Having just finished Graham Priest’s book, in contradiction, a book central to the logical tradition Priest calls dialetheism, I’ve decided to post some thoughts to the blog. Logic is not my area of specialization, nor do I have much to say at the moment about how Priest puts dialetheism to use in reformulating classical logic and semantic theory, but there are some lines of convergence between dialetheism and the metaphysical project I’m engaged in at the moment. Stated baldly, dialetheism accepts that some contradictions are true, or some statements such as the liar’s paradox have two truth values, true and false (hence the term dialetheism); and this position is one I have echoed in various places in my published writings by referring to both/ands, paradoxa, and double articulation. There is a crucial difference, however: whereas I begin my understanding of both/and, paradoxa, and double articulation through a reading of Deleuze, Priest begins with Hegel, and in particular with Hegel’s account of motion. Priest cites Hegel:
[M]otion itself is contradiction’s immediate existence. Something moves not because at one moment of time it is here and at another there, but because at one and the same moment it is here and not here.
A true account of motion, therefore, implies that something both is and is not, or it implies a contradiction. Deleuze, by contrast, rejects Hegelian contradiction because it places difference in subservience to identity, to the identities that constitute the contradictory pair: “α ∧ not-α.” A both/and is thus not a difference subservient to both α and not-α but rather it is the forward-slash, “/”, the “and” between identities that involves a difference deeper than any contradiction – it is the transcendental condition for such contradictions. This is the project Deleuze calls transcendental empiricism and which I detail in my book Deleuze’s Hume.
There is much to be gained from Priest’s work and I’ll return to it frequently as I work through my theory of concepts. Priest shows for example how a consistent semantic theory that rules out paradoxes would lack the expressive power of natural languages and thus an adequate semantic theory should embrace the paradoxes that give languages their expressive power, and dialetheism does just this. Similarly, dialetheism is able to account for many of the other findings of classical logic, set theory, and the philosophies of time and language, and it is able to do so while resolving a number of their residual puzzles. Dialetheism also provides for an interesting critique of Kant’s claim that our moral obligations presuppose our abilities, or what it is we can do – “ought implies can.” This is not simply a matter of showing how we can have contradictory obligations that are thus impossible to fully realize – I might for example promise to bring a friend to the doctor’s having forgotten that I had already promised another friend I would take them to lunch at the same time. More importantly with respect to deontology dialetheism can be used to argue that the very emergence of obligation entails the encounter with realities that undermine and contradict our very will, our very capacity to determine what we can or cannot do (there’s a tie-in here with Levi’s discussions of anti-praxis). In short, it is the impossible that makes moral obligation possible. This will have repercussions with respect to a theory of concepts as well. As Priest himself sums up what he takes to be the significance of dialetheism, he argues that it subsumes classical logic, or dialetheism and its embrace of the inconsistent is to classical logic as Einstein’s special theory of relativity is to Newtonian theory. The former, in both cases, can account for the results of the latter, and they can account for much more besides, including things that were inconceivable by the lights of classical logic and Newtonian theory.
With this last point I return to the difference between dialetheism and a Deleuzian theory of concepts as I understand it. For Priest dialetheism is compatible with science and in fact enhances the capacities of science as he understands it. When Priest adopts a Hegelian understanding of motion, which he develops as the ‘spread hypothesis’, he does so because he is able to reconcile a Russellean (i.e., scientific) view of motion as change of place and the functions that best represent this motion with Hegel’s dialethic understanding of motion as contradiction. Dialetheism, therefore, can supplement the traditional work of science and provide it with tools that can expand its representational power. The role of paradox in this context is thus as a spur to the development of our scientific understanding. Priest therefore sees his development of dialetheism as being analogous to Cantor’s development of the transfinite. The common feature between Cantor and Priest is the affirmation of paradox. In the case of Cantor, the paradoxes associated with infinity had been recognized since Galileo and probably before – namely, the paradox that a subset, such as the set of even numbers, is equal to the set as a whole, the set of natural numbers, when taken to infinity. The moral Priest draws from this is that
with hindsight we can see that they [the paradoxes] pointed the way to the study of a new realm, the recognition of whose existence emerged in the nineteenth century as the result of the work of Bolzano and Dedekind. The paradoxical objects became paradigms of the behaviour of infinities, their paradoxical properties being, indeed, a definition of infinity…[and this culminated with] Cantor’s discovery of the beautiful structure existing beyond the finite, the transfinite, [which] was one of the most exciting steps in the development of human thought. (208-9).
Similarly, for Priest, the inconsistent was thought to ‘have no interesting or important structure,’ but it was precisely the point of in contradiction to show that it does indeed have a structure, and one that expands upon the established structures of classical logic, semantic theory, etc. There is thus a place in our understanding for the transconsistent.
With this move the difference between Hegelian contradiction and Deleuzian difference-in-itself becomes clearer. From the perspective of representational thought and understanding difference is subservient to identity, and in the hands of science, as Deleuze and Guattari discuss this in What is Philosophy?, differences become subsumed by the scientific problematic of formulating functions that allow for the representation of the actualization of the actual (what they call functives). With a Russellian/Galilean understanding of motion, for example, one can use a function to determine where a particular object will be at any given moment on its trajectory. From this perspective, a paradox or contradiction may very well entail the undoing of a function’s ability to represent the actualization of the actual. The function will fail to locate the object, to predict its trajectory. Priest, after all, does not embrace all contradictions and paradoxes, and those that do undermine our abilities to develop fruitful concepts he calls dysfunctional. Some paradoxes, however, such as the paradoxes of infinity, do spur the development of new concepts, and concepts with greater representational power. For Deleuze and Guattari, however, philosophical concepts are not to be confused with the representational functions of science. Rather than represent the actualization of the actual, a philosophical concept problematizes established functions and representations. They problematize the actual. Priest’s work on the inconsistent has done just this, for dialetheism does indeed challenge and problematize long-standing assumptions regarding the law of non-contradiction that has been passed down to us largely unchallenged since Aristotle. Yet in the end the result of this problematizing effort was an enhancement of our representational powers (if not an enhancement of science itself), and it was for this reason that Priest applauds dialetheism, whereas the philosophical effort to create concepts is instead one of problematizing representational functions without seeking to further enhance these very functions. And in doing this philosophical concepts may provide for the problematics that science will develop but their value is not limited by this outcome, nor are they to be judged by their ability to do this. Philosophical concepts, to adopt a phrase from Deleuze and Guattari, must be done with the judgment of God, or the judgments of science. How this is to be understood is the focus of my current project and likely the subject of future posts.