## TPTP Problem File: AGT027^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : AGT027^2 : TPTP v7.0.0. Released v5.2.0.
% Domain   : Agents
% Problem  : Two different degrees of belief
% Version  : [Ben11] axioms : Reduced > Complete.
% English  :

% Refs     : [Ben11] Benzmueller (2011), Email to Geoff Sutcliffe
%          : [Ben11] Benzmueller (2011), Combining and Automating Classical
% Source   : [Ben11]
% Names    : Ex_10_1_KI4s [Ben11]

% Status   : Theorem
% Rating   : 0.88 v7.0.0, 0.86 v6.4.0, 0.83 v6.3.0, 0.80 v6.2.0, 0.71 v6.1.0, 0.57 v6.0.0, 0.86 v5.5.0, 0.83 v5.4.0, 0.80 v5.3.0, 1.00 v5.2.0
% Syntax   : Number of formulae    :   81 (   0 unit;  40 type;  31 defn)
%            Number of atoms       :  320 (  36 equality; 148 variable)
%            Maximal formula depth :   11 (   6 average)
%            Number of connectives :  211 (   4   ~;   4   |;   8   &; 187   @)
%                                         (   0 <=>;   8  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :  185 ( 185   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   44 (  40   :;   0   =)
%            Number of variables   :   93 (   3 sgn;  29   !;   6   ?;  58   ^)
%                                         (  93   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

%------------------------------------------------------------------------------
%----Include embedding of quantified multimodal logic in simple type theory
include('Axioms/LCL013^0.ax').
%------------------------------------------------------------------------------
thf(a1,type,(
a1: \$i > \$i > \$o )).

thf(a2,type,(
a2: \$i > \$i > \$o )).

thf(a,type,(
a: mu )).

thf(tom,type,(
tom: mu )).

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

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

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

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

thf(axiom_a1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mbox @ a2 @ ( mimplies @ ( mdia @ a2 @ ( q @ X ) ) @ ( p @ X ) ) ) ) )).

thf(axiom_a2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mbox @ a1 @ ( mimplies @ ( mand @ ( r @ X ) @ ( s @ X ) ) @ ( q @ X ) ) ) ) )).

thf(axiom_a3,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mbox @ a1 @ ( mimplies @ ( s @ X ) @ ( mbox @ a1 @ ( r @ X ) ) ) ) ) )).

thf(axiom_a4,axiom,
( mvalid @ ( mdia @ a1 @ ( s @ a ) ) )).

thf(axiom_I_for_a2_a1,axiom,
( mvalid
@ ( mforall_prop
@ ^ [Phi: \$i > \$o] :
( mimplies @ ( mbox @ a2 @ Phi ) @ ( mbox @ a1 @ Phi ) ) ) )).

thf(axiom_4s_for_a1_a1,axiom,
( mvalid
@ ( mforall_prop
@ ^ [Phi: \$i > \$o] :
( mimplies @ ( mbox @ a1 @ Phi ) @ ( mbox @ a1 @ ( mbox @ a1 @ Phi ) ) ) ) )).

thf(axiom_4s_for_a1_a2,axiom,
( mvalid
@ ( mforall_prop
@ ^ [Phi: \$i > \$o] :
( mimplies @ ( mbox @ a1 @ Phi ) @ ( mbox @ a2 @ ( mbox @ a1 @ Phi ) ) ) ) )).

thf(axiom_4s_for_a2_a1,axiom,
( mvalid
@ ( mforall_prop
@ ^ [Phi: \$i > \$o] :
( mimplies @ ( mbox @ a2 @ Phi ) @ ( mbox @ a1 @ ( mbox @ a2 @ Phi ) ) ) ) )).

thf(axiom_4s_for_a2_a2,axiom,
( mvalid
@ ( mforall_prop
@ ^ [Phi: \$i > \$o] :
( mimplies @ ( mbox @ a2 @ Phi ) @ ( mbox @ a2 @ ( mbox @ a2 @ Phi ) ) ) ) )).

thf(conj,conjecture,
( mvalid
@ ( mexists_ind
@ ^ [X: mu] :
( mbox @ a1 @ ( p @ X ) ) ) )).

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