modern ternary computer, using three-valued ternary logic

Contents

[edit] Ada

We first specify a package "Logic" for three-valued logic. Observe that predefined Boolean functions, "and" "or" and "not" are overloaded:

package Logic is
type Ternary is (True, Unknown, False);
 
-- logic functions
function "and"(Left, Right: Ternary) return Ternary;
function "or"(Left, Right: Ternary) return Ternary;
function "not"(T: Ternary) return Ternary;
function Equivalent(Left, Right: Ternary) return Ternary;
function Implies(Condition, Conclusion: Ternary) return Ternary;
 
-- conversion functions
function To_Bool(X: Ternary) return Boolean;
function To_Ternary(B: Boolean) return Ternary;
function Image(Value: Ternary) return Character;
end Logic;

Next, the implementation of the package:

package body Logic is
-- type Ternary is (True, Unknown, False);
 
function Image(Value: Ternary) return Character is
begin
case Value is
when True => return 'T';
when False => return 'F';
when Unknown => return '?';
end case;
end Image;
 
function "and"(Left, Right: Ternary) return Ternary is
begin
return Ternary'max(Left, Right);
end "and";
 
function "or"(Left, Right: Ternary) return Ternary is
begin
return Ternary'min(Left, Right);
end "or";
 
function "not"(T: Ternary) return Ternary is
begin
case T is
when False => return True;
when Unknown => return Unknown;
when True => return False;
end case;
end "not";
 
function To_Bool(X: Ternary) return Boolean is
begin
case X is
when True => return True;
when False => return False;
when Unknown => raise Constraint_Error;
end case;
end To_Bool;
 
function To_Ternary(B: Boolean) return Ternary is
begin
if B then
return True;
else
return False;
end if;
end To_Ternary;
 
function Equivalent(Left, Right: Ternary) return Ternary is
begin
return To_Ternary(To_Bool(Left) = To_Bool(Right));
exception
when Constraint_Error => return Unknown;
end Equivalent;
 
function Implies(Condition, Conclusion: Ternary) return Ternary is
begin
return (not Condition) or Conclusion;
end Implies;
 
end Logic;

Finally, a sample program:

with Ada.Text_IO, Logic;
 
procedure Test_Tri_Logic is
 
use Logic;
 
type F2 is access function(Left, Right: Ternary) return Ternary;
type F1 is access function(Trit: Ternary) return Ternary;
 
procedure Truth_Table(F: F1; Name: String) is
begin
Ada.Text_IO.Put_Line("X | " & Name & "(X)");
for T in Ternary loop
Ada.Text_IO.Put_Line(Image(T) & " | " & Image(F(T)));
end loop;
end Truth_Table;
 
procedure Truth_Table(F: F2; Name: String) is
begin
Ada.Text_IO.New_Line;
Ada.Text_IO.Put_Line("X | Y | " & Name & "(X,Y)");
for X in Ternary loop
for Y in Ternary loop
Ada.Text_IO.Put_Line(Image(X) & " | " & Image(Y) & " | " & Image(F(X,Y)));
end loop;
end loop;
end Truth_Table;
 
begin
Truth_Table(F => "not"'Access, Name => "Not");
Truth_Table(F => "and"'Access, Name => "And");
Truth_Table(F => "or"'Access, Name => "Or");
Truth_Table(F => Equivalent'Access, Name => "Eq");
Truth_Table(F => Implies'Access, Name => "Implies");
end Test_Tri_Logic;
The output:
X | Not(X)
T |  F
? |  ?
F |  T

X | Y | And(X,Y)
T | T |  T
T | ? |  ?
T | F |  F
? | T |  ?
? | ? |  ?
? | F |  F
F | T |  F
F | ? |  F
F | F |  F

... (and so on)

[edit] ALGOL 68

Works with: ALGOL 68 version Revision 1 - one minor extension to language used - PRAGMA READ, like C's #include directive.
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny.

File: Ternary_logic.a68

# -*- coding: utf-8 -*- #
 
INT trit width = 1, trit base = 3;
MODE TRIT = STRUCT(BITS trit);
CO FORMAT trit fmt = $c("?","⌈","⌊",#|"~"#)$; CO
 
# These values treated are as per "Balanced ternary" #
# eg true=1, maybe=0, false=-1 #
TRIT true =INITTRIT 4r1, maybe=INITTRIT 4r0,
false=INITTRIT 4r2;
 
# Warning: redefines standard builtins flip & flop #
LONGCHAR flap="?", flip="⌈", flop="⌊";
 
OP REPR = (TRIT t)LONGCHAR:
[]LONGCHAR(flap, flip, flop)[1+ABS trit OF t];
 
############################################
# Define some OPerators for coercing MODES #
############################################
OP INITTRIT = (BOOL in)TRIT:
(in|true|false);
 
OP INITBOOL = (TRIT in)BOOL:
(trit OF in=trit OF true|TRUE|:trit OF in=trit OF false|FALSE|
raise value error(("vague TRIT to BOOL coercion: """, REPR in,""""));~
);
OP B = (TRIT in)BOOL: INITBOOL in;
 
# These values treated are as per "Balanced ternary" #
# n.b true=1, maybe=0, false=-1 #
# Warning: BOOL ABS FALSE (0) is not the same as TRIT ABS false (-1) #
 
OP INITINT = (TRIT t)INT:
CASE 1+ABS trit OF t
IN #maybe# 0, #true # 1, #false#-1
OUT raise value error(("invalid TRIT value",REPR t)); ~
ESAC;
 
OP INITTRIT = (INT in)TRIT: (
TRIT out;
trit OF out:= trit OF
CASE 2+in
IN false, maybe, true
OUT raise value error(("invalid TRIT value",in)); ~
ESAC;
out
);
 
OP INITTRIT = (BITS b)TRIT:
(TRIT out; trit OF out:=b; out);
 
##################################################
# Define the LOGICAL OPerators for the TRIT MODE #
##################################################
MODE LOGICAL = TRIT;
PR READ "Template_operators_logical_mixin.a68" PR
 
COMMENT
Kleene logic truth tables:
END COMMENT
 
OP AND = (TRIT a,b)TRIT: (
[,]TRIT(
# ∧ ## false, maybe, true #
#false# (false, false, false),
#maybe# (false, maybe, maybe),
#true # (false, maybe, true )
)[@-1,@-1][INITINT a, INITINT b]
);
 
OP OR = (TRIT a,b)TRIT: (
[,]TRIT(
# ∨ ## false, maybe, true #
#false# (false, maybe, true),
#maybe# (maybe, maybe, true),
#true # (true, true, true)
)[@-1,@-1][INITINT a, INITINT b]
);
 
PRIO IMPLIES = 1; # PRIO = 1.9 #
OP IMPLIES = (TRIT a,b)TRIT: (
[,]TRIT(
# ⊃ ## false, maybe, true #
#false# (true, true, true),
#maybe# (maybe, maybe, true),
#true # (false, maybe, true)
)[@-1,@-1][INITINT a, INITINT b]
);
 
PRIO EQV = 1; # PRIO = 1.8 #
OP EQV = (TRIT a,b)TRIT: (
[,]TRIT(
# ≡ ## false, maybe, true #
#false# (true, maybe, false),
#maybe# (maybe, maybe, maybe),
#true # (false, maybe, true )
)[@-1,@-1][INITINT a, INITINT b]
);
File: Template_operators_logical_mixin.a68
# -*- coding: utf-8 -*- #
 
OP & = (LOGICAL a,b)LOGICAL: a AND b;
CO # not included as they are treated as SCALAR #
OP EQ = (LOGICAL a,b)LOGICAL: a = b,
NE = (LOGICAL a,b)LOGICAL: a /= b,
= (TRIT a,b)TRIT: a /= b,
¬= = (TRIT a,b)TRIT: a /= b;
END CO
 
#IF html entities possible THEN
¢ "parked" operators for completeness ¢
OP ¬ = (LOGICAL a)LOGICAL: NOT a,
∧ = (LOGICAL a,b)LOGICAL: a AND b,
/\ = (LOGICAL a,b)LOGICAL: a AND b,
∨ = (LOGICAL a,b)LOGICAL: a OR b,
\/ = (LOGICAL a,b)LOGICAL: a OR b,
⊃ = (TRIT a,b)TRIT: a IMPLIES b,
≡ = (TRIT a,b)TRIT: a EQV b;
FI#

 
#IF algol68c THEN
OP ~ = (LOGICAL a)LOGICAL: NOT a,
~= = (LOGICAL a,b)LOGICAL: a /= b; SCALAR!
FI#
File: test_Ternary_logic.a68
#!/usr/local/bin/a68g --script #
# -*- coding: utf-8 -*- #
 
PR READ "prelude/general.a68" PR
PR READ "Ternary_logic.a68" PR
 
[]TRIT trits = (false, maybe, true);
 
FORMAT
col fmt = $" "g" "$,
row fmt = $l3(f(col fmt)"|")f(col fmt)$,
row sep fmt = $l3("---+")"---"l$;
 
PROC row sep = VOID:
printf(row sep fmt);
 
PROC title = (STRING op name, LONGCHAR op char)VOID:(
print(("Operator: ",op name));
printf((row fmt,op char,REPR false, REPR maybe, REPR true))
);
 
PROC print trit op table = (LONGCHAR op char, STRING op name, PROC(TRIT,TRIT)TRIT op)VOID: (
printf($l$);
title(op name, op char);
FOR i FROM LWB trits TO UPB trits DO
row sep;
TRIT ti = trits[i];
printf((col fmt, REPR ti));
FOR j FROM LWB trits TO UPB trits DO
TRIT tj = trits[j];
printf(($"|"$, col fmt, REPR op(ti,tj)))
OD
OD;
printf($l$)
);
 
printf((
$"Comparitive table of coercions:"l$,
$" TRIT BOOL INT"l$
));
 
FOR it FROM LWB trits TO UPB trits DO
TRIT t = trits[it];
printf(( $" "g" "$, REPR t,
IF trit OF t = trit OF maybe THEN " " ELSE B t FI,
INITINT t, $l$))
OD;
 
printf((
$l"Specific test of the IMPLIES operator:"l$,
$" "g" implies "g" is "b("not ","")"a contradiction!"l$,
B false, B false, B(false IMPLIES false),
B false, B true, B(false IMPLIES true),
B false, REPR maybe, B(false IMPLIES maybe),
B true, B false, B(true IMPLIES false),
B true, B true, B(true IMPLIES true),
REPR maybe, Btrue, B(maybe IMPLIES true),
$" "g" implies "g" is "g" a contradiction!"l$,
B true, REPR maybe, REPR (true IMPLIES maybe),
REPR maybe, B false, REPR (maybe IMPLIES false),
REPR maybe, REPR maybe, REPR (maybe IMPLIES maybe),
$l$
));
 
printf($"Kleene logic truth table samples:"l$);
 
print trit op table("≡","EQV", (TRIT a,b)TRIT: a EQV b);
print trit op table("⊃","IMPLIES", (TRIT a,b)TRIT: a IMPLIES b);
print trit op table("∧","AND", (TRIT a,b)TRIT: a AND b);
print trit op table("∨","OR", (TRIT a,b)TRIT: a OR b)

Output:

Comparitive table of coercions:
  TRIT BOOL         INT
  ⌊    F             -1  
  ?                  +0  
  ⌈    T             +1  

Specific test of the IMPLIES operator:
  F implies F is not a contradiction!
  F implies T is not a contradiction!
  F implies ? is not a contradiction!
  T implies F is a contradiction!
  T implies T is not a contradiction!
  ? implies T is not a contradiction!
  T implies ? is ? a contradiction!
  ? implies F is ? a contradiction!
  ? implies ? is ? a contradiction!

Kleene logic truth table samples:

Operator: EQV
 ≡ | ⌊ | ? | ⌈ 
---+---+---+---
 ⌊ | ⌈ | ? | ⌊ 
---+---+---+---
 ? | ? | ? | ? 
---+---+---+---
 ⌈ | ⌊ | ? | ⌈ 

Operator: IMPLIES
 ⊃ | ⌊ | ? | ⌈ 
---+---+---+---
 ⌊ | ⌈ | ⌈ | ⌈ 
---+---+---+---
 ? | ? | ? | ⌈ 
---+---+---+---
 ⌈ | ⌊ | ? | ⌈ 

Operator: AND
 ∧ | ⌊ | ? | ⌈ 
---+---+---+---
 ⌊ | ⌊ | ⌊ | ⌊ 
---+---+---+---
 ? | ⌊ | ? | ? 
---+---+---+---
 ⌈ | ⌊ | ? | ⌈ 

Operator: OR
 ∨ | ⌊ | ? | ⌈ 
---+---+---+---
 ⌊ | ⌊ | ? | ⌈ 
---+---+---+---
 ? | ? | ? | ⌈ 
---+---+---+---
 ⌈ | ⌈ | ⌈ | ⌈ 

[edit] AutoHotkey

Ternary_Not(a){
SetFormat, Float, 2.1
return Abs(a-1)
}
 
Ternary_And(a,b){
return a<b?a:b
}
 
Ternary_Or(a,b){
return a>b?a:b
}
 
Ternary_IfThen(a,b){
return a=1?b:a=0?1:a+b>1?1:0.5
}
 
Ternary_Equiv(a,b){
return a=b?1:a=1?b:b=1?a:0.5
}
Examples:
aa:=[1,0.5,0]
bb:=[1,0.5,0]
 
for index, a in aa
Res .= "`tTernary_Not`t" a "`t=`t" Ternary_Not(a) "`n"
Res .= "-------------`n"
 
for index, a in aa
for index, b in bb
Res .= a "`tTernary_And`t" b "`t=`t" Ternary_And(a,b) "`n"
Res .= "-------------`n"
 
for index, a in aa
for index, b in bb
Res .= a "`tTernary_or`t" b "`t=`t" Ternary_Or(a,b) "`n"
Res .= "-------------`n"
 
for index, a in aa
for index, b in bb
Res .= a "`tTernary_then`t" b "`t=`t" Ternary_IfThen(a,b) "`n"
Res .= "-------------`n"
 
for index, a in aa
for index, b in bb
Res .= a "`tTernary_equiv`t" b "`t=`t" Ternary_Equiv(a,b) "`n"
 
StringReplace, Res, Res, 1, true, all
StringReplace, Res, Res, 0.5, maybe, all
StringReplace, Res, Res, 0, false, all
MsgBox % Res
return
Outputs:
	Ternary_Not	true	=	false
	Ternary_Not	maybe	=	maybe
	Ternary_Not	false	=	true
-------------
true	Ternary_And	true	=	true
true	Ternary_And	maybe	=	maybe
true	Ternary_And	false	=	false
maybe	Ternary_And	true	=	maybe
maybe	Ternary_And	maybe	=	maybe
maybe	Ternary_And	false	=	false
false	Ternary_And	true	=	false
false	Ternary_And	maybe	=	false
false	Ternary_And	false	=	false
-------------
true	Ternary_or	true	=	true
true	Ternary_or	maybe	=	true
true	Ternary_or	false	=	true
maybe	Ternary_or	true	=	true
maybe	Ternary_or	maybe	=	maybe
maybe	Ternary_or	false	=	maybe
false	Ternary_or	true	=	true
false	Ternary_or	maybe	=	maybe
false	Ternary_or	false	=	false
-------------
true	Ternary_then	true	=	true
true	Ternary_then	maybe	=	maybe
true	Ternary_then	false	=	false
maybe	Ternary_then	true	=	true
maybe	Ternary_then	maybe	=	maybe
maybe	Ternary_then	false	=	maybe
false	Ternary_then	true	=	true
false	Ternary_then	maybe	=	true
false	Ternary_then	false	=	true
-------------
true	Ternary_equiv	true	=	true
true	Ternary_equiv	maybe	=	maybe
true	Ternary_equiv	false	=	false
maybe	Ternary_equiv	true	=	maybe
maybe	Ternary_equiv	maybe	=	true
maybe	Ternary_equiv	false	=	maybe
false	Ternary_equiv	true	=	false
false	Ternary_equiv	maybe	=	maybe
false	Ternary_equiv	false	=	true


