when you’re given a new mathematical statement that doesn’t seem to be immediately provable or disprovable from your axioms, how do you assign it a probability value?
50%? Sometimes you can infer its likely truth value from the conditions in which you’ve come upon it, but given that there are about as many false statements as true ones of any form, this default seems right. Then, there could be lots of syntactic heuristics that allow you to adjust this initial estimate.
“Syntactic heuristics” is a nice turn of phrase, but you could have just as well said “you can always say 50%, or maybe use some sort of clever algorithm”. Not very helpful.
I don’t expect there is any clever simple trick to it. (But I also don’t think that assigning probabilities to logical statements is a terribly useful activity, or one casting foundational light on decision theory.)
But I also don’t think that assigning probabilities to logical statements is a terribly useful activity, or one casting foundational light on decision theory.
Can you explain this a bit more? Do you have any reasons for this suspicion?
I don’t have any reasons for the suspicion that assigning probabilities to logical statements casts foundational light on decision theory. I don’t see how having such an assignment helps any.
50%? Sometimes you can infer its likely truth value from the conditions in which you’ve come upon it, but given that there are about as many false statements as true ones of any form, this default seems right. Then, there could be lots of syntactic heuristics that allow you to adjust this initial estimate.
“Syntactic heuristics” is a nice turn of phrase, but you could have just as well said “you can always say 50%, or maybe use some sort of clever algorithm”. Not very helpful.
I don’t expect there is any clever simple trick to it. (But I also don’t think that assigning probabilities to logical statements is a terribly useful activity, or one casting foundational light on decision theory.)
Can you explain this a bit more? Do you have any reasons for this suspicion?
I don’t have any reasons for the suspicion that assigning probabilities to logical statements casts foundational light on decision theory. I don’t see how having such an assignment helps any.