Priest, (8) ‘Dialectic and Dialetheic’, section 8, “Identity in Difference”, summary

 

by Corry Shores
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[The following is summary. All boldface, underlying and bracketed commentary are my own, unless otherwise indicated.]

Graham Priest

“Dialectic and Dialetheic”

8 Identity in Difference

Brief Summary:

Hegel’s dialectic takes the form of identity in difference, formulable as (a=b)&(a≠b). This is a variation on the dialetheic formulation A&~A.

Summary 

At the end of section 4, Priest noted that although dialetheic logic says that there are true cases of A&~A, this formulation might not apply to Hegel’s dialectics, which calls for a more unified, intimate, internal, intensional relation/contradiction between the terms. He returns now to this issue so to better describe “the exact nature of dialectical contradictions” (410). [First recall what Priest says regarding the dialetheias of nature and spirit in Hegel:

For spirit, s, is | then both spirit and not spirit. In the notation of section 3: (s=s)&(s≠s). Alternatively, nature, n, which is not spirit, is spirit: (n≠s)&(n=s). The existence (truth) of this contradiction allows spirit to think (understand) what it is: spirit and nature, spirit and not spirit; and thus to achieve its telos, in which form it is the Absolute.
(401-402)

] We previously have encountered many contradictions taking the form (a=b)&(a≠b), “something’s being identical with, and different from something (else?)” (410). This is Hegel’s notion of identity in difference. Priest “will now argue that this is the form of a dialectical contradiction, to which all others reduce” (410).

We have also encountered contradictions of the form: (a=a)&(a≠a). “They are obviously of this form” (411) [(a=b)&(a≠b), the form of ‘identity in difference’.] We also encountered the contradiction of something

being identical with its opposite: ^A=^~A. Thus for example, that something, a is free (Fa) is identical to its being bound (not free): ^Fa=^~Fa. This, too is a special form of identity in difference. For, as we noted in section 3, it is always true that ^A≠^~A. Thus, identity of opposites is just the identity in difference (^A=^~A)&(^A≠^~A).
(411)

[Perhaps Priest understands “identity in difference” to mean that both something equals its negation and it does not equal its negation.]

[In this next paragraph, Priest refers to the Tarski T-scheme. Recall that Tarski uses quotations around a sentence to mean the name for that sentence. For example:

“snow is white” is true if and only if snow is white.

In our article here, (see sect.3) Priest is doing something similar with the ^ symbol, meaning we would put ‘that’ in front of the sentence. The basic operations Priest seems he is doing is that if we have a sentence, let us say: snow is white; we can then make it: that snow is white is true iff snow is white. Since snow is white, then we can just say: that snow is white is true. However, we can substitute the  ‘quoted’ or ‘that-ed’ term with its negation, since we equated the two. Thus this means that the negation is true as well.]

In fact, the identity of opposites ^A=^~A is doubly contradictory, since it also gives rise to the contradiction A&~A. For either A or ~A; without loss of generality, suppose the former. Then ^A is true, by the T-scheme (^A is true iff A). But if ^A=^~A, ^A is true implies ^~A is true  (by the substitutivity of identicals). Hence ^~A is true too. It follows that ~A, again by the T-scheme. Thus, both A and ~A.
(411)

[In the above, we began by assuming either only A or not-A. No matter which we assume, the other holds by consequence (when one is also equated to the other). We did not assume a third possibility. It is not completely clear to me how this factors in. It might have something to do with truth gaps, in which a proposition is neither true nor false, and so if A were valueless, we would not say A or not-A. For, it need not be that one is true, since A is valueless. See discussion of valueless statements, and Priest’s defense of the law of excluded middle in these situations, in section 1.3 and section 4.7 of In Contradiction.] One objection to the above reasoning is that it uses the law of excluded middle, but Hegel spoke against it. Priest replies that it is still valid given our semantics (outlined in section 3 of this article), and also that Hegel’s objection to the law is not that it is untrue but rather that it is trivial; “it may be false (as well)!” (411).

Priest will now address two particular cases of identity in difference. The first is “something’s being identical with itself is its being different from itself. This is just the identity of opposites (^a=a)=(^a≠a)” (411). The second is the dialectics of motion that he discussed in section 4. This as well can be understood as the identity of opposites. [[The reasoning here seems to be as follows. Consider an instantaneous state of contradiction. For an object to be in one position is for it to be in another. If it were only in the same position, it would not be in motion. Thus we can equate its being in a position with its not being in that position. Because this affirms both terms, we can conjoin them even though one can be seen as the negation of the other. Priest furthermore says that any such equation of a term with its negation can be seen as one term ‘changing’ into its opposite, perhaps because the equation brings one term upon the other. This is important, because it could explain the collapsing of one term upon the other. In logic this seems to be a mode of substitution. ~A passes into A in the sense that it supplants A, or overlays it. It is also important to note that Priest here distinguishes the equality of opposites with the conjunction. The conjunction does not imply the passing of one into the other, even though the passing of one into the other implies the conjunction of the terms.]]

we may take the instantaneous contradiction produced in a state of motion to be that the body’s being in a certain place is its not being in that place, ^A=^~A. This will imply that it both is and is not in that place, A&~A, as I have just observed. Moreover, because this type of contradiction is identified as a state of change, it is natural to describe any state of the form ^A=^~A as a state where ^A is changing into its opposite ^~A, or vice versa. Thus, the identity of opposites is frequently described in this way, as, for example, the opposites going over into each other.
(411)

So recall again the objection that A&~A does not do justice to the intimacy of the terms in dialectical contradiction. Priest shows how dialectical contradictions take the form (a=b)&(a≠b) [which is a variation on A&~A]. The intimacy here is that the terms are identical.

We have now seen that all the dialectical contradictions we have met are instances of identity in difference: (a=b)&(a≠b). We may therefore take this to be the general form of a dialectical contradiction. This is an excellent way of doing justice to the point we noted in section 4, that the poles of a dialectical contradiction must have a tighter relation than mere extensional conjunction. For the poles of the identity in difference (a=b)&(a≠b), a and b, are actually identical with (though different from) each other; (dialectical) identity is therefore the relationship between the poles of a dialectical contradiction.
(412)

Citations from:

Priest, Graham. ‘Dialectic and Dialetheiç’. Science & Society, 1989/1990, 53 (4) 388–415.