I can reason as follows: There is 0.5 chance that it is Heads. Let P represent the actual, unknown, state of the outcome of the toss (Heads or Tails); and let Q represent the other state. If Q, then anything follows. For example, Q implies that I will win $1 billion. Therefore the value of this bet is at least $500,000,000, which is 0.5 * $1,000,000, and I should be willing to pay that much to take the bet.
This doesn’t go through, what you have are two separate propositions “H → (T → [insert absurdity here]” and “T → (H → [insert absurdity here]” [1], and actually deriving a contradiction from the consequent requires proving which antecedent obtains, which you can’t do since neither is a theorem.
The distinction then with logical uncertainty is supposedly that you do already have a proof of the analogue of H or T, so you can derive the consequent that either H or T derives a contradiction
You don’t really have these either, unless you can prove NOT(H AND T) i.e. can you definitively rule out a coin landing both heads and tails? But that’s kinda pedantic
Thank you for addressing specifically the example I raised!
This doesn’t go through, what you have are two separate propositions “H → (T → [insert absurdity here]” and “T → (H → [insert absurdity here]” [1], and actually deriving a contradiction from the consequent requires proving which antecedent obtains, which you can’t do since neither is a theorem.
So what changes if H and T are theorems? Let O mean “googolth digit of pi is odd” and E mean “googolth digit of pi is even”. I have two separate propositions:
O → ( E → Absurdity )
E → (O → Absurdity)
Now its possible to prove either E or O. How does it allow me to derive a contradition?
This doesn’t go through, what you have are two separate propositions “H → (T → [insert absurdity here]” and “T → (H → [insert absurdity here]” [1], and actually deriving a contradiction from the consequent requires proving which antecedent obtains, which you can’t do since neither is a theorem.
The distinction then with logical uncertainty is supposedly that you do already have a proof of the analogue of H or T, so you can derive the consequent that either H or T derives a contradiction
You don’t really have these either, unless you can prove NOT(H AND T) i.e. can you definitively rule out a coin landing both heads and tails? But that’s kinda pedantic
Thank you for addressing specifically the example I raised!
So what changes if H and T are theorems? Let O mean “googolth digit of pi is odd” and E mean “googolth digit of pi is even”. I have two separate propositions:
O → ( E → Absurdity )
E → (O → Absurdity)
Now its possible to prove either E or O. How does it allow me to derive a contradition?