TPTP Problem File: MGT055+1.p
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% File : MGT055+1 : TPTP v9.0.0. Released v2.4.0.
% Domain : Management (Organisation Theory)
% Problem : Conditions for a constant then jumping hazard of mortality 1
% Version : [Han98] axioms.
% English : When (`eta' < `sigma') in a drifting environment, an endowed
% organization's hazard of mortality remains constant until age
% reaches `eta', then jumps to a higher level, then jumps again at
% age `sigma'.
% Refs : [Kam00] Kamps (2000), Email to G. Sutcliffe
% : [CH00] Carroll & Hannan (2000), The Demography of Corporation
% : [Han98] Hannan (1998), Rethinking Age Dependence in Organizati
% Source : [Kam00]
% Names : LEMMA 8 [Han98]
% Status : Theorem
% Rating : 0.18 v9.0.0, 0.25 v8.2.0, 0.22 v8.1.0, 0.19 v7.4.0, 0.13 v7.3.0, 0.21 v7.2.0, 0.17 v6.4.0, 0.23 v6.3.0, 0.21 v6.2.0, 0.20 v6.1.0, 0.27 v6.0.0, 0.30 v5.5.0, 0.37 v5.4.0, 0.36 v5.3.0, 0.41 v5.2.0, 0.35 v5.1.0, 0.38 v4.1.0, 0.39 v4.0.1, 0.35 v4.0.0, 0.33 v3.7.0, 0.35 v3.5.0, 0.26 v3.4.0, 0.32 v3.3.0, 0.43 v3.2.0, 0.45 v3.1.0, 0.44 v2.7.0, 0.33 v2.6.0, 0.67 v2.5.0, 0.50 v2.4.0
% Syntax : Number of formulae : 15 ( 0 unt; 0 def)
% Number of atoms : 62 ( 8 equ)
% Maximal formula atoms : 12 ( 4 avg)
% Number of connectives : 54 ( 7 ~; 4 |; 26 &)
% ( 7 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-3 aty)
% Number of functors : 6 ( 6 usr; 3 con; 0-2 aty)
% Number of variables : 40 ( 40 !; 0 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : See MGT042+1.p for the mnemonic names.
%--------------------------------------------------------------------------
include('Axioms/MGT001+0.ax').
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%----Problem Axioms
%----An endowment provides an immunity that lasts until an
%----organization's age exceeds `eta'.
fof(definition_1,axiom,
! [X] :
( has_endowment(X)
<=> ! [T] :
( organization(X)
& ( smaller_or_equal(age(X,T),eta)
=> has_immunity(X,T) )
& ( greater(age(X,T),eta)
=> ~ has_immunity(X,T) ) ) ) ).
%----An organization's hazard of mortality is constant during periods
%----in which it has immunity.
fof(assumption_2,axiom,
! [X,T0,T] :
( ( organization(X)
& has_immunity(X,T0)
& has_immunity(X,T) )
=> hazard_of_mortality(X,T0) = hazard_of_mortality(X,T) ) ).
%----An organization's hazard of mortality is lower during periods in
%----which it has immunity than in periods in which it does not.
fof(assumption_3,axiom,
! [X,T0,T] :
( ( organization(X)
& has_immunity(X,T0)
& ~ has_immunity(X,T) )
=> greater(hazard_of_mortality(X,T),hazard_of_mortality(X,T0)) ) ).
%----Two states of the environment are dissimilar for an organization
%----if and only if the organization cannot be aligned to both.
%----
%----Added quantification over X.
fof(definition_2,axiom,
! [X,T0,T] :
( dissimilar(X,T0,T)
<=> ( organization(X)
& ~ ( is_aligned(X,T0)
<=> is_aligned(X,T) ) ) ) ).
%----An organization is aligned with the state of the environment at
%----its time of founding.
fof(assumption_13,axiom,
! [X,T] :
( ( organization(X)
& age(X,T) = zero )
=> is_aligned(X,T) ) ).
%----An organization's capability is higher in the state of the
%----environment to which it is aligned.
%----
%----Changed dissimilar(X,T0,T) to ~ is_aligned(X,T).
%----This makes theorem 5,9,10,11 and lemma 8 derivable.
fof(assumption_14,axiom,
! [X,T0,T] :
( ( organization(X)
& is_aligned(X,T0)
& ~ is_aligned(X,T) )
=> greater(capability(X,T0),capability(X,T)) ) ).
%----Environmental drift: the environments at times separated by more
%----than `sigma' are dissimilar.
fof(assumption_15,axiom,
! [X,T0,T] :
( ( organization(X)
& age(X,T0) = zero )
=> ( greater(age(X,T),sigma)
<=> dissimilar(X,T0,T) ) ) ).
%----Superiority in capability lowers the hazard of mortality when an
%----organization lacks immunity.
fof(assumption_16,axiom,
! [X,T0,T] :
( ( organization(X)
& ~ has_immunity(X,T0)
& ~ has_immunity(X,T)
& greater(capability(X,T),capability(X,T0)) )
=> greater(hazard_of_mortality(X,T0),hazard_of_mortality(X,T)) ) ).
%----Problem theorems
%----When (`eta' < `sigma') in a drifting environment, an endowed
%----organization's hazard of mortality remains constant until age
%----reaches `eta', then jumps to a higher level, then jumps again at
%----age `sigma'.
%----From D1, D2, A2, A3, A13-16 (texts says D1-2, A1-3, 13-16; also needs
%----D<, D<=, MP>str, MP>tra).
fof(lemma_8,conjecture,
! [X,T0,T1,T2,T3] :
( ( organization(X)
& has_endowment(X)
& age(X,T0) = zero
& smaller_or_equal(age(X,T1),eta)
& greater(age(X,T2),eta)
& smaller_or_equal(age(X,T2),sigma)
& greater(age(X,T3),sigma)
& greater(sigma,eta)
& greater(eta,zero) )
=> ( greater(hazard_of_mortality(X,T3),hazard_of_mortality(X,T2))
& greater(hazard_of_mortality(X,T2),hazard_of_mortality(X,T1))
& hazard_of_mortality(X,T1) = hazard_of_mortality(X,T0) ) ) ).
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