TPTP Problem File: LCL634^1.p

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% File     : LCL634^1 : TPTP v8.0.0. Bugfixed v5.2.0.
% Domain   : Logical Calculi
% Problem  : Goedel's ontological argument on the existence of God
% Version  : [Ben08] axioms.
% English  :

% Refs     : [Fit00] Fitting (2000), Higher-Order Modal Logic - A Sketch
%          : [Ben08] Benzmueller (2008), Email to G. Sutcliffe
% Source   : [Ben08]
% Names    : Fitting-HOLML-Ex-God-alternative-b [Ben08]

% Status   : CounterSatisfiable
% Rating   : 0.80 v7.4.0, 0.75 v7.2.0, 0.67 v5.4.0, 1.00 v5.2.0
% Syntax   : Number of formulae    :   26 (  10 unt;  12 typ;   9 def)
%            Number of atoms       :   48 (   9 equ;   0 cnn)
%            Maximal formula atoms :    7 (   3 avg)
%            Number of connectives :   64 (   1   ~;   1   |;   1   &;  58   @)
%                                         (   0 <=>;   3  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   3 avg)
%            Number of types       :    3 (   1 usr)
%            Number of type conns  :   69 (  69   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   12 (  11 usr;   0 con; 1-3 aty)
%            Number of variables   :   34 (  24   ^  10   !;   0   ?;  34   :)
% SPC      : TH0_CSA_EQU_NAR

% Bugfixes : v4.0.0 - Complete overhaul.
%          : v5.2.0 - Added missing types
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%----Some higher-order multimodal operators are needed
%----Base type individuals; corresponds to type 0 in [Fit00]
thf(individuals_decl,type,
individuals: \$tType ).

thf(e_type,type,
e: ( individuals > \$i > \$o ) > individuals > \$i > \$o ).

thf(g_type,type,
g: individuals > \$i > \$o ).

thf(mall_ind_type,type,
mall_ind: ( individuals > \$i > \$o ) > \$i > \$o ).

thf(mall_indset_type,type,
mall_indset: ( ( individuals > \$i > \$o ) > \$i > \$o ) > \$i > \$o ).

thf(mbox_type,type,
mbox: ( \$i > \$i > \$o ) > ( \$i > \$o ) > \$i > \$o ).

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

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

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

thf(mvalid_type,type,
mvalid: ( \$i > \$o ) > \$o ).

%----mnot corresponds to set complement
thf(mnot,definition,
( mnot
= ( ^ [Phi: \$i > \$o,W: \$i] :
~ ( Phi @ W ) ) ) ).

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

%----mimplies defined via mnot and mor
thf(mimplies,definition,
( mimplies
= ( ^ [Phi: \$i > \$o,Psi: \$i > \$o] : ( mor @ ( mnot @ Phi ) @ Psi ) ) ) ).

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

%----mall_ind (for individuals)
thf(mall_ind,definition,
( mall_ind
= ( ^ [Phi: individuals > \$i > \$o,W: \$i] :
! [X: individuals] : ( Phi @ X @ W ) ) ) ).

%----mall_indset (for sets of individuals)
thf(mall_indset,definition,
( mall_indset
= ( ^ [Phi: ( individuals > \$i > \$o ) > \$i > \$o,W: \$i] :
! [X: individuals > \$i > \$o] : ( Phi @ X @ W ) ) ) ).

%----Validity of a multi modal logic formula can now be encoded as
thf(mvalid,definition,
( mvalid
= ( ^ [Phi: \$i > \$o] :
! [W: \$i] : ( Phi @ W ) ) ) ).

%----The encoding of the example
%----r is an S5 accessibility relation: reflexive, transitive and symmetric
thf(r_type,type,
r: \$i > \$i > \$o ).

thf(r_refl,axiom,
! [X: \$i] : ( r @ X @ X ) ).

thf(r_trans,axiom,
! [X: \$i,Y: \$i,Z: \$i] :
( ( ( r @ X @ Y )
& ( r @ Y @ Z ) )
=> ( r @ X @ Z ) ) ).

thf(r_sym,axiom,
! [X: \$i,Y: \$i] :
( ( r @ X @ Y )
=> ( r @ Y @ X ) ) ).

%----The axioms of the example
%----Positiveness: p
thf(p_type,type,
p: ( individuals > \$i > \$o ) > \$i > \$o ).

thf(positiveness,axiom,
( mvalid
@ ( mall_indset
@ ^ [X: individuals > \$i > \$o] :
( mimplies @ ( mnot @ ( p @ X ) )
@ ( p
@ ^ [Z: individuals] : ( mnot @ ( X @ Z ) ) ) ) ) ) ).

%----Being God: g
thf(g,definition,
( g
= ( ^ [Z: individuals] :
( mall_indset
@ ^ [X: individuals > \$i > \$o] : ( mimplies @ ( p @ X ) @ ( X @ Z ) ) ) ) ) ).

%----Essence: e
thf(e,definition,
( e
= ( ^ [X: individuals > \$i > \$o,Z: individuals] :
( mall_indset
@ ^ [Y: individuals > \$i > \$o] :
( mimplies @ ( Y @ Z )
@ ( mbox @ r
@ ( mall_ind
@ ^ [U: individuals] : ( mimplies @ ( X @ U ) @ ( Y @ U ) ) ) ) ) ) ) ) ).

%----Conjecture: Being God is the essence of anything that is, in fact, God.
thf(thm,conjecture,
( mvalid
@ ( mall_ind
@ ^ [Z: individuals] : ( mimplies @ ( g @ Z ) @ ( e @ g @ Z ) ) ) ) ).

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