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Logically, when I think about p unless q I want to say that it is equivalent to q -> ~p, but the only equivalence is ~q -> p. I verified by truth table that my intuition is wrong.

An example of why I want to think this way: I will go golfing tomorrow unless it rains in my mind is equivalent to If it rains tomorrow then I will not go golfing.

Is this basically a similar case to how when we state implications in general English that we imply the biconditional, even though it is a illogic thing to do?

How can I think about this when telling myself not to follow my intuition in this case? Is the reason that q -> ~p is wrong that p unless q doesn't say anything about what will happen if q is true?

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3 Answers

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"I will go golfing tomorrow unless it rains" says what you will do tomorrow when it isn't raining. It doesn't actually say anything about what you will do when it is raining.

"If it rains tomorrow, then I will not go golfing" says what you will do when it is raining. It doesn't actually say anything about what you will do when it isn't raining.

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Regarding your golfing activity:

I will go golfing tomorow (without a doubt) unless it rains (then it is not sure that I will go golfing, but I may go).

Now the only thing you can extract from that is that if you did not go golfing, then it had to rain. (Or else you would have gone golfing).

So $p$ unless $q$ has to be understood as : $p$ is true is $q$ is false. But if $q$ is true, then maybe $p$ is true, maybe it is false.

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An example of why I want to think this way: I will go golfing tomorrow unless it rains in my mind is equivalent to If it rains tomorrow then I will not go golfing.

I will go golfing tomorrow unless it rains says that you will go golfing if it does not rain.

If it rains tomorrow then I will not go golfing says nothing about what you will(or won't) do if it does not rain.

So, clearly they are not equivalent statements. They don't convey the same information. The actual equivalent statement is:

If it does not rain tomorrow then I will go golfing

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"I will $P$ unless it $Q$" is equivalent to "If it not $Q$, then I will $P$", hence to: $\neg Q\to P$

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