## TPTP Axioms File: LCL017^0.ax

```%------------------------------------------------------------------------------
% File     : LCL017^0 : TPTP v7.5.0. Released v6.1.0.
% Domain   : Logic Calculi (Second Order Modal Logic)
% Axioms   : Embedding of second order modal logic S5 with universal access
% Version  : [Ben16] axioms.
% English  : Embedding of second order modal logic S5 (with a universal
%            accessibility) relation in simple type theory. In this case, the
%            guarding accessibility constraints in the definition of box and
%            diamond can be dropped.

% Refs     : [Ben16] Benzmueller (2016), Email to Geoff Sutcliffe
% Source   : [Ben16]
% Names    : QML_S5U.ax [Ben16]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   41 (   1 unit;  21 type;  20 defn)
%            Number of atoms       :  200 (  21 equality;  58 variable)
%            Maximal formula depth :    9 (   5 average)
%            Number of connectives :   43 (   4   ~;   2   |;   3   &;  31   @)
%                                         (   1 <=>;   2  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&;   0  !!;   0  ??)
%            Number of type conns  :  119 ( 119   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   26 (  21   :)
%            Number of variables   :   53 (   5 sgn;   6   !;   4   ?;  43   ^)
%                                         (  53   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_SAT_EQU

%------------------------------------------------------------------------------
%----Declaration of additional base type mu
thf(mu_type,type,(
mu: \$tType )).

%----Equality on individuals
thf(meq_ind_type,type,(
meq_ind: mu > mu > \$i > \$o )).

thf(meq_ind,definition,
( meq_ind
= ( ^ [X: mu,Y: mu,W: \$i] : ( X = Y ) ) )).

%----Modal operators mtrue, mfalse, mnot, mor, mand, mimplies, mequiv, ...
thf(mtrue_type,type,(
mtrue: \$i > \$o )).

thf(mtrue,definition,
( mtrue
= ( ^ [W: \$i] : \$true ) )).

thf(mfalse_type,type,(
mfalse: \$i > \$o )).

thf(mfalse,definition,
( mfalse
= ( ^ [W: \$i] : \$false ) )).

thf(mnot_type,type,(
mnot: ( \$i > \$o ) > \$i > \$o )).

thf(mnot,definition,
( mnot
= ( ^ [Phi: \$i > \$o,W: \$i] :
~ ( Phi @ W ) ) )).

thf(mor_type,type,(
mor: ( \$i > \$o ) > ( \$i > \$o ) > \$i > \$o )).

thf(mor,definition,
( mor
= ( ^ [Phi: \$i > \$o,Psi: \$i > \$o,W: \$i] :
( ( Phi @ W )
| ( Psi @ W ) ) ) )).

thf(mand_type,type,(
mand: ( \$i > \$o ) > ( \$i > \$o ) > \$i > \$o )).

thf(mand,definition,
( mand
= ( ^ [Phi: \$i > \$o,Psi: \$i > \$o,W: \$i] :
( ( Phi @ W )
& ( Psi @ W ) ) ) )).

thf(mimplies_type,type,(
mimplies: ( \$i > \$o ) > ( \$i > \$o ) > \$i > \$o )).

thf(mimplies,definition,
( mimplies
= ( ^ [Phi: \$i > \$o,Psi: \$i > \$o,W: \$i] :
( ( Phi @ W )
=> ( Psi @ W ) ) ) )).

thf(mimplied_type,type,(
mimplied: ( \$i > \$o ) > ( \$i > \$o ) > \$i > \$o )).

thf(mimplied,definition,
( mimplied
= ( ^ [Phi: \$i > \$o,Psi: \$i > \$o,W: \$i] :
( ( Psi @ W )
=> ( Phi @ W ) ) ) )).

thf(mequiv_type,type,(
mequiv: ( \$i > \$o ) > ( \$i > \$o ) > \$i > \$o )).