[edit] BBC BASIC

      INSTALL @lib$ + "CLASSLIB"
 
REM Create a ternary class:
DIM trit{tor, tand, teqv, tnot, tnor, s, v}
DEF PRIVATE trit.s (t&) LOCAL t$():DIM t$(2):t$()="FALSE","MAYBE","TRUE":=t$(t&)
DEF PRIVATE trit.v (t$) = INSTR("FALSE MAYBE TRUE", t$) DIV 6
DEF trit.tnot (t$) = FN(trit.s)(2 - FN(trit.v)(t$))
DEF trit.tor (a$,b$) LOCAL t:t=FN(trit.v)(a$)ORFN(trit.v)(b$):=FN(trit.s)(t+(t>2))
DEF trit.tnor (a$,b$) = FN(trit.tnot)(FN(trit.tor)(a$,b$))
DEF trit.tand (a$,b$) = FN(trit.tnor)(FN(trit.tnot)(a$),FN(trit.tnot)(b$))
DEF trit.teqv (a$,b$) = FN(trit.tor)(FN(trit.tand)(a$,b$),FN(trit.tnor)(a$,b$))
PROC_class(trit{})
 
PROC_new(mytrit{}, trit{})
 
REM Test it:
PRINT "Testing NOT:"
PRINT "NOT FALSE = " FN(mytrit.tnot)("FALSE")
PRINT "NOT MAYBE = " FN(mytrit.tnot)("MAYBE")
PRINT "NOT TRUE = " FN(mytrit.tnot)("TRUE")
 
PRINT '"Testing OR:"
PRINT "FALSE OR FALSE = " FN(mytrit.tor)("FALSE","FALSE")
PRINT "FALSE OR MAYBE = " FN(mytrit.tor)("FALSE","MAYBE")
PRINT "FALSE OR TRUE = " FN(mytrit.tor)("FALSE","TRUE")
PRINT "MAYBE OR MAYBE = " FN(mytrit.tor)("MAYBE","MAYBE")
PRINT "MAYBE OR TRUE = " FN(mytrit.tor)("MAYBE","TRUE")
PRINT "TRUE OR TRUE = " FN(mytrit.tor)("TRUE","TRUE")
 
PRINT '"Testing AND:"
PRINT "FALSE AND FALSE = " FN(mytrit.tand)("FALSE","FALSE")
PRINT "FALSE AND MAYBE = " FN(mytrit.tand)("FALSE","MAYBE")
PRINT "FALSE AND TRUE = " FN(mytrit.tand)("FALSE","TRUE")
PRINT "MAYBE AND MAYBE = " FN(mytrit.tand)("MAYBE","MAYBE")
PRINT "MAYBE AND TRUE = " FN(mytrit.tand)("MAYBE","TRUE")
PRINT "TRUE AND TRUE = " FN(mytrit.tand)("TRUE","TRUE")
 
PRINT '"Testing EQV (similar to EOR):"
PRINT "FALSE EQV FALSE = " FN(mytrit.teqv)("FALSE","FALSE")
PRINT "FALSE EQV MAYBE = " FN(mytrit.teqv)("FALSE","MAYBE")
PRINT "FALSE EQV TRUE = " FN(mytrit.teqv)("FALSE","TRUE")
PRINT "MAYBE EQV MAYBE = " FN(mytrit.teqv)("MAYBE","MAYBE")
PRINT "MAYBE EQV TRUE = " FN(mytrit.teqv)("MAYBE","TRUE")
PRINT "TRUE EQV TRUE = " FN(mytrit.teqv)("TRUE","TRUE")
 
PROC_discard(mytrit{})

Output:

Testing NOT:
NOT FALSE = TRUE
NOT MAYBE = MAYBE
NOT TRUE  = FALSE

Testing OR:
FALSE OR FALSE = FALSE
FALSE OR MAYBE = MAYBE
FALSE OR TRUE  = TRUE
MAYBE OR MAYBE = MAYBE
MAYBE OR TRUE  = TRUE
TRUE  OR TRUE  = TRUE

Testing AND:
FALSE AND FALSE = FALSE
FALSE AND MAYBE = FALSE
FALSE AND TRUE  = FALSE
MAYBE AND MAYBE = MAYBE
MAYBE AND TRUE  = MAYBE
TRUE  AND TRUE  = TRUE

Testing EQV (similar to EOR):
FALSE EQV FALSE = TRUE
FALSE EQV MAYBE = MAYBE
FALSE EQV TRUE  = FALSE
MAYBE EQV MAYBE = MAYBE
MAYBE EQV TRUE  = MAYBE
TRUE  EQV TRUE  = TRUE

[edit] C

[edit] Implementing logic using lookup tables

#include <stdio.h>
 
typedef enum {
TRITTRUE, /* In this enum, equivalent to integer value 0 */
TRITMAYBE, /* In this enum, equivalent to integer value 1 */
TRITFALSE /* In this enum, equivalent to integer value 2 */
} trit;
 
/* We can trivially find the result of the operation by passing
the trinary values as indeces into the lookup tables' arrays. */

trit tritNot[3] = {TRITFALSE , TRITMAYBE, TRITTRUE};
trit tritAnd[3][3] = { {TRITTRUE, TRITMAYBE, TRITFALSE},
{TRITMAYBE, TRITMAYBE, TRITFALSE},
{TRITFALSE, TRITFALSE, TRITFALSE} };
 
trit tritOr[3][3] = { {TRITTRUE, TRITTRUE, TRITTRUE},
{TRITTRUE, TRITMAYBE, TRITMAYBE},
{TRITTRUE, TRITMAYBE, TRITFALSE} };
 
trit tritThen[3][3] = { { TRITTRUE, TRITMAYBE, TRITFALSE},
{ TRITTRUE, TRITMAYBE, TRITMAYBE},
{ TRITTRUE, TRITTRUE, TRITTRUE } };
 
trit tritEquiv[3][3] = { { TRITTRUE, TRITMAYBE, TRITFALSE},
{ TRITMAYBE, TRITMAYBE, TRITMAYBE},
{ TRITFALSE, TRITMAYBE, TRITTRUE } };
 
/* Everything beyond here is just demonstration */
 
const char* tritString[3] = {"T", "?", "F"};
 
void demo_binary_op(trit operator[3][3], const char* name)
{
trit operand1 = TRITTRUE; /* Declare. Initialize for CYA */
trit operand2 = TRITTRUE; /* Declare. Initialize for CYA */
 
/* Blank line */
printf("\n");
 
/* Demo this operator */
for( operand1 = TRITTRUE; operand1 <= TRITFALSE; ++operand1 )
{
for( operand2 = TRITTRUE; operand2 <= TRITFALSE; ++operand2 )
{
printf("%s %s %s: %s\n", tritString[operand1],
name,
tritString[operand2],
tritString[operator[operand1][operand2]]);
}
}
 
}
 
int main()
{
trit op1 = TRITTRUE; /* Declare. Initialize for CYA */
trit op2 = TRITTRUE; /* Declare. Initialize for CYA */
 
/* Demo 'not' */
for( op1 = TRITTRUE; op1 <= TRITFALSE; ++op1 )
{
printf("Not %s: %s\n", tritString[op1], tritString[tritNot[op1]]);
}
demo_binary_op(tritAnd, "And");
demo_binary_op(tritOr, "Or");
demo_binary_op(tritThen, "Then");
demo_binary_op(tritEquiv, "Equiv");
 
 
return 0;
}

Output:

Not T: F
Not ?: ?
Not F: T
 
T And T: T
T And ?: ?
T And F: F
? And T: ?
? And ?: ?
? And F: F
F And T: F
F And ?: F
F And F: F
 
T Or T: T
T Or ?: T
T Or F: T
? Or T: T
? Or ?: ?
? Or F: ?
F Or T: T
F Or ?: ?
F Or F: F
 
T Then T: T
T Then ?: ?
T Then F: F
? Then T: T
? Then ?: ?
? Then F: ?
F Then T: T
F Then ?: T
F Then F: T
 
T Equiv T: T
T Equiv ?: ?
T Equiv F: F
? Equiv T: ?
? Equiv ?: ?
? Equiv F: ?
F Equiv T: F
F Equiv ?: ?
F Equiv F: T

[edit] Using functions

#include <stdio.h>
 
typedef enum { t_F = -1, t_M, t_T } trit;
 
trit t_not (trit a) { return -a; }
trit t_and (trit a, trit b) { return a < b ? a : b; }
trit t_or (trit a, trit b) { return a > b ? a : b; }
trit t_eq (trit a, trit b) { return a * b; }
trit t_imply(trit a, trit b) { return -a > b ? -a : b; }
char t_s(trit a) { return "F?T"[a + 1]; }
 
#define forall(a) for(a = t_F; a <= t_T; a++)
void show_op(trit (*f)(trit, trit), const char *name) {
trit a, b;
printf("\n[%s]\n F ? T\n -------", name);
forall(a) {
printf("\n%c |", t_s(a));
forall(b) printf(" %c", t_s(f(a, b)));
}
puts("");
}
 
int main(void)
{
trit a;
 
puts("[Not]");
forall(a) printf("%c | %c\n", t_s(a), t_s(t_not(a)));
 
show_op(t_and, "And");
show_op(t_or, "Or");
show_op(t_eq, "Equiv");
show_op(t_imply, "Imply");
 
return 0;
}
output
[Not]
F | T
? | ?
T | F
 
[And]
F ? T
-------
F | F F F
? | F ? ?
T | F ? T
 
[Or]
F ? T
-------
F | F ? T
? | ? ? T
T | T T T
 
[Equiv]
F ? T
-------
F | T ? F
? | ? ? ?
T | F ? T
 
[Imply]
F ? T
-------
F | T T T
? | ? ? T
T | F ? T

[edit] Variable truthfulness

Represent each possible truth value as a floating point value x, where the var has x chance of being true and 1 - x chance of being false. When using if3 conditional on a potential truth varible, the result is randomly sampled to true or false according to the chance. (This description is definitely very confusing perhaps).

#include <stdio.h>
#include <stdlib.h>
 
typedef double half_truth, maybe;
 
inline maybe not3(maybe a) { return 1 - a; }
 
inline maybe
and3(maybe a, maybe b) { return a * b; }
 
inline maybe
or3(maybe a, maybe b) { return a + b - a * b; }
 
inline maybe
eq3(maybe a, maybe b) { return 1 - a - b + 2 * a * b; }
 
inline maybe
imply3(maybe a, maybe b) { return or3(not3(a), b); }
 
#define true3(x) ((x) * RAND_MAX > rand())
#define if3(x) if (true3(x))
 
int main()
{
maybe roses_are_red = 0.25; /* they can be white or black, too */
maybe violets_are_blue = 1; /* aren't they just */
int i;
 
puts("Verifying flowery truth for 40 times:\n");
 
puts("Rose is NOT red:"); /* chance: .75 */
for (i = 0; i < 40 || !puts("\n"); i++)
printf( true3( not3(roses_are_red) ) ? "T" : "_");
 
/* pick a rose and a violet; */
puts("Rose is red AND violet is blue:");
/* chance of rose being red AND violet being blue is .25 */
for (i = 0; i < 40 || !puts("\n"); i++)
printf( true3( and3(roses_are_red, violets_are_blue) )
? "T" : "_");
 
/* chance of rose being red OR violet being blue is 1 */
puts("Rose is red OR violet is blue:");
for (i = 0; i < 40 || !puts("\n"); i++)
printf( true3( or3(roses_are_red, violets_are_blue) )
? "T" : "_");
 
/* pick two roses; chance of em being both red or both not red is .625 */
puts("This rose is as red as that rose:");
for (i = 0; i < 40 || !puts("\n"); i++)
if3(eq3(roses_are_red, roses_are_red)) putchar('T');
else putchar('_');
 
return 0;
}

[edit] C#

using System;
 
/// <summary>
/// Extension methods on nullable bool.
/// </summary>
/// <remarks>
/// The operators !, & and | are predefined.
/// </remarks>
public static class NullableBoolExtension
{
public static bool? Implies(this bool? left, bool? right)
{
return !left | right;
}
 
public static bool? IsEquivalentTo(this bool? left, bool? right)
{
return left.HasValue && right.HasValue ? left == right : default(bool?);
}
 
public static string Format(this bool? value)
{
return value.HasValue ? value.Value.ToString() : "Maybe";
}
}
 
public class Program
{
private static void Main()
{
var values = new[] { true, default(bool?), false };
 
foreach (var left in values)
{
Console.WriteLine("¬{0} = {1}", left.Format(), (!left).Format());
foreach (var right in values)
{
Console.WriteLine("{0} & {1} = {2}", left.Format(), right.Format(), (left & right).Format());
Console.WriteLine("{0} | {1} = {2}", left.Format(), right.Format(), (left | right).Format());
Console.WriteLine("{0} → {1} = {2}", left.Format(), right.Format(), left.Implies(right).Format());
Console.WriteLine("{0} ≡ {1} = {2}", left.Format(), right.Format(), left.IsEquivalentTo(right).Format());
}
}
}
}

Output:

¬True = False
True & True = True
True | True = True
True → True = True
True ≡ True = True
True & Maybe = Maybe
True | Maybe = True
True → Maybe = Maybe
True ≡ Maybe = Maybe
True & False = False
True | False = True
True → False = False
True ≡ False = False
¬Maybe = Maybe
Maybe & True = Maybe
Maybe | True = True
Maybe → True = True
Maybe ≡ True = Maybe
Maybe & Maybe = Maybe
Maybe | Maybe = Maybe
Maybe → Maybe = Maybe
Maybe ≡ Maybe = Maybe
Maybe & False = False
Maybe | False = Maybe
Maybe → False = Maybe
Maybe ≡ False = Maybe
¬False = True
False & True = False
False | True = True
False → True = True
False ≡ True = False
False & Maybe = False
False | Maybe = Maybe
False → Maybe = True
False ≡ Maybe = Maybe
False & False = False
False | False = False
False → False = True
False ≡ False = True

[edit] Common Lisp

(defun tri-not (x) (- 1 x))
(defun tri-and (&rest x) (apply #'* x))
(defun tri-or (&rest x) (tri-not (apply #'* (mapcar #'tri-not x))))
(defun tri-eq (x y) (+ (tri-and x y) (tri-and (- 1 x) (- 1 y))))
(defun tri-imply (x y) (tri-or (tri-not x) y))
 
(defun tri-test (x) (< (random 1e0) x))
(defun tri-string (x) (if (= x 1) "T" (if (= x 0) "F" "?")))
 
