I recently started Geometry. The class said that if a premise was true in an if statement, then the conclusion had to be true. This makes sense on a basic level. Take this for example:
If a = b and c = b, then a = c if a = 2, b = 2, and c = 2.
We have a true premise (a = b and c = b), so we conclude that the conclusion is true (a = c).
However, what if I did something like this:
If a = b and c = b, then a = c + 2 if a = 2, b = 2, and c = 2.
I still have a true premise, but my conclusion isn't true.
My question, then, is how can someone be sure that their conclusion is true? Are all basic premises and conclusions based on inductive reasoning and experience?
In summary, is there a mathematical way to ensure that a conclusion is true?
Excellent question! You have just the right mindset here of a good logician. And yes, the whole area of logic, in particular formal logic (or symbolic logic, or mathematical logic) is devoted to laying out and distinguishing between just those inferences that you can make, and those that you cannot. Or, as logicians say: those inferences that are valid versus those that are not.
There are many online resources to learn more about this (here is one, and in particular follow their page on symbolic logic), and I highly encourage you to do so, given your question. Just do a search on 'logic' or 'formal logic', and go from there. I think you'll enjoy it! And, as a bonus, you'll learn about how logic is at the foundation of computers and a good bit of artificial intelligence.
Finally, just to give you a taste, here is a proof created in the Fitch proof system that shows that you can indeed infer $a=c$ from $a=b$ and $c=b$:
The two rules used here are:
$= \ Intro$ ('Identity Introduction'): every object is identical to itself, and so we can always say that $a=a$ (or $b=b$ or ...)
$= \ Elim$ ('Identity Elimination'): If we know that two objects are identical, then whatever is true of the one, has to be true of the other thing as well. Mathemtaically/formally: if we have $a=b$, and we have some statement involving $a$, then we can substitute any of those $a$'s with $b$'s.
On the other hand, to disprove some inference, i.e. to show that you cannot infer $a = c+2$ from $a=b$ and $c=b$, all we need to do is provide a counterexample ... and you yourself provided one: just assume $a=b=c=2$: that will make the premises true, but the conclusion false, and so that tells us that that is an invalid inference.