## TPTP Problem File: PHI005^2.p

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```%------------------------------------------------------------------------------
% File     : PHI005^2 : TPTP v7.2.0. Released v6.1.0.
% Domain   : Philosophy
% Problem  : Necessarily, God exists
% Version  : [Ben13] axioms : Reduced > Especial.
% English  :

% Refs     : [Ben13] Benzmueller (2009), Email to Geoff Sutcliffe
% Source   : [Ben13]
% Names    : T3 [Ben13]

% Status   : Theorem
% Rating   : 0.56 v7.2.0, 0.50 v7.1.0, 0.62 v7.0.0, 0.57 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.57 v6.1.0
% Syntax   : Number of formulae    :   59 (   0 unit;  29 type;  25 defn)
%            Number of atoms       :  165 (  26 equality;  81 variable)
%            Maximal formula depth :   10 (   5 average)
%            Number of connectives :   88 (   5   ~;   3   |;   4   &;  72   @)
%                                         (   1 <=>;   3  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :  159 ( 159   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   34 (  29   :;   0   =)
%            Number of variables   :   66 (   3 sgn;   8   !;   4   ?;  54   ^)
%                                         (  66   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

%------------------------------------------------------------------------------
%----Axioms for Quantified Modal Logic KB.
include('Axioms/LCL016^0.ax').
include('Axioms/LCL016^1.ax').
% include('Axioms/PHI001^0.ax').
%------------------------------------------------------------------------------
%----Signature
thf(positive_tp,type,(
positive: ( mu > \$i > \$o ) > \$i > \$o )).

thf(god_tp,type,(
god: mu > \$i > \$o )).

%----Constant symbol for essence: ess
thf(essence_tp,type,(
essence: ( mu > \$i > \$o ) > mu > \$i > \$o )).

%----Constant symbol for necessary existence: ne
thf(necessary_existence_tp,type,(
necessary_existence: mu > \$i > \$o )).

%----D1: A God-like being possesses all positive properties.
thf(defD1,definition,
( god
= ( ^ [X: mu] :
( mforall_indset
@ ^ [Phi: mu > \$i > \$o] :
( mimplies @ ( positive @ Phi ) @ ( Phi @ X ) ) ) ) )).

%----D3: Necessary existence of an entity is the exemplification of all its
%----essences.
thf(defD3,definition,
( necessary_existence
= ( ^ [X: mu] :
( mforall_indset
@ ^ [Phi: mu > \$i > \$o] :
( mimplies @ ( essence @ Phi @ X )
@ ( mbox
@ ( mexists_ind
@ ^ [Y: mu] :
( Phi @ Y ) ) ) ) ) ) )).

%----A5: Necessary existence is positive.
thf(axA5,axiom,
( mvalid @ ( positive @ necessary_existence ) )).

%----C: Possibly, God exists.
thf(corC,lemma,
( mvalid
@ ( mdia
@ ( mexists_ind
@ ^ [X: mu] :
( god @ X ) ) ) )).

%----T2: Being God-like is an essence of any God-like being
thf(thmT2,lemma,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mimplies @ ( god @ X ) @ ( essence @ god @ X ) ) ) )).

%----T3: Necessarily God exists.
thf(thmT3,conjecture,
( mvalid
@ ( mbox
@ ( mexists_ind
@ ^ [X: mu] :
( god @ X ) ) ) )).
%------------------------------------------------------------------------------
```