;; to say (tri-if (condition) (yes) (no))
(defmacro tri-if (tri ifcase &optional elsecase)
`(if (tri-test ,tri) ,ifcase ,elsecase))
 
(defun print-table (func header)
(let ((vals '(1 .5 0)))
(format t "~%~a:~%" header)
(format t " ~{~a ~^~}~%---------~%" (mapcar #'tri-string vals))
(loop for row in vals do
(format t "~a | " (tri-string row))
(loop for col in vals do
(format t "~a " (tri-string (funcall func row col))))
(write-line ""))))
 
(write-line "NOT:")
(loop for row in '(1 .5 0) do
(format t "~a | ~a~%" (tri-string row) (tri-string (tri-not row))))
 
(print-table #'tri-and "AND")
(print-table #'tri-or "OR")
(print-table #'tri-imply "IMPLY")
(print-table #'tri-eq "EQUAL")
output
NOT:
T | F
? | ?
F | T
 
AND:
T ? F
---------
T | T ? F
? | ? ? F
F | F F F
 
OR:
T ? F
---------
T | T T T
? | T ? ?
F | T ? F
 
IMPLY:
T ? F
---------
T | T ? F
? | T ? ?
F | T T T
 
EQUAL:
T ? F
---------
T | T ? F
? | ? ? ?
F | F ? T

[edit] D

Partial translation of a C entry:

import std.stdio;
 
struct Trit {
private enum Val : byte { F = -1, M, T }
private Val t;
alias t this;
static immutable Trit[3] vals = [{Val.F}, {Val.M}, {Val.T}];
static immutable F = Trit(Val.F); // Not necessary but handy.
static immutable M = Trit(Val.M);
static immutable T = Trit(Val.T);
 
string toString() const pure nothrow {
return "F?T"[t + 1 .. t + 2];
}
 
Trit opUnary(string op)() const pure nothrow
if (op == "~") {
return Trit(-t);
}
 
Trit opBinary(string op)(in Trit b) const pure nothrow
if (op == "&") {
return t < b ? this : b;
}
 
Trit opBinary(string op)(in Trit b) const pure nothrow
if (op == "|") {
return t > b ? this : b;
}
 
Trit opBinary(string op)(in Trit b) const pure nothrow
if (op == "^") {
return ~(this == b);
}
 
Trit opEquals(in Trit b) const pure nothrow {
return Trit(cast(Val)(t * b));
}
 
Trit imply(in Trit b) const pure nothrow {
return -t > b ? ~this : b;
}
}
 
void showOperation(string op)(in string opName) {
writef("\n[%s]\n F ? T\n -------", opName);
foreach (immutable a; Trit.vals) {
writef("\n%s |", a);
foreach (immutable b; Trit.vals)
static if (op == "==>")
writef(" %s", a.imply(b));
else
writef(" %s", mixin("a " ~ op ~ " b"));
}
writeln();
}
 
void main() {
writeln("[Not]");
foreach (const a; Trit.vals)
writefln("%s | %s", a, ~a);
 
showOperation!"&"("And");
showOperation!"|"("Or");
showOperation!"^"("Xor");
showOperation!"=="("Equiv");
showOperation!"==>"("Imply");
}
Output:
[Not]
F | T
? | ?
T | F

[And]
    F ? T
  -------
F | F F F
? | F ? ?
T | F ? T

[Or]
    F ? T
  -------
F | F ? T
? | ? ? T
T | T T T

[Xor]
    F ? T
  -------
F | F ? T
? | ? ? ?
T | T ? F

[Equiv]
    F ? T
  -------
F | T ? F
? | ? ? ?
T | F ? T

[Imply]
    F ? T
  -------
F | T T T
? | ? ? T
T | F ? T

[edit] Delphi

unit TrinaryLogic;
 
interface
 
//Define our own type for ternary logic.
//This is actually still a Boolean, but the compiler will use distinct RTTI information.
type
TriBool = type Boolean;
 
const
TTrue:TriBool = True;
TFalse:TriBool = False;
TMaybe:TriBool = TriBool(2);
 
function TVL_not(Value: TriBool): TriBool;
function TVL_and(A, B: TriBool): TriBool;
function TVL_or(A, B: TriBool): TriBool;
function TVL_xor(A, B: TriBool): TriBool;
function TVL_eq(A, B: TriBool): TriBool;
 
implementation
 
Uses
SysUtils;
 
function TVL_not(Value: TriBool): TriBool;
begin
if Value = True Then
Result := TFalse
else If Value = False Then
Result := TTrue
else
Result := Value;
end;
 
function TVL_and(A, B: TriBool): TriBool;
begin
Result := TriBool(Iff(Integer(A * B) > 1, Integer(TMaybe), A * B));
end;
 
function TVL_or(A, B: TriBool): TriBool;
begin
Result := TVL_not(TVL_and(TVL_not(A), TVL_not(B)));
end;
 
function TVL_xor(A, B: TriBool): TriBool;
begin
Result := TVL_and(TVL_or(A, B), TVL_not(TVL_or(A, B)));
end;
 
function TVL_eq(A, B: TriBool): TriBool;
begin
Result := TVL_not(TVL_xor(A, B));
end;
 
end.

And that's the reason why you never on no account ever should compare against the values of True or False unless you intent ternary logic!

An alternative version would be using an enum type

type TriBool = (tbFalse, tbMaybe, tbTrue);

and defining a set of constants implementing the above tables:

const
tvl_not: array[TriBool] = (tbTrue, tbMaybe, tbFalse);
tvl_and: array[TriBool, TriBool] = (
(tbFalse, tbFalse, tbFalse),
(tbFalse, tbMaybe, tbMaybe),
(tbFalse, tbMaybe, tbTrue),
);
tvl_or: array[TriBool, TriBool] = (
(tbFalse, tbMaybe, tbTrue),
(tbMaybe, tbMaybe, tbTrue),
(tbTrue, tbTrue, tbTrue),
);
tvl_xor: array[TriBool, TriBool] = (
(tbFalse, tbMaybe, tbTrue),
(tbMaybe, tbMaybe, tbMaybe),
(tbTrue, tbMaybe, tbFalse),
);
tvl_eq: array[TriBool, TriBool] = (
(tbTrue, tbMaybe, tbFalse),
(tbMaybe, tbMaybe, tbMaybe),
(tbFalse, tbMaybe, tbTrue),
);
 

That's no real fun, but lookup can then be done with

Result := tvl_and[A, B];

[edit] Erlang

% Implemented by Arjun Sunel
-module(ternary).
-export([main/0, nott/1, andd/2,orr/2, then/2, equiv/2]).
 
main() ->
{ok, [A]} = io:fread("Enter A: ","~s"),
{ok, [B]} = io:fread("Enter B: ","~s"),
andd(A,B).
 
nott(S) ->
if
S=="T" ->
io : format("F\n");
 
S=="F" ->
io : format("T\n");
 
true ->
io: format("?\n")
end.
 
andd(A, B) ->
if
A=="T", B=="T" ->
io : format("T\n");
 
A=="F"; B=="F" ->
io : format("F\n");
 
true ->
io: format("?\n")
end.
 
 
orr(A, B) ->
if
A=="T"; B=="T" ->
io : format("T\n");
 
A=="?"; B=="?" ->
io : format("?\n");
 
true ->
io: format("F\n")
end.
 
 
then(A, B) ->
if
B=="T" ->
io : format("T\n");
 
A=="?" ->
io : format("?\n");
 
A=="F" ->
io :format("T\n");
B=="F" ->
io:format("F\n");
true ->
io: format("?\n")
end.
 
equiv(A, B) ->
if
A=="?" ->
io : format("?\n");
 
A=="F" ->
io : format("~s\n", [nott(B)]);
 
true ->
io: format("~s\n", [B])
end.
 

[edit] Factor

For boolean logic, Factor uses t and f with the words >boolean, not, and, or, xor. For ternary logic, we add m and define the words >trit, tnot, tand, tor, txor and t=. Our new class, trit, is the union class of t, m and f.

! rosettacode/ternary/ternary.factor
! http://rosettacode.org/wiki/Ternary_logic
USING: combinators kernel ;
IN: rosettacode.ternary
 
SINGLETON: m
UNION: trit t m POSTPONE: f ;
 
GENERIC: >trit ( object -- trit )
M: trit >trit ;
 
: tnot ( trit1 -- trit )
>trit { { t [ f ] } { m [ m ] } { f [ t ] } } case ;
 
: tand ( trit1 trit2 -- trit )
>trit {
{ t [ >trit ] }
{ m [ >trit { { t [ m ] } { m [ m ] } { f [ f ] } } case ] }
{ f [ >trit drop f ] }
} case ;
 
: tor ( trit1 trit2 -- trit )
>trit {
{ t [ >trit drop t ] }
{ m [ >trit { { t [ t ] } { m [ m ] } { f [ m ] } } case ] }
{ f [ >trit ] }
} case ;
 
: txor ( trit1 trit2 -- trit )
>trit {
{ t [ tnot ] }
{ m [ >trit drop m ] }
{ f [ >trit ] }
} case ;
 
: t= ( trit1 trit2 -- trit )
{
{ t [ >trit ] }
{ m [ >trit drop m ] }
{ f [ tnot ] }
} case ;

Example use:

( scratchpad ) CONSTANT: trits { t m f }
( scratchpad ) trits [ tnot ] map .
{ f m t }
( scratchpad ) trits [ trits swap [ tand ] curry map ] map .
{ { t m f } { m m f } { f f f } }
( scratchpad ) trits [ trits swap [ tor ] curry map ] map .
{ { t t t } { t m m } { t m f } }
( scratchpad ) trits [ trits swap [ txor ] curry map ] map .
{ { f m t } { m m m } { t m f } }
( scratchpad ) trits [ trits swap [ t= ] curry map ] map .
{ { t m f } { m m m } { f m t } }

[edit] Fortran

Please find the demonstration and compilation with gfortran at the start of the code. A module contains the ternary logic for easy reuse. Consider input redirection from unixdict.txt as vestigial. Or I could delete it.

 
!-*- mode: compilation; default-directory: "/tmp/" -*-
!Compilation started at Mon May 20 23:05:46
!
!a=./f && make $a && $a < unixdict.txt
!gfortran -std=f2003 -Wall -ffree-form f.f03 -o f
!
!ternary not
! 1.0 0.5 0.0
!
!
!ternary and
! 0.0 0.0 0.0
! 0.0 0.5 0.5
! 0.0 0.5 1.0
!
!
!ternary or
! 0.0 0.5 1.0
! 0.5 0.5 1.0
! 1.0 1.0 1.0
!
!
!ternary if
! 1.0 1.0 1.0
! 0.5 0.5 1.0
! 0.0 0.5 1.0
!
!
!ternary eq
! 1.0 0.5 0.0
! 0.5 0.5 0.5
! 0.0 0.5 1.0
!
!
!Compilation finished at Mon May 20 23:05:46
 
 
!This program is based on the j implementation
!not=: -.
!and=: <.
!or =: >.
!if =: (>. -.)"0~
!eq =: (<.&-. >. <.)"0
 
module trit
 
real, parameter :: true = 1, false = 0, maybe = 0.5
 
contains
 
real function tnot(y)
real, intent(in) :: y
tnot = 1 - y
end function tnot
 
real function tand(x, y)
real, intent(in) :: x, y
tand = min(x, y)
end function tand
 
real function tor(x, y)
real, intent(in) :: x, y
tor = max(x, y)
end function tor
 
real function tif(x, y)
real, intent(in) :: x, y
tif = tor(y, tnot(x))
end function tif
 
real function teq(x, y)
real, intent(in) :: x, y
teq = tor(tand(tnot(x), tnot(y)), tand(x, y))
end function teq
 
end module trit
 
program ternaryLogic
use trit
integer :: i
real, dimension(3) :: a = [false, maybe, true] ! (/ ... /)
write(6,'(/a)')'ternary not' ; write(6, '(3f4.1/)') (tnot(a(i)), i = 1 , 3)
write(6,'(/a)')'ternary and' ; call table(tand, a, a)
write(6,'(/a)')'ternary or' ; call table(tor, a, a)
write(6,'(/a)')'ternary if' ; call table(tif, a, a)
write(6,'(/a)')'ternary eq' ; call table(teq, a, a)
 
contains
 
subroutine table(u, x, y) ! for now, show the table.
real, external :: u
real, dimension(3), intent(in) :: x, y
integer :: i, j
write(6, '(3(3f4.1/))') ((u(x(i), y(j)), j=1,3), i=1,3)
end subroutine table
 
end program ternaryLogic
 

[edit] Go

Go has four operators for the bool type: ==, &&, ||, and !.

package main
 
import "fmt"
 
type trit int8
 
const (
trFalse trit = iota - 1
trMaybe
trTrue
)
 
func (t trit) String() string {
switch t {
case trFalse:
return "False"
case trMaybe:
return "Maybe"
case trTrue:
return "True "
}
panic("Invalid trit")
}
 
func trNot(t trit) trit {
return -t
}
 
func trAnd(s, t trit) trit {
if s < t {
return s
}
return t
}
 
func trOr(s, t trit) trit {
if s > t {
return s
}
return t
}
 
func trEq(s, t trit) trit {
return s * t
}
 
func main() {
trSet := []trit{trFalse, trMaybe, trTrue}
 
fmt.Println("t not t")
for _, t := range trSet {
fmt.Println(t, trNot(t))
}
 
fmt.Println("\ns t s and t")
for _, s := range trSet {
for _, t := range trSet {
fmt.Println(s, t, trAnd(s, t))
}
}
 
fmt.Println("\ns t s or t")
for _, s := range trSet {
for _, t := range trSet {
fmt.Println(s, t, trOr(s, t))
}
}
 
fmt.Println("\ns t s eq t")
for _, s := range trSet {
for _, t := range trSet {
fmt.Println(s, t, trEq(s, t))
}
}
}

Output:

t     not t
False True 
Maybe Maybe
True  False

s     t     s and t
False False False
False Maybe False
False True  False
Maybe False False
Maybe Maybe Maybe
Maybe True  Maybe
True  False False
True  Maybe Maybe
True  True  True 

s     t     s or t
False False False
False Maybe Maybe
False True  True 
Maybe False Maybe
Maybe Maybe Maybe
Maybe True  True 
True  False True 
True  Maybe True 
True  True  True 

s     t     s eq t
False False True 
False Maybe Maybe
False True  False
Maybe False Maybe
Maybe Maybe Maybe
Maybe True  Maybe
True  False False
True  Maybe Maybe
True  True  True 

[edit] Groovy

Solution:

enum Trit {
TRUE, MAYBE, FALSE
 
private Trit nand(Trit that) {
switch ([this,that]) {
case { FALSE in it }: return TRUE
case { MAYBE in it }: return MAYBE
default  : return FALSE
}
}
private Trit nor(Trit that) { this.or(that).not() }
 
Trit and(Trit that) { this.nand(that).not() }
Trit or(Trit that) { this.not().nand(that.not()) }
Trit not() { this.nand(this) }
Trit imply(Trit that) { this.nand(that.not()) }
Trit equiv(Trit that) { this.and(that).or(this.nor(that)) }
}

Test:

printf 'AND\n         '
Trit.values().each { b -> printf ('%6s', b) }
println '\n ----- ----- -----'
Trit.values().each { a ->
printf ('%6s | ', a)
Trit.values().each { b -> printf ('%6s', a.and(b)) }
println()
}
 
printf '\nOR\n '
Trit.values().each { b -> printf ('%6s', b) }
println '\n ----- ----- -----'
Trit.values().each { a ->
printf ('%6s | ', a)
Trit.values().each { b -> printf ('%6s', a.or(b)) }
println()
}
 
println '\nNOT'
Trit.values().each {
printf ('%6s | %6s\n', it, it.not())
}
 
printf '\nIMPLY\n '
Trit.values().each { b -> printf ('%6s', b) }
println '\n ----- ----- -----'
Trit.values().each { a ->
printf ('%6s | ', a)
Trit.values().each { b -> printf ('%6s', a.imply(b)) }
println()
}
 
printf '\nEQUIV\n '
Trit.values().each { b -> printf ('%6s', b) }
println '\n ----- ----- -----'
Trit.values().each { a ->
printf ('%6s | ', a)
Trit.values().each { b -> printf ('%6s', a.equiv(b)) }
println()
}

Output:

AND
           TRUE MAYBE FALSE
          ----- ----- -----
  TRUE |   TRUE MAYBE FALSE
 MAYBE |  MAYBE MAYBE FALSE
 FALSE |  FALSE FALSE FALSE

OR
           TRUE MAYBE FALSE
          ----- ----- -----
  TRUE |   TRUE  TRUE  TRUE
 MAYBE |   TRUE MAYBE MAYBE
 FALSE |   TRUE MAYBE FALSE

