Heraclitus famously believed in the equality of opposites, as do I. Would the truth of the equality of opposites have any significant implications for reason and logic?
Did Heraclitus believe in the identity of opposites? I do not think so. He is popularly quoted as having said,
No man ever steps in the same river twice.
But what he actually said is quite different,
We both step and do not step in the same rivers. We are and are not.
This points to a very different point of view; it is not the case that opposites are identical, but that opposites coincide. This isn't contradictory with the law of identity, which is also often misstated as:
Nothing can both be and not be.
Of course, if that misapprehension was true, nobody would ever die, for if so, somebody would be alive, and then not be alive. But the actual text is,
one cannot say of something that it is and that it is not in the same relation and at the same time.
So, yes, we can enter the same river - in the relation in which the river is the same, regardless of what individual atoms compose it - and never enter the same river - in the relation in which the individual atoms of the "same" river never flow through in the same order and position, if ever.
If so, the opposites
- We often step into the same river.
- We never step into the same river.
are both true, but they are not identical: they are true concerning different relations. "River" means different things in each sentence, different meanings that are fused in most ordinary uses of the word.
Heraclitus is known for his doctrine of flux, which is a special case of the unity of opposites, and is symbolically or otherwise suggested in his ontology of fire (this ontology was quoted approvingly by Heisenberg).
Three fragments are known to support this view:
B61: Sea is the purest and the most polluted water; for fish, drinkable and healthy; for men, undrinkable and unhealthy
B10: Collections, wholes and not wholes; brought together, and pulled apart; sung in unison, sung apart; from all things, one; from one, all things.
B88: As the same thing in us is living and dead, waking and sleeping, young and old; for these things having changed around us are those, and those in turn having changed around are these.
One way to make sense of these oracular statements is to seek different senses, and this means looking at these statements through normal logic predicated on the law of non-contradiction.
Another way, is to take the doctrine of the unity of opposites seriously, as in:
B45: They do not understand: how that which seperates unites with itself. It is a harmony of oppositions, as in the case of the bow & lyre
And this line of thought is taken up, for example, in Hegels Logic; he begins with positing the equality of Being & Non-Being, which are opposites, but in his description are also equal and unequal - but this moment doesn't last, and is driven forward.
A suggestion of it is also in his description of the infinitesimal: it is here, and it is not; that is an infinitesimal is not equivalent to an extensionless point; this perspective is formally supported by intuitionistic logic, dual to paraconsistent logic - where an inconsistency does not render the whole logic invalid.
Going back to Antiquity, It's also in Aristotle: he posits three forms of change of things that are (which are classified in his categories); and these are motion, alteration, generation/corruption - but he takes motion to be the most exemplary form of change; this is motion in a wide sense (not the narrow sense of merely physical change), and what drives motion is contradiction; this is not logical contradiction, but the onto-logical unity of opposites; the usual notion of motion, which is the contemporary sense, A calls merely adequate.
Maybe you have something in mind along the lines of Heyting semantics, e.g., http://plato.stanford.edu/entries/proof-theoretic-semantics/ The aim, briefly, is to model not the denotations of statements, but their proofs, i.e., meaning=proofs. Then a proof of proposition P would simultaneously be a proof of .not.(opposite-P) And since, a la Heyting, meaning=proofs, P and opposite-P mean the same thing.
There are a few unmentioned subtleties, e.g., incompleteness, whereby the fact that I can construct a proof of P doesn't necesarily mean I can construct a corresponding proof of .not.(opposite-P), i.e., even though it's true, I may not be able to prove it. Both-way-proofs correspond to what's called "recursive", one-way to "recursively enumerable". So I could have a P-proof but not a corresponding .not.(opposite-P)-proof. And then they wouldn't be "equal".
Though I agree you have mistaken your source, others have clarified that more directly, and the impulse you are following also arises from other sources. So I am going to focus on the question per se, out of context.
