## TPTP Axioms File: LCL013^4.ax

```%------------------------------------------------------------------------------
% File     : LCL013^4 : TPTP v7.5.0. Released v4.0.0.
% Domain   : Logic Calculi (Modal logic)
% Axioms   : Modal logic B
% Version  : [Ben09] axioms.
% English  : Embedding of monomodal logic B 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      :

%------------------------------------------------------------------------------
%----We reserve an accessibility relation constant rel_b
thf(rel_b_type,type,(
rel_b: \$i > \$i > \$o )).

%----We define mbox_b and mdia_b based on rel_b
thf(mbox_b_type,type,(
mbox_b: ( \$i > \$o ) > \$i > \$o )).

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

thf(mdia_b_type,type,(
mdia_b: ( \$i > \$o ) > \$i > \$o )).

thf(mdia_b,definition,
( mdia_b
= ( ^ [Phi: \$i > \$o] :
( mnot @ ( mbox_b @ ( 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_b )).

thf(a2,axiom,
( msymmetric @ rel_b )).

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