NOT
  TRUE |  FALSE
 MAYBE |  MAYBE
 FALSE |   TRUE

IMPLY
           TRUE MAYBE FALSE
          ----- ----- -----
  TRUE |   TRUE MAYBE FALSE
 MAYBE |   TRUE MAYBE MAYBE
 FALSE |   TRUE  TRUE  TRUE

EQUIV
           TRUE MAYBE FALSE
          ----- ----- -----
  TRUE |   TRUE MAYBE FALSE
 MAYBE |  MAYBE MAYBE MAYBE
 FALSE |  FALSE MAYBE  TRUE

[edit] Haskell

All operations given in terms of NAND, the functionally-complete operation.

import Prelude hiding (Bool(..), not, (&&), (||), (==))
 
main = mapM_ (putStrLn . unlines . map unwords)
[ table "not" $ unary not
, table "and" $ binary (&&)
, table "or" $ binary (||)
, table "implies" $ binary (=->)
, table "equals" $ binary (==)
]
 
data Trit = False | Maybe | True deriving (Show)
 
False `nand` _ = True
_ `nand` False = True
True `nand` True = False
_ `nand` _ = Maybe
 
not a = nand a a
 
a && b = not $ a `nand` b
 
a || b = not a `nand` not b
 
a =-> b = a `nand` not b
 
a == b = (a && b) || (not a && not b)
 
inputs1 = [True, Maybe, False]
inputs2 = [(a,b) | a <- inputs1, b <- inputs1]
 
unary f = map (\a -> [a, f a]) inputs1
binary f = map (\(a,b) -> [a, b, f a b]) inputs2
 
table name xs = map (map pad) . (header :) $ map (map show) xs
where header = map (:[]) (take ((length $ head xs) - 1) ['A'..]) ++ [name]
 
pad s = s ++ replicate (5 - length s) ' '

Output:

A     not
True  False
Maybe Maybe
False True

A     B     and
True  True  True
True  Maybe Maybe
True  False False
Maybe True  Maybe
Maybe Maybe Maybe
Maybe False False
False True  False
False Maybe False
False False False

A     B     or
True  True  True
True  Maybe True
True  False True
Maybe True  True
Maybe Maybe Maybe
Maybe False Maybe
False True  True
False Maybe Maybe
False False False

A     B     implies
True  True  True
True  Maybe Maybe
True  False False
Maybe True  True
Maybe Maybe Maybe
Maybe False Maybe
False True  True
False Maybe True
False False True

A     B     equals
True  True  True
True  Maybe Maybe
True  False False
Maybe True  Maybe
Maybe Maybe Maybe
Maybe False Maybe
False True  False
False Maybe Maybe
False False True

[edit] Icon and Unicon

The following example works in both Icon and Unicon. There are a couple of comments on the code that pertain to the task requirements:

  • Strictly speaking there are no binary values in Icon and Unicon. There are a number of flow control operations that result in expression success (and a result) or failure which affects flow. As a result there really isn't a set of binary operators to map into ternary. The example provides the minimum required by the task plus xor.
  • The code below does not define a data type as it doesn't really make sense in this case. Icon and Unicon can create records which would be overkill and clumsy in this case. Unicon can create objects which would also be overkill. The only remaining option is to reinterpret one of the existing types as ternary values. The code below implements balanced ternary values as integers in order to simplify several of the functions.
  • The use of integers doesn't really support strings of trits well. While there is a function showtrit to ease display a converse function to decode character trits in a string is not included.


$define TRUE    1
$define FALSE -1
$define UNKNOWN 0
 
invocable all
link printf
 
procedure main() # demonstrate ternary logic
 
ufunc := ["not3"]
bfunc := ["and3", "or3", "xor3", "eq3", "ifthen3"]
 
every f := !ufunc do { # display unary functions
printf("\nunary function=%s:\n",f)
every t1 := (TRUE | FALSE | UNKNOWN) do
printf(" %s : %s\n",showtrit(t1),showtrit(not3(t1)))
}
 
 
every f := !bfunc do { # display binary functions
printf("\nbinary function=%s:\n ",f)
every t1 := (&null | TRUE | FALSE | UNKNOWN) do {
printf(" %s : ",showtrit(\t1))
every t2 := (TRUE | FALSE | UNKNOWN | &null) do {
if /t1 then printf("  %s",showtrit(\t2)|"\n")
else printf("  %s",showtrit(f(t1,\t2))|"\n")
}
}
}
end
 
procedure showtrit(a) #: return printable trit of error if invalid
return case a of {TRUE:"T";FALSE:"F";UNKNOWN:"?";default:runerr(205,a)}
end
 
procedure istrit(a) #: return value of trit or error if invalid
return (TRUE|FALSE|UNKNOWN|runerr(205,a)) = a
end
 
procedure not3(a) #: not of trit or error if invalid
return FALSE * istrit(a)
end
 
procedure and3(a,b) #: and of two trits or error if invalid
return min(istrit(a),istrit(b))
end
 
procedure or3(a,b) #: or of two trits or error if invalid
return max(istrit(a),istrit(b))
end
 
procedure eq3(a,b) #: equals of two trits or error if invalid
return istrit(a) * istrit(b)
end
 
procedure ifthen3(a,b) #: if trit then trit or error if invalid
return case istrit(a) of { TRUE: istrit(b) ; UNKNOWN: or3(a,b); FALSE: TRUE }
end
 
procedure xor3(a,b) #: xor of two trits or error if invalid
return not3(eq3(a,b))
end

printf.icn provides support for the printf family of functions

Output:
unary function=not3:
 T : F
 F : T
 ? : ?

binary function=and3:
       T  F  ?
 T :   T  F  ?
 F :   F  F  F
 ? :   ?  F  ?

binary function=or3:
       T  F  ?
 T :   T  T  T
 F :   T  F  ?
 ? :   T  ?  ?

binary function=xor3:
       T  F  ?
 T :   F  T  ?
 F :   T  F  ?
 ? :   ?  ?  ?

binary function=eq3:
       T  F  ?
 T :   T  F  ?
 F :   F  T  ?
 ? :   ?  ?  ?

binary function=ifthen3:
       T  F  ?
 T :   T  F  ?
 F :   T  T  T
 ? :   T  ?  ?

[edit] J

The designers of J felt that user defined types were harmful, so that part of the task will not be supported here.

Instead:

true: 1 false: 0 maybe: 0.5

not=: -.
and=: <.
or =: >.
if =: (>. -.)"0~
eq =: (<.&-. >. <.)"0

Example use:

   not 0 0.5 1
1 0.5 0
 
0 0.5 1 and/ 0 0.5 1
0 0 0
0 0.5 0.5
0 0.5 1
 
0 0.5 1 or/ 0 0.5 1
0 0.5 1
0.5 0.5 1
1 1 1
 
0 0.5 1 if/ 0 0.5 1
1 1 1
0.5 0.5 1
0 0.5 1
 
0 0.5 1 eq/ 0 0.5 1
1 0.5 0
0.5 0.5 0.5
0 0.5 1

Note that this implementation is a special case of "fuzzy logic" (using a limited set of values).

Note that while >. and <. could be used for boolean operations instead of J's +. and *., the identity elements for >. and <. are not boolean values, but are negative and positive infinity. See also: Boolean ring

Note that we might instead define values between 0 and 1 to represent independent probabilities:

not=: -.
and=: *
or=: *&.-.
if =: (or -.)"0~
eq =: (*&-. or *)"0

However, while this might be a more intellectually satisfying approach, this gives us some different results from the task requirement, for the combination of two "maybe" values:

   not 0 0.5 1
1 0.5 0
 
0 0.5 1 and/ 0 0.5 1
0 0 0
0 0.25 0.5
0 0.5 1
 
0 0.5 1 or/ 0 0.5 1
0 0.5 1
0.5 0.75 1
1 1 1
 
0 0.5 1 if/ 0 0.5 1
1 1 1
0.5 0.75 1
0 0.5 1
 
0 0.5 1 eq/ 0 0.5 1
1 0.5 0
0.5 0.4375 0.5
0 0.5 1

Another interesting possibility would involve using George Boole's original operations. This leaves us without any "not", (if we include the definition of logical negation which was later added to the definition of Boolean algebra, then the only numbers which can be used with Boolean algebra are 1 and 0). So, it's not clear how we would implement "if" or "eq". However, "and" and "or" would look like this:

and=: *.
or=: +.

And, the boolean result tables would look like this:

   0 0.5 1 and/ 0 0.5 1
0 0 0
0 0.5 1
0 1 1
 
0 0.5 1 or/ 0 0.5 1
0 0.5 1
0.5 0.5 0.5
1 0.5 1

[edit] Java

Works with: Java version 1.5+
public class Logic{
public static enum Trit{
TRUE, MAYBE, FALSE;
 
public Trit and(Trit other){
if(this == TRUE){
return other;
}else if(this == MAYBE){
return (other == FALSE) ? FALSE : MAYBE;
}else{
return FALSE;
}
}
 
public Trit or(Trit other){
if(this == TRUE){
return TRUE;
}else if(this == MAYBE){
return (other == TRUE) ? TRUE : MAYBE;
}else{
return other;
}
}
 
public Trit tIf(Trit other){
if(this == TRUE){
return other;
}else if(this == MAYBE){
return (other == TRUE) ? TRUE : MAYBE;
}else{
return TRUE;
}
}
 
public Trit not(){
if(this == TRUE){
return FALSE;
}else if(this == MAYBE){
return MAYBE;
}else{
return TRUE;
}
}
 
public Trit equals(Trit other){
if(this == TRUE){
return other;
}else if(this == MAYBE){
return MAYBE;
}else{
return other.not();
}
}
}
public static void main(String[] args){
for(Trit a:Trit.values()){
System.out.println("not " + a + ": " + a.not());
}
for(Trit a:Trit.values()){
for(Trit b:Trit.values()){
System.out.println(a+" and "+b+": "+a.and(b)+
"\t "+a+" or "+b+": "+a.or(b)+
"\t "+a+" implies "+b+": "+a.tIf(b)+
"\t "+a+" = "+b+": "+a.equals(b));
}
}
}
}

Output:

not TRUE: FALSE
not MAYBE: MAYBE
not FALSE: TRUE
TRUE and TRUE: TRUE	 TRUE or TRUE: TRUE	 TRUE implies TRUE: TRUE	 TRUE = TRUE: TRUE
TRUE and MAYBE: MAYBE	 TRUE or MAYBE: TRUE	 TRUE implies MAYBE: MAYBE	 TRUE = MAYBE: MAYBE
TRUE and FALSE: FALSE	 TRUE or FALSE: TRUE	 TRUE implies FALSE: FALSE	 TRUE = FALSE: FALSE
MAYBE and TRUE: MAYBE	 MAYBE or TRUE: TRUE	 MAYBE implies TRUE: TRUE	 MAYBE = TRUE: MAYBE
MAYBE and MAYBE: MAYBE	 MAYBE or MAYBE: MAYBE	 MAYBE implies MAYBE: MAYBE	 MAYBE = MAYBE: MAYBE
MAYBE and FALSE: FALSE	 MAYBE or FALSE: MAYBE	 MAYBE implies FALSE: MAYBE	 MAYBE = FALSE: MAYBE
FALSE and TRUE: FALSE	 FALSE or TRUE: TRUE	 FALSE implies TRUE: TRUE	 FALSE = TRUE: FALSE
FALSE and MAYBE: FALSE	 FALSE or MAYBE: MAYBE	 FALSE implies MAYBE: TRUE	 FALSE = MAYBE: MAYBE
FALSE and FALSE: FALSE	 FALSE or FALSE: FALSE	 FALSE implies FALSE: TRUE	 FALSE = FALSE: TRUE

[edit] jq

jq itself does not have an extensible type system, so we'll use false, "maybe", and true as the three values since ternary logic agrees with Boolean logic for true and false, and because jq prints these three values consistently.

For consistency, all the ternary logic operators are defined here with the prefix "ternary_", but such a prefix is only needed for "not", "and", and "or", as these are jq keywords.
def ternary_nand(a; b):
if a == false or b == false then true
elif a == "maybe" or b == "maybe" then "maybe"
else false
end ;
 
def ternary_not(a): ternary_nand(a; a);
 
def ternary_or(a; b): ternary_nand( ternary_not(a); ternary_not(b) );
 
def ternary_nor(a; b): ternary_not( ternary_or(a;b) );
 
def ternary_and(a; b): ternary_not( ternary_nand(a; b) );
 
def ternary_imply(this; that):
ternary_nand(this, ternary_not(that));
 
def ternary_equiv(this; that):
ternary_or( ternary_and(this; that); ternary_nor(this; that) );
 
def display_and(a; b):
a as $a | b as $b
| "\($a) and \($b) is \( ternary_and($a; $b) )";
def display_equiv(a; b):
a as $a | b as $b
| "\($a) equiv \($b) is \( ternary_equiv($a; $b) )";
# etc etc
 
# Invoke the display functions:
display_and( (false, "maybe", true ); (false, "maybe", true) ),
display_equiv( (false, "maybe", true ); (false, "maybe", true) ),
"etc etc"
 
Output:
"false and false is false"
"false and maybe is false"
"false and true is false"
"maybe and false is false"
"maybe and maybe is maybe"
"maybe and true is maybe"
"true and false is false"
"true and maybe is maybe"
"true and true is true"
"false equiv false is true"
"false equiv maybe is maybe"
"false equiv true is false"
"maybe equiv false is maybe"
"maybe equiv maybe is maybe"
"maybe equiv true is maybe"
"true equiv false is false"
"true equiv maybe is maybe"
"true equiv true is true"
"etc etc"
 

[edit] Liberty BASIC

 
'ternary logic
'0 1 2
'F ? T
'False Don't know True
'LB has NOT AND OR XOR, so we implement them.
'LB has no EQ, but XOR could be expressed via EQ. In 'normal' boolean at least.
 
global tFalse, tDontKnow, tTrue
tFalse = 0
tDontKnow = 1
tTrue = 2
 
print "Short and long names for ternary logic values"
for i = tFalse to tTrue
print shortName3$(i);" ";longName3$(i)
next
print
 
print "Single parameter functions"
print "x";" ";"=x";" ";"not(x)"
for i = tFalse to tTrue
print shortName3$(i);" ";shortName3$(i);" ";shortName3$(not3(i))
next
print
 
print "Double parameter fuctions"
print "x";" ";"y";" ";"x AND y";" ";"x OR y";" ";"x EQ y";" ";"x XOR y"
for a = tFalse to tTrue
for b = tFalse to tTrue
print shortName3$(a);" ";shortName3$(b);" "; _
shortName3$(and3(a,b));" "; shortName3$(or3(a,b));" "; _
shortName3$(eq3(a,b));" "; shortName3$(xor3(a,b))
next
next
 
function and3(a,b)
and3 = min(a,b)
end function
 
function or3(a,b)
or3 = max(a,b)
end function
 
function eq3(a,b)
select case
case a=tDontKnow or b=tDontKnow
eq3 = tDontKnow
case a=b
eq3 = tTrue
case else
eq3 = tFalse
end select
end function
 
function xor3(a,b)
xor3 = not3(eq3(a,b))
end function
 
function not3(b)
not3 = 2-b
end function
 
'------------------------------------------------
function shortName3$(i)
shortName3$ = word$("F ? T", i+1)
end function
 
function longName3$(i)
longName3$ = word$("False,Don't know,True", i+1, ",")
end function
 

Output:

Short and long names for ternary logic values
F False
? Don't know
T True

Single parameter functions
x =x  not(x)
F  F    T
?  ?    ?
T  T    F

Double  parameter fuctions
x y  x AND y  x OR y  x EQ y  x XOR y
F F     F       F       T        F
F ?     F       ?       ?        ?
F T     F       T       F        T
? F     F       ?       ?        ?
? ?     ?       ?       ?        ?
? T     ?       T       ?        ?
T F     F       T       F        T
T ?     ?       T       ?        ?
T T     T       T       T        F

[edit] Mathematica

Type definition is not allowed in Mathematica. We can just use the build-in symbols "True" and "False", and add a new symbol "Maybe".

