This might be a weird question but I was trying to distinguish the difference between an implication and modus ponens. I think the distinction is clear in my head (but I have something missing), modus ponens is just a rule of inference that produces more true statement based on $A$ and $A \implies B$. An implication is a truth function that when given statements with truth values, i.e. the truth values of $A \implies B$ can be known only from the truth values of $A$ and $B$ (no other info is needed). It can be easily evaluated by looking up the truth table. These make sense. However, I reached a confusion/contradiction in the model of how I thought logic (and mathematics!) worked. It seems to me that according to modus ponens we need to know already the value of the implication $A \implies B$ before we can known if $B$ is true. However, according to the table to evaluate the truth function we need the values of both $A$ and $B$. So it seems like a chicken and egg problem. How is it actually possible to know $A \implies B$ without knowing $B$? It seems odd to me. I do understand however, how the inference is suppose to work. Since we know the output of the function (=implication) and one of its inputs, then it should be trivial, to know the other input because of the way the truth table for implication is defined. I think that part makes sense. However, what I don't understand is how in practice we are able to know $A \implies B$ is true in the first.
I think the main issue I had is that in my head what I thought is that for $A \implies B$ to be known to be true, we actually proceeded to apply rules of inference to our statements and then reached $B$. Thats I think how I thought I did maths in practice. I started with $A$ and applied valid maths rules and inference rules until I reached $B$. Thus, it seems that I never actually used the truth functional implication to do any maths, only maths facts that produced step by step another maths step until the final $B$ was produced. I assume there must be some confusion in how I thought mathematics worked, so I wanted to clarify, does someone know where my confusion is?
What does a "maths step" even mean? I thought it was modus ponens but now I realized I need to know $A \implies B$ for that to be true but that can't be true because that's what I am trying to figure out how to get the truth value of in the first place to be able to even use modus ponens.
In the end it all seems to boil down to, how do we actually conclude $A \implies B$ is true in a proof?
I always thought that we started with $A$, the just mechanically moved from $A$ to the next step and the next step until $A$ arrived at $B$ and then at that point we'd know $B$ was true. Is that not correct? I'm not sure if what I am asking is what a "step" means in mathematics. It seems like it because I would have thought intuitively that a series of steps like that must be a set of implications OR alternatively a series of steps of Modus Ponens. Regardless, it seems to me I understand what the difference between a modus ponens and implications are but I can't seem how to figure out how an implication is even known to be true in the first place without resulting in circular logic.
How does one know if $A \implies B$ (an implication) is true without knowing if $B$ (the consequent is true) is true?
my apologies I wasn't quite sure how to compress my question.
Best Answer
Consider the following statements:
$A$: Adam lives in Boston; $B$: Adam lives in Massachusetts.
You've never met Adam before, and you have no idea whether statements $A$ or $B$ are true. But you do know that the statement $A \Rightarrow B$ is true. If I now presented to you a way to prove that Adam lived in Boston, you could reasonably conclude that he lives in Massachusetts.