If-then logic can be tricky. So hopefully this is an explanation to clear it up. Terminology here is that both p (premise) and q (conclusion) are statements that are either true or false, the arrow (->) can be read as implies or as a whole statement ‘If p, then q.’ Here is the truth table for if then logic:
|p||q||p -> q|
I’m going to use these statements for examples of p and q.
p: “The weather is clear”
q: “I will walk to school”
If the weather is clear, then I will walk to school.
Good weather and walking go together p->q is true.
If the weather is clear, then I will not walk to school.
The weather being clear isn’t a good reason not to walk so p->q is false.
If the weather is not clear, then I will not walk to school.
For most people this is a given truth as not clear usually means rain, and who would walk to school in the rain. p->q is true.
If the weather is not clear, then I will walk to school.
Here’s where people get tripped up. How can the weather not be clear and still walk to school? It’s cloudy. There’s a storm coming, but not here yet. The person absolutely must be at school whether rain or shine.
This brings up 2 very important conclusions. First, since a false premise (p) always leads to a true statement, using one for reasoning most likely brings no clarity to the problem. Second, a true conclusion (q) always leads to a true statement, whether the premise is actually true or not.