Maybe /: ! Maybe = Maybe;
Maybe /: (And | Or | Nand | Nor | Xor | Xnor | Implies | Equivalent)[Maybe, Maybe] = Maybe;

Example:

trits = {True, Maybe, False};
Print@Grid[
ArrayFlatten[{{{{Not}}, {{Null}}}, {List /@ trits,
List /@ Not /@ trits}}]];
Do[Print@Grid[
ArrayFlatten[{{{{operator}}, {{Null, Null,
Null}}}, {{{Null}}, {trits}}, {List /@ trits,
Outer[operator, trits, trits]}}]], {operator, {And, Or, Nand,
Nor, Xor, Xnor, Implies, Equivalent}}]

Output:

Not	
True	False
Maybe	Maybe
False	True



And			
	True	Maybe	False
True	True	Maybe	False
Maybe	Maybe	Maybe	False
False	False	False	False



Or			
	True	Maybe	False
True	True	True	True
Maybe	True	Maybe	Maybe
False	True	Maybe	False



Nand			
	True	Maybe	False
True	False	Maybe	True
Maybe	Maybe	Maybe	True
False	True	True	True



Nor			
	True	Maybe	False
True	False	False	False
Maybe	False	Maybe	Maybe
False	False	Maybe	True



Xor			
	True	Maybe	False
True	False	Maybe	True
Maybe	Maybe	Maybe	Maybe
False	True	Maybe	False



Xnor			
	True	Maybe	False
True	True	Maybe	False
Maybe	Maybe	Maybe	Maybe
False	False	Maybe	True



Implies			
	True	Maybe	False
True	True	Maybe	False
Maybe	True	Maybe	Maybe
False	True	True	True



Equivalent			
	True	Maybe	False
True	True	Maybe	False
Maybe	Maybe	Maybe	Maybe
False	False	Maybe	True

[edit] МК-61/52

П0	Сx	С/П	^	1	+	3	*	+	1
+ 3 x^y ИП0 <-> / [x] ^ ^ 3
/ [x] 3 * - 1 - С/П 1 5
6 3 3 БП 00 1 9 5 6 9
БП 00 1 5 9 2 9 БП 00 1
5 6 6 5 БП 00 /-/ ЗН С/П

Instruction:

БП XX С/П a ^ b С/П,

where XX = 28 for AND; 35 for OR; 42 for implies; 49 for equivalent; 56 for NOT;

a, b ∈ {-1, 0, 1}.

[edit] Nimrod

type Trit* = enum ttrue, tmaybe, tfalse
 
proc `$`*(a: Trit): string =
case a
of ttrue: "T"
of tmaybe: "?"
of tfalse: "F"
 
proc `not`*(a: Trit): Trit =
case a
of ttrue: tfalse
of tmaybe: tmaybe
of tfalse: ttrue
 
proc `and`*(a, b: Trit): Trit =
const t: array[Trit, array[Trit, Trit]] =
[ [ttrue, tmaybe, tfalse]
, [tmaybe, tmaybe, tfalse]
, [tfalse, tfalse, tfalse] ]
t[a][b]
 
proc `or`*(a, b: Trit): Trit =
const t: array[Trit, array[Trit, Trit]] =
[ [ttrue, ttrue, ttrue]
, [ttrue, tmaybe, tmaybe]
, [ttrue, tmaybe, tfalse] ]
t[a][b]
 
proc then*(a, b: Trit): Trit =
const t: array[Trit, array[Trit, Trit]] =
[ [ttrue, tmaybe, tfalse]
, [ttrue, tmaybe, tmaybe]
, [ttrue, ttrue, ttrue] ]
t[a][b]
 
proc equiv*(a, b: Trit): Trit =
const t: array[Trit, array[Trit, Trit]] =
[ [ttrue, tmaybe, tfalse]
, [tmaybe, tmaybe, tmaybe]
, [tfalse, tmaybe, ttrue] ]
t[a][b]
 
import strutils
 
var
op1 = ttrue
op2 = ttrue
 
for t in Trit:
echo "Not ", t , ": ", not t
 
for op1 in Trit:
for op2 in Trit:
echo "$# and $#: $#".format(op1, op2, op1 and op2)
echo "$# or $#: $#".format(op1, op2, op1 or op2)
echo "$# then $#: $#".format(op1, op2, op1.then op2)
echo "$# equiv $#: $#".format(op1, op2, op1.equiv op2)

Output:

Not T: F
Not ?: ?
Not F: T
T and   T: T
T or    T: T
T then  T: T
T equiv T: T
T and   ?: ?
T or    ?: T
T then  ?: ?
T equiv ?: ?
T and   F: F
T or    F: T
T then  F: F
T equiv F: F
? and   T: ?
? or    T: T
? then  T: T
? equiv T: ?
? and   ?: ?
? or    ?: ?
? then  ?: ?
? equiv ?: ?
? and   F: F
? or    F: ?
? then  F: ?
? equiv F: ?
F and   T: F
F or    T: T
F then  T: T
F equiv T: F
F and   ?: F
F or    ?: ?
F then  ?: T
F equiv ?: ?
F and   F: F
F or    F: F
F then  F: T
F equiv F: T

[edit] OCaml

type trit = True | False | Maybe
 
let t_not = function
| True -> False
| False -> True
| Maybe -> Maybe
 
let t_and a b = match (a,b) with
| (True,True) -> True
| (False,_) | (_,False) -> False
| _ -> Maybe
 
let t_or a b = t_not (t_and (t_not a) (t_not b))
 
let t_eq a b = match (a,b) with
| (True,True) | (False,False) -> True
| (False,True) | (True,False) -> False
| _ -> Maybe
 
let t_imply a b = t_or (t_not a) b
 
let string_of_trit = function
| True -> "True"
| False -> "False"
| Maybe -> "Maybe"
 
let () =
let values = [| True; Maybe; False |] in
let f = string_of_trit in
Array.iter (fun v -> Printf.printf "Not %s: %s\n" (f v) (f (t_not v))) values;
print_newline ();
let print op str =
Array.iter (fun a ->
Array.iter (fun b ->
Printf.printf "%s %s %s: %s\n" (f a) str (f b) (f (op a b))
) values
) values;
print_newline ()
in
print t_and "And";
print t_or "Or";
print t_imply "Then";
print t_eq "Equiv";
;;

Output:

Not True: False
Not Maybe: Maybe
Not False: True

True And True: True
True And Maybe: Maybe
True And False: False
Maybe And True: Maybe
Maybe And Maybe: Maybe
Maybe And False: False
False And True: False
False And Maybe: False
False And False: False

True Or True: True
True Or Maybe: True
True Or False: True
Maybe Or True: True
Maybe Or Maybe: Maybe
Maybe Or False: Maybe
False Or True: True
False Or Maybe: Maybe
False Or False: False

True Then True: True
True Then Maybe: Maybe
True Then False: False
Maybe Then True: True
Maybe Then Maybe: Maybe
Maybe Then False: Maybe
False Then True: True
False Then Maybe: True
False Then False: True

True Equiv True: True
True Equiv Maybe: Maybe
True Equiv False: False
Maybe Equiv True: Maybe
Maybe Equiv Maybe: Maybe
Maybe Equiv False: Maybe
False Equiv True: False
False Equiv Maybe: Maybe
False Equiv False: True

[edit] Using a general binary -> ternary transform

Instead of writing all of the truth-tables by hand, we can construct a general binary -> ternary transform and apply it to any logical function we want:

type trit = True | False | Maybe
 
let to_bin = function True -> [true] | False -> [false] | Maybe -> [true;false]
 
let eval f x =
List.fold_left (fun l c -> List.fold_left (fun m d -> ((d c) :: m)) l f) [] x
 
let rec from_bin =
function [true] -> True | [false] -> False
| h :: t -> (match (h, from_bin t) with
(true,True) -> True | (false,False) -> False | _ -> Maybe)
| _ -> Maybe
 
let to_ternary1 uop = fun x -> from_bin (eval [uop] (to_bin x))
let to_ternary2 bop = fun x y -> from_bin (eval (eval [bop] (to_bin x)) (to_bin y))
 
let t_not = to_ternary1 (not)
let t_and = to_ternary2 (&&)
let t_or = to_ternary2 (||)
let t_equiv = to_ternary2 (=)
let t_imply = to_ternary2 (fun p q -> (not p) || q)
 
let str = function True -> "True " | False -> "False" | Maybe -> "Maybe"
let iterv f = List.iter f [True; False; Maybe]
 
let table1 s u =
print_endline ("\n"^s^":");
iterv (fun v -> print_endline (" "^(str v)^" -> "^(str (u v))));;
 
let table2 s b =
print_endline ("\n"^s^":");
iterv (fun u ->
iterv (fun v ->
print_endline (" "^(str u)^" "^(str v)^" -> "^(str (b u v)))));;
 
table1 "not" t_not;;
table2 "and" t_and;;
table2 "or" t_or;;
table2 "equiv" t_equiv;;
table2 "implies" t_imply;;

Output:

not:
  True  -> False
  False -> True 
  Maybe -> Maybe

and:
  True  True  -> True 
  True  False -> False
  True  Maybe -> Maybe
  False True  -> False
  False False -> False
  False Maybe -> False
  Maybe True  -> Maybe
  Maybe False -> False
  Maybe Maybe -> Maybe

or:
  True  True  -> True 
  True  False -> True 
  True  Maybe -> True 
  False True  -> True 
  False False -> False
  False Maybe -> Maybe
  Maybe True  -> True 
  Maybe False -> Maybe
  Maybe Maybe -> Maybe

equiv:
  True  True  -> True 
  True  False -> False
  True  Maybe -> Maybe
  False True  -> False
  False False -> True 
  False Maybe -> Maybe
  Maybe True  -> Maybe
  Maybe False -> Maybe
  Maybe Maybe -> Maybe

implies:
  True  True  -> True 
  True  False -> False
  True  Maybe -> Maybe
  False True  -> True 
  False False -> True 
  False Maybe -> True 
  Maybe True  -> True 
  Maybe False -> Maybe
  Maybe Maybe -> Maybe

[edit] ooRexx

 
tritValues = .array~of(.trit~true, .trit~false, .trit~maybe)
tab = '09'x
 
say "not operation (\)"
loop a over tritValues
say "\"a":" (\a)
end
 
say
say "and operation (&)"
loop aa over tritValues
loop bb over tritValues
say (aa" & "bb":" (aa&bb))
end
end
 
say
say "or operation (|)"
loop aa over tritValues
loop bb over tritValues
say (aa" | "bb":" (aa|bb))
end
end
 
say
say "implies operation (&&)"
loop aa over tritValues
loop bb over tritValues
say (aa" && "bb":" (aa&&bb))
end
end
 
say
say "equals operation (=)"
loop aa over tritValues
loop bb over tritValues
say (aa" = "bb":" (aa=bb))
end
end
 
::class trit
-- making this a private method so we can control the creation
-- of these. We only allow 3 instances to exist
::method new class private
forward class(super)
 
::method init class
expose true false maybe
-- delayed creation
true = .nil
false = .nil
maybe = .nil
 
-- read only attribute access to the instances.
-- these methods create the appropriate singleton on the first call
::attribute true class get
expose true
if true == .nil then true = self~new("True")
return true
 
::attribute false class get
expose false
if false == .nil then false = self~new("False")
return false
 
::attribute maybe class get
expose maybe
if maybe == .nil then maybe = self~new("Maybe")
return maybe
 
-- create an instance
::method init
expose value
use arg value
 
-- string method to return the value of the instance
::method string
expose value
return value
 
-- "and" method using the operator overload
::method "&"
use strict arg other
if self == .trit~true then return other
else if self == .trit~maybe then do
if other == .trit~false then return .trit~false
else return .trit~maybe
end
else return .trit~false
 
-- "or" method using the operator overload
::method "|"
use strict arg other
if self == .trit~true then return .trit~true
else if self == .trit~maybe then do
if other == .trit~true then return .trit~true
else return .trit~maybe
end
else return other
 
-- implies method...using the XOR operator for this
::method "&&"
use strict arg other
if self == .trit~true then return other
else if self == .trit~maybe then do
if other == .trit~true then return .trit~true
else return .trit~maybe
end
else return .trit~true
 
-- "not" method using the operator overload
::method "\"
if self == .trit~true then return .trit~false
else if self == .trit~maybe then return .trit~maybe
else return .trit~true
 
-- "equals" using the "=" override. This makes a distinction between
-- the "==" operator, which is real equality and the "=" operator, which
-- is trinary equality.
::method "="
use strict arg other
if self == .trit~true then return other
else if self == .trit~maybe then return .trit~maybe
else return \other
 
not operation (\)
\True: False
\False: True
\Maybe: Maybe

and operation (&)
True & True: True
True & False: False
True & Maybe: Maybe
False & True: False
False & False: False
False & Maybe: False
Maybe & True: Maybe
Maybe & False: False
Maybe & Maybe: Maybe

or operation (|)
True | True: True
True | False: True
True | Maybe: True
False | True: True
False | False: False
False | Maybe: Maybe
Maybe | True: True
Maybe | False: Maybe
Maybe | Maybe: Maybe

implies operation (&&)
True && True: True
True && False: False
True && Maybe: Maybe
False && True: True
False && False: True
False && Maybe: True
Maybe && True: True
Maybe && False: Maybe
Maybe && Maybe: Maybe

equals operation (=)
True = True: True
True = False: False
True = Maybe: Maybe
False = True: False
False = False: True
False = Maybe: Maybe
Maybe = True: Maybe
Maybe = False: Maybe
Maybe = Maybe: Maybe

