the validity of an argument is determined by its logical form. The logical form of a sentence (or proposition or statement or truthbearer) or set of sentences is the form obtained by abstracting from the subject matter of its content terms or by regarding the content terms as mere placeholders or blanks on a form.
This definition does not had anything to the standard notion of form. We might just as well look up the word "form" in an English dictionary.
The idea is that logic concerns the relations between terms or their referents, rather than the terms or their referent themselves. Take for example:
(C) If a = 2, then if x = a, then x > 0
This clearly doesn't say anything about a or x because it doesn't say whether a = 2, x = a and x > 0 are true or false. Yet, it does say something, and something rather interesting, and what it says is easily understood by anyone who is proficient in, in this case, English. Broadly, what conditional (C) above says only concerns the relation between the possible truth values of a = 2, x = a and x > 0. These possible truth values depend on the semantic of the language used, and therefore, crucially, on the semantic of the symbols =, >, 2 and 0. If we don't equivocate, then (C) is true, and it is true irrespective of whether a = 2 and x = a would be actually true or not. Given that we don't assume anything as to a = 2 and x = a prior to considering conditional (C), we actually don't assume that a = 2 and x = a are true. Yet, conditional (C) is undoubtedly true.
But why does it work? Why is only the form important and not the subject matter? I mean, is it a fact? Is there a proof that shows conclusively that this is the case? Or is it just something that we found to work?
It is definitely something we find works. Conditional (C) certainly works, given the semantic we use, and I don't think anyone could reasonably disagree.
The broader explanation is that logic is a cognitive capacity, somewhat like vision or memory. And these work because they are the result of natural selection and of the evolution of species. If human logic didn't work, our ancestors presumably wouldn't have bene able to survive, prosper and reproduce, and then we would not be here to talk about it. So, they did survive, which suggests human logic works fine. That is, logic is a selective advantage, it is an adaptation to our environment, and we inherited it from our ancestors through our DNA, and our species probably inherited it from its ancestor species. This is the only scientific explanation as to why humans are logical.
Something else is the fact that we cannot seem to be able (reasonably) to fault logical truths. This is only the case because human logic is the only logic native to our own brain. We are unable to imagine any colour which would be outside the spectrum of colours we normally experience, and this even though other animals seem to experience other colours (ultraviolet). Some people also do, but they are in a tiny minority, and they themselves cannot imagine any colour outside the ones already familiar to them. The colours we perceive seem to be genetically determined. And so seems our logic.
This does not mean that there is no alternative logic, but there is certainly no alternative logic that we know of and validated by natural selection over the entire biosphere over a period of possibly 525 million years since the first proto-neuron cells, and possibly longer, broadly 2 or 3 billion years, if we go back to the first unicellular organism with something resembling a capacity to learn from experience.