Main title: The Ancient Greeks on paradox and contradiction

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Demetrios Bastiras, University of Adelaide, Department of Philosophy, MA (Logic) Postgraduate, email:

Title: The logic of Herakleitos & the Law of Non-Contradiction

Summary of contribution

First section of talk: Placing the logic of Herakleitos in context

  1. An outline of a very modern debate focusing on the support or denial of the Law of Non Contradiction

  2. This modern debate has an ancient parallel.

  3. Question: When did the denial of LNC originate in ancient Greece?

I will argue the origin of the denial of LNC can be found in the 5th century text Dissoi Logoi

Second section of the talk, short version of text below on pages 2-5

  1. I will argue against a modern claim that Herakleitos is the first to deny LNC.

Principles of Thought

Law of Non Contradiction (LNC) : no statement can be both true and false

Law of Excluded Middle (LEM) : all statements are either true or false

Aristotle (384-322 BC) and Chrysippus (280-208 BC) questioned the validity of LEM.

Eubulides (380-310 BC) and the text Dissoi Logoi (5TH C BC) rejected the validity of LNC.

Did Herakleitos deny LNC?

Graham Priest (current paraconsistent logician), and Sextus Empiricus (160-210 AD) support the claim that Herakleitos denied LNC.

Bertrand Russell, and Diogenes Laertius (3rd C AD) argue Herakleitos did not reject LNC.
I abstain from judgement on the matter, since both sides of the debate on Herakleitos seem to be supported somewhat equally well. Herakleitos most definitely was an influence on those who denied LNC, and also those who did not deny LNC. The fragment Dissoi Logoi (5th C BC) is however, a clear example of the denial of LNC.
Main Section
Herakleitos as a denier of LNC
On the one hand, Graham Priest and Richard Routley claim that Herakleitos was the first to deny LNC, and this view seems to be supported by some of the ancient sources and commentators, for example the ancient sceptic Sextus Empiricus. On the other hand, Bertrand Russell finds only apparent inconsistency in Herakleitos, arguing there was no denial of LNC in Herakleitos. And this view seems to be supported by some of the ancient sources, for example Diogenes Laertius. Lets examine their arguments in a little more detail.

Paraconsistent logicians

  1. Priest/Routley1

Priest/Routley suggest that Herakleitos is much more easily interpreted as a Paraconsistentist (as did Hegel). The Unity of Opposites means that opposites are united, and sometimes even identical. Note this does not mean all opposites are identical, and that all things are identical, xRy  x = y.

“The way up and down are the same” (this is a contradiction)
“ One cannot step into the same river twice, nor can one grasp a mortal substance in a stable condition, but it scatters again and gathers; it forms and dissolves, and approaches and departs.”
“ It rests by changing”
Priest/Routley suggest this is best described by the logical form (h,x) hx & hx.
This view is strengthened by Priests argument on motion being inconsistent. In change (where motion is a change in position), there is at each stage a moment where the changing item is both in a given state, because it has reached that state, but also no tin that state, because it is not stationary but moving through and beyond that state. A more detailed account can be found in Priests book ‘In Contradiction’.
The unity of opposites is part of the wider cosmological theme, that some opposites are not only connected but are identical. This would suffice for dialethism.
The paraconsistent argument that Herakleitos is strongly paraconsistent [14] :

  1. Some (suitable) opposites are identical. Let f and –f be among such opposites (a predicate representation of opposites is convenient but not essential). Then f = -f. (If I was to pick at Priest/Routleys argument, I would not agree to this last move: On what basis does an opposite assume the “=” type of identity?) Now let x be some item that has f; then fx. For all ordinary predicates this follows from a suitable theory of objects (for let x be kzfz, i.e. an arbitrary object which is f). However, a less exotic route will serve. By excluded middle, fx or else –fx, for any object x. Whichever alternative is assumed the argument continues in the same fashion. Now since fx and –f = f, also –fx, so fx and –fx. While this gives dialethism of a sort, it could be contended that this is only “predicate dialethism”, which is compatible with non-paraconsistent positions, indeed with an extension of classical logic (For example “(z)(hz)  (z)(hz)”). To reach dialethism proper it needs also to be granted

(2) among the suitable opposites is some pair, h and –h, such that –hz iff –(hz), i.e. for which predicate and sentence negation coincide (for suitable z) (but has this been established?). Then indeed dialethism proper follows: for hz & -(hz). (why does it follow?).
Sextus Empiricus

Although Priest and Routley did not mention this, it turn out that Sextus Empiricus2 is one of the ancient sources suggesting that Herakleitos did believe in contradictions actually existing, hence denying LNC.

