I would propose that a logic is anything which can be modelled with a piece and parcel of brain.
I will show, surprisingly enough, that this is a further step in Cartesian Logic.
At first sight, it may look as if I were answering a riddle, by further mysteries. Indeed, but with mysteries which can be subjected to experimental inquiry (now or tomorrow).
What is a brain? A type of Quantum Computer! And what is Computing, and the Quantum? Well, works in progress. There is something called Quantum Logic, but it does not necessarily defines the world, as exactly what Quantum Physics is, is still obscure.
In practice? Logic is what works, a set of rules to go from a set A of statements to a set B of statements.
In this perspective, Medieval logic did not decline. Instead it transmutated into mathematics.
An article in Aeon, “The Rise And Fall And Rise Of Logic”,
Reflects on the importance on the history of the notion of logic:
“Reflecting on the history of logic forces us to reflect on what it means to be a reasonable cognitive agent, to think properly. Is it to engage in discussions with others? Is it to think for ourselves? Is it to perform calculations?
In the Critique of Pure Reason (1781), Immanuel Kant stated that no progress in logic had been made since Aristotle. He therefore concludes that the logic of his time had reached the point of completion. There was no more work to be done. Two hundred years later, after the astonishing developments in the 19th and 20th centuries, with the mathematisation of logic at the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Arabic logic is.)”
The old racist Prussian, Kant, a fascist, enslaving cog in the imperial machine turned false philosopher was unsurprisingly incorrect.
The author of the referenced article, Catarina Dutilh Novaes, is professor of philosophy and the Rosalind Franklin fellow in the Department of Theoretical Philosophy at the University of Groningen in the Netherlands. Her work focuses on the philosophy of logic and mathematics, and she is broadly interested in philosophy of mind and science. Her latest book is The Cambridge Companion to Medieval Logic (2016).
She attributes the decline of logic, in the post-medieval period known as the Renaissance and the Enlightenment, to the rise of printed books, self-study and the independent thinker. She rolls out Descartes, and his break from formal logic:
Catarina writes: “Another reason logic gradually lost its prominence in the modern period was the abandonment of predominantly dialectical modes of intellectual enquiry. A passage by René Descartes – yes, the fellow who built a whole philosophical system while sitting on his own by the fireplace in a dressing gown – represents this shift in a particularly poignant way.”
Speaking of how the education of a young pupil should proceed, in Principles of Philosophy (1644) René Descartes writes:
After that, he should study logic. I do not mean the logic of the Schools, for this is strictly speaking nothing but a dialectic which teaches ways of expounding to others what one already knows or even of holding forth without judgment about things one does not know. Such logic corrupts good sense rather than increasing it. I mean instead the kind of logic which teaches us to direct our reason with a view to discovering the truths of which we are ignorant.
Catarina adds: “Descartes hits the nail on the head when he claims that the logic of the Schools (scholastic logic) is not really a logic of discovery. Its chief purpose is justification and exposition.”
Instead, Descartes claims and I claim that a new sort of logic arose: Medieval Logic transmuted itself into mathematics (Descartes does not say this, but he means it). And mathematics is not really logical in the strictest sense. As it has too many rules to be strictly logical.
Buridan, a great logician who studied well the Liar Paradox (which gave the Incompleteness Theorems) had students such as (bishop) Oresme, who demonstrated what, it turned out, were the first practical theorems in calculus (more than 2 centuries before the formal invention of calculus by Fermat, and Fermat’s discovery of the Fundamental Theorem of Calculus, that integration and differentiation are inverse to each other).
For example, under the influence of Buridan and then Oresme, graphs and later equations themselves were invented. So logic became mathematics. That was blatant by the time Descartes invented Algebraic Geometry. Algebraic Geometry gave ways to deduce, to go from a set A to a set B, using a completely new method never seen before.
In turn, by the Nineteenth Century, mathematical methods contributed to old questions in Logic (the most striking being the use of Cantor Diagonalization to show incompleteness, thanks to the Liar Paradox, self-referential method.
In this spirit, not only Set Theory, naive or not, but Category Theory can be viewed as types of logic. So is, of course, computer science. Logic is whatever enables to deduce. Thus even poetry is a form of logic.
Logic is everywhere there is mental activity, and it is never complete.
If logic is just pieces of brain, then what? Well, some progress in pure logic can be made, just paying attention to how the brain works. The brain works sequentially, temporally, with local linear logics (axonal and dendritic systems). The brain tends to be deprived of contradictions (but not always, and nothing infuriates people more, than to be exposed to their own contradictions and gaps in… logic). Also all these pieces of brain, these logics, are not just temporally ordered, but finite.
As we try to use logic to look forward, as a bunch of monkeys messing up our space rock, it is important to realize that what logic is, has not been properly defined, let alone circumscribed. Indeed, if, surprise, surprise, logic has not been properly defined, let alone circumscribed, much more is logically possible than people suspect!