## TPTP Axioms File: LCL013^6.ax

```%------------------------------------------------------------------------------
% File     : LCL013^6 : TPTP v7.5.0. Released v4.0.0.
% Domain   : Logic Calculi (Modal logic)
% Axioms   : Modal logic S5
% Version  : [Ben09] axioms.
% English  : Embedding of monomodal logic S5 in simple type theory.

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

% Status   : Satisfiable
% Syntax   : Number of formulae    :    8 (   0 unit;   3 type;   2 defn)
%            Number of atoms       :   36 (   2 equality;   5 variable)
%            Maximal formula depth :    9 (   4 average)
%            Number of connectives :   11 (   1   ~;   1   |;   0   &;   9   @)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&;   0  !!;   0  ??)
%            Number of type conns  :   10 (  10   >;   0   *;   0   +)
%            Number of symbols     :    9 (   3   :;   0  :=)
%            Number of variables   :    4 (   0 sgn;   1   !;   0   ?;   3   ^)
%                                         (   4   :;   0  :=;   0  !>;   0  ?*)
% SPC      :

%------------------------------------------------------------------------------
%----We reserve an accessibility relation constant rel_s5
thf(rel_s5_type,type,(
rel_s5: \$i > \$i > \$o )).

%----We define mbox_s5 and mdia_s5 based on rel_s5
thf(mbox_s5_type,type,(
mbox_s5: ( \$i > \$o ) > \$i > \$o )).

thf(mbox_s5,definition,
( mbox_s5
= ( ^ [Phi: \$i > \$o,W: \$i] :
! [V: \$i] :
( ~ ( rel_s5 @ W @ V )
| ( Phi @ V ) ) ) )).

thf(mdia_s5_type,type,(
mdia_s5: ( \$i > \$o ) > \$i > \$o )).

thf(mdia_s5,definition,
( mdia_s5
= ( ^ [Phi: \$i > \$o] :
( mnot @ ( mbox_s5 @ ( 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_s5 )).

thf(a2,axiom,
( mtransitive @ rel_s5 )).

thf(a3,axiom,
( msymmetric @ rel_s5 )).

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