Logic is a fascinating study in its own right, no less so for a student of language. Indeed, I think the relationship between language and logic is an especially interesting topic. But studying logic is a bit like learning how to ride a bike for the first time, even though reasoning itself is something that comes quite naturally to us.
There are some pitfalls which you should know about from the outset:
Fundamentally, logic is about what follows from what. When we use logic to evaluate arguments, the real issue is not the content of the argument in question, but simply its form.
Any argument of the following form is considered valid:
(1) If P, then Q.
(3) Therefore, Q.
There is no way that you can have the premises (1) and (2) be true but the conclusion (3) false.
Here is an example of such an argument:
(1) If Newton was a physicist, then he was a scientist.
(2) Newton was a physicist.
(3) Therefore, Newton was a scientist.
Since the premises happen to be true, the argument is also sound. But now consider the following:
(1) If Chardonnay is a car, then Paris is the capital of Spain.
(2) Chardonnay is a car.
(3) Therefore, Paris is the capital of Spain.
Something must have gone wrong, because the conclusion is obviously wrong. Paris is the capital of France, not Spain. And quite clearly, Chardonnay is a wine rather than a car. But not only is the argument valid (because it has exactly the same form as the previous argument), but the first premise is true! How can this be?
It’s important here to understand something about how conditional sentences work in logic. A statement like “if P, then Q” is only false if the antecedent (P) is true but the consequent (Q) is false. Since they are both false, there is no way for that to be the case. This is known as material implication.
Another odd thing is that you can infer anything from a contradiction (known in technical terms as the principle of explosion).
For example (note that “or” here is inclusive, not exclusive):
(1) Water is wet and water is not wet. (Assumption)
(2) Water is wet, from (1) by conjunction elimination.
(3) Water is wet or Elvis is alive, from (2) by disjunction introduction.
(4) But water is not wet, from (1) by conjunction elimination.
(5) Therefore, Elvis is alive, from (3) and (4) by disjunctive syllogism.
If it’s any consolation (I’m sure it isn’t), these things get less confusing with practice.