What is logic in the context of Classical Logic?

Question

How mathematicians define the concept of logic for the purposes of Classical Logic?

Also, how do philosophers and mathematicians at different period in history defined the word "logic", if they did?

What is logic for academics?

EDIT

I put here various claims about Classical Logic which can be found on this forum:

The convention about the material conditional is one of the things that makes classical logic classical.

"By the convention about the material conditional, when the premise is false the conditional is true." It is not merely a convention; it is a provable fact of classical logic.

What is generally now called Classical logic is the framework that is thought to encapsulate both the Predicate calculus formalism and the compositional aspect that a structured, two-valued algebra of logic gave us in the propositional case.

LEM (Law of Excluded Middle) (...) is central to classical logic

in Classical Propositional Logic, True and False are understood as forming a two-valued Boolean Algebra

Also, Wikipedia says that Classical logic is also called “standard logic” and that it is the one formal system which is the most studied and used. It gives its defining characteristics as follows:

1. Law of excluded middle (LEM)
2. Double-negation elimination
3. Law of non-contradiction (LNC)
4. Principle of Explosion (PoE)
5. Monotonicity of entailment (MoE)
6. Idempotency of entailment (IoE)
7. Commutativity of conjunction
8. De Morgan’s Laws

MoE says that if a sentence follows from a set of sentences, then it also follows from any superset of these sentences. A corollary is that a valid argument cannot become invalid by the addition of new premises.

IoE says that if a consequence follows, it also follows when any number of instances of an existing premise are added.

Answer 1

This question is so broad, you would need a lengthy article to do it justice. I can offer some comments, but by keeping it fairly short, some knowledgeable readers will squirm at the oversimplification.

For Plato, logic was the study of Logos, or Reason with a capital R. Aristotle codified rules of syllogistic logic and Diodorus and others formulated stoic logic. This emphasis on rules and forms shifted the understanding of logic in the direction of being substantially formal. But formal or not, logic for the ancients was concerned with distinguishing good arguments from bad arguments, with the implication that one ought to strive to reason well and avoid error. As such, logic was closely linked to rhetoric and dialectic. In terms of how we think of such matters today, it included a lot of what we would now call epistemology.

The medieval logicians mostly followed Aristotle's formal approach and continued to develop the syllogistic and stoic logics. Then with Descartes, things took a turn. Logic came to be seen as intimately linked to how we think. According to Descartes, we have innate ideas, e.g. of God, of mind, and of body, and what we perceive clearly and distinctly must be true. Logic here is concerned with what must be true because of our innate ideas and our perceptions.

This rationalistic approach persisted until the mid 19th century, with the possible exception of Leibniz who held a rather more modern understanding of logic and language. Kant described logic as "the absolutely necessary rules of thought without which there can be no employment whatsoever of the understanding" (Critique of Pure Reason, A52/B76). Boole used the expression "Laws of Thought" in his book on logic.

Frege and others disagreed with this psychologistic understanding of logic and returned to a more Aristotelian and scholastic study of the formal relationships between propositions. This is sometimes expressed by saying that logic is concerned with the laws of truth, not the laws of thought. How human beings think and reason is the subject matter of cognitive psychology; logic is the study of what follows from what. This has remained the dominant view within analytical philosophy and within mathematics to the present day. If you pick up a textbook of logic you can expect it to follow this understanding.

That said, it is fair to say that some philosophers of logic are willing to make room for some cognitive elements within our understanding of logic, and also there has been at least one project to attempt to model how ordinary human reasoning works. (Dov Gabbay and John Woods, "The New Logic", Logic Journal of the IGPL, 2001, Vol. 9, No. 2, pp. 141-174.)

As to 'classical logic', Frege's logic is called classical because it retains several core features of ancient logics. It includes bivalence, the law of non-contradiction, the law of excluded middle and the law of identity. It is characterised by other properties too, including the reflexivity, transitivity, idempotency and monotonicity of entailment; all four forms of de Morgan's rules; and structural rules such as contraction, weakening and permutation. It differs from ancient logics in other respects, including existential import. There are many non-classical logics, and in recent years there has been increasing support for the idea of logical pluralism, i.e. that there is no single uniquely correct logic.

Answer 2

How mathematicians define the concept of logic for the purposes of Classical Logic?

Mathematical logic has been, for the most part, coextensive with "classical logic" in that it is inconsistency intolerant i.e. it has no room whatsoever for contradictions (the method of proof reductio ad absurdum is fully legit in mathematics). However, this has resulted in some inverse consequences for mathematics (re Kurt Gödel's Incompleteness Theorems, which to my reckoning is a reductio ad absurdum). That said, there's Fuzzy Logic, formalized mathematically, I'm told, using interestingly named fuzzy sets.

Also, how do philosophers and mathematicians at different period in history defined the word "logic", if they did?

Philosophers, included in that category are logicians, are generally more experimental in their approach, tinkering around with the laws/principles/etc. of classical logic. The motivation for that ranges from simple curiosity to absolute necessity I suppose. Paraconsistent logic, Dialetheism are some of the results. The takeaway? For one, Inconsistency tolerant logics and more realistic logics (e.g. those that have unknown as a "truth" value).

What is logic for academics?

For academics, classical logic is the mainstay. As advertised, classical logic offers a system of thinking which takes you from some truths (true premises) to further truths (true conclusions).

Food for thought: Logical Nihilism (in a nutshell, briefly, there is no logic)