Can anybody please explain to me how "if p, then q" is logically equivalent to "q unless ~p" ? My problem is with all truth possibilities of p and q except for when both are true It seems in those situations they would not have the same logical meaning For e.g: "if p, then q" is true if p is false and q is true. However, shouldn't this possible combination render "q unless ~p" false? Feel free to use truth tables or whatever method that would help in explanation.
Usually, "P unless Q" is "symbolized as P ∨ Q. See :
- Stephen Cole Kleene, Mathematical logic (1967 - Dover ed 2002), page 64.
According to the truth-functional definition of conncetives (see truth tables), we hvae that:
P ∨ Q is equivalento to ¬P → Q.
Thus, the answer to your question is: NO, for P → Q we get: "¬P unless Q"
Now that you've edited the question, the two are indeed equivalent. Both are equivalent to "¬P or Q".
The issue with understanding "unless" is that often when we hear it in ordinary English we interpret it to mean "unless and only unless", i.e. equivalent to the exclusive or. We often do the same with "if", i.e. we 'perfect' it and understand it to mean "if and only if". For example, if I say, "if you wash my car I'll give you ten dollars", the obvious assumption is that if you don't then I won't. I haven't said I won't, and maybe I'll give you the ten dollars anyway, but the implicature is that I won't. Similarly with "unless": if I say, "unless you leave now I'll call the police", I may just call them anyway, I didn't say I wouldn't, but the implicature is that I won't if you leave.
So, in your example, "Q unless ¬P" is still true when P is false and Q is true. In classical propositional logic, "unless" is equivalent to the inclusive "or".