[edit] Pascal

Program TernaryLogic (output);
 
type
trit = (terTrue, terMayBe, terFalse);
 
function terNot (a: trit): trit;
begin
case a of
terTrue: terNot := terFalse;
terMayBe: terNot := terMayBe;
terFalse: terNot := terTrue;
end;
end;
 
function terAnd (a, b: trit): trit;
begin
terAnd := terMayBe;
if (a = terFalse) or (b = terFalse) then
terAnd := terFalse
else
if (a = terTrue) and (b = terTrue) then
terAnd := terTrue;
end;
 
function terOr (a, b: trit): trit;
begin
terOr := terMayBe;
if (a = terTrue) or (b = terTrue) then
terOr := terTrue
else
if (a = terFalse) and (b = terFalse) then
terOr := terFalse;
end;
 
function terEquals (a, b: trit): trit;
begin
if a = b then
terEquals := terTrue
else
if a <> b then
terEquals := terFalse;
if (a = terMayBe) or (b = terMayBe) then
terEquals := terMayBe;
end;
 
function terIfThen (a, b: trit): trit;
begin
terIfThen := terMayBe;
if (a = terTrue) or (b = terFalse) then
terIfThen := terTrue
else
if (a = terFalse) and (b = terTrue) then
terIfThen := terFalse;
end;
 
function terToStr(a: trit): string;
begin
case a of
terTrue: terToStr := 'True ';
terMayBe: terToStr := 'Maybe';
terFalse: terToStr := 'False';
end;
end;
 
begin
writeln('Ternary logic test:');
writeln;
writeln('NOT ', ' True ', ' Maybe', ' False');
writeln(' ', terToStr(terNot(terTrue)), ' ', terToStr(terNot(terMayBe)), ' ', terToStr(terNot(terFalse)));
writeln;
writeln('AND ', ' True ', ' Maybe', ' False');
writeln('True ', terToStr(terAnd(terTrue,terTrue)), ' ', terToStr(terAnd(terMayBe,terTrue)), ' ', terToStr(terAnd(terFalse,terTrue)));
writeln('Maybe ', terToStr(terAnd(terTrue,terMayBe)), ' ', terToStr(terAnd(terMayBe,terMayBe)), ' ', terToStr(terAnd(terFalse,terMayBe)));
writeln('False ', terToStr(terAnd(terTrue,terFalse)), ' ', terToStr(terAnd(terMayBe,terFalse)), ' ', terToStr(terAnd(terFalse,terFalse)));
writeln;
writeln('OR ', ' True ', ' Maybe', ' False');
writeln('True ', terToStr(terOR(terTrue,terTrue)), ' ', terToStr(terOR(terMayBe,terTrue)), ' ', terToStr(terOR(terFalse,terTrue)));
writeln('Maybe ', terToStr(terOR(terTrue,terMayBe)), ' ', terToStr(terOR(terMayBe,terMayBe)), ' ', terToStr(terOR(terFalse,terMayBe)));
writeln('False ', terToStr(terOR(terTrue,terFalse)), ' ', terToStr(terOR(terMayBe,terFalse)), ' ', terToStr(terOR(terFalse,terFalse)));
writeln;
writeln('IFTHEN', ' True ', ' Maybe', ' False');
writeln('True ', terToStr(terIfThen(terTrue,terTrue)), ' ', terToStr(terIfThen(terMayBe,terTrue)), ' ', terToStr(terIfThen(terFalse,terTrue)));
writeln('Maybe ', terToStr(terIfThen(terTrue,terMayBe)), ' ', terToStr(terIfThen(terMayBe,terMayBe)), ' ', terToStr(terIfThen(terFalse,terMayBe)));
writeln('False ', terToStr(terIfThen(terTrue,terFalse)), ' ', terToStr(terIfThen(terMayBe,terFalse)), ' ', terToStr(terIfThen(terFalse,terFalse)));
writeln;
writeln('EQUAL ', ' True ', ' Maybe', ' False');
writeln('True ', terToStr(terEquals(terTrue,terTrue)), ' ', terToStr(terEquals(terMayBe,terTrue)), ' ', terToStr(terEquals(terFalse,terTrue)));
writeln('Maybe ', terToStr(terEquals(terTrue,terMayBe)), ' ', terToStr(terEquals(terMayBe,terMayBe)), ' ', terToStr(terEquals(terFalse,terMayBe)));
writeln('False ', terToStr(terEquals(terTrue,terFalse)), ' ', terToStr(terEquals(terMayBe,terFalse)), ' ', terToStr(terEquals(terFalse,terFalse)));
writeln;
end.

Output:

:> ./TernaryLogic
Ternary logic test:

NOT  True  Maybe False
     False Maybe True 

AND    True  Maybe False
True   True  Maybe False
Maybe  Maybe Maybe False
False  False False False

OR     True  Maybe False
True   True  True  True 
Maybe  True  Maybe Maybe
False  True  Maybe False

IFTHEN True  Maybe False
True   True  Maybe False
Maybe  True  Maybe Maybe
False  True  True  True 

EQUAL  True  Maybe False
True   True  Maybe False
Maybe  Maybe Maybe Maybe
False  False Maybe True 

[edit] Perl

File Trit.pm:

package Trit;
 
# -1 = false ; 0 = maybe ; 1 = true
 
use Exporter 'import';
 
our @EXPORT_OK = qw(TRUE FALSE MAYBE is_true is_false is_maybe);
our %EXPORT_TAGS = (
all => \@EXPORT_OK,
const => [qw(TRUE FALSE MAYBE)],
bool => [qw(is_true is_false is_maybe)],
);
 
use List::Util qw(min max);
 
use overload
'=' => sub { $_[0]->clone() },
'<=>'=> sub { $_[0]->cmp($_[1]) },
'cmp'=> sub { $_[0]->cmp($_[1]) },
'==' => sub { ${$_[0]} == ${$_[1]} },
'eq' => sub { $_[0]->equiv($_[1]) },
'>' => sub { ${$_[0]} > ${$_[1]} },
'<' => sub { ${$_[0]} < ${$_[1]} },
'>=' => sub { ${$_[0]} >= ${$_[1]} },
'<=' => sub { ${$_[0]} <= ${$_[1]} },
'|' => sub { $_[0]->or($_[1]) },
'&' => sub { $_[0]->and($_[1]) },
'!' => sub { $_[0]->not() },
'~' => sub { $_[0]->not() },
'""' => sub { $_[0]->tostr() },
'0+' => sub { $_[0]->tonum() },
;
 
sub new
{
my ($class, $v) = @_;
my $ret =
!defined($v) ? 0 :
$v eq 'true' ? 1 :
$v eq 'false'? -1 :
$v eq 'maybe'? 0 :
$v > 0 ? 1 :
$v < 0 ? -1 :
0;
return bless \$ret, $class;
}
 
sub TRUE() { new Trit( 1) }
sub FALSE() { new Trit(-1) }
sub MAYBE() { new Trit( 0) }
 
sub clone
{
my $ret = ${$_[0]};
return bless \$ret, ref($_[0]);
}
 
sub tostr { ${$_[0]} > 0 ? "true" : ${$_[0]} < 0 ? "false" : "maybe" }
sub tonum { ${$_[0]} }
 
sub is_true { ${$_[0]} > 0 }
sub is_false { ${$_[0]} < 0 }
sub is_maybe { ${$_[0]} == 0 }
 
sub cmp { ${$_[0]} <=> ${$_[1]} }
sub not { new Trit(-${$_[0]}) }
sub and { new Trit(min(${$_[0]}, ${$_[1]}) ) }
sub or { new Trit(max(${$_[0]}, ${$_[1]}) ) }
 
sub equiv { new Trit( ${$_[0]} * ${$_[1]} ) }

File test.pl:

use Trit ':all';
 
my @a = (TRUE(), MAYBE(), FALSE());
 
print "\na\tNOT a\n";
print "$_\t".(!$_)."\n" for @a; # Example of use of prefix operator NOT. Tilde ~ also can be used.
 
 
print "\nAND\t".join("\t",@a)."\n";
for my $a (@a) {
print $a;
for my $b (@a) {
print "\t".($a & $b); # Example of use of infix & (and)
}
print "\n";
}
 
print "\nOR\t".join("\t",@a)."\n";
for my $a (@a) {
print $a;
for my $b (@a) {
print "\t".($a | $b); # Example of use of infix | (or)
}
print "\n";
}
 
print "\nEQV\t".join("\t",@a)."\n";
for my $a (@a) {
print $a;
for my $b (@a) {
print "\t".($a eq $b); # Example of use of infix eq (equivalence)
}
print "\n";
}
 
print "\n==\t".join("\t",@a)."\n";
for my $a (@a) {
print $a;
for my $b (@a) {
print "\t".($a == $b); # Example of use of infix == (equality)
}
print "\n";
}

Output:

a	NOT a
true	false
maybe	maybe
false	true

AND	true	maybe	false
true	true	maybe	false
maybe	maybe	maybe	false
false	false	false	false

OR	true	maybe	false
true	true	true	true
maybe	true	maybe	maybe
false	true	maybe	false

EQV	true	maybe	false
true	true	maybe	false
maybe	maybe	maybe	maybe
false	false	maybe	true

==	true	maybe	false
true	1		
maybe		1	
false			1

[edit] Perl 6

Works with: niecza

Implementation:

enum Trit <Foo Moo Too>;
 
sub prefix:<¬> (Trit $a) { Trit(1-($a-1)) }
 
sub infix:<> is equiv(&infix:<*>) (Trit $a, Trit $b) { $a min $b }
sub infix:<> is equiv(&infix:<+>) (Trit $a, Trit $b) { $a max $b }
 
sub infix:<> is equiv(&infix:<..>) (Trit $a, Trit $b) { ¬$a max $b }
sub infix:<> is equiv(&infix:<eq>) (Trit $a, Trit $b) { Trit(1 + ($a-1) * ($b-1)) }

The precedence of each operator is specified as equivalent to an existing operator. We've taken the liberty of using an arrow for implication, to avoid confusing it with , (U+2283 SUPERSET OF).

To test, we use this code:

say '¬';
say "Too {¬Too}";
say "Moo {¬Moo}";
say "Foo {¬Foo}";
 
sub tbl (&op,$name) {
say '';
say "$name Too Moo Foo";
say " ╔═══════════";
say "Too║{op Too,Too} {op Too,Moo} {op Too,Foo}";
say "Moo║{op Moo,Too} {op Moo,Moo} {op Moo,Foo}";
say "Foo║{op Foo,Too} {op Foo,Moo} {op Foo,Foo}";
}
 
tbl(&infix:<>, '∧');
tbl(&infix:<>, '∨');
tbl(&infix:<>, '→');
tbl(&infix:<>, '≡');
 
say '';
say 'Precedence tests should all print "Too":';
say ~(
Foo ∧ Too ∨ Too ≡ Too,
Foo ∧ (Too ∨ Too) ≡ Foo,
Too ∨ Too ∧ Foo ≡ Too,
(Too ∨ Too) ∧ Foo ≡ Foo,
 
¬Too ∧ Too ∨ Too ≡ Too,
¬Too ∧ (Too ∨ Too) ≡ ¬Too,
Too ∨ Too ∧ ¬Too ≡ Too,
(Too ∨ Too) ∧ ¬Too ≡ ¬Too,
 
Foo ∧ Too ∨ Foo → Foo ≡ Too,
Foo ∧ Too ∨ Too → Foo ≡ Foo,
);

Output:

¬
Too Foo
Moo Moo
Foo Too

∧   Too Moo Foo
   ╔═══════════
Too║Too Moo Foo
Moo║Moo Moo Foo
Foo║Foo Foo Foo

∨   Too Moo Foo
   ╔═══════════
Too║Too Too Too
Moo║Too Moo Moo
Foo║Too Moo Foo

→   Too Moo Foo
   ╔═══════════
Too║Too Moo Foo
Moo║Too Moo Moo
Foo║Too Too Too

≡   Too Moo Foo
   ╔═══════════
Too║Too Moo Foo
Moo║Moo Moo Moo
Foo║Foo Moo Too

Precedence tests should all print "Too":
Too Too Too Too Too Too Too Too Too Too

[edit] PicoLisp

In addition for the standard T (for "true") and NIL (for "false") we define 0 (zero, for "maybe").

(de 3not (A)
(or (=0 A) (not A)) )
 
(de 3and (A B)
(cond
((=T A) B)
((=0 A) (and B 0)) ) )
 
(de 3or (A B)
(cond
((=T A) T)
((=0 A) (or (=T B) 0))
(T B) ) )
 
(de 3impl (A B)
(cond
((=T A) B)
((=0 A) (or (=T B) 0))
(T T) ) )
 
(de 3equiv (A B)
(cond
((=T A) B)
((=0 A) 0)
(T (3not B)) ) )

Test:

(for X '(T 0 NIL)
(println 'not X '-> (3not X)) )
 
(for Fun '((and . 3and) (or . 3or) (implies . 3impl) (equivalent . 3equiv))
(for X '(T 0 NIL)
(for Y '(T 0 NIL)
(println X (car Fun) Y '-> ((cdr Fun) X Y)) ) ) )

Output:

not T -> NIL
not 0 -> 0
not NIL -> T
T and T -> T
T and 0 -> 0
T and NIL -> NIL
0 and T -> 0
0 and 0 -> 0
0 and NIL -> NIL
NIL and T -> NIL
NIL and 0 -> NIL
NIL and NIL -> NIL
T or T -> T
T or 0 -> T
T or NIL -> T
0 or T -> T
0 or 0 -> 0
0 or NIL -> 0
NIL or T -> T
NIL or 0 -> 0
NIL or NIL -> NIL
T implies T -> T
T implies 0 -> 0
T implies NIL -> NIL
0 implies T -> T
0 implies 0 -> 0
0 implies NIL -> 0
NIL implies T -> T
NIL implies 0 -> T
NIL implies NIL -> T
T equivalent T -> T
T equivalent 0 -> 0
T equivalent NIL -> NIL
0 equivalent T -> 0
0 equivalent 0 -> 0
0 equivalent NIL -> 0
NIL equivalent T -> NIL
NIL equivalent 0 -> 0
NIL equivalent NIL -> T

[edit] Python

In Python, the keywords 'and', 'not', and 'or' are coerced to always work as boolean operators. I have therefore overloaded the boolean bitwise operators &, |, ^ to provide the required functionality.

class Trit(int):
def __new__(cls, value):
if value == 'TRUE':
value = 1
elif value == 'FALSE':
value = 0
elif value == 'MAYBE':
value = -1
return super(Trit, cls).__new__(cls, value // (abs(value) or 1))
 
def __repr__(self):
if self > 0:
return 'TRUE'
elif self == 0:
return 'FALSE'
return 'MAYBE'
 
def __str__(self):
return repr(self)
 
def __bool__(self):
if self > 0:
return True
elif self == 0:
return False
else:
raise ValueError("invalid literal for bool(): '%s'" % self)
 
def __or__(self, other):
if isinstance(other, Trit):
return _ttable[(self, other)][1]
else:
try:
return _ttable[(self, Trit(bool(other)))][1]
except:
return NotImplemented
 
def __ror__(self, other):
if isinstance(other, Trit):
return _ttable[(self, other)][1]
else:
try:
return _ttable[(self, Trit(bool(other)))][1]
except:
return NotImplemented
 
def __and__(self, other):
if isinstance(other, Trit):
return _ttable[(self, other)][0]
else:
try:
return _ttable[(self, Trit(bool(other)))][0]
except:
return NotImplemented
 
def __rand__(self, other):
if isinstance(other, Trit):
return _ttable[(self, other)][0]
else:
try:
return _ttable[(self, Trit(bool(other)))][0]
except:
return NotImplemented
 
def __xor__(self, other):
if isinstance(other, Trit):
return _ttable[(self, other)][2]
else:
try:
return _ttable[(self, Trit(bool(other)))][2]
except:
return NotImplemented
 
def __rxor__(self, other):
if isinstance(other, Trit):
return _ttable[(self, other)][2]
else:
try:
return _ttable[(self, Trit(bool(other)))][2]
except:
return NotImplemented
 
def __invert__(self):
return _ttable[self]
 
def __getattr__(self, name):
if name in ('_n', 'flip'):
# So you can do x._n == x.flip; the inverse of x
# In Python 'not' is strictly boolean so we can't write `not x`
# Same applies to keywords 'and' and 'or'.
return _ttable[self]
else:
raise AttributeError
 
 
 
TRUE, FALSE, MAYBE = Trit(1), Trit(0), Trit(-1)
 
_ttable = {
# A: -> flip_A
TRUE: FALSE,
FALSE: TRUE,
MAYBE: MAYBE,
# (A, B): -> (A_and_B, A_or_B, A_xor_B)
(MAYBE, MAYBE): (MAYBE, MAYBE, MAYBE),
(MAYBE, FALSE): (FALSE, MAYBE, MAYBE),
(MAYBE, TRUE): (MAYBE, TRUE, MAYBE),
(FALSE, MAYBE): (FALSE, MAYBE, MAYBE),
(FALSE, FALSE): (FALSE, FALSE, FALSE),
(FALSE, TRUE): (FALSE, TRUE, TRUE),
( TRUE, MAYBE): (MAYBE, TRUE, MAYBE),
( TRUE, FALSE): (FALSE, TRUE, TRUE),
( TRUE, TRUE): ( TRUE, TRUE, FALSE),
}
 
 
values = ('FALSE', 'TRUE ', 'MAYBE')
 
print("\nTrit logical inverse, '~'")
for a in values:
expr = '~%s' % a
print('  %s = %s' % (expr, eval(expr)))
 
for op, ophelp in (('&', 'and'), ('|', 'or'), ('^', 'exclusive-or')):
print("\nTrit logical %s, '%s'" % (ophelp, op))
for a in values:
for b in values:
expr = '%s %s %s' % (a, op, b)
print('  %s = %s' % (expr, eval(expr)))
Output
Trit logical inverse, '~'
  ~FALSE = TRUE
  ~TRUE  = FALSE
  ~MAYBE = MAYBE

