by Carl Pierer
Now, paraconsistent logics deny this explosion principle. There are different ways of doing this, but we will stick with Priest's way in his (Priest, 1989). His is a dialethic interpretation, meaning that he thinks there are sentences that are both true and false. This has some interesting consequences. Note, first of all, that this does not mean that all sentences are true and false. Most importantly, most classical notions are indeed preserve. So, we have, for propositions A and B:
- ~A is true implies A is false
- A is true implies ~A is false
- A&B is true only if A is true and B is true
- A&B is false only if at least one of A or B is false
These are quite orthodox. Now, of course, on the dialethic point of view, A could be both true and false, and suppose B is true. Then A&B is both true and false. Next, we need the notion of logical consequence, which Priest defines also quite classically:
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 make B (at least) true makes A (at least) true.(Priest, 1989)
What does this mean exactly?
Well, let A = P or ~P. Now, suppose P is true. Then ~P is false. Hence A is true. Suppose then that P is false. Then ~P is true. Hence A is true. (This again is classical). Suppose finally that P is both true and false. Then ~P is both true and false. So A is both true and false. This shows that for all assignments of truth values (true, false, both) A is (at least) true. Hence A (the law of the excluded middle) is a logical truth, even for this dialethic logic.
Where things get interesting, however, is with the notion of logical consequence. We run into trouble if we attempt to show that anything can be deduced from a contradiction. To see why, let's proceed in the same manner as before:
Suppose we start with a proposition A which is both true and false. We can then form the disjunct A or B. Since A is both true and false, ~A is also true and false. In particular, we have that ~A is true.
At this point, however, we run into a problem. We cannot deduce that B is true. For everything so far mentioned can be true, and B could still take the value false. Hence, by the definition of logical consequence, B is not a logical consequence of A&~A.
In the article under discussion, Priest uses this sketch of a dialethic logic to argue that Hegel can consistently be read in a dialethic fashion. That is, that Hegel does indeed claim that there are true contradictions. While Hegel, according to Priest, is quite clear about this, subsequent interpreters have attempted to interpret Hegel's use of "contradiction" in a non-logical sense. Priest wishes to show that Hegel means what he says, i.e. logical contradictions.
With this sketch of a more powerful formal logic developed above, we can see that there is at least no immediate (logical) objection to proceeding in this way. Priest, then gives an example how dialethic logic could be applied to dialectics. Consider movement. For Hegel, for an object b to be in a state of motion means to be at position s at time t. Yet, this is not all. If it were, then b's being in motion would be no different from its being at rest. Instead, Hegel suggest, b is also not at position s at time t. Hence, to be in motion is both to be and not to be at a point at a certain time. So, the difference between rest and movement is that the sentence "b is at s at time t" is true only (i.e. consistent) when b is at rest. It is true and false (i.e. paraconsistent) when b is moving. We can see here the nice link that Priest makes with Hegel. Classical formal logic does have its place in the static, in the consistent. However, as soon as change enters the picture it becomes necessary to think paraconsistently.
Of course, many objections can be raised to this characterisation. An important one is that this brand of formal logic seems quite happy with the fact that this is merely an external contradiction. What Priest means with this is that "b is at s at time t" and "b is not at s at time t" are perfectly meaningful in their own right and can be asserted independently. For the dialectician, however, the contradictions of interest are internal. That is, they are provoked by the concepts themselves. They are not accidental and depend on each other.
Priest has quite a nifty reply to this. First, he introduces a logical operator ^, which turns a proposition into a noun phrase. What this means is that if A is the proposition "Mozart is a great composer", ^A would be "Mozart's being a great composer". Priest notes in particular that ^A is an object. While it is not entirely clear what he means by an object, it seems that it conforms more or less with the mathematical notion of an object. He writes: "Which object it denotes we may assume very little about". Furthermore, Priest needs the T-scheme: ^A is true if and only if A is true. There is some intuitive sense as to why this should be true and for the present purposes we shall contend with that. Secondly, identity statements (of the form a = b) will play a special role. Priest points out that they behave in quite the conventional way, with the law of identity (a = a) holding and the substitutivity of identicals being preserved. Of course, a statement like a = b can be true, false or both, like all statements for the dialectician. Note that Priest suggests it is always true that ^A ≠ ^~A. Since ^A can be thought of as an object and ^~A its opposite (for example "Mozart's being a great composer" and "Mozart's not being a great composer"), and since an object is not the same as its opposite, Priest considers this a natural requirement.
Now, this allows us to illustrate the formal rendition of the intrinsic contradiction. If we suppose that A stands for "b is at s at time t", then Priest claims "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". Now however, this gives rise to the contradiction "A&~A" in the following way: Since we have the law of the excluded middle (A or ~A), we may assume without loss of generality that A is true. Hence, by the T-scheme, we have ^A is true. Since ^A = ^~A, by the substitutivity of identicals, we have that ^~A is true and hence – T-scheme again – that ~A is true. Therefore A&~A is true.
We hence see that the kind of contradiction we have here is of the form (a = b) & (a ≠ b). The former is the contradiction of motion ^A = ^~A. The latter is always true, as was mentioned before. Priest thinks that this captures the more intimate relation of the internal contradiction:
(…) 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 dialectical contradiction. (Priest, 1989)
Of course, there are several ways to attack this proposal and to reject it. First of all, the dialethic claim of true contradictions is outrageous to many. What kind of truth, exactly, are we talking about here? Priest addresses this point in particular in his book Doubt Truth To Be A Liar. Secondly, dialectics is often framed a logic of concepts. How does the dialethic interpretation deal with this? (Ficara, 2013) replies to this.
Yet, instead of going into the nitty-gritty of the debate, I would like to sketch why such an approach is interesting and why it would be good to understand its limitations. There are three main reasons:
First, attempting to give a formal interpretation of Hegel allows to open a dialog. People feeling ill at ease with Hegel's use of contradictions can come to see that not all hell breaks loose and that the view need not amount to trivialism (where everything is true). Conversely, scepticism towards formal logic and mathematical thinking could be redeemed in the eyes of dialecticians. But this is more of a hope and perhaps not even the most interesting aspect.
Second, while it is all fine and good to suggest that when Hegel criticised formal logic he only meant the consistent type, it would be interesting to see in what way dialethic logic can render the movement that dialectic describes. Formal logic (whether consistent or paraconsistent) seems to fix states, through definitions, through formalisation, thereby denying their internal development. To formulate it provocatively: to understand if this poses a deeper obstruction (deeper than the consistent/inconsistent divide) would help appreciate whether there is a need for a non-formal thinking.
Third, Hegel's dialectic has subsequently found political interpretations in such thinkers as Marx, Engels, or Adorno. If his dialectic can be rendered in dialethic logic, how about the political ones? Understanding how politics enters into formalisation would allow us to appreciate if indeed there is a link between the kind of formal logic and the kind of politics it allows to express.
Of course, all of these are big questions that are beyond the scope of the present essay. However, I hope that this sketch of Priest's argument helps to appreciate that formal logic and dialectic thinking need not be opposed. Indeed, that exploring the way in which they might work together, gives rise to many deep questions.
References
Ficara, E. (2013). Dialectic and Dialethism. History and Philosophy of Logic, 35-52.
Priest, G. (1989). Dialectic and Dialethic. Science and Society, 388-415.