As far as probability theory is concerned, there is a 1⁄2 chance that the coin came up heads, and conditioned on Beauty being interviewed the probability is still 1⁄2.
Here we see an illustration of OP’s argument that stating things in terms of probability can be ambiguous. But this ambiguity could be cleared up by instead saying either,
“As far as probability theory is concerned, there is a 1⁄2 chance that the coin came up heads, and conditioned on Beauty being interviewed at least once during the week, the probability is still 1⁄2.”
or,
“As far as probability theory is concerned, there is a 1⁄2 chance that the coin came up heads; but conditioned on the observation that today is either Monday or Tuesday, and Beauty is being interviewed, the probability is 2⁄3.”
Or, a commentator could be equally ambiguous using the language of wagers by saying,
“Beauty will be asked whether the coin came up heads or tails, and given $1 if she is correct.”
Using the language of probability does not make one suddenly unable to speak clearly; and using the language of wagers doesn’t guarantee precision.
Maybe the language of probability makes it easier to shoot yourself in the foot, but that doesn’t mean there’s something nonsensical about it.
Here we see an illustration of OP’s argument that stating things in terms of probability can be ambiguous. But this ambiguity could be cleared up by instead saying either,
“As far as probability theory is concerned, there is a 1⁄2 chance that the coin came up heads, and conditioned on Beauty being interviewed at least once during the week, the probability is still 1⁄2.”
or,
“As far as probability theory is concerned, there is a 1⁄2 chance that the coin came up heads; but conditioned on the observation that today is either Monday or Tuesday, and Beauty is being interviewed, the probability is 2⁄3.”
Or, a commentator could be equally ambiguous using the language of wagers by saying,
“Beauty will be asked whether the coin came up heads or tails, and given $1 if she is correct.”
Using the language of probability does not make one suddenly unable to speak clearly; and using the language of wagers doesn’t guarantee precision.
Maybe the language of probability makes it easier to shoot yourself in the foot, but that doesn’t mean there’s something nonsensical about it.