As for your comments on Parmenides, well I think this is where we get into a chain of infinite regression. We are bound to find echoes of all kinds of things in the presocratics, and further echoes probably exist before them. But if you do a web search right now for the Principle of Non Contradiction which has many formulations in the Aristotelian corpus, even beyond those offered in the Metaphysics, you'll find that the principle has been attributed by scholars and classicists to Aristotle.
Do that same web search but throw in the word "Parmenides" and you'll find a number of folks who attribute the principle to him as well. This was all a side note anyway. Parmenides poem is quite obscure and I don't think I'd get much out of it unless I read a lot of secondary literature on it. (The only thing I really have read was a paper by Anscombe which was interesting to say that least.) I will agree that Aristotle gave a pretty clear formulation of it.
Again, it would be unusual for him to use the kinds of operators we would typically use today, i.e., -(p&-p) etc But these operators often simply reflect what is in the surface or deeper meaning of our utterances. Aristotle was obviously dealing with principles in long hand, but many of them 'convert'...look up some of the many other formulations available. A number of times he says the linguistic equivalent of it cannot be the case that p and not p are simultaneously true
I certainly don't have a problem with the operators having "deeper meanings" but then we're not dealing with them "formally". There isn't a whole lot of "deep meaning" in the formalization.
Descartes' arguments in his Meditations in first philosophy can pretty easily be reduced down to a series of Aristotelian syllogisms.
Now that I'd like to see. You have a journal reference?
I would suggest, however, that he wasn't using it. His argument was pretty informal. The main purpose of formalization is to ensure that there aren't unstated assumptions or leaps of reasoning. That was Hilbert's program for mathematics.
I had heard that someone actually did a dissertation codifying Hegel's book on logic into contemporary predicate logic. I actually haven't read the book, but the professor who brought it up suggested it wasn't any good.
I've heard the same of Spinoza's Ethics but there are far too many leaps in reasoning in that piece...
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