Less Wrong has a number of participants who endorse the idea of assigning probability values to beliefs. Less Wrong also seems to have a number of participants who broadly fall into the "New Atheist" group, many of the members of which insist that there is an important semantic distinction to be made between "lack of belief in God" and "belief that God does not exist."
I'm not sure how to translate this distinction into probabilistic terms, assuming it is possible to do so-- it is a basic theorem in standard probability theory (e.g. starting from the Kolmogorov axioms) that P(X) + P(not(X)) = 1 for any event X. In particular, if you take "lack of belief in God" to mean that you assign a value very close to 0 for P("God exists"), then you must assign a value very close to 1 for P(not("God exists")). I would have thought (perhaps naively) that not("God exists") and "God does not exist" are equivalent, and that what it means to say that you believe in some proposition X is that you assign it a probability that is close to 1 (though not exactly 1, if you're following the advice to never assign probabilities of exactly 0 or 1 to anything). This would imply that that a lack of belief in God implies a belief that God does not exist.
Am I misunderstanding something about translating these statements into probabilistic language? Or am I just wrong to think that there are people who simultaneously endorse both assigning probabilities to beliefs and the distinction between "lack of belief that God exists" and "belief that God does not exist?"
Less Wrong has a number of participants who endorse the idea of assigning probability values to beliefs. Less Wrong also seems to have a number of participants who broadly fall into the "New Atheist" group, many of the members of which insist that there is an important semantic distinction to be made between "lack of belief in God" and "belief that God does not exist."
I'm not sure how to translate this distinction into probabilistic terms, assuming it is possible to do so-- it is a basic theorem in standard probability theory (e.g. starting from the Kolmogorov axioms) that P(X) + P(not(X)) = 1 for any event X. In particular, if you take "lack of belief in God" to mean that you assign a value very close to 0 for P("God exists"), then you must assign a value very close to 1 for P(not("God exists")). I would have thought (perhaps naively) that not("God exists") and "God does not exist" are equivalent, and that what it means to say that you believe in some proposition X is that you assign it a probability that is close to 1 (though not exactly 1, if you're following the advice to never assign probabilities of exactly 0 or 1 to anything). This would imply that that a lack of belief in God implies a belief that God does not exist.
Am I misunderstanding something about translating these statements into probabilistic language? Or am I just wrong to think that there are people who simultaneously endorse both assigning probabilities to beliefs and the distinction between "lack of belief that God exists" and "belief that God does not exist?"