TPTP Problem File: MGT063+1.p
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%--------------------------------------------------------------------------
% File : MGT063+1 : TPTP v8.2.0. Released v2.4.0.
% Domain : Management (Organisation Theory)
% Problem : Conditions for increasing then decreasing hazard of mortality
% Version : [Han98] axioms.
% English : If environmental drift destroys alignment before advantage can
% be gained from occupancy of a robust position, then the hazard
% of mortality for an unendowed organization with a robust
% position initially increases with age, then decreases with
% further aging and falls below the initial level.
% 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 9 [Han98]
% Status : Theorem
% Rating : 0.44 v8.2.0, 0.42 v8.1.0, 0.39 v7.5.0, 0.41 v7.4.0, 0.33 v7.3.0, 0.38 v7.2.0, 0.34 v7.1.0, 0.35 v7.0.0, 0.33 v6.4.0, 0.31 v6.3.0, 0.33 v6.2.0, 0.32 v6.1.0, 0.47 v6.0.0, 0.52 v5.5.0, 0.59 v5.4.0, 0.61 v5.3.0, 0.63 v5.2.0, 0.45 v5.1.0, 0.48 v5.0.0, 0.46 v4.1.0, 0.48 v4.0.0, 0.46 v3.7.0, 0.45 v3.5.0, 0.42 v3.4.0, 0.37 v3.3.0, 0.43 v3.2.0, 0.55 v3.1.0, 0.67 v2.7.0, 0.50 v2.6.0, 0.83 v2.5.0, 1.00 v2.4.0
% Syntax : Number of formulae : 20 ( 6 unt; 0 def)
% Number of atoms : 77 ( 12 equ)
% Maximal formula atoms : 16 ( 3 avg)
% Number of connectives : 69 ( 12 ~; 4 |; 29 &)
% ( 8 <=>; 16 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 12 ( 11 usr; 0 prp; 1-3 aty)
% Number of functors : 11 ( 11 usr; 9 con; 0-2 aty)
% Number of variables : 34 ( 34 !; 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 unendowed organization never possesses immunity.
fof(assumption_1,axiom,
! [X,T] :
( ( organization(X)
& ~ has_endowment(X) )
=> ~ has_immunity(X,T) ) ).
%----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) ) ).
%----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) ) ) ).
%----An organization's position is robust if and only if it provides
%----positional advantage only after age `tau'.
%----
%----Text says fragile_position(X) instead of robust_position(X).
%----Interchanged ~ positional_advantage(X,T) and positional_advantage(X,T).
fof(definition_4,axiom,
! [X] :
( robust_position(X)
<=> ! [T] :
( ( smaller_or_equal(age(X,T),tau)
=> ~ positional_advantage(X,T) )
& ( greater(age(X,T),tau)
=> positional_advantage(X,T) ) ) ) ).
%----An organization's immunity. alignment of capability with the
%----current state of the environment and positional advantage jointly
%----affect the hazard of mortality with the following ordinal scaling:
fof(assumption_17,axiom,
! [X,T] :
( organization(X)
=> ( ( has_immunity(X,T)
=> hazard_of_mortality(X,T) = very_low )
& ( ~ has_immunity(X,T)
=> ( ( ( is_aligned(X,T)
& positional_advantage(X,T) )
=> hazard_of_mortality(X,T) = low )
& ( ( ~ is_aligned(X,T)
& positional_advantage(X,T) )
=> hazard_of_mortality(X,T) = mod1 )
& ( ( is_aligned(X,T)
& ~ positional_advantage(X,T) )
=> hazard_of_mortality(X,T) = mod2 )
& ( ( ~ is_aligned(X,T)
& ~ positional_advantage(X,T) )
=> hazard_of_mortality(X,T) = high ) ) ) ) ) ).
%----The levels of hazard of mortality are ordered:
%----
%----Split over 5 separate formulas because TPTP gives an error on top
%----level occurrences of `&'.
fof(assumption_18a,axiom,
greater(high,mod1) ).
fof(assumption_18b,axiom,
greater(mod1,low) ).
fof(assumption_18c,axiom,
greater(low,very_low) ).
fof(assumption_18d,axiom,
greater(high,mod2) ).
fof(assumption_18e,axiom,
greater(mod2,low) ).
%----Position dominates alignment:
fof(assumption_19,axiom,
greater(mod2,mod1) ).
%----Problem theorems
%----Robust position without endowment when (`sigma' < `tau'): If
%----environmental drift destroys alignment before advantage can
%----be gained from occupancy of a robust position (`sigma' < `tau'), then
%----the hazard of mortality for an unendowed organization with a
%----robust position initially increases with age, then decreases with
%----further aging and falls below the initial level.
%----From D2, D4, A1, A13, A15, A17, A18 (text says D1,2,4 and A1,2,13-15,
%----17-19; also needs D<, D<=, MP>str, MP>com, MP>tra).
%----
%----Added (`sigma' < `tau') in antecedent
%----and (hazard_of_mortality(X,T1) = hazard_of_mortality(X,T0)).
fof(theorem_9,conjecture,
! [X,T0,T1,T2,T3] :
( ( organization(X)
& robust_position(X)
& ~ has_endowment(X)
& age(X,T0) = zero
& greater(sigma,zero)
& greater(tau,zero)
& smaller(sigma,tau)
& smaller_or_equal(age(X,T1),sigma)
& greater(age(X,T2),sigma)
& smaller_or_equal(age(X,T2),tau)
& greater(age(X,T3),tau) )
=> ( smaller(hazard_of_mortality(X,T3),hazard_of_mortality(X,T1))
& smaller(hazard_of_mortality(X,T1),hazard_of_mortality(X,T2))
& hazard_of_mortality(X,T1) = hazard_of_mortality(X,T0) ) ) ).
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