A=A

is untrue because the A on the left is in a different place: it appears elsewhere (being-there) and for those of us who read left to right the second A appears later.
There is a relation of equivalence, not of identity

The crucial part of the statement is then '='. The equivalence of two appearances states that they stem from the one being. To ascribe Being is to ascribe equivalence so that the sender and the receiver can be fixed in their different places, the '=' thus referring to the equivalence of the perfect message leaving the left A and arriving in the same form at the right A.

A≠A

"you can never step into the same river twice", but the river at the left A is still equivalent to its later appearing as the river at the right-hand A, except that the equivalence is under erasure. The '≠' thus indicates not only non-equivalence, but the oblique stroke of noise: a fourth party in the formula. The glitch in the equivalence emphasises the time the transmission from left to right takes, so emphasising becoming over being.

A=A is a necessary upshot of the transition from mediation to communication. Communication, which must divide the flux to secure survival, distinguishes sender from receiver (prior to subject/object, human/natural, male/female, colonist/colonised etcetera since all those distinctions require communication in order to be stated and enforced).

A=A imagines communication as perfect: left and right equal one another in the sense that they are equally informed, in this case minimally informed by the relation each enters into with the other. The content of the message '=' transmitted between the two 'A's is "the term on the left is equal to the term on the right".

'A≠A' imagines the communication as imperfect. The content of '≠' is not simply that the terms are unequal but that the message concerning their equality has been interrupted. Nonetheless, it transmits a message that connects the terms, whose content therefore is 'there is a relation between the two terms, but it is not a relation of equivalence'. (This is not startling to Lacanians, for whom the statement 'I think therefore I am' concerns two distinct and non-equivalent uses of the word 'I').

Translating '≠' as the statement of a relation other than equivalence can also be read as the statement 'the two terms are unequal'. It is impossible to tell, however, as in the master-slave dialectic, which of the two is the lesser. This is because the terms themselves are products of the initial non-identity that arises from the difference in relative location constructed by the sign of communication that divides them. This in turn is because the inequality is not inherent in either but rather in the oblique stroke which, in striking through the sign of communication between equivalents, achieves eminence over them both.

Three possible solutions:

i) the '=' is not the sign of communication but, taken in isolation, the sign of mediation, denoting the primacy of interconnection. It only becomes the sign of communication when it is placed between and divides terms. The noise of the oblique stroke that undoes the equivalence of being would then be the mark of repressed primal mediation re-asserting the pre-eminence of mediation over communication. This form of noise has environmental origins like sunspots, cosmic radiation or electrical storms. It belongs to the Real.

ii) the oblique stroke arises as an affordance of the communicative, and denotes in its most primitive form GĂ¶del's theorem: the statement 'A=A' is either incomplete or incoherent. In this case the source of the noise is the technique of mathematics. It arises from the functioning of the Symbolic and is systemic.

iii) The diagram of 'A=A' runs left to right. A second axiom is required to complete the feedback loop ('If A=B then it is also the case that B=A'). This secondary axiom however removes the temporal distinction of the original statement ('If A then A=A'), where the word 'then' or its synonyms ('it follows', for example) indicate a direction in time. If 'A≠A' reads only as 'there is a relation (other than equivalence) between left and right terms', there is no time involved. This would indicate that solution (i) is correct, and that the repressed primality of timeless flux has erased the temporality implicit in logic. It is also the case that the secondary axiom permits its own reversal, that is 'If A=A then A'. This reversal can then itself be reversed, so that the feedback mechanism introduced by the secondary axiom enters into endless recurrence. This repetition of the same then marks not the erasure of time but its measure in terms of repetitions of the same. This is abstract or clock time. Placing the '=' under erasure, by admitting the possibility that the two A's are not equal, makes it impossible to reverse the formula A≠A since one or other of these A's, we do not know which, is the greater. This inequality would could not arise from the return of the primal '=' without terms, since the terms remain, and remain distinct from one another (that is, the expression 'A≠A may be translated as ~ A=~ A, not-A equals not-A, in which case the modes of not-being-A on either side of the sign of communication might be entirely unlike one another). A≠A is then a double negation, undoing both the perfect communication of equivalence with its tendency to infinite repetition, and the timelessness of the '=' sign when it appears without terms. True, the '≠' might still be a technical glitch in the infinitely repetitive measure of time in feedback loops. However, this glitch must itself occur within the logic of time, that is as a historical event of interruption. Thus the third solution would be that what strikes obliquely through both primal flux and infinite measure is an event. Neither the perpetual motion of primal mediation nor the infinite recurrence of feedback permit events. Therefore a third mode of time emerges in the sign '≠'. This third mode of time, the time that permits events to occur, we call history.

The third solution belongs to the Imaginary: it is structured by the division of the self into the one that thinks and the one that is; by the experience of the mirror stage where the child misrecognises its ideal image in its reflection, more potent and coherent than it feels itself to be; and which enables the child to identify with other bodies beyond itself, the first building blocks of socialisation. History belongs to the social rather than the perpetual flux of the Real or the administered repetitions of the Symbolic. Non-equivalence, non-identity belong to events, and thence to the possibility of becoming, the possibility of changing the world, to potential and therefore to action.

(note added 16.10.18): syntagmatic/paradigmatic structures in logic: that 2+2=4 and 4=2+2 are not identical, though equivalent; that 4=3+1 indicates the identity of 4 with the previous statement but non-identical partition into a different set of components; thus too 4=4+0 and 4=5-1. The mediation matters.