# What would be the implications if the equality of opposites was true?

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?

• If by opposite you mean negation, then yes, the implications would involve violating the law of non-contradiction. Aug 15 '16 at 10:11
• @EliranH I'm not sure that's true because under the equality of opposites, I think contradictory statements would both be true in some opposite sense at the same time, not in the same sense. Aug 15 '16 at 11:17
• That would violate bivalence (which simply leads to a different kind of logic). In any case, you might be interested reading about Dialetheism; that entry explores ideas similar to your suggestion. Aug 15 '16 at 11:27
• Also paraconsistent logic. Aug 15 '16 at 14:05
• "Everything in my logic is indebted to Heraclitus", that would be Hegel. "The truth only is as the unity of distinct opposites... the Absolute is the unity of being and non-being. When we understand that proposition as that “Being is and yet is not,” this does not seem to make much sense... But we have another sentence that gives the meaning of the principle better. For Heraclitus says: “Everything is in a state of flux; nothing subsists nor does it ever remain the same.”" marxists.org/reference/archive/hegel/works/hp/hpheraclitus.htm Aug 15 '16 at 20:39

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.

and

• 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.

• "We both step and do not step in the same rivers. We are and are not." For me, this quote means that all things are ever-changing. If we step in the same river twice, we do not truly do so because it was a different "us" that stepped in before. And we also do not truly do so because it's a different river; the river is ever-changing. I'm not so sure he was getting at the equality of opposites with this statement though. Aug 15 '16 at 14:35
• @RobertFrost - I fear that your interpretation is only compatible with the mangled version I described in the anwser: if we never step into the same river, because the river is ever-changing, then only one side of Heraclitus' paradox is true. But it is a paradox because it makes two opposite claims, which are both true; the river is never the same, because it is ever -changing, but it is also always the same, as long as it exists - otherwise it wouldn't "be ever-changing"; there would be nothing to describe as the subject of such perpetual change. The paradox is not only true, it is necessary. Aug 15 '16 at 14:48
• @Robert "A thing is... its relationship with the remainder" sounds like Adorno's negative dialectic, "by colliding with its own boundary, unitary thought surpasses itself" plato.stanford.edu/entries/adorno/#5 Also Wittgenstein, "In order to draw a limit to thought, one would have to find both sides of the limit thinkable". For a more formal logical way of arriving at true contradictions take a look at dialetheism plato.stanford.edu/entries/dialetheism Aug 15 '16 at 21:29
• @Robert Sorry, but I see only an analogy, and it is unclear to me why thought can be analogized to such an image, or how we would go about testing it even as a hypothesis. I am also not sure what "it turns out they are the same" would mean in the thought context, or how it could turn out so. Aug 19 '16 at 2:28
• @Conifold ok thanks for discussing though and for your comments. Actually I think my analogy is more applicable to the limit of knowledge. Where I'm coming from is that for a universe to contain its own complete model then the model must contain the entire universe, so they both surround each other in the way i described. If thought is equated to knowledge then a thought only can be equal in limit to the universe, if the analogy I described above is true. Aug 19 '16 at 7:57

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".

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.

And

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.

• Intuitionism is not paraconsistent, it is consistent, but incomplete. Not everything warrants a truth-value, but conflicting established truth-values really constitute reductio-ad-absurdum.
– user9166
Aug 15 '16 at 15:30
• @jobermark:ok, I've changed it to 'dual to'. Aug 15 '16 at 15:46
• @MoziburUllah if we were to draw a boundary, however transient, between existence and nonexistence, with all things that exist on one side and all things that do not exist on the other, then we find that the same line defines both things. The boundary is like a single line, in the shape of a fractal over the entire surface of a sphere. All points on the sphere are one side or other of the line, but which is the inside the boundary and which is outside? Aug 26 '16 at 8:16
• @robert frost: I suggest you ask this as a new question...thats whats the sites for. Aug 26 '16 at 8:17
• That way you're addressing the community, rather than just me... Aug 26 '16 at 8:20

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.

• A very helpful analysis thanks, although (and this is clearly a subjective matter), I disagree that the various forms of opposite cannot be be combined into some overriding sense of "opposite" in much the same way as the function $f(x)=-\frac{1}{x}$ combines two forms of negation into a single "opposite" that can allow us to understand $-n$ and $n$ as opposites, as well as $n$ and $\frac{1}{n}$, yet preserving the notion that neither of those pairs are true opposites in the full sense of the function $-1\frac{1}{n}$. Aug 16 '16 at 12:25
• I agree there is damp in-between dry and wet, but two things must, to be different, be not the same in some property and therefore all "differences" can ultimately be reduced to some aggregation of minimal properties, which either are or are not. I'm sure the laws of physics (read nature) boil down to some giant involution. Aug 16 '16 at 12:25
• Whoops no MathJax! Aug 16 '16 at 12:26
• So you have got to the 'orthogonal' point in understanding the five layers, that means you read half of it. But the other layers of opposition apply in other times and places. You can insist that our arbitrary linguistic processes just must work out somehow, but they don't. See Quite on Natural Kinds, or something. "I am sure it all works out in the end" is just not an argument, it is an article of religious faith.
– user9166
Aug 16 '16 at 13:04
• That depends upon what you consider rigorous. The idea that a thing is the aggregation of its properties fails to be rigorous in the way required by Quine. Really do look at the paper on Natural Kinds, or the questions asked about it here. Language and reality do not fit together that well, if you want things to actually be rigorous. You already have this complex 'net' talking about similarity, and opposition just multiplies it, and adds some additional quirks like Russel's Paradox and all the other paradoxes of simple negation.
– user9166
Aug 16 '16 at 18:24