A logic is called
classical if it has certain basic properties that are shared by standard propositional logic and the predicate calculus. Exactly which properties get focused on depends on the context, and usually what the contrast case is. Classical logics are distinguished from modal logics in being purely truth-functional, for instance, while they are distinguished from other logics by being monotonic, from some logics by accepting the law of the excluded middle, and from some logics by being bivalent (only dealing with true and false), and so forth. The irony with the name is that Aristotelian logic, which is the 'classical' logic in our usual sense of the term 'classical' is nonclassical in the modern sense, as a number of people have started pointing out. Indeed, Aristotelian logic is a sprawling thing with a complete indifference to the notion that there should be only one system of logic, and a perfect willingness to assume that the properties of classical logic, in the modern sense, fail for some domains. This is true even in Aristotle himself, since Aristotle's full logic -- that is, if you don't stick only to the strict and bare syllogistic formalism, which Aristotle himself does not -- has modal components, dialogical components, paraconsistent components, and relevance components. (That it has the first two is undeniable; the other two are somewhat more controversial, but it's not hard to find passages in Aristotle that can at least be interpreted in these ways.) This is unsurprising; building the One True Logic is utterly off of Aristotle's radar, and it is clear from a number of places that Aristotle thinks that what logical principles you should use depends entirely on what you are doing. Aristotle simply doesn't try to fit everything into formal syllogisms. Indeed, in the context of the full Organon, the formal theory of syllogisms found in the
Prior Analytics is almost a secondary matter, important as it is in its own right; Aristotle's interest in the formal theory of syllogisms appears entirely subordinate to his attempt to establish a general theory of demonstration, which itself is quite clearly not reducible to the formal theory of syllogistic validity.
This carefree pluralism is in fact endemic to the Aristotelian tradition at least up to the late Middle Ages, although different thinkers go in different directions on different topics. And again, the reason seems pragmatic: people are not interested in abstract systems simply in themselves -- they are into logic in order to do things with it.
ADDED LATER: AT
branemrys.blogspot.kr, pseudonoma adds the following, which I thought worthwhile enough to put here, since (given the current mismatch between my commenting system and Blogger's division-by-country) it would be invisible to most people:
One asterisk I would want to add to your final point, when you write
"And again, the reason seems pragmatic: people are not interested in abstract systems simply in themselves -- they are into logic in order to do things with it."
It seems to me important --if one is to achieve a fundamental clarity concerning this fine point --to stress not only the practical or propadeutic character of Aristotelian logic, but also to its origin in the Metaphysics. Aristotle is not treating of anything like "a priori formal laws" whose intrinsic systematicity is , e.g., a necessary consequence of their source in the unity of transcendental apperception. One *may* not need the Metaphysics to be convinced of the truth of Aristotle's formal logic, but he can in no wise discover this logic himself without making metaphysical assumptions (or overtly engaging in metaphysics, god forbid!). The categories are predicables second and ways of saying being first, and the principle of non-contradiction is, as Book Gamma reminds us, a metaphysical principle before it is a logical one.
