TPTP Problem File: PHI006^4.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : PHI006^4 : TPTP v9.0.0. Released v6.4.0.
% Domain : Philosophy
% Problem : Inconsistency of the axioms in Goedel's original manuscript
% Version : [Ben16] axioms : Biased.
% English :
% Refs : [Ben16] Benzmueller (2016), Email to Geoff Sutcliffe
% Source : [Ben16]
% Names : Inconsistency_S5U_direct_fullaxioms.p [Ben16]
% Status : ContradictoryAxioms
% Rating : 0.25 v9.0.0, 0.30 v8.2.0, 0.46 v8.1.0, 0.45 v7.5.0, 0.29 v7.4.0, 0.67 v7.2.0, 0.62 v7.0.0, 0.57 v6.4.0
% Syntax : Number of formulae : 55 ( 24 unt; 25 typ; 23 def)
% Number of atoms : 99 ( 24 equ; 0 cnn)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 110 ( 4 ~; 2 |; 3 &; 98 @)
% ( 1 <=>; 2 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 2 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 150 ( 150 >; 0 *; 0 +; 0 <<)
% Number of symbols : 35 ( 32 usr; 10 con; 0-3 aty)
% Number of variables : 70 ( 60 ^; 6 !; 4 ?; 70 :)
% SPC : TH0_CAX_EQU_NAR
% Comments : This problem file has been generated by Sledgehammer.
% : The $false conjecture makes this correspond to the Isabelle
% sources. Otherwise it could be omitted and the status would be
% Unsatisfiable.
%------------------------------------------------------------------------------
include('Axioms/LCL017^0.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 ).
%----A1: Either the property or its negation are positive, but not both.
%----(Remark: only the left to right is needed for proving T1)
thf(axA1a,axiom,
( mvalid
@ ( mforall_indset
@ ^ [Phi: mu > $i > $o] :
( mimplies
@ ( positive
@ ^ [X: mu] : ( mnot @ ( Phi @ X ) ) )
@ ( mnot @ ( positive @ Phi ) ) ) ) ) ).
thf(axA1b,axiom,
( mvalid
@ ( mforall_indset
@ ^ [Phi: mu > $i > $o] :
( mimplies @ ( mnot @ ( positive @ Phi ) )
@ ( positive
@ ^ [X: mu] : ( mnot @ ( Phi @ X ) ) ) ) ) ) ).
%----A2: A property necessarily implied by a positive property is positive.
thf(axA2,axiom,
( mvalid
@ ( mforall_indset
@ ^ [Phi: mu > $i > $o] :
( mforall_indset
@ ^ [Psi: mu > $i > $o] :
( mimplies
@ ( mand @ ( positive @ Phi )
@ ( mbox
@ ( mforall_ind
@ ^ [X: mu] : ( mimplies @ ( Phi @ X ) @ ( Psi @ X ) ) ) ) )
@ ( positive @ Psi ) ) ) ) ) ).
%----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 ) ) ) ) ) ).
%----A3: The property of being God-like is positive.
thf(axA3,axiom,
mvalid @ ( positive @ god ) ).
%----A4: Positive properties are necessary positive properties.
thf(axA4,axiom,
( mvalid
@ ( mforall_indset
@ ^ [Phi: mu > $i > $o] : ( mimplies @ ( positive @ Phi ) @ ( mbox @ ( positive @ Phi ) ) ) ) ) ).
%----D2: An essence of an individual is a property that is
%----necessarily implying any of its properties.
thf(defD2,definition,
( essence
= ( ^ [Phi: mu > $i > $o,X: mu] :
( mforall_indset
@ ^ [Psi: mu > $i > $o] :
( mimplies @ ( Psi @ X )
@ ( mbox
@ ( mforall_ind
@ ^ [Y: mu] : ( mimplies @ ( Phi @ Y ) @ ( Psi @ Y ) ) ) ) ) ) ) ) ).
%----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 ) ).
%---Inconsistency
thf(conj,conjecture,
$false ).
%------------------------------------------------------------------------------