Formal logic, which was logic prior to Hegel, saw its field of study as restricted to the laws by means of which the truth of one proposition followed from that of another. Ilyenkov explains in his essay on Hegel, that Hegel's revolution in logic was effected by widening the scope of logic and the field of observation upon which the validity of logic could be tested, from logic manifested in the articulation of propositions to the manifestation of logic in all aspects of human practice.
Formal logic also put outside of its scope, the proof of primary or axiomatic truths or the derivation of the categories by which means of which propositions indicated reality. Hegel also broadened logic to include critique or derivation of these categories.
Either way, logic is concerned only with truth, that is, with thinking which corresponds to or reflects the world outside of thought, outside of individual consciousness, and further, the criterion of truth for logic is the extent to which it provides an adequate guide to practice.
Just as thought reflects the material world and can contain nothing that does not already exist in the material world, or at least the conditions for its formation, human practice is practice of material human beings in the material world, and there can be nothing in human practice which fundamentally contradicts Nature.
Thus, in elaborating the most general laws governing the development of the social practice, Hegel necessarily uncovered laws which are not unique or special to the human condition, but are objective, material laws of Nature. So, in looking not just at what thought thought of itself, but at what it did, Hegel not only and not so much discovered a far richer means of understanding individual scientific consciousness, but more importantly, the laws governing the development of collective spritual, cultural or social activity and of the material world in general.
Hegel did not disprove or eradicate formal logic at all, he merely defined its immanent limits and uncovered its inner contradictions, its origin and its own limits, beyond which it necessarily passed over into something else, its life and its death; he negated it; he sublated it:
To sublate [aufhaben], and the sublated (that which exists ideally as a moment) constitute one of the most important notions in philosophy. It is a fundamental determination which repeatedly occurs throughout the whole of philosophy, the meaning of which is to be clearly grasped and especially distinguished from nothing. What is sublated is not thereby reduced to nothing. ...
'To sublate' has a twofold meaning in the language: on the one hand it means to preserve, to maintain, and equally it also means to cause to cease, to put an end to. Even 'to preserve' includes a negative element, namely, that something is removed from its influence, in order to preserve it. Thus what is sublated is at the same time preserved; it has only lost its immediacy but is not on that account annihilated. [Science of Logic, Book I, Section One, Chapter 1, 2 Moments of Becoming]
Formal logic is at its most powerful, not at all when it is treated as something of minor value and little use, but on the contrary, when it is utilised with the maximum consistency and thoroughness, but with consciousness of its immanent limits and an understanding of when and how it supersedes itself. Nothing is more valueless than uncritical playing with logical contradiction and inconsistency justified by thoughtless and shallow reference to dialectics.
Although I think George Novack is completely wrong in his treatment of formal logic in The Logic of Marxism, his basic initial proposition on the validity of formal logic is profoundly correct:
What characteristics of material reality are reflected and conceptually reproduced in these formal laws of thought? The law of identity formulates the material fact that definite things, and traits of things, persist and maintain recognisable similarity amidst all their phenomenal changes. Wherever essential continuity exists in reality, the law of identity is applicable.
We could neither act nor think correctly without consciously or unconsciously obeying this law. If we couldn't recognise ourselves as the same person from moment to moment and from day to day - and there are people who cannot, who through amnesia or some other mental disturbance have lost their consciousness of self-identity - we would be lost. But the law of identity is no less valid for the rest of the universe than for human consciousness. It applies every day and everywhere to social life. If we couldn't recognise the same piece of metal through all its various operations, we couldn't get very far with production. If a farmer couldn't follow the corn he sows from the seed to the ear and then on to the meal, agriculture would be impossible.
The infant takes a great step forward in understanding the nature of the world when he grasps for the first time the fact that the mother who feeds him remains the same person throughout various acts of feeding. The recognition of this truth is nothing but a particular instance of the recognition of the law of identity. [The Logic of Marxism, Lecture 1, Part 4]
Thus, (continuing our theme of approaching Hegel's Logic from the point of view of a theory of cognition) perception begins when we we recognise something, when we perceive something as persistent in the stream of "one damn thing after another" of immediate perception, when we can say "A = A". The whole of formal logic rests on this identity of a thing with itself, with recognition of the continuity of something. The whole of formal logic falls to pieces when "A not = A".
It is manifestly obvious that identity is an abstraction, and:
If it be the office of comparison to reduce existing differences to Identity, the science which most perfectly fulfils that end is mathematics. The reason of that is that quantitative difference is only the difference which is quite external. ... (Shorter Logic, § 117n) ... If quantity is not reached through the action of thought, but taken uncritically from our generalised image of it, we are liable to exaggerate the range of its validity, or even to raise it to the height of an absolute category. And that such a danger is real, we see when the title of exact science is restricted to those sciences the objects of which can be submitted to mathematical calculation. (Shorter Logic, § 99n)
But it is equally obvious that identity and quantity and mathematics are abstractions which reflect material reality, and identity is an abstraction which, as Novack points out, is the fundamental basis of human practice. While in a certain sense the world is recreated anew every minute, we act, in practice, for most of the time, on the basis first of all of continuity.
As referred to in the above quote from The Shorter Logic, the science of Identity is Mathematics. Mathematics is the science in which formal logic is applied in a specific sense, and in this special sense, is adhered to inflexibly and with unquestionable heuristic value.
