Next time somebody asks you “What’s the Probability of X”, where X is any proposition, say this: there is no such probability. It doesn’t exist.
“Briggs, what’s the probability of this die coming up 6?”
It doesn’t exist.
“Briggs, and am deep sixed?”
I don’t know: there is no such probability.
This is your periodic reminder that no probability exists unconditionally. You can never have a probability without reference to some evidence, premises, supposeds, model, whatever.
That is, you must never write
but you should write
where E is the evidence brought to the problem. Change E, change the probability of X with respect to E.
Now you will see textbooks write probability the first way. Usually this is shorthand, because authors are lazy or the text appears busy when writing it properly. Do not let this fool you. Or sometimes the authors believe such probabilities exist. Frequentists believe this. They are mistaken.
You can conceive of no probability without reference to some evidence—and that evidence always includes the knowledge of the words and grammar of the proposition X. That evidence is implicitly in E, even if you fail to write it down.
This should not come as a surprise, since no proposition (in logic) is either true or false without references to the evidence used to judge is true or false. The evidence used to decide truth or falsity always includes the word knowledge and grammar, too. I emphasize this, because it often forgotten.
Now in real life if you ask me X = “What is the probability I die?” I will say it is certain. This is because I am using shorthand, as are you. The E with which we are using to judge the proposition is tacitly understood and agreed to by both of us. There is no reason to belabor or elaborate something so obvious.
This commonality is present in most mundane probability questions. But it’s only there because of shared cultural experience. If there is any doubt, the premises E must be set down explicitly.
This is why (as we have often discussed) there is no probability of being struck by lightning, or having a car accident, or even of winning the lottery. No probability exists. Probability is always a deduction from accepted evidence.
That includes all scientific probabilities, too. Including all the medical trials reported in the news. Nobody has a probability of cancer, or even of health. The reason is there are no such things, because there are no referents. Always insist on one!
Update I’m elevating a comment to the main post, because it highlights a common and devastating error.
A commenter below wrote “If P is a proposition, then the proposition P AND ~P is false, always, not matter the ‘evidence’ or anything else.”
This is false, and easily seen to be false as long as you keep in mind the cautions above. Every proposition that is evaluated is conditional on stated and unstated or implicit conditions, some of which include the word or symbol definition and grammar. This cannot be stressed highly enough, for even after it is said, it is forgotten, as this commenter proves.
For instance, is this proposition (enclosed in quotes) true or false?
“♠ ⌈ ‰”
You can’t tell. (The probability it is true given these unknown symbols, as proven in this award-eligible peer-reviewed book is “(0, 1)”.) Why? Because you have no idea what the symbols mean, nor how to manipulate them (the grammar).
But if I tell you ♠ = P, where P is a proposition, and ⌈ = AND (logical) and ‰ = not-P (but only when preceded by ♠), then we are back to “P AND ~P”, which we still can’t say is true or false until we recognize the implicit unspoken assumed premises; i.e.
Pr(P AND ~P | what I know about logic, the symbols P, etc.) = 0.
Proving, as claimed, that no proposition stands alone. Just because you’re not writing the right hand side down, doesn’t mean it isn’t there. Recall, too, probability is a matter of epistemology, and not causality. We are not saying why any proposition is true, but how we know it is.