Hegel on Identity by Double-Negation
The law of identity in its classical form states that everything is identical (only) to itself. It does not purport to tell us what any particular thing is, but defines the sense-constraint inherent in the concept of identity of a thing that makes any thing meaningful as a thing. The qualifier ‘only’ is bracketed because it is already implied by the term ‘identical’, or self-sameness, or =; that which is not itself is necessarily not identical to itself. Every individual, by virtue of being identifiable, implies either constitutive or contextual uniqueness, which is to say, the quality of being a one that is differentiable from every other one, “for it is impossible to think of anything if we do not think of one thing; but if this is possible, one name might be assigned to this thing” (Aristotle 1006b). “To single x out is to isolate x in experience; to determine or fix upon x in particular by drawing spatio-temporal boundaries and distinguishing it in its environment from other things and unlike kinds...” (Wiggins, David. Sameness and Substance Renewed. 2001, 6) Another way, it is impossible to formulate the law of identity without already applying it, since it is the objective law of meaning itself.
The law of identity excludes everything else by implication of the term ‘identical’. We can also employ the law of non-contradiction to prove this point: https://michaelkowalik.substack.com/p/the-law-of-identity. Hegel erroneously assumed that self-identity does not imply difference from anything else, which resulted in a cascade of false inferences: “If everything is identical with itself, then it is not different, is not opposed, has no ground. Or if it is assumed that there are no two things alike, that is, that all things are different from each other, then A is not equal to A, nor is A in opposition”. (Science of Logic 11.260)
The law of identity implies that identity consists in being different from everything else.
This alternative way of formulating the law of identity employs double negation (negation of the negation), signified by the terms ‘different’ and ‘else’; that which is different from everything else is itself or the same. In eliminating this double negation and rearranging terms we would return to the classical form of the law. We can further formalise the alternative form of the law in a way that approximates Hegel's thesis of the ‘unity of identity and difference’:
Identity (of a thing) is identical to difference from everything else.
The intended sense of the term ‘identical’ is the logical relation of identity (=):
Identity (of a thing) = difference from everything else.
On this interpretation, the term “abstract identity” in Hegel signifies the identity of a thing in relation to other things, or what a thing is, as in ‘this is a table’, which is different from everything that is not a table, whereas “essential identity” is the logical relation of identity (=/is): “Essence is therefore simple self-identity. This self-identity is the immediacy of reflection. It is not that self-equality which being is, or also nothing, but a self-equality which, in producing itself as unity, does not produce itself over again, as from another, but is a pure production, from itself and in itself, essential identity. It is not, therefore, abstract identity or an identity which is the result of a relative negation preceding it, one that separates indeed what it distinguishes from it but, for the rest, leaves it existing outside it, the same after as before.” (SL 11.260)
It is crucial to note that ‘X is/= different from not-X (everything else)’ implies that ‘not-X (everything else) is/= different from X’. The not-X is logically related to X, indeed constitutively necessary to both X and not-X insofar as X exists only in relation to not-X, and vice versa. This is not a contradiction (in the sense of not-X = X) as erroneously concluded by Hegel; both forms have the same meaning as the law of identity: (X = X) <-> (X = not-not-X) <-> (not-X = not-X). Hegel was evidently not aware of the logical equivalence of these allegedly opposite forms: “each [thing] is self-unlike and contradictory in its equality with itself, and each self-identical in its difference, in its contradiction: that everything intrinsically is this movement of transition of one of these determinations to the other, and that everything is this transition because each determination is itself, within it, the opposite of itself.” (SL 11.261)”All things are in themselves contradictory.” (SL 11.286)
It could be argued that Hegel's notion of “contradiction” allegedly contained ‘in all things’ is a misnomer, not a formal contradiction, since it is ‘resolved/sublated’ at the meta-level, or what Hegel calls “ground” (SL 11.280-83), which amounts to a distinction of logical types: 1) identities of the relata; 2) identity of the relation between the relata. Another way, the relation of difference is logically distinct from the relation of identity. The difference/opposition between the relations of identity and difference is a relation of a higher order (different logical type). That relation has identity in its own right, to which the opposites belong (as parts) without being identical with the meta-relation (the whole) or with one another. Every identity is related to all other identities, by virtue of being different from them.
The Hegelian notion of the ‘unity’ of identity and difference is therefore at best trivial, applicable to everything and always, because everything is constitutively related in the realm of meaning, every word is defined by all other words, every object is related to all other objects in space and time. In this sense, Hegelian ‘unity’ is just the property of being in the same world; it neither complements nor transcends the law of identity. Otherwise, it amounts to a negation of all meaning.
ADDENDUM
An alternative formulation of the law of non-contradiction
The classical formulations of the law are:
1) 'a proposition cannot be true and false at the same time and in the same respect'
2) 'the same attribute cannot at the same time belong and not belong to the same subject in the same respect’
The first version is grounded in the concept of truth, which can only be defined by appealing to consistency with all other true propositions, anchored in the evolved, existentially integrated meanings with which we describe our physical reality and causal relations.
The second definition is grounded in what consciousness can meaningfully attribute to the same identity: it is impossible to mentally integrate a ‘thing’ as a definite something without this law.
I propose a third definition of the law of non-contradiction:
3) Mutually exclusive meanings cannot be expressed in the same thought, nor can they be simultaneously intended for action.
It is impossible to think a thought that both affirms and denies the same meaning, or, a hypothetical thought that affirms and denies the same meaning has no meaning, therefore is not a thought but two successive or independent thoughts of which one must be denied for the other to be meaningful.