Trit logical and, '&'
  FALSE & FALSE = FALSE
  FALSE & TRUE  = FALSE
  FALSE & MAYBE = FALSE
  TRUE  & FALSE = FALSE
  TRUE  & TRUE  = TRUE
  TRUE  & MAYBE = MAYBE
  MAYBE & FALSE = FALSE
  MAYBE & TRUE  = MAYBE
  MAYBE & MAYBE = MAYBE

Trit logical or, '|'
  FALSE | FALSE = FALSE
  FALSE | TRUE  = TRUE
  FALSE | MAYBE = MAYBE
  TRUE  | FALSE = TRUE
  TRUE  | TRUE  = TRUE
  TRUE  | MAYBE = TRUE
  MAYBE | FALSE = MAYBE
  MAYBE | TRUE  = TRUE
  MAYBE | MAYBE = MAYBE

Trit logical exclusive-or, '^'
  FALSE ^ FALSE = FALSE
  FALSE ^ TRUE  = TRUE
  FALSE ^ MAYBE = MAYBE
  TRUE  ^ FALSE = TRUE
  TRUE  ^ TRUE  = FALSE
  TRUE  ^ MAYBE = MAYBE
  MAYBE ^ FALSE = MAYBE
  MAYBE ^ TRUE  = MAYBE
  MAYBE ^ MAYBE = MAYBE
Extra doodling in the Python shell
>>> values = (TRUE, FALSE, MAYBE)
>>> for a in values:
	for b in values:
		assert (a & ~b) | (b & ~a) == a ^ b

		
>>> 

[edit] Racket

#lang typed/racket
 
; to avoid the hassle of adding a maybe value that is as special as
; the two standard booleans, we'll use symbols to make our own
(define-type trit (U 'true 'false 'maybe))
 
(: not (trit -> trit))
(define (not a)
(case a
[(true) 'false]
[(maybe) 'maybe]
[(false) 'true]))
 
(: and (trit trit -> trit))
(define (and a b)
(case a
[(false) 'false]
[(maybe) (case b
[(false) 'false]
[else 'maybe])]
[(true) (case b
[(true) 'true]
[(maybe) 'maybe]
[(false) 'false])]))
 
(: or (trit trit -> trit))
(define (or a b)
(case a
[(true) 'true]
[(maybe) (case b
[(true) 'true]
[else 'maybe])]
[(false) (case b
[(true) 'true]
[(maybe) 'maybe]
[(false) 'false])]))
 
(: ifthen (trit trit -> trit))
(define (ifthen a b)
(case b
[(true) 'true]
[(maybe) (case a
[(false) 'true]
[else 'maybe])]
[(false) (case a
[(true) 'false]
[(maybe) 'maybe]
[(false) 'true])]))
 
(: iff (trit trit -> trit))
(define (iff a b)
(case a
[(maybe) 'maybe]
[(true) b]
[(false) (not b)]))
 
(for: : Void ([a (in-list '(true maybe false))])
(printf "~a ~a = ~a~n" (object-name not) a (not a)))
(for: : Void ([proc (in-list (list and or ifthen iff))])
(for*: : Void ([a (in-list '(true maybe false))]
[b (in-list '(true maybe false))])
(printf "~a ~a ~a = ~a~n" a (object-name proc) b (proc a b))))

Output:

not true = false
not maybe = maybe
not false = true
true and true = true
true and maybe = maybe
true and false = false
maybe and true = maybe
maybe and maybe = maybe
maybe and false = false
false and true = false
false and maybe = false
false and false = false
true or true = true
true or maybe = true
true or false = true
maybe or true = true
maybe or maybe = maybe
maybe or false = maybe
false or true = true
false or maybe = maybe
false or false = false
true ifthen true = true
true ifthen maybe = maybe
true ifthen false = false
maybe ifthen true = true
maybe ifthen maybe = maybe
maybe ifthen false = maybe
false ifthen true = true
false ifthen maybe = true
false ifthen false = true
true iff true = true
true iff maybe = maybe
true iff false = false
maybe iff true = maybe
maybe iff maybe = maybe
maybe iff false = maybe
false iff true = false
false iff maybe = maybe
false iff false = true

[edit] REXX

This REXX program is a re-worked version of the REXX program used for truth table.

/*REXX program displays a  ternary truth table   [true,  false,  maybe] */
/* for the variables and one or more expressions. */
/*Infix notation is supported with one character propositional constants*/
/*variables (propositional constants) allowed: A──►Z, a──►z except u. */
/*All propositional constants are case insensative (except lowercase v).*/
 
parse arg expression /*get expression from the C. L. */
if expression\='' then do /*Got one? Then show user's stuff*/
call truthTable expression /*show and tell T.T.*/
exit /*we're all done with truth table*/
end
 
call truthTable "a & b ; AND"
call truthTable "a | b ; OR"
call truthTable "a ^ b ; XOR"
call truthTable "a ! b ; NOR"
call truthTable "a ¡ b ; NAND"
call truthTable "a xnor b ; XNOR" /*XNOR is the same as NXOR. */
exit /*stick a fork in it, we're done.*/
/*─────────────────────────────────────truthTable subroutine────────────*/
truthTable: procedure; parse arg $ ';' comm 1 $o; $o=strip($o)
$=translate(strip($),'|',"v"); $u=$; upper $u
$u=translate($u,'()()()',"[]{}«»"); $$.=0; PCs=; hdrPCs=
@abc='abcdefghijklmnopqrstuvwxyz'; @abcU=@abc; upper @abcU
 
@='ff'x /*─────────infix operators───────*/
op.= /*a single quote (') wasn't */
/* implemented for negation. */
op.0 ='false boolFALSE' /*unconditionally FALSE */
op.1 ='and and & *' /* AND, conjunction */
op.2 ='naimpb NaIMPb' /*not A implies B */
op.3 ='boolb boolB' /*B (value of) */
op.4 ='nbimpa NbIMPa' /*not B implies A */
op.5 ='boola boolA' /*A (value of) */
op.6 ='xor xor && % ^' /* XOR, exclusive OR */
op.7 ='or or | + v' /* OR, disjunction */
op.8 ='nor nor ! ↓' /* NOR, not OR, Pierce operator */
op.9 ='xnor xnor nxor' /*NXOR, not exclusive OR, not XOR*/
op.10='notb notB' /*not B (value of) */
op.11='bimpa bIMPa' /* B implies A */
op.12='nota notA' /*not A (value of) */
op.13='aimpb aIMPb' /* A implies B */
op.14='nand nand ¡ ↑' /*NAND, not AND, Sheffer operator*/
op.15='true boolTRUE' /*unconditionally TRUE */
/*alphabetic names need changing.*/
op.16='\ NOT ~ ─ . ¬' /* NOT, negation */
op.17='> GT' /*conditional */
op.18='>= GE ─> => ──> ==>' "1a"x /*conditional */
op.19='< LT' /*conditional */
op.20='<= LE <─ <= <── <==' /*conditional */
op.21='\= NE ~= ─= .= ¬=' /*conditional */
op.22='= EQ EQUAL EQUALS =' "1b"x /*biconditional */
op.23='0 boolTRUE' /*TRUEness */
op.24='1 boolFALSE' /*FALSEness */
 
op.25='NOT NOT NEG' /*not, neg */
 
do jj=0 while op.jj\=='' | jj<16 /*change opers──►what REXX likes.*/
new=word(op.jj,1)
do kk=2 to words(op.jj) /*handle each token separately. */
_=word(op.jj,kk); upper _
if wordpos(_,$u)==0 then iterate /*no such animal in this string. */
if datatype(new,'m') then new!=@ /*expresion needs transcribing. */
else new!=new
$u=changestr(_,$u,new!) /*transcribe the function (maybe)*/
if new!==@ then $u=changeFunc($u,@,new) /*use internal bool name.*/
end /*kk*/
end /*jj*/
 
$u=translate($u, '()', "{}") /*finish cleaning up transcribing*/
do jj=1 for length(@abcU) /*see what variables are used. */
_=substr(@abcU,jj,1) /*use available upercase alphabet*/
if pos(_,$u)==0 then iterate /*found one? No, keep looking. */
$$.jj=2 /*found: set upper bound for it.*/
PCs=PCs _ /*also, add to propositional cons*/
hdrPCs=hdrPCS center(_,length('false')) /*build a PC header.*/
end /*jj*/
$u=PCs '('$u")" /*separate PCs from expression. */
ptr='_────►_' /*a pointer for the truth table. */
hdrPCs=substr(hdrPCs,2) /*create a header for the PCs. */
say hdrPCs left('',length(ptr)-1) $o /*display PC header + expression.*/
say copies('───── ',words(PCs)) left('',length(ptr)-2) copies('─',length($o))
/*Note: "true"s: right─justified*/
do a=0 to $$.1
do b=0 to $$.2
do c=0 to $$.3
do d=0 to $$.4
do e=0 to $$.5
do f=0 to $$.6
do g=0 to $$.7
do h=0 to $$.8
do i=0 to $$.9
do j=0 to $$.10
do k=0 to $$.11
do l=0 to $$.12
do m=0 to $$.13
do n=0 to $$.14
do o=0 to $$.15
do p=0 to $$.16
do q=0 to $$.17
do r=0 to $$.18
do s=0 to $$.19
do t=0 to $$.20
do u=0 to $$.21
do !=0 to $$.22
do w=0 to $$.23
do x=0 to $$.24
do y=0 to $$.25
do z=0 to $$.26
interpret '_=' $u /*evaluate truth T.*/
_=changestr(0,_,'false') /*convert 0──►false*/
_=changestr(1,_,'_true') /*convert 1──►_true*/
_=changestr(2,_,'maybe') /*convert 2──►maybe*/
_=insert(ptr,_,wordindex(_,words(_))-1) /*──►*/
say translate(_,,'_') /*display truth tab*/
end /*z*/
end /*y*/
end /*x*/
end /*w*/
end /*v*/
end /*u*/
end /*t*/
end /*s*/
end /*r*/
end /*q*/
end /*p*/
end /*o*/
end /*n*/
end /*m*/
end /*l*/
end /*k*/
end /*j*/
end /*i*/
end /*h*/
end /*g*/
end /*f*/
end /*e*/
end /*d*/
end /*c*/
end /*b*/
end /*a*/
 
say; return
/*─────────────────────────────────────SCAN subroutine──────────────────*/
scan: procedure; parse arg x,at; L=length(x); t=L; lp=0; apost=0; quote=0
if at<0 then do; t=1; x=translate(x,'()',")("); end
do j=abs(at) to t by sign(at); _=substr(x,j,1); __=substr(x,j,2)
if quote then do; if _\=='"' then iterate
if __=='""' then do; j=j+1; iterate; end
quote=0; iterate
end
if apost then do; if _\=="'" then iterate
if __=="''" then do; j=j+1; iterate; end
apost=0; iterate
end
if _=='"' then do; quote=1; iterate; end
if _=="'" then do; apost=1; iterate; end
if _==' ' then iterate
if _=='(' then do; lp=lp+1; iterate; end
if lp\==0 then do; if _==')' then lp=lp-1; iterate; end
if datatype(_,'U') then return j-(at<0)
if at<0 then return j+1
end /*j*/
return min(j,L)
/*─────────────────────────────────────changeFunc subroutine────────────*/
changeFunc: procedure; parse arg z,fC,newF; funcPos=0
do forever
funcPos=pos(fC,z,funcPos+1); if funcPos==0 then return z
origPos=funcPos
z=changestr(fC,z,",'"newF"',")
funcPos=funcPos+length(newF)+4
where=scan(z, funcPos)  ; z=insert( '}', z, where)
where=scan(z, 1-origPos) ; z=insert('trit{', z, where)
end /*forever*/
/*─────────────────────────────────────TRIT subroutine──────────────────*/
trit: procedure; arg a,$,b; v=\(a==2|b==2); o= a==1|b==1; z= a==0|b==0
select
when $=='FALSE' then return 0
when $=='AND' then if v then return a & b; else return 2
when $=='NAIMPB' then if v then return \(\a & \b); else return 2
when $=='BOOLB' then return b
when $=='NBIMPA' then if v then return \(\b & \a); else return 2
when $=='BOOLA' then return a
when $=='XOR' then if v then return a && b  ; else return 2
when $=='OR' then if v then return a | b  ; else
if o then return 1; else return 2
when $=='NOR' then if v then return \(a | b)  ; else return 2
when $=='XNOR' then if v then return \(a && b) ; else return 2
when $=='NOTB' then if v then return \b  ; else return 2
when $=='NOTA' then if v then return \a  ; else return 2
when $=='AIMPB' then if v then return \(a & \b) ; else return 2
when $=='NAND' then if v then return \(a & b) ; else
if z then return 1; else return 2
when $=='TRUE' then return 1
otherwise return -13 /*error, unknown function.*/
end /*select*/

Some older REXXes don't have a   changestr   BIF, so one is included here ──► CHANGESTR.REX.

output

  A     B          a & b ; AND
───── ─────        ───────────
false false  ────► false
false  true  ────► false
false maybe  ────► maybe
 true false  ────► false
 true  true  ────►  true
 true maybe  ────► maybe
maybe false  ────► maybe
maybe  true  ────► maybe
maybe maybe  ────► maybe

  A     B          a | b ; OR
───── ─────        ──────────
false false  ────► false
false  true  ────►  true
false maybe  ────► maybe
 true false  ────►  true
 true  true  ────►  true
 true maybe  ────►  true
maybe false  ────► maybe
maybe  true  ────►  true
maybe maybe  ────► maybe

  A     B          a ^ b ; XOR
───── ─────        ───────────
false false  ────► false
false  true  ────►  true
false maybe  ────► maybe
 true false  ────►  true
 true  true  ────► false
 true maybe  ────► maybe
maybe false  ────► maybe
maybe  true  ────► maybe
maybe maybe  ────► maybe

  A     B          a ! b ; NOR
───── ─────        ───────────
false false  ────►  true
false  true  ────► false
false maybe  ────► maybe
 true false  ────► false
 true  true  ────► false
 true maybe  ────► maybe
maybe false  ────► maybe
maybe  true  ────► maybe
maybe maybe  ────► maybe

  A     B          a ¡ b ; NAND
───── ─────        ────────────
false false  ────►  true
false  true  ────►  true
false maybe  ────►  true
 true false  ────►  true
 true  true  ────► false
 true maybe  ────► maybe
maybe false  ────►  true
maybe  true  ────► maybe
maybe maybe  ────► maybe

  A     B          a xnor b ; XNOR
───── ─────        ───────────────
false false  ────►  true
false  true  ────► false
false maybe  ────► maybe
 true false  ────► false
 true  true  ────►  true
 true maybe  ────► maybe
maybe false  ────► maybe
maybe  true  ────► maybe
maybe maybe  ────► maybe

[edit] Ruby

Ruby, like Smalltalk, has two boolean classes: TrueClass for true and FalseClass for false. We add a third class, MaybeClass for MAYBE, and define ternary logic for all three classes.

We keep !a, a & b and so on for binary logic. We add !a.trit, a.trit & b and so on for ternary logic. The code for !a.trit uses def !, which works with Ruby 1.9, but fails as a syntax error with Ruby 1.8.