The notion of the unity of opposites is a strange one, from the point of view of nature, because our natural intuition of opposition is not one that maps reasonably onto real things.
To 'unify opposites', first we need to unpack the notion of opposition: One list of ways in which things can seem opposed has five aspects, at five different levels of organization: complementarity, polarity, orthogonality, reciprocity, and paradox.
We think first of dichotomies, but most of those simply are misunderstandings of the nature of what we are looking at. Things are wet or dry -- but they aren't, they are also damp.
So we fall back on polarities. Surely wet and dry are ends of a continuum, polar opposites -- but they aren't, things can be damp with water or oil, soap or alcohol...
So we think we can separate these things by substance or types of substance. Surely things are saturated in each way to a different degree according to the saturating. You add more of something and the situation becomes more saturated in that kind of thing.
But they aren't. Things can be less saturated with water and with oil if you add soap. Soap makes the medium hold more of the two things by allowing them to combine. So the medium is less saturated with either of those things than they would be if you didn't add the soap. At the same time, they don't completely combine into something else with which our situation is or is not saturated to a given degree -- each sometimes makes more room for each of the others, even though they all remain in the mixture.
But even there, we have to back off and look at what we mean by saturated or filled. We mean space in the containing rag or box is occupied. We fall back on our notion of space. But we know, in the end that space itself is not a clear and reasonable concept because of things like Kant's antinomies. We can't conceive of the boundlessness of infinite space, but neither can we really conceive of a bounded space that is not just part of an unbounded space. So what on earth do we mean by the idea of space, is it bounded or boundless? What do we mean by full or empty if empty space naturally pulls the molecules of all of our substances out of our rag and they can just wander off into infinity over time, especially if infinity is something we can't really 'get' (while at the same time it is perfectly clear)?
So things can be opposed by being complementary, by being relative but distant, by being unrelated and thus contrasting ways of being distant, by being part of a balance between distinguished things through a higher order of relation, or even by being so constituted that the mind can only hold one of them clearly in focus at a time. Who says we haven't missed several more?
To unify any of these ways of being opposite simply shifts focus to one of the others.
At the same time, the ways in which things can be similar is mapped out in terms of this complex of ways things can be different. And we can apply the different modes of differing to these derived modes of similarity in all these ways, and probably others too subtle to make our list.
So we can't start our grand simplification process of unifying opposites. It cannot be directed or analyzed. It is too simple to be simple, even though it is clearly not complex.
Already given answers having adequately addressed various aspects of the question, I would like to offer a complementary answer.
According to Hegel (and relatives) the dialectic process can be demonstrated (thus proved) at a fundamental level. That of pure Being itself. Thus, at least according to Hegel (drawing inspiration from Heraclitus), the unity of opposites, as necessary part of the dialectic process, holds true.
So assuming that unity of opposites holds true what does this mean for logic?
In contrast to some opinions, dialectic, as used by Hegel and relatives, does not violate the law of non-contradiction (thus requiring some paraconsistent logic) because the "opposites" need not both hold in the same sense and at the same time.
It is important to understand that dialectic is a process that unfolds in time. There are dialectical "moments" for the thesis and the antithesis. They do not happen instantaneously at the same time. This is an important aspect of Hegelian dialectic. For example, the fundamental dialectic of Being unfolds in time (as Becoming).
Furthermore, the dialectic process (unity of opposites) entails the fluidity of identity. Identity now is a process, not static.
Thus the dialectic concept of unity of opposites need not violate the law of non-contradiction. In case the unity of opposites does violate the law of non-contradiction then some form of paraconsistent logic would be needed and classical logic would be a subset of that logic. See Dialectic logic.
That being said, it is important to point out that what is the "opposite" in a given case is not always clear nor unique. This has been pointed out even for Hegel's own more complex examples. So it is important to avoid simplistic analyses and dichotomies (as sometimes made by proponents of the theory of "opposites").