I will call a question “underdetermined” if we do not have any certain answer to it. Examples of underdetermined questions include:
(Q.1) Is physicalism true?
Physicalism would be falsified only if it were discovered that there is some type of existing entity which is nonphysical. Physicalism would be proven only if it were proven that there is no type of existing entity which is nonphysical. We have no evidence sufficient to prove the existence or non-existence of such entities. Therefore, (Q.1) is underdetermined.
(Q.2) Does P = NP?
Most people think that P != NP. However, this has not been proven to be true or false. In principle it ought to be perfectly possible to prove either that P = NP, or that P != NP, but this has not yet been done. Therefore (Q.2) is underdetermined.
(Q.3) Will the sun rise tomorrow?
This question is underdetermined because we only have evidence to the effect that the sun has risen at the start of every day up until today. There is no way of proving with certainty, from the available data, that the sun will rise tomorrow. It may not rise tomorrow.
(Q.4) Is mathematics consistent?
This question is underdetermined because it has not been proven either that mathematics is consistent, or that mathematics is inconsistent. It cannot be proven that mathematics is consistent, because any theory which contains a proof of its own consistency is inconsistent. There are consistent subsets of mathematics, whose consistency can be proven within theories which are themselves inconsistent or unable to prove their own consistency.
(Q.5) Is Jesus Christ our Lord and Savior?
It is difficult to imagine what proof that this was true or false would look like. Therefore, (Q.5) is underdetermined.
(Q.6) Are humans in telepathic contact with higher intelligences?
There are various texts which were allegedly received from higher intelligences. Either these are all invalid, or some of them are valid. We do not have proof either that any such text is valid, or that all of them are invalid. Therefore, (Q.6) is underdetermined.
Examples of questions that are not underdetermined include:
(Q.7) What is the sum of two and three?
(Q.8) What am I wearing right now?
(Q.9) Does there exist a mountain surrounded entirely by higher ground?
Now I suggest:
(H.1) There are cases in which it is rational to believe in a particular answer to an underdetermined question.
In other words, there are cases in which it is rational to believe beyond the evidence, or to believe without proof.
(H.1) is suggested by questions such as (Q.3). It seems obviously rational to believe that the sun will rise tomorrow, even though (Q.3) is an underdetermined question. If we accept (H.1), then the question arises of when it is rational to believe one side of an underdetermined binary question. I suggest three circumstances:
(C.1) When one side is more probable then the other side.
(C.2) When one side fits with the rest of what we believe, and the other does not.
(C.3) When one side has more utility than the other side.
Thus, for instance:
* It is rational for a scientist to believe in the positive side of (Q.1), because of (C.2).
* It is rational for a Christian to believe in the negative side of (Q.1), because of (C.2).
* It is rational to believe in the negative side of (Q.2), because of (C.1).
* It is rational to believe in the positive side of (Q.3), because of (C.1) and (C.3). (C.1) because the sun has always risen. (C.3) because I have various affairs to which I will have to attend if the sun rises tomorrow, and I would not give these affairs any attention if I believed that the sun would not rise. Because the sun probably will rise, there is utility in my believing that it will do so and arranging my plans according to this assumption.
* It is rational to believe in the positive side of (Q.5), because of (C.3).
* It is rational for a physicalist to believe in the negative side of (Q.6), because of (C.2).
* It is rational for me to believe in the positive side of (Q.6), because of (C.2).
One of the interesting implications of (H.1) is that:
(H.2) Rational people may hold different beliefs.
* Two people may differ in how they evaluate the probabilities of the two sides of an underdetermined binary question. Thus (C.1) leaves room for rational people to hold different beliefs.
* A given belief may have better belief coherence for one person than for another; thus (C.2) leaves room for rational people to hold different beliefs.
* A given belief may have more utility for one person than for another person; thus (C.3) leaves room for rational people to hold different beliefs.