## TPTP Axioms File: LCL016^1.ax

```%------------------------------------------------------------------------------
% File     : LCL016^1 : TPTP v7.5.0. Released v6.1.0.
% Domain   : Logic Calculi (Second Order Modal Logic)
% Axioms   : Embedding of second order modal logic in simple type theory
% Version  : [Ben13] axioms.
% English  : Extends K to KB by adding symmetric of rel.

% Refs     : [Ben13] Benzmueller (2009), Email to Geoff Sutcliffe
%          : [BP13]  Benzmueller & Paulson (2013), Quantified Multimodal Lo
% Source   : [Ben13]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :    3 (   0 unit;   1 type;   1 defn)
%            Number of atoms       :   14 (   1 equality;   6 variable)
%            Maximal formula depth :    8 (   5 average)
%            Number of connectives :    6 (   0   ~;   0   |;   0   &;   5   @)
%                                         (   0 <=>;   1  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&;   0  !!;   0  ??)
%            Number of type conns  :    5 (   5   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    4 (   1   :)
%            Number of variables   :    3 (   0 sgn;   2   !;   0   ?;   1   ^)
%                                         (   3   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_SAT_EQU

%------------------------------------------------------------------------------
%----Definition of properties of accessibility relations
thf(msymmetric_type,type,(
msymmetric: ( \$i > \$i > \$o ) > \$o )).

thf(msymmetric,definition,
( msymmetric
= ( ^ [R: \$i > \$i > \$o] :
! [S: \$i,T: \$i] :
( ( R @ S @ T )
=> ( R @ T @ S ) ) ) )).

thf(sym,axiom,
( msymmetric @ rel )).

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