thf(mequiv,definition,
( mequiv
= ( ^ [Phi: \$i > \$o,Psi: \$i > \$o,W: \$i] :
( ( Phi @ W )
<=> ( Psi @ W ) ) ) )).

thf(mxor_type,type,(
mxor: ( \$i > \$o ) > ( \$i > \$o ) > \$i > \$o )).

thf(mxor,definition,
( mxor
= ( ^ [Phi: \$i > \$o,Psi: \$i > \$o,W: \$i] :
( ( ( Phi @ W )
& ~ ( Psi @ W ) )
| ( ~ ( Phi @ W )
& ( Psi @ W ) ) ) ) )).

%----Universal quantification: individuals
thf(mforall_ind_type,type,(
mforall_ind: ( mu > \$i > \$o ) > \$i > \$o )).

thf(mforall_ind,definition,
( mforall_ind
= ( ^ [Phi: mu > \$i > \$o,W: \$i] :
! [X: mu] :
( Phi @ X @ W ) ) )).

%----Universal quantification: sets of individuals (properties)
thf(mforall_indset_type,type,(
mforall_indset: ( ( mu > \$i > \$o ) > \$i > \$o ) > \$i > \$o )).

thf(mforall_indset,definition,
( mforall_indset
= ( ^ [Phi: ( mu > \$i > \$o ) > \$i > \$o,W: \$i] :
! [X: mu > \$i > \$o] :
( Phi @ X @ W ) ) )).

%----Universal quantification: propositions
thf(mforall_prop_type,type,(
mforall_prop: ( ( \$i > \$o ) > \$i > \$o ) > \$i > \$o )).

thf(mforall_prop,definition,
( mforall_prop
= ( ^ [Phi: ( \$i > \$o ) > \$i > \$o,W: \$i] :
! [P: \$i > \$o] :
( Phi @ P @ W ) ) )).

%----Existential quantification: individuals
thf(mexists_ind_type,type,(
mexists_ind: ( mu > \$i > \$o ) > \$i > \$o )).

thf(mexists_ind,definition,
( mexists_ind
= ( ^ [Phi: mu > \$i > \$o,W: \$i] :
? [X: mu] :
( Phi @ X @ W ) ) )).

%----Existential quantification: sets of individuals (properties)
thf(mexists_indset_type,type,(
mexists_indset: ( ( mu > \$i > \$o ) > \$i > \$o ) > \$i > \$o )).

thf(mexists_indset,definition,
( mexists_indset
= ( ^ [Phi: ( mu > \$i > \$o ) > \$i > \$o,W: \$i] :
? [X: mu > \$i > \$o] :
( Phi @ X @ W ) ) )).

%----Existential quantification: propositions
thf(mexists_prop_type,type,(
mexists_prop: ( ( \$i > \$o ) > \$i > \$o ) > \$i > \$o )).

thf(mexists_prop,definition,
( mexists_prop
= ( ^ [Phi: ( \$i > \$o ) > \$i > \$o,W: \$i] :
? [P: \$i > \$o] :
( Phi @ P @ W ) ) )).

%----Box operator (exploting universal accessibility)
thf(mbox_type,type,(
mbox: ( \$i > \$o ) > \$i > \$o )).

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

%----Diamond operator
thf(mdia_type,type,(
mdia: ( \$i > \$o ) > \$i > \$o )).

thf(mdia,definition,
( mdia
= ( ^ [Phi: \$i > \$o,W: \$i] :
? [V: \$i] :
( Phi @ V ) ) )).

%----The notion of validity
thf(mvalid_type,type,(
mvalid: ( \$i > \$o ) > \$o )).

thf(mvalid,definition,
( mvalid
= ( ^ [Phi: \$i > \$o] :
! [W: \$i] :
( Phi @ W ) ) )).

%----Definition of invalidity
thf(minvalid_type,type,(
minvalid: ( \$i > \$o ) > \$o )).

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

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