In mathematics proper, it should be remembered, "A" indicates absolutely anything; it is quite meaningless (Being = Nothing). For mathematics, in the proposition "A = A" the operative symbol is the "=". The proposition becomes an empty tautology or obvious falsehood only when it is interpreted in a non-mathematical way, for example as meaning "this A is the same as that A".
This is not at all to say that dialectics is absent from mathematics. Unthinkable. The movement from one proposition to another is alway dialectical and only sometimes and in a certain respect formal. But without formal logic, there is no mathematics. While it is nonsense to elevate mathematics to be a model for all sciences (as was common in past centuries), it is even more nonsensical to devalue mathematics as a science.
But formal logic is not at all limited to mathematics. The first condition for the validity or relevance of formal logic is the relative validity of identity in respect of the precise movement of cognition in question.
In practice we regard the world as not only subject to change, but subject to our change. In scientific thinking we regard objects critically, as subject to analysis and synthesis. To the extent that we regard an object critically, regard reality as something to be changed, then we specifically reject the law of identity, and assert that "A not = A", and formal logic takes a break. Here we cannot grasp an "A" with the aim of carrying it forward to use in other relations, but aim to revolutionise it and uncover from its clothes a new A, A-, if you like.
The Law of Excluded Middle states that if a proposition A is not true then its denial "not-A" is true. Even within the narrow limits of formal logic this "law" is unreliable, and common sense will confirm the view that this line of reasoning is unreliable. The Dutch logician Bruuwer reconstructed mathematical logic by eliminating the law of Excluded Middle from the rule book, and showed that mathematics is little the worse for the loss.
The Law of Non-Contradiction states that both a proposition, A, and its denial, not-A, cannot be true within the domain of a single "theory", within the domain of validity of the law of identity, "A = A". This law is indeed fundamental to formal logic.
It is well known that the consistent application of the basic set of formal logical principles leads to "antimonies", or flat contradictions. This discovery contributed to Hegel's revolution in logic, but also led to further development within formal logic. Nowadays, the conditions which give rise to such contradictions are well known, and formal logic is able to proceed while reliably avoiding such "bad" contradictions by the introduction of a number of proscriptions on the categories.
At first glance, it would seem that an explanation of these limitations would be of great significance, but personally I think that it is of genuine interest only to professional logicians or pedants.
In so far as the Law of Sufficient Ground may be said to exist in formal logic, it is the so-called law of decibability, that any proposition which is valid within a given theory, may be proved or disproved. In 1931, Kurt Gödel disproved this thesis in his famous Gödel's Theorem. This discovery brought about a huge crisis in the world of mathematics and logic, but it can hardly be said that it reduced formal logic to a nullity.
The other component of formal logic is the syllogism, which Hegel subjects to criticism in the Doctrine of the Notion. Hegel's critique is very profound. In the first place, his understanding of the Notion is fundamentally at odds with that of formal logic. In the second place, his Notion of the Notion demonstrates in practice this more profound approach and provides the archtypal demonstration of his method of deriving a concept from its own immanent nature rather than by external definition as an abstract universal. In the third place, he anticipates the materialist critique of logic by demonstrating that the syllogism and its categories of universal, particular and individual are "forms of the notion, the vital spirit of the actual world".
The Logic of the Notion is usually treated as a science of form only, and understood to deal with the form of notion, judgement, and syllogism as form, without in the least touching the question whether anything is true. The answer to that question is supposed to depend on the content only. .... On the contrary they really are, as forms of the notion, the vital spirit of the actual world. That only is true of the actual which is true in virtue of these forms, through them and in them. As yet, however, the truth of these forms has never been considered or examined on their own account any more than their necessary interconnection. ...
The Notion as Notion contains the three following 'moments' or functional parts. (1) The first is Universality - meaning that it is in free equality with itself in its specific character. (2) The second is Particularity - that is, the specific character, in which the universal continues serenely equal to itself. (3) The third is Individuality - meaning the reflection-into-self of the specific characters of universality and particularity; which negative self-unity has complete and original determinateness, without any loss to its self-identity or universality.
For formal logic the form of the category is considered to lie outisde its domain, with a minor exception in relation to limits which are prescribed in order to avoid antimonies. The nearest formal logic can come to conceiving of the Notion is the "class", which indicates by some effective means, individuals to which can be attributed an "abstract universal". An "abstract universal" is that property which is common to many individuals.
No wonder Hegel regards all this as simply a "yawn"! It is very trivial stuff. When elaborated to an exceedingly high level of complexity as in developed branches of mathematics it can provide a substance of some interest, but is of a very restricted domain of truth, which is the same as the domain marked out by the extent of validity of the Law of Identity, and broadly recognisable as what is known in mathematics as Set Theory, the Theory of Groups, and so on.
Classes (abstract universals) in mathematics lead a kind of "double existence", once as a "collection" of "elements" having a given property and secondly as the property which constitutes the notion of the class or set.
Again, as an indispensable, though far from exhaustive, component of mathematics, the abstract universal has its place. But in the context of creative social or natural theory that is a very small place.
But again, the same comments as above apply. In so far as "A = A", as the relations between things remain unchanged, as the things conceived of remain separate and distinct, etc., etc., the notion of abstract universal retains validity. The whole of Hegel's Logic constitutes a more profound concept of "notion", and from the standpoint of Hegel's Notion, it s easy to see how limited is formal logic's notion of Abstract Universal.
The issue is, how to understand and recognise the boundaries of formal logic.