I'm wondering if "If p then q" is equivalent to "p unless q" or "p or not q" in regards to philosophy?
In classical propositional logic, "if P then Q" is equivalent to "not P or Q" and to "not (P and not Q) and to "P only if Q". 'Unless' is taken to be equivalent to the inclusive 'or'. So in your two examples, "if P then Q" is not equivalent to "P unless Q" nor is it equivalent to "P or not Q".
In classical logic this kind of conditional is called 'material implication' and it is a truth function, which is to say that the truth of the conditional depends only on the truth values of P and Q. In practice, ordinary English conditionals do not always, perhaps do not often, behave like this. Conditionals usually express some kind of connection between P and Q, so their truth depends in some deeper way on what P and Q mean. Treating conditionals as truth functions runs into highly counterintuitive examples that are sometimes called the paradoxes of material implication. In reality, they are not paradoxes at all, just examples of conditionals that are not material implications.
Another point to notice is that conditionals often carry an implicature that the antecedent part is a precondition for the consequent part. For example, we hear a difference between:
- If you learn to play the cello, I'll buy you a cello.
- You'll learn to play the cello only if I buy you a cello.
- Mary will continue to love John unless he goes bald.
- John will go bald unless Mary continues to love him.
These examples are from David Sanford's book "If P then Q".
if P then Q is not equivalent to
P unless Q.
... Unless Q literally means : If not-Q then ...
In propositional logic,
P unless Q can be translated :
If not-Q then P.
Using Modus Tollens we can derive the following conditional :
If not-P then Q.
P unless Q is not equivalent to
if P then Q, but to
if not-P then Q.
Always use examples :
- You breathe unless you are dead (P unless Q)
- If you are not dead then you breathe (if not-Q then P)
- If you do not breathe then you are dead (if not-P then Q)
- Make the negation of what comes after the
unlessan antecedent of the conditional if..then.. (after if)
- Make what comes before
unlessa consequent of the conditional if..then.. (after then)
if p then q should be equivalent to q unless not p
To extend on the answer below and give an example:
p = your're alive q = you breathe.
So: If (you're alive) then (you breathe) or: (You breathe) unless not (you're alive)
Or in more common words: Your breathe unless you're not alive.
There is another interesting thing you might consider. But there's a warning: The following thoughts are true in a programming context, but mey well be false in a philosopy context. So it is up to you to decide.
In programming you can replace "if p then q" by "p and q". This is because the term is evaluated from left to right. So if p evaluates to false, then q does not need to be evaluated. Because no matter to what q would evaluate, the result would be false anyway.
Now if you would consider this as "equivalent", then you may apply all kind of boolean algebra to it. So f.ex. negate inputs and outputs and replace the and with or.
But, as already said, in philosophy this may well be different.