“It is true that Aenesidemus and his followers used to say the Sceptic Way is a road leading up to Herakleitan philosophy, since to hold that the same thing is the subject of appearances is a preliminary to holding that it is the subject of opposite realities, and while the Sceptics say that the same thing is subject of opposite appearances, the Herakleitans go on from this to assert their reality.”3
So, the sceptic view that the same thing apparently possesses opposite attributes or qualities is regarded as a step on the road to the Herakleitan view that it really posses such qualities. Sextus argues that honey appears to be sweet to a healthy person, bitter to some people who are ill; We can assume from the previous argument therefore, that the Herakleitans must have thought that honey actually was both bitter and sweet at the same time in essence.
However, the paraconsistentist cannot assume from this that the Herakleitans did hold some contradictions to be true, because this account is given by Sextus, and Sextus clearly did not man “contradictory” when using the word “opposite”. For example4, Sextus says “The phrase “opposed judgements” we do not employ in the sense of negations and affirmations only but simply as equivalent to “conflicting judgements”. We can infer that “opposites” includes, for the Sceptics, “contraries” (For example “All are wise”, “None are wise), as well as contradictories (“for example “Some are wise”, “None are wise”), whereas the Stoic and Megarian logicians used it as the later only.

Herakleitos as a supporter of LNC

Bertrand Russell

Russell suggests that the best way to interpret Herakleitos is to distinguish to different parts to his theory, one of a balanced adjustment of opposing tendencies, not actually contradictory. Behind the apparent strife between opposites, there lies a hidden harmony or attunement, which is the world. He suggests that this if the meaning behind the Theory of Flux, yet admits that this appears to conflict with the fragment type “ we step and do not step, we are and are not”. This is the second part of Herakleitos’ theory, the more contradictory fragment type “we are and we are not” “the upward path and the downward path is the same”. The sloping road allows the potential for both up and down, it depends on where you decide to go. Herakleitos’ theory of opposites reminds us that what appear to be conflicting features are really essential parts of a consistent situation. Thus, just as one could not conceive of an upward path without a downward path, and no one could conceive a concept of good without a concept of evil, the world is full of consistent, yet opposing forces – in the sense just implied.

This is how Russell glosses over the inconsistencies in Herakleitos essentially by claiming “(z)(hz)  (z)(hz) type of opposition. Russell contrasts the flux fragments with another fragment from Herakleitos which says that ‘We step and do not step into the same river we are and are not’. At first sight, Russell claims, it seems this cannot be reconciled with the previous statement. Priest/Routley identify such fragments as true contradictions. However, according to Russell, this present saying belongs to a different aspect of the theory. The clue lies in the second half. We are and we are not is a somewhat cryptic way of saying that the unity of our existence consists in perpetual change, or to express it in the language later forged by Plato, our being is a perpetual becoming.

Diogenes Laertius

According to Diogenes5, Herakleitos’ treatise On Nature was divided into three discourses: On the Universe, On Politics, On Theology. Generally, Herakleitos held that all things are composed of fire, and into fire again they are resolved. All things are caused by destiny, and existent things are brought into harmony by the clash of opposing currents.

On some of Herakleitos’ particular tenets, no clear explanation is given6. For example, fire is the elements, all things are exchange for fire and come into being by rarefaction and condensation. Also, All things come into being by conflict of opposites, and the sum of things flows like a stream. At least some explanation on the intended use of the word opposite is given; the opposite which tends towards birth or creation is called war and strife, and that which tends to destruction is called concord and peace.
The fragment “The path up is the same as the path down” is interpreted in an interesting way by Diogenes7. The downward path is when fire turns into moisture, moisture condenses into water, and water again turns into earth. The upward path is when the earth is liquefied, giving rise to water, the water evaporates into moisture, and the moisture rarefies into fire.
Aristotle also thought Herakleitos writing were unclear8. Aristotle states9, “ Surely nature yearns for contraries and effects harmony from them and not from similars…That was also said by Herakleitos the Obscure: Combinations- wholes and not wholes, concurring differing, concordant discordant, from all things one and from one all things. [B10]. In this way the structure of the universe, I mean, of the heavens and the earth and the whole world- was arranged by one harmony through blending of the most contrary principles. Notably, Aristotle, refers to these opposites as contraries, not contradictories. Plotinus10, similarly says “For Herakleitos…posits necessary exchanges between contraries, and talks of a path both up and down [B 60], and changing it rests…”

1 Priest/Routley (Paraconsistent Logic: essays on the inconsistent)

2 Sextus Empiricus, Against The Mathematicians, VII 132-133

3 ibid, XXIX.

4 Ibid, IV.

5 Diogenes Laertius, page 413

6 Diogenes Laertius, page 415

7 Diogenes Laertius, page 417

8 Aristotle, Rhetoric, 1407b14-18

9 Aristotle, On The World 369b7-8, 20-25.

10 Plotinus, Enneads IV vii 1.8-17

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