This post follows others about logic, such as here.
The purpose of logic is to ensure that one’s discourse makes sense. The most important part of making sense is avoiding contradictions, which would both affirm and deny a proposition. Some propositions may be affirmed in part and denied in part; that is different. The point is that if a specific proposition is fully affirmed, then one should not deny it as well because that ends up saying nothing. The only exception would be if the speaker or author changed their mind, so that the time of utterance became part of the proposition.
It is not always easy to spot a contradiction. The speaker may be unaware of it or may even be hiding it. Politicians are infamous for “talking out of both sides of their mouth”. They may say one thing to one audience, but say something contradictory to another audience. If someone doesn’t compare what they said on both occasions, people may not be aware of the contradiction. If an author has written long works or many works over the years, there may be contradictions hidden in them that a specialist would have to discover.
Another aspect of making sense is keeping on topic. If one’s speech wanders over multiple topics without having a main topic, that might be entertaining for the moment but it doesn’t make good sense. Stand-up comedians tend to do this; we don’t expect them to make sense as long as they make us laugh. A politician may talk around a topic without asserting anything significant about it; that may serve their purpose to avoid upsetting anyone but it makes little sense.
Another aspect of making sense is asserting something definite. If one’s speech is full of ambiguity, the audience does not know what they speaker is trying to say. Many words have multiple meanings, and if the speaker does not ensure the audience knows which meaning is intended, their speech is ambiguous. Poets tend to do this; we expect ambiguity because it leads to multiple meanings. But prose that lacks specificity doesn’t make specific assertions, and so doesn’t make much sense.
Logic begins with the making of a distinction. An elementary distinction is what I call a unary distinction, which is a one-way distinction. That is, one thing is distinguished from another but not vice versa. For example, out of all the animals, one may distinguish human beings. That distinction does not deny that human beings are animals; it only indicates a particular kind of animal. One could make another unary distinction that distinguishes non-human animals from human animals. These two unary distinctions together would make a binary distinction. We tend to make binary distinctions so that distinctions have two sides, but it is good to remember that they are made up of two unary distinctions.
Once distinctions are made, then one’s discourse makes assertions by indicating sides of distinctions. The conclusion should follow deductively from the assertions made. There is a whole calculus here that has been formalized by Aristotle, Boole, Frege, and others. I’ve written about them before; see here and here.