[dpflan]

Thanks for this, though I may not be fully comprehending. If earthquakes can set off burglar alarms, and if burglars can set off burglar alarms , why would there be no -|ALARM->EARTHQUAKE theorem? Understandably the ALARM is for detecting burglars not earthquakes, but it does function as a detector for both.

[orodley]

It doesn't function as a detector for both, it functions as a detector for either. I'm not being pedantic, there is a subtle difference. ALARM -> EARTHQUAKE is false: the correct theorem is ALARM -> (EARTHQUAKE | BURGLAR). If the alarm goes off then you have no certain knowledge about which one caused it, only probabilities.

[scrupulusalbion]

It would seem that an additional source of information (e.g. a nearby seismograph bolted to bedrock) would be the solution to this problem. This is analogous to a blind person trying to identify whether a sound is a recording or the Real McCoy. The solution is to either somehow make him able to see or to allow him to use some other sense (e.g. touch) by which to measure the origin of the sound. If he hears a dog barking, then if he sees/feels a dog, he is almost completely assured [0] that it is a real dog that was barking and not a recording playing from a machine.

Thus, adding another sense (a seismograph and sightedness/touch, respectively) would seem to eliminate the problem. If this is correct, then such problems are more ones of a lack of relevant, heterogeneous information and less of a lack of logical expressiveness of probabilities.

[0] = I say "almost", because knowledge is a fundamental philosophical problem. The usual means to "almost" assurance is, as I am arguing, to employ more heterogeneous sources of information until the only logical alternatives to what you believe are absurdities (e.g. impossible).

[dpflan]

Ah, I wasn't picking up on the subtlety; thanks for adding clarity.