Belief Theorem, March 12, 2012

Why it is rational to believe in a Creator: A simplied analytical framework

Overall Utility, U = U_E + U_H

U_E is utility in life on earth

U_H is Utility in the life hereafter if there is one

Utility in life on earth

It is possible that a human agent derives utility from consumption goods A [Code Permissible] and B [Sin Goods]. There is, however, a physically limited maximum consumption (due to finite lifetime) of these on earth. Utility is increasing in A and B and there is no saturation point.

Say, effort is misery and leads to disutility. Believing takes effort and consumption of A and B does not require effort.

To make it easy, if an agent does not believe, enjoys maximum earthly consumptions of A and B, and derives maximum earthly utility of U_Max,E. An agent who believes in Divinity System and also does not consume B (sin goods) on earth, would have the lowest utility on earth, U_Min,E due to no consumption of utility enhancing good B as well as maximum earthly misery/disutility from maximum effort toward belief. Someone believing but not practicing as well and not putting as much effort toward belief, would have earthly utility U_Mid,E, with U_Min,E < U_Mid,E < U_Max,E . 

Magnitude of earthly utility U_E is known ex ante once the belief and effort decisions are made.

Utility in life hereafter if there is one

Utility in life hereafter is random ex ante. States realize ex-post and state contingent utilities follow.

There are just two states: State 1: Divinity System (Creator, Code, Reward /Penalty in life hereafter) exists, ex ante probability p; State 2: No Divinity System, ex ante probability 1-p.

If State 1 realizes ex post

i. Agent believed in Divinity and Lived with Maximum Piousness (no consumption of sin goods)

          An infinite time in heaven with unlimited consumption of A and B

          Utility in Hereafter = U_Max,H

ii. Agent believed in Divinity but consumed some sin goods

 A limited time in hell with limited misery, related to consumption of B on earth.

An infinite time in heaven with unlimited consumption of A and B

                   Utility in Hereafter = U_Mid,H

iii. Agent did not believe in Divinity

                   An infinite time in hell meaning unlimited misery

                             Utility in Hereafter = U_Min,H

If State 2 realizes ex post

For all agents, utility is 0.0, as there is no life hereafter.

Expected Total Utility [with risk neutrality]

i. Agent believe in Divinity with maximum Piousness: U_Min,E + p  U_Max,H
ii. Agent believe in Divinity with low Piousness: U_Mid,E + p  U_Mid,H
iii. Agent did not believe in Divinity: U_Max,E + p  U_Min,H

Belief Theorem
Belief dominates if  U_Mid,H is sufficiently large, that is, p  U_Mid,H > [U_Max,E  ─ U_Mid,E + p  U_Min,H ], or equivalently,
p > [U_Max,E  ─ U_Mid,E ] / [U_Mid,H  ─ U_Min,H ]

Comment: By most belief systems (Divine Codes), the reward in life hereafter for just believing and not practicing Divine Code is clearly larger than the infinite misery from not believing (U_Min,H = ─Infinity). As such, for any p>0, belief dominates.