TPTP Axioms File: LCL017^1.ax


%------------------------------------------------------------------------------
% File     : LCL017^1 : TPTP v7.5.0. Released v7.5.0.
% Domain   : Logic Calculi (Modal Logic)
% Axioms   : Variable Domain Quantifiers for Modal Logic
% Version  : [Gus20] axioms.
% English  : 

% Refs     : [Gus20] Gustafsson (2020), Email to Geoff Sutcliffe
% Source   : [Gus20]
% Names    : 

% Status   : Satisfiable
% Syntax   : Number of formulae    :    6 (   0 unit;   3 type;   2 defn)
%            Number of atoms       :   18 (   2 equality;   9 variable)
%            Maximal formula depth :    8 (   6 average)
%            Number of connectives :   11 (   0   ~;   0   |;   0   &;  10   @)
%                                         (   0 <=>;   1  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   14 (  14   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    7 (   3   :;   0   =)
%            Number of variables   :    7 (   0 sgn;   2   !;   1   ?;   4   ^)
%                                         (   7   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_SAT_EQU_NAR

% Comments : Combine with LCL016^0 for Quantified Modal Logic K wth variable 
%            domain.
%          : Combine with LCL016^0 and LCL016^1 for Quantified Modal Logic KB 
%            with variable domain.
%          : Combine with LCL017^0 for Quantified Modal Logic S5 with variable 
%            domain.
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thf(eiw_ind,type,(
    eiw_ind: $i > mu > $o )).

thf(nonempty_ind,axiom,(
    ! [V: $i] :
    ? [X: mu] :
      ( eiw_ind @ V @ X ) )).

thf(mforall_eiw_ind_type,type,(
    mforall_eiw_ind: ( mu > $i > $o ) > $i > $o )).

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

thf(mexists_eiw_ind_type,type,(
    mexists_eiw_ind: ( mu > $i > $o ) > $i > $o )).

thf(mexists_eiw_ind,definition,
    ( mexists_eiw_ind
    = ( ^ [Phi: mu > $i > $o] :
          ( mnot
          @ ( mforall_eiw_ind
            @ ^ [X: mu] :
                ( mnot @ ( Phi @ X ) ) ) ) ) )).

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