TPTP Problem File: MGT051+1.p
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% File : MGT051+1 : TPTP v9.0.0. Released v2.4.0.
% Domain : Management (Organisation Theory)
% Problem : Conditions for constant then increasing hazard of mortality
% Version : [Han98] axioms.
% English : An endowed organization's hazard of mortality remains constant
% during the period of immunity and increases monotonically with
% age once immunity ends.
% 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 : THEOREM 4 [Han98]
% Status : Theorem
% Rating : 0.30 v9.0.0, 0.36 v8.2.0, 0.33 v8.1.0, 0.31 v7.4.0, 0.30 v7.3.0, 0.31 v7.2.0, 0.28 v7.1.0, 0.26 v7.0.0, 0.27 v6.4.0, 0.31 v6.3.0, 0.33 v6.2.0, 0.40 v6.1.0, 0.47 v6.0.0, 0.39 v5.5.0, 0.48 v5.4.0, 0.50 v5.3.0, 0.56 v5.2.0, 0.45 v5.1.0, 0.48 v5.0.0, 0.50 v4.1.0, 0.43 v4.0.1, 0.39 v4.0.0, 0.38 v3.7.0, 0.40 v3.5.0, 0.37 v3.4.0, 0.32 v3.3.0, 0.43 v3.2.0, 0.55 v3.1.0, 0.56 v2.7.0, 0.67 v2.6.0, 0.50 v2.4.0
% Syntax : Number of formulae : 16 ( 0 unt; 0 def)
% Number of atoms : 73 ( 15 equ)
% Maximal formula atoms : 12 ( 4 avg)
% Number of connectives : 62 ( 5 ~; 4 |; 29 &)
% ( 4 <=>; 20 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 7 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-2 aty)
% Number of functors : 8 ( 8 usr; 1 con; 0-2 aty)
% Number of variables : 44 ( 44 !; 0 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : See MGT042+1.p for the mnemonic names.
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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)) ) ).
%----When an organization lacks immunity, superior capability and
%----position imply a lower hazard of mortality.
fof(assumption_4,axiom,
! [X,T0,T] :
( ( organization(X)
& ~ has_immunity(X,T0)
& ~ has_immunity(X,T) )
=> ( ( ( greater(capability(X,T),capability(X,T0))
& greater_or_equal(position(X,T),position(X,T0)) )
=> smaller(hazard_of_mortality(X,T),hazard_of_mortality(X,T0)) )
& ( ( greater_or_equal(capability(X,T),capability(X,T0))
& greater(position(X,T),position(X,T0)) )
=> smaller(hazard_of_mortality(X,T),hazard_of_mortality(X,T0)) )
& ( ( capability(X,T) = capability(X,T0)
& position(X,T) = position(X,T0) )
=> hazard_of_mortality(X,T) = hazard_of_mortality(X,T0) ) ) ) ).
%----Increased knowledge elevates an organization's capability; and
%----increased accumulation of organizational internal frictions
%----diminishes its capability.
fof(assumption_5,axiom,
! [X,T0,T] :
( organization(X)
=> ( ( ( greater(stock_of_knowledge(X,T),stock_of_knowledge(X,T0))
& smaller_or_equal(internal_friction(X,T),internal_friction(X,T0)) )
=> greater(capability(X,T),capability(X,T0)) )
& ( ( smaller_or_equal(stock_of_knowledge(X,T),stock_of_knowledge(X,T0))
& greater(internal_friction(X,T),internal_friction(X,T0)) )
=> smaller(capability(X,T),capability(X,T0)) )
& ( ( stock_of_knowledge(X,T) = stock_of_knowledge(X,T0)
& internal_friction(X,T) = internal_friction(X,T0) )
=> capability(X,T) = capability(X,T0) ) ) ) ).
%----Improved ties with external actors enhance an organization's position.
fof(assumption_6,axiom,
! [X,T0,T] :
( organization(X)
=> ( ( greater(external_ties(X,T),external_ties(X,T0))
=> greater(position(X,T),position(X,T0)) )
& ( external_ties(X,T) = external_ties(X,T0)
=> position(X,T) = position(X,T0) ) ) ) ).
%----Case: liability of senescence (Ass. 10-12 replacing A7-9)!
%----
%----An organization's stock of knowledge does not vary with its age
%----(contra assumption 7).
fof(assumption_10,axiom,
! [X,T0,T] :
( organization(X)
=> stock_of_knowledge(X,T) = stock_of_knowledge(X,T0) ) ).
%----The quality of an organization's external ties does not vary with
%----its age (contra assumption 8).
fof(assumption_11,axiom,
! [X,T0,T] :
( organization(X)
=> external_ties(X,T) = external_ties(X,T0) ) ).
%----The quality of an organization's internal friction increases
%----monotonically with its age (contra assumption 9).
fof(assumption_12,axiom,
! [X,T0,T] :
( ( organization(X)
& greater(age(X,T),age(X,T0)) )
=> greater(internal_friction(X,T),internal_friction(X,T0)) ) ).
%----Problem theorems
%----The senescence theorem for unendowed organizations: An endowed
%----organization's hazard of mortality remains constant during the
%----period of immunity and increases monotonically with age once
%----immunity ends.
%----From D1, A2-6, and A10-12 (text says D1, A1-4, L5-6; also needs D<,
%----D<=, D>=, MP>str, MP>com, MP>tra).
fof(theorem_4,conjecture,
! [X,T0,T1,T2,T3] :
( ( organization(X)
& has_endowment(X)
& smaller_or_equal(age(X,T0),age(X,T1))
& smaller_or_equal(age(X,T1),eta)
& greater(age(X,T2),eta)
& greater(age(X,T3),age(X,T2)) )
=> ( 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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