Works with: Ruby version 1.9
# trit.rb - ternary logic
# http://rosettacode.org/wiki/Ternary_logic
 
require 'singleton'
 
# MAYBE, the only instance of MaybeClass, enables a system of ternary
# logic using TrueClass#trit, MaybeClass#trit and FalseClass#trit.
#
#  !a.trit # ternary not
# a.trit & b # ternary and
# a.trit | b # ternary or
# a.trit ^ b # ternary exclusive or
# a.trit == b # ternary equal
#
# Though +true+ and +false+ are internal Ruby values, +MAYBE+ is not.
# Programs may want to assign +maybe = MAYBE+ in scopes that use
# ternary logic. Then programs can use +true+, +maybe+ and +false+.
class MaybeClass
include Singleton
 
# maybe.to_s # => "maybe"
def to_s; "maybe"; end
end
 
MAYBE = MaybeClass.instance
 
class TrueClass
TritMagic = Object.new
class << TritMagic
def index; 0; end
def !; false; end
def & other; other; end
def | other; true; end
def ^ other; [false, MAYBE, true][other.trit.index]; end
def == other; other; end
end
 
# Performs ternary logic. See MaybeClass.
#  !true.trit # => false
# true.trit & obj # => obj
# true.trit | obj # => true
# true.trit ^ obj # => false, maybe or true
# true.trit == obj # => obj
def trit; TritMagic; end
end
 
class MaybeClass
TritMagic = Object.new
class << TritMagic
def index; 1; end
def !; MAYBE; end
def & other; [MAYBE, MAYBE, false][other.trit.index]; end
def | other; [true, MAYBE, MAYBE][other.trit.index]; end
def ^ other; MAYBE; end
def == other; MAYBE; end
end
 
# Performs ternary logic. See MaybeClass.
#  !maybe.trit # => maybe
# maybe.trit & obj # => maybe or false
# maybe.trit | obj # => true or maybe
# maybe.trit ^ obj # => maybe
# maybe.trit == obj # => maybe
def trit; TritMagic; end
end
 
class FalseClass
TritMagic = Object.new
class << TritMagic
def index; 2; end
def !; true; end
def & other; false; end
def | other; other; end
def ^ other; other; end
def == other; [false, MAYBE, true][other.trit.index]; end
end
 
# Performs ternary logic. See MaybeClass.
#  !false.trit # => true
# false.trit & obj # => false
# false.trit | obj # => obj
# false.trit ^ obj # => obj
# false.trit == obj # => false, maybe or true
def trit; TritMagic; end
end

This IRB session shows ternary not, and, or, equal.

$ irb
irb(main):001:0> require './trit'
=> true
irb(main):002:0> maybe = MAYBE
=> maybe
irb(main):003:0> !true.trit
=> false
irb(main):004:0> !maybe.trit
=> maybe
irb(main):005:0> maybe.trit & false
=> false
irb(main):006:0> maybe.trit | true
=> true
irb(main):007:0> false.trit == true
=> false
irb(main):008:0> false.trit == maybe
=> maybe

This program shows all 9 outcomes from a.trit ^ b.

require 'trit'
maybe = MAYBE
 
[true, maybe, false].each do |a|
[true, maybe, false].each do |b|
printf "%5s ^ %5s => %5s\n", a, b, a.trit ^ b
end
end
$ ruby -I. trit-xor.rb
 true ^  true => false
 true ^ maybe => maybe
 true ^ false =>  true
maybe ^  true => maybe
maybe ^ maybe => maybe
maybe ^ false => maybe
false ^  true =>  true
false ^ maybe => maybe
false ^ false => false

[edit] Run BASIC

testFalse	= 0  ' F
testDoNotKnow = 1 ' ?
testTrue = 2 ' T
 
print "Short and long names for ternary logic values"
for i = testFalse to testTrue
print shortName3$(i);" ";longName3$(i)
next i
print
 
print "Single parameter functions"
print "x";" ";"=x";" ";"not(x)"
for i = testFalse to testTrue
print shortName3$(i);" ";shortName3$(i);" ";shortName3$(not3(i))
next
print
 
print "Double parameter fuctions"
html "<table border=1><TR align=center bgcolor=wheat><TD>x</td><td>y</td><td>x AND y</td><td>x OR y</td><td>x EQ y</td><td>x XOR y</td></tr>"
for a = testFalse to testTrue
for b = testFalse to testTrue
html "<TR align=center><td>"
html shortName3$(a); "</td><td>";shortName3$(b); "</td><td>"
html shortName3$(and3(a,b));"</td><td>";shortName3$(or3(a,b)); "</td><td>"
html shortName3$(eq3(a,b)); "</td><td>";shortName3$(xor3(a,b));"</td></tr>"
next
next
html "</table>"
function and3(a,b)
and3 = min(a,b)
end function
 
function or3(a,b)
or3 = max(a,b)
end function
 
function eq3(a,b)
eq3 = testFalse
if a = tDontKnow or b = tDontKnow then eq3 = tDontKnow
if a = b then eq3 = testTrue
end function
 
function xor3(a,b)
xor3 = not3(eq3(a,b))
end function
 
function not3(b)
not3 = 2-b
end function
 
'------------------------------------------------
function shortName3$(i)
shortName3$ = word$("F ? T", i+1)
end function
 
function longName3$(i)
longName3$ = word$("False,Don't know,True", i+1, ",")
end function
Short and long names for ternary logic values

F False ? Don't know T True

Single parameter functions x =x not(x) F F T ?  ?  ? T T F

Double parameter fuctions
xyx AND yx OR yx EQ yx XOR y
FFFFTF
F?F?FT
FTFTFT
?FF?FT
????TF
?T?TFT
TFFTFT
T??TFT
TTTTTF

[edit] Scala

sealed trait Trit { self =>
def nand(that:Trit):Trit=(this,that) match {
case (TFalse, _) => TTrue
case (_, TFalse) => TTrue
case (TMaybe, _) => TMaybe
case (_, TMaybe) => TMaybe
case _ => TFalse
}
 
def nor(that:Trit):Trit = this.or(that).not()
def and(that:Trit):Trit = this.nand(that).not()
def or(that:Trit):Trit = this.not().nand(that.not())
def not():Trit = this.nand(this)
def imply(that:Trit):Trit = this.nand(that.not())
def equiv(that:Trit):Trit = this.and(that).or(this.nor(that))
}
case object TTrue extends Trit
case object TMaybe extends Trit
case object TFalse extends Trit
 
object TernaryLogic extends App {
val v=List(TTrue, TMaybe, TFalse)
println("- NOT -")
for(a<-v) println("%6s => %6s".format(a, a.not))
println("\n- AND -")
for(a<-v; b<-v) println("%6s : %6s => %6s".format(a, b, a and b))
println("\n- OR -")
for(a<-v; b<-v) println("%6s : %6s => %6s".format(a, b, a or b))
println("\n- Imply -")
for(a<-v; b<-v) println("%6s : %6s => %6s".format(a, b, a imply b))
println("\n- Equiv -")
for(a<-v; b<-v) println("%6s : %6s => %6s".format(a, b, a equiv b))
}

Output:

- NOT -
 TTrue => TFalse
TMaybe => TMaybe
TFalse =>  TTrue

- AND -
 TTrue :  TTrue =>  TTrue
 TTrue : TMaybe => TMaybe
 TTrue : TFalse => TFalse
TMaybe :  TTrue => TMaybe
TMaybe : TMaybe => TMaybe
TMaybe : TFalse => TFalse
TFalse :  TTrue => TFalse
TFalse : TMaybe => TFalse
TFalse : TFalse => TFalse

- OR -
 TTrue :  TTrue =>  TTrue
 TTrue : TMaybe =>  TTrue
 TTrue : TFalse =>  TTrue
TMaybe :  TTrue =>  TTrue
TMaybe : TMaybe => TMaybe
TMaybe : TFalse => TMaybe
TFalse :  TTrue =>  TTrue
TFalse : TMaybe => TMaybe
TFalse : TFalse => TFalse

- Imply -
 TTrue :  TTrue =>  TTrue
 TTrue : TMaybe => TMaybe
 TTrue : TFalse => TFalse
TMaybe :  TTrue =>  TTrue
TMaybe : TMaybe => TMaybe
TMaybe : TFalse => TMaybe
TFalse :  TTrue =>  TTrue
TFalse : TMaybe =>  TTrue
TFalse : TFalse =>  TTrue

- Equiv -
 TTrue :  TTrue =>  TTrue
 TTrue : TMaybe => TMaybe
 TTrue : TFalse => TFalse
TMaybe :  TTrue => TMaybe
TMaybe : TMaybe => TMaybe
TMaybe : TFalse => TMaybe
TFalse :  TTrue => TFalse
TFalse : TMaybe => TMaybe
TFalse : TFalse =>  TTrue

[edit] Seed7

The type boolean does not define separate xor, implies and equiv operators. But there are replacements for them:

Instead of Use
p xor q p <> q
p implies q p <= q
p equiv q p = q

Since ternary logic needs xor, implies and equiv with a trit result they are introduced as the operators xor, -> and ==. The trit operators and and or are defined as short circuit operators. A short circuit operator evaluates the second parameter only when necessary. This is analogous to the boolean operators and and or, which use also short circuit evaluation.

$ include "seed7_05.s7i";
 
const type: trit is new enum
False, Maybe, True
end enum;
 
# Enum types define comparisons (=, <, >, <=, >=, <>) and
# the conversions ord and conv.
 
const func string: str (in trit: aTrit) is
return [] ("False", "Maybe", "True")[succ(ord(aTrit))];
 
enable_output(trit); # Allow writing trit values
 
const array trit: tritNot is [] (True, Maybe, False);
const array array trit: tritAnd is [] (
[] (False, False, False),
[] (False, Maybe, Maybe),
[] (False, Maybe, True ));
const array array trit: tritOr is [] (
[] (False, Maybe, True ),
[] (Maybe, Maybe, True ),
[] (True, True, True ));
const array array trit: tritXor is [] (
[] (False, Maybe, True ),
[] (Maybe, Maybe, Maybe),
[] (True, Maybe, False));
const array array trit: tritImplies is [] (
[] (True, True, True ),
[] (Maybe, Maybe, True ),
[] (False, Maybe, True ));
const array array trit: tritEquiv is [] (
[] (True, Maybe, False),
[] (Maybe, Maybe, Maybe),
[] (False, Maybe, True ));
 
const func trit: not (in trit: aTrit) is
return tritNot[succ(ord(aTrit))];
 
const func trit: (in trit: aTrit1) and (in trit: aTrit2) is
return tritAnd[succ(ord(aTrit1))][succ(ord(aTrit2))];
 
const func trit: (in trit: aTrit1) and (ref func trit: aTrit2) is func
result
var trit: res is False;
begin
if aTrit1 = True then
res := aTrit2;
elsif aTrit1 = Maybe and aTrit2 <> False then
res := Maybe;
end if;
end func;
 
const func trit: (in trit: aTrit1) or (in trit: aTrit2) is
return tritOr[succ(ord(aTrit1))][succ(ord(aTrit2))];
 
const func trit: (in trit: aTrit1) or (ref func trit: aTrit2) is func
result
var trit: res is True;
begin
if aTrit1 = False then
res := aTrit2;
elsif aTrit1 = Maybe and aTrit2 <> True then
res := Maybe;
end if;
end func;
 
$ syntax expr: .().xor.() is -> 15;
const func trit: (in trit: aTrit1) xor (in trit: aTrit2) is
return tritImplies[succ(ord(aTrit1))][succ(ord(aTrit2))];
 
const func trit: (in trit: aTrit1) -> (in trit: aTrit2) is
return tritImplies[succ(ord(aTrit1))][succ(ord(aTrit2))];
 
const func trit: (in trit: aTrit1) == (in trit: aTrit2) is
return tritEquiv[succ(ord(aTrit1))][succ(ord(aTrit2))];
 
const func trit: rand (in trit: low, in trit: high) is
return trit conv (rand(ord(low), ord(high)));
 
# Begin of test code
 
var trit: operand1 is False;
var trit: operand2 is False;
 
const proc: writeTable (ref func trit: tritExpr, in string: name) is func
begin
writeln;
writeln(" " <& name rpad 7 <& " | False Maybe True");
writeln("---------+---------------------");
for operand1 range False to True do
write(" " <& operand1 rpad 7 <& " | ");
for operand2 range False to True do
write(tritExpr rpad 7);
end for;
writeln;
end for;
end func;
 
const proc: main is func
begin
writeln(" not" rpad 8 <& " | False Maybe True");
writeln("---------+---------------------");
write(" | ");
for operand1 range False to True do
write(not operand1 rpad 7);
end for;
writeln;
writeTable(operand1 and operand2, "and");
writeTable(operand1 or operand2, "or");
writeTable(operand1 xor operand2, "xor");
writeTable(operand1 -> operand2, "->");
writeTable(operand1 == operand2, "==");
end func;

Output:

 not     | False  Maybe  True
---------+---------------------
         | True   Maybe  False  

 and     | False  Maybe  True
---------+---------------------
 False   | False  False  False  
 Maybe   | False  Maybe  Maybe  
 True    | False  Maybe  True   

 or      | False  Maybe  True
---------+---------------------
 False   | False  Maybe  True   
 Maybe   | Maybe  Maybe  True   
 True    | True   True   True   

 xor     | False  Maybe  True
---------+---------------------
 False   | True   True   True   
 Maybe   | Maybe  Maybe  True   
 True    | False  Maybe  True   

 ->      | False  Maybe  True
---------+---------------------
 False   | True   True   True   
 Maybe   | Maybe  Maybe  True   
 True    | False  Maybe  True   

 ==      | False  Maybe  True
---------+---------------------
 False   | True   Maybe  False  
 Maybe   | Maybe  Maybe  Maybe  
 True    | False  Maybe  True   

[edit] Tcl

The simplest way of doing this is by constructing the operations as truth tables. The code below uses an abbreviated form of truth table.

package require Tcl 8.5
namespace eval ternary {
# Code generator
proc maketable {name count values} {
set sep ""
for {set i 0; set c 97} {$i<$count} {incr i;incr c} {
set v [format "%c" $c]
lappend args $v; append key $sep "$" $v
set sep ","
}
foreach row [split $values \n] {
if {[llength $row]>1} {
lassign $row from to
lappend table $from [list return $to]
}
}
proc $name $args \
[list ckargs $args]\;[concat [list switch -glob --] $key [list $table]]
namespace export $name
}
# Helper command to check argument syntax
proc ckargs argList {
foreach var $argList {
upvar 1 $var v
switch -exact -- $v {
true - maybe - false {
continue
}
default {
return -level 2 -code error "bad ternary value \"$v\""
}
}
}
}
 
# The "truth" tables; “*” means “anything”
maketable not 1 {
true false
maybe maybe
false true
}
maketable and 2 {
true,true true
false,* false
*,false false
* maybe
}
maketable or 2 {
true,* true
*,true true
false,false false
* maybe
}
maketable implies 2 {
false,* true
*,true true
true,false false
* maybe
}
maketable equiv 2 {
*,maybe maybe
maybe,* maybe
true,true true
false,false true
* false
}
}

Demonstrating:

namespace import ternary::*
puts "x /\\ y == x \\/ y"
puts " x | y || result"
puts "-------+-------++--------"
foreach x {true maybe false} {
foreach y {true maybe false} {
set z [equiv [and $x $y] [or $x $y]]
puts [format " %-5s | %-5s || %-5s" $x $y $z]
}
}

Output:

x /\ y == x \/ y
 x     | y     || result
-------+-------++--------
 true  | true  || true 
 true  | maybe || maybe
 true  | false || false
 maybe | true  || maybe
 maybe | maybe || maybe
 maybe | false || maybe
 false | true  || false
 false | maybe || maybe
 false | false || true 
Personal tools
Namespaces

Variants
Actions
Community
Explore
Misc
Toolbox