Monday, February 20, 2012

Probably not important

Consider this sentence:

            PN       PN is probably not true.

Sentences can be true even if they’re probably not, at least on some readings of ‘probably’. I’m intending ‘probably’ in one of those ways. It sounds Moore-paradoxical to say ‘P but probably not-P’, but Moore-paradoxicality isn’t inconsistency. So the instance of the sloppy T-schema for PN...

            T-PN   PN is true iff PN is probably not true.

...doesn’t entail a contradiction. Maybe it’s true but probably not true. Contrast the instance of the sloppy T-schema for L:

            L          L is not true.

            T-L      L is true iff L is not true.

T-L does entail a contradiction, so L is paradoxical. T-PN doesn’t, but that’s not the end of it. First, note that there’s another way PN could go: it could be untrue though not probably. This might be because it was probably true, or because its truth and untruth were equiprobable. So it might be true, and it might be untrue. Which is it?

A fairly reasonable-sounding principle of indifference might say its truth and untruth were equiprobable. In that case it isn’t probably untrue, which means it isn’t true. But if the equiprobability was itself probable, then its consequence that PN isn’t true would seem to be probable, which means PN is true, which is a contradiction. So the equiprobability isn’t probable. What are the other credible options?

Maybe PN is probably not true. But this means it’s true. So if PN is probably probably not true, then it’s probably true. That doesn’t sound a very delicious combination.

The other possibility is that PN is probably true. In that case it’s not true. So if it’s probably probably true then it’s probably not true. Yuk again.

Can this lot even be made consistent? Here are the entailments:

            Probably true untrue
            Probably not true true
            Equiprobable untrue

If the disjunction of probably and equiprobable is probable, then PN is probably untrue, but this contradicts the disjunction being probable, since the options are exclusive. So the disjunction is not probable, so probably not must be at least 50% probable. But if it’s exactly 50% probable, then given the entailments, PN is equiprobably true and not true. But this makes equiprobable true, which we’ve established is not probable. So suppose probably not is probable. This means PN is probably true, which is a contradiction. So the improbable equiprobable must be true. But if something must be true, then it’s probable. So the disjunction is probable after all, and that led to a contradiction. So dialetheism must be true. But if dialetheism must be true, we can reject some of the above reductio reasoning. So some contradictions are true, but possibly not this one.

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