Priest, (3) ‘Dialectic and Dialetheic’, section 3, “Dialetheic Logic”, 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”

3 Dialetheic Logic

Brief Summary:

Dialetheic logic is just like orthodox logic except that it allows for true contradictions, and when there are true contradictions, we cannot infer from them any other proposition we want.

Summary 

Previously Priest explained that those who argue Hegel’s and Marx’s dialectics do not involve self-contradiction (or that they do and for this reason are flawed) are adhering to the modern Frege/Russell logic, which forbids contradiction. However, this is merely an assumption made in this kind of logic, and other assumptions have been called into question. Now Priest will give an informal overview of what dialetheic logic is. In Frege/Russell logic, a sentence can be assigned only one of two truth values, T (true) and F (false). However, in dialetheic logic, a sentence can also take both values. “(Thus, technically, semantic values are non-empty subsets of {T,F}.)” (393). So if a sentence is true, that does not rule out that it is also false, and vice versa. Priest then explains the “truth table” conditions for computing truth values of more complex sentences. The conditions for negation, conjunction, and disjunction are all orthodox. So ~A is true iff A is false, and ~A is false iff A is true. Everything is as you know it for conjunction (true only if both conjuncts are true), and disjunction. [In the next part, Priest seems to be explaining why when something is both true and false, the conjunction of it with its negation is also true and false, and thus at least true.] “Notice that if A is true and false, so is ~A; and so, moreover, is A&~A. In particular, it is true, as dialetheists claim” (393). [The next definitions seem to establish validity and implication:]

 

Logical truth and logical consequence are also defined in the orthodox fashion:

A is a logical truth just if A is (at least) true under all assignments of values

A is a logical consequence of B just if every assignment of values that makes B (at least) true makes A (at least) true
(393-394)

[Priest’s next point seems to be that dialetheic logic and classical logic give the same set of logical truths. I am not exactly sure how this is so. I can understand that all logical truths in classical logic are also true in dialetheic logic. But there are true formulations, like A&~A which are true in dialetheic and not true in classical. So you will have to read the following to interpret it for yourself.]

It may be interesting to note that A is a logical truth if A is a two-valued tautology. Thus, these semantics give the same set of logical truths as does orthodox logic. Thus, both Av~A and ~(A&~A) are logical truths. The second of these may seem surprising initially. But if a certain contradiction, A&~A, may be true, there is no reason why the “secondary contradiction” (A&~A)&~(A&~A) should not also be true.
(394)

[Priest’s next point seems to be that the principle of explosion does not apply in dialetheic logic. So an inference is valid if there is no situation where all the premisses are true and the conclusion false. If we have A&~A, in classical logic we can derive B, because there is no way to make the premise true. But since it can be true in dialetheic logic, we can say A&~A is true and B is false, and thus we cannot validly infer B from the conjunction. Let me quote from Priest’s Logic: A Short Introduction.

an inference is valid provided that there is no situation which makes all the premisses true, and the conclusion untrue (false). That is, it is valid if there is no way of assigning Ts and Fs to the relevant sentences, which results in all the premisses having the value T and the conclusion having the value F.
(Priest, Logic: A Short Introduction, p.13)

 

It certainly doesn't seem valid. The wealth of the Queen – great or not – would seem to have no bearing on the aviatory abilities of pigs.
(Priest, Logic: A Short Introduction, p.8)

What about the inference with which we started: q, ¬q/p? Proceeding as before, we get:

Again, the inference is valid; and now we see why. There is no row in which both of the premisses are true and the conclusion is false. Indeed , there is no row in which both of the premisses are true. The conclusion doesn’t really matter at all! Sometimes, logicians describe this situation by saying that the inference is vacuously valid, just because the premisses could never be true together.
(Priest, Logic: A Short Introduction, p.14)

{Later, Priest discusses the ‘new assumptions’ that a formulation can be both true and false.}

it is worth returning to the inference with which we started in Chapter 2: q, ¬q/p. As we saw in that chapter, given the assumptions made there, this inference is valid. But given the new assumptions, things are different. To see why, just take a situation where q has the values T and F, but p has just the value F. Since q is both T and F, ¬q is also both | T and F. Hence, both premisses are T (and F as well, but that is not relevant), and the conclusion, p, is not T. This gives us another diagnosis of why we find the inference intuitively invalid. It is invalid.
(Priest, Logic: A Short Introduction, p.35)

I continue by quoting from the Dialectic paper. The ‘as might be expected’ in the following perhaps refers to our intuition that any sentence whatsoever does not validly result from a contradiction.]

The semantics do, however, give a notion of logical consequence different from the orthodox one. In particular, and as might be expected, B is not a consequence of A&~A, as may be seen by simply assigning B the value F, while assigning A both T and F.
(394)

Priest shows how we can “extend these propositional semantics to a semantics for full first-order logic” in another text [Chapter 5 of In Contradiction]. Priest also says that we can say that identity statements taking the form a = b can be both true and false, if enough care is taken “concerning how, exactly, truth values are assigned” [he does not go into those details here] (394). However, if that care is taken, then “all the principles of identity, such as the law of identity (a = a) and the substitutivity of identicals, are assured. As usual, I will write a ≠ b for ~a = b.” (394)

Priest will need another operator which will turn sentences into objects. In English we can use ‘that’ to turn ‘John is happy’ into ‘That John is happy…’ or we can use a gerund, ‘John’s being happy.’ Priest will use the ^ symbol to mean “that”, such that

if A is any sentence, ^A is a noun phrase, and therefore denotes an object. […] it is fairly clear that in some sense ^A and ^~A are opposites. (Think of John’s being happy and his not being happy […].) Since an object is not the same as its opposite, it is natural to require that ^A ≠ ^~A.
(395)

As we can see, all this is orthodox logic, except for the condition that some formulations can be assigned both T and F. Hence “orthodox logic is just a special case of these semantics which ignores a dialectically important case” and “Thus we may stretch Hegel's claim a little as follows:(Frege/Russell) formal logic is perfectly valid in its domain, but dialectical (dialetheic) logic is more general” (395).

Classical logic’s domain is ‘the consistent’, which is what? “dialecticians have had a standard line here: the static is consistent; only when change enters the picture do contradictions arise” (395). Thus we can argue if want that dialects is based on dialetheism. (395)

But “Dialetheic logic is certainly not dialectics”, yet we can at least say that dialetheic logic is a rigorous non-trivial formal logic, and dialectics is compatible with it. [Priest writes: “Dialetheic logic is certainly not dialectics; but it is quite sufficient to show that dialetheism is compatible with the rigor of a non-trivial formal logic” (396).] Priest will move onto dialectics, now that we have established that it can be logically valid to have contradictions in one’s systems.

 

 

 

Citations from:

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

Or if otherwise noted, from:

Priest, Graham. Logic: A Very Short Introduction. Oxford / New York: Oxford, 2000.