TPTP Axioms File: LCL013^5.ax


%------------------------------------------------------------------------------
% File     : LCL013^5 : TPTP v7.4.0. Released v4.0.0.
% Domain   : Logic Calculi (Modal logic)
% Axioms   : Modal logic S4
% Version  : [Ben09] axioms.
% English  : Embedding of monomodal logic S4 in simple type theory.

% Refs     : [Ben09] Benzmueller (2009), Email to Geoff Sutcliffe
% Source   : [Ben09]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :    7 (   0 unit;   3 type;   2 defn)
%            Number of atoms       :   34 (   2 equality;   5 variable)
%            Maximal formula depth :    9 (   4 average)
%            Number of connectives :   10 (   1   ~;   1   |;   0   &;   8   @)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&;   0  !!;   0  ??)
%            Number of type conns  :   10 (  10   >;   0   *;   0   +)
%            Number of symbols     :    8 (   3   :;   0  :=)
%            Number of variables   :    4 (   0 sgn;   1   !;   0   ?;   3   ^)
%                                         (   4   :;   0  :=;   0  !>;   0  ?*)
% SPC      : 

% Comments : Requires LCL013^0 or (LCL015^0 and LCL015^1)
%------------------------------------------------------------------------------
%----We reserve an accessibility relation constant rel_s4
thf(rel_s4_type,type,(
    rel_s4: $i > $i > $o )).

%----We define mbox_s4 and mdia_s4 based on rel_s4
thf(mbox_s4_type,type,(
    mbox_s4: ( $i > $o ) > $i > $o )).

thf(mbox_s4,definition,
    ( mbox_s4
    = ( ^ [Phi: $i > $o,W: $i] :
        ! [V: $i] :
          ( ~ ( rel_s4 @ W @ V )
          | ( Phi @ V ) ) ) )).

thf(mdia_s4_type,type,(
    mdia_s4: ( $i > $o ) > $i > $o )).

thf(mdia_s4,definition,
    ( mdia_s4
    = ( ^ [Phi: $i > $o] :
          ( mnot @ ( mbox_s4 @ ( mnot @ Phi ) ) ) ) )).

%----We have now two options for stating the B conditions: 
%----We can (i) directly formulate conditions for the accessibility relation
%----constant or we can (ii) state corresponding axioms. We here prefer (i)
thf(a1,axiom,
    ( mreflexive @ rel_s4 )).

thf(a2,axiom,
    ( mtransitive @ rel_s4 )).

%------------------------------------------------------------------------------