I reject 0 and 1 for non logical facts as well. ”I think therefore I am” is a logical proof of my own existence, and as such, I assign a probability of 1 to the proposition: “I exist”.
“I think therefore I am” is a logical proof of my own existence, and as such, I assign a probability of 1 to the proposition: “I exist”.
I’m not at all sure it’s any such thing. It depends a little on how broadly you’re prepared to construe “my own existence”.
You aren’t really entitled to say “I think”. You know that some thought is happening, but you don’t really know that what’s having that thought is the right sort of thing to be labelled “I” because that word carries a lot of baggage (e.g., the assumption of persistence through time) that you aren’t entitled to when all you have is the knowledge that some thinking is going on. So, for instance, if you go on—as Descartes does—to draw further inferences involving “I” from “I exist”, and if you assume at different times that you’re referring to the same “I”, then you are cheating.
Also I mentioned that even if you actually have a logical proof of something, you cannot assign a probability of 1 to the conclusion, because you might have made a mistake in the argument. You are pointing out some ways that might have happened here. Even if it did not, no one can reasonably assign a probability of 1 to the claim that they did not make such a mistake, and hence to the conclusion.
Right; sorry for not phrasing that in a way that sounded like agreement with you. We should be less that totally certain about mathematical statements in real life, but when setting up the formalism for probability, we’re “inside” math rather than outside of it; there isn’t going to be a good argument for assigning less than probability 1 to logical truths. Only bad things happen when you try.
This does change a bit when we take logical uncertainty into account, but although we understand logical uncertainty better these days, there’s not a super strong argument one way or the other in that setting—you can formulate versions of logical induction which send probabilities to zero immediately when things get ruled out, and you can also formulate versions in which probabilities rapidly approach zero once something has been logically ruled out. The version which jumps to zero is a bit better, but no big theoretical advantage comes out of it afaik. And, in some abstract sense, the version which merely rapidly approaches zero is more prepared for “mistakes” from the deductive system—it could handle a deductive system which occasionally withdrew faulty proofs.
I reject 0 and 1 for non logical facts as well.
”I think therefore I am” is a logical proof of my own existence, and as such, I assign a probability of 1 to the proposition: “I exist”.
I’m not at all sure it’s any such thing. It depends a little on how broadly you’re prepared to construe “my own existence”.
You aren’t really entitled to say “I think”. You know that some thought is happening, but you don’t really know that what’s having that thought is the right sort of thing to be labelled “I” because that word carries a lot of baggage (e.g., the assumption of persistence through time) that you aren’t entitled to when all you have is the knowledge that some thinking is going on. So, for instance, if you go on—as Descartes does—to draw further inferences involving “I” from “I exist”, and if you assume at different times that you’re referring to the same “I”, then you are cheating.
For more about this stuff, see the Stanford Encyclopedia of Philosophy.
Also I mentioned that even if you actually have a logical proof of something, you cannot assign a probability of 1 to the conclusion, because you might have made a mistake in the argument. You are pointing out some ways that might have happened here. Even if it did not, no one can reasonably assign a probability of 1 to the claim that they did not make such a mistake, and hence to the conclusion.
Right; sorry for not phrasing that in a way that sounded like agreement with you. We should be less that totally certain about mathematical statements in real life, but when setting up the formalism for probability, we’re “inside” math rather than outside of it; there isn’t going to be a good argument for assigning less than probability 1 to logical truths. Only bad things happen when you try.
This does change a bit when we take logical uncertainty into account, but although we understand logical uncertainty better these days, there’s not a super strong argument one way or the other in that setting—you can formulate versions of logical induction which send probabilities to zero immediately when things get ruled out, and you can also formulate versions in which probabilities rapidly approach zero once something has been logically ruled out. The version which jumps to zero is a bit better, but no big theoretical advantage comes out of it afaik. And, in some abstract sense, the version which merely rapidly approaches zero is more prepared for “mistakes” from the deductive system—it could handle a deductive system which occasionally withdrew faulty proofs.