%-------------------------------------------------------------------------- % File : MGT063+1 : TPTP v6.4.0. 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 : 0.33 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 unit) % Number of atoms : 77 ( 12 equality) % Maximal formula depth : 17 ( 5 average) % Number of connectives : 69 ( 12 ~ ; 4 |; 29 &) % ( 8 <=>; 16 =>; 0 <=) % ( 0 <~>; 0 ~|; 0 ~&) % Number of predicates : 12 ( 0 propositional; 1-3 arity) % Number of functors : 11 ( 9 constant; 0-2 arity) % Number of variables : 34 ( 0 singleton; 34 !; 0 ?) % Maximal term depth : 2 ( 1 average) % SPC : FOF_THM_RFO_SEQ % Comments : See MGT042+1.p for the mnemonic names. %-------------------------------------------------------------------------- include('Axioms/MGT001+0.ax'). %-------------------------------------------------------------------------- %----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) ) ) )). %--------------------------------------------------------------------------