## TPTP Axioms File: LCL002-1.ax

```%--------------------------------------------------------------------------
% File     : LCL002-1 : TPTP v7.5.0. Released v1.0.0.
% Domain   : Logic Calculi (Wajsberg Algebras)
% Axioms   : Alternative Wajsberg algebra definitions
% Version  : [AB90] (equality) axioms.
% English  :

% Refs     : [FRT84] Font et al. (1984), Wajsberg Algebras
%          : [AB90]  Anantharaman & Bonacina (1990), An Application of the
%          : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic
% Source   : [Bon91]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of clauses    :    6 (   0 non-Horn;   6 unit;   1 RR)
%            Number of atoms      :    6 (   6 equality)
%            Maximal clause size  :    1 (   1 average)
%            Number of predicates :    1 (   0 propositional; 2-2 arity)
%            Number of functors   :    7 (   2 constant; 0-2 arity)
%            Number of variables  :   11 (   0 singleton)
%            Maximal term depth   :    4 (   2 average)
% SPC      :

% Comments : Requires LCL001-0.ax LCL001-2.ax
%--------------------------------------------------------------------------
%----Definitions of and_star and xor, where and_star is AC and xor is C
cnf(xor_definition,axiom,
( xor(X,Y) = or(and(X,not(Y)),and(not(X),Y)) )).

cnf(xor_commutativity,axiom,
( xor(X,Y) = xor(Y,X) )).

cnf(and_star_definition,axiom,
( and_star(X,Y) = not(or(not(X),not(Y))) )).

%---I guess the next two can be derived from the AC of and
cnf(and_star_associativity,axiom,
( and_star(and_star(X,Y),Z) = and_star(X,and_star(Y,Z)) )).

cnf(and_star_commutativity,axiom,
( and_star(X,Y) = and_star(Y,X) )).

%----Definition of false in terms of truth
cnf(false_definition,axiom,
( not(truth) = falsehood )).

%--------------------------------------------------------------------------
```