In the lottery paradox, we are asked to consider a fair lottery. In such a case, we can easily demonstrate for each ticket that it is not likely to win. However, we also see that there must be a winning ticket, since that is the situation described.

It has been claimed that it is justifiable to believe that ticket A will not win, ticket B will not win, ticket C will not win… and so on, but also to believe that there is a ticket which will win. This is clearly an inconsistent set of beliefs, and supposedly justified.

Whence the belief that ticket A will not win? This fact is not given directly by the situation. The best our mathematical theories can produce is that ticket A might not win. As a rational man, I cannot accept that A cannot win simply because it is not likely to. Whence that ticket B will not win? This fact is not given… and so on.

In the preface paradox, we are to consider a book wherein the author’s views are heavily argued in a rational manner and therefore are justified. On the other hand, the author realizes himself capable of a mistake and so takes responsibility in his preface and admits that there are likely to be errors in the book.

It is claimed again, and along the same lines, that this set of beliefs, those expressed in both the body and preface, is inconsistent. We are offered a choice between two claims: the overly brash “All of my views are right, since they are justified” and the overly shy “At least one of my views is wrong, though they be justified.”

If we were using simple predicate logic, one, and only one, of these views must be correct, but must we use that logic? What prevents me from sensibly asserting “All of my views are merely likely to be right.” using an appropriate modal logic?

And yet, the idea that beliefs might be both inconsistent and justified has infected our philosophical work, made its way past peer review and into the minds of once-rational men. The coherentist will readily admit that coherence does not admit inconsistency, and so the supposed existence of these beliefs is meant to eliminate coherentism. Yet I have not been given a shred of evidence that these beliefs exist, nor can I find it when I search.

I’ll admit, the man who believes his ticket definitely won’t win, or his book to certainly have an error, they would be inconsistent to also believe they might have a chance. Then again, they will never be described as rational men.

The man who thinks his lottery ticket will probably not win, the man who thinks he probably made a mistake in his massive book, they are hardly inconsistent for believing they might win, might not have made a mistake. “Might win” only requires the slimmest of possibilities, and that possibility is given in the situation and in their beliefs.

When did philosophy fall so low as to require a text like this? Perhaps when they became enamored of formal systems and forgot their mathematics…