In my previous post on this topic, I used conditional derivation to try to prove that one statement entailed another statement, to show that ‘There is a person who is omniscient and perfectly free’ entails ‘There is a person who is perfectly good’.

But because I’m a bit unclear on how the logic of conditional statements relates to entailment, I’m not sure that conditional derivation can be used this way.

In any case, implication (the logical relationship in a true conditional statement) is similar to entailment in that both logical relationships have the characteristic of transitivity. Since I was basically using this feature of conditional statements, I can revise my reasoning to make use of the transitivity of entailment, rather than the transitivity of implication:

(PFO1*) ‘There is a person who is both perfectly free and omniscient’ entails the statement ‘There is a person P who is so constituted that if P is in circumstance C and A is the morally best action for P in C, then P is perfectly free, P knows that P is in C, P is in C, P knows that A is the morally best action for P in C, P knows everything that is entailed by the A being the morally best action for P in C, and P is able to do A in C’.

[a necessary truth based on the meaning of ‘omniscient’ and on the principle that ‘ought implies can’]

(PFO2*) ‘There is a person P who is so constituted that if P is in circumstance C and A is the morally best action for P in C, then P is perfectly free, P knows that P is in C, P is in C, P knows that A is the morally best action for P in C, P knows everything that is entailed by the A being the morally best action for P in C, and P is able to do A in C’ entails the statement ‘There is a person P who is so constituted that if P is in circumstance C and A is the morally best action for P in C, then P is perfectly free, P knows that P is in C, P is in C, P knows that A is the most rational action for P in C, and P is able to do A in C.’

[a necessary truth based on the fact that ‘A is the morally best action for P in C’ entails ‘A is the most rational action for P in C’]

(PFO3*) ‘There is a person P who is so constituted that if P is in circumstance C and A is the morally best action for P in C, then P is perfectly free, P knows that P is in C, P is in C, P knows that A is the most rational action for P in C, and P is able to do A in C.’

entails the statement ‘There is a person P who is so constituted that if P is in circumstance C and A is the morally best action for P in C, then P is perfectly free, P believes that P is in C, P is in C, P believes that A is the most rational action for P in C, and P is able to do A in C.’

[a necessary truth based on the fact that ‘P knows that X is the case’ entails ‘P believes that X is the case’]

Therefore:

(PFO4*) ‘There is a person who is both perfectly free and omniscient’ entails the statement ‘There is a person P who is so constituted that if P is in circumstance C and A is the morally best action for P in C, then P is perfectly free, P believes that P is in C, P is in C, P believes that A is the most rational action for P in C, and P is able to do A in C.’

[a deduction from (PFO1*), (PFO2*), and (PFO3*) based on the transitivity of entailment: A entails B; B entails C; C entails D; therefore: A entails D.]

(PFO5*) ‘There is a person P who is so constituted that if P is in circumstance C and A is the morally best action for P in C, then P is perfectly free, P believes that P is in C, P is in C, P believes that A is the most rational action for P in C, and P is able to do A in C.’ entails the statement ‘There is a person P who is so constituted that if P is in circumstance C and A is the morally best action for P in C, then P will do A in C.’

[a necessary truth based on the meaning of ‘perfectly free person’ which entails being a person who always does what he/she believes to be the most rational action, when there is such an action that he/she is able to do]

Therefore:

(PFO6*) ‘There is a person who is both perfectly free and omniscient’ entails the statement ‘There is a person P who is so constituted that if P is in circumstance C and A is the morally best action for P in C, then P will do A in C.’

[ a deduction from (PFO4*) and (PFO5*) based on the transitivity of entailment: A entails B; B entails C; therefore: A entails C.]