# Deleuze’s Timed Logic (I)

Gilles Deleuze’s philosophy can accommodate any number of logics in a subspace of its metaphysics. By accommodate, I mean that any given logic can be observed, operated and considered for consistency within that subspace.

The fact that we can isolate part of Deleuze’s system and consider its logics does not imply that they hold for the full system. The isolated logic may contradict important principles of the wider metaphysics, be inconsistent with its logics, or operate with them in ways that breakdown, either necessarily or contingently.

It is possible to conceive of existential quantifiers in the subspace Deleuze calls ‘actual’, to the point of saying they operate in it. However, it is also correct to say that individual things do not exist for Deleuze’s metaphysics – when taken at its fullest. At the wider level, the quantification ‘there exists an x’ should be replaced by ‘there becomes multiplicity’. There is no ‘x’ and no existence at this level, because multiplicities cannot be identified and becoming cannot be reduced to existence at time t, or over a stretch of time.

In Difference and Repetition, the ideas of completeness and condition do a lot of work to account for different subspaces and their relations. The first explains how there can be logical subspaces inconsistent with the full metaphysics, because they do not hold for the complete system, where complete does not mean closed or finished, but rather taken across all its processes.

The second tells how different subspaces are necessarily in touch with one another as conditions, that is, some feature of one space can only be explained by referring to a feature of another space. Since spaces are distinguished according to these features, condition does not indicate smooth or consistent passage following a shared law or principle.

Condition is a concept for relating heterogeneous spaces, rather than unifying a homogeneous one. For instance, a condition is brought in when a principle of identity is explained according to a principle of becoming, because the principle of identity is incapable of accounting for changes between identities.

The questions ‘Is this complete?’ and ‘What are the conditions for this?’ disrupt a subspace, its contents and its logic, by connecting them to another space, where neither space can be reduced to the other and where they cannot be synthesised into a new space that includes both.

When combined, completeness and condition hinder the construction of logic for a system, since a logic standardly implies homogeneity through axioms such as transitivity. One of the ways of looking at the problem of logic in Deleuze’s philosophy is therefore whether there can be a logic for heterogeneity: for conditions and completeness.

Deleuze’s middle-period metaphysics (Difference and Repetition and Logic of Sense) has three main subspaces: actual, virtual and surface of intensities. Roughly, they can be thought of as the world we can identify, multiplicities of pure changes and degrees of intensities. There is a space of intensities because, while not belonging to them, they play out in both other spaces, as degrees of change.

Existential quantification holds in some way in the actual. It does not hold for the virtual, because virtual multiplicities cannot be completely identified, but only partially expressed. It also does not hold for intensities, since they are only varying degrees partly manifested in the other spaces. Virtual multiplicities are conditions for actual identities and intensities are conditions for the determination of actual identities and virtual multiplicities.

‘What is the condition for x identified at time t becoming y at time t+1?’ is a Deleuzian version of the familiar philosophical question ‘How can x’ at time t and x” at time t+1 both be called x?’ The contrast is between the question ‘What is the metaphysical structure required for explaining James becoming different?’ and ‘How can James at two different ages be the same person?’ For the first question, the problem is about explaining change and difference. For the second, the problem is about how to return change and difference to manageable identity. I want to draw attention to three points about these questions.

First, note how the Deleuzian question implicitly relies on existential quantification (x, or James, at time t) to then move to a condition counter to existential quantification (the time-changing multiplicity explaining the becoming of x into y). This is one of the reasons why I’ve talked of subspaces where a logic holds and wider connected spaces where it doesn’t.

Second, Deleuze’s argument takes the familiar problem of a changing thing over time and claims it can only be addressed satisfactorily by moving to a different metaphysics. Familiar concepts of existence and time depend on conditions that cannot be reduced to the existence of things, or to time as a reference for when things happen (either as past, now and future, or as before and after). Solutions such as taking time intervals rather than points (James over interval t to t+1) are unsatisfactory for Deleuze, thereby pushing him towards a process metaphysics prioritising change.

Third, these questions are very odd when taken from some science-based viewpoints. It is strange to ask for metaphysical explanations about becoming or about identity, when we have scientific explanations using causality or association, and when we have empirical data about entities and relations leading to precise definitions of entities.

‘Should we avoid metaphysics?’ is an important question, but not one concerning me in these posts, because I’m trying to get at the logic of Deleuze’s wider metaphysics, right or wrong. Findings about logic might inform answers to questions about metaphysics and science; they have in the cases of many-valued logics for quantum mechanics and temporal logics for computing science.

In Deleuze’s metaphysics, the real is incomplete unless actual processes, those where we can identify things, are connected to virtual ones, where we are dealing with multiplicities in the process of becoming. However, the opposite is also the case. Without existential quantification in the actual, multiple becoming lacks determination and remains chaotic.

These reciprocal determinations are asymmetrical, the determination of the virtual by the actual (‘differenciation’) is a condition for picking out things and determining multiplicities, whereas the determination of the actual by the virtual (‘differentiation’) is a condition for change in actual things and increasing chaos in virtual multiplicities. For Deleuze, the real is both of these, plus the added determination afforded by intensities defining an individuation, a take on the real determined by intensities of actual change and virtual multiplicities.

The same can be said for propositions as for existential quantifiers. In Logic of Sense, Deleuze discusses propositions by taking traditional categories such as designation (the thing picked out), signification (the meaning) and manifestation (the utterance) and adding the category of sense. He doesn’t deny that the first three are features of propositions, but rather aims to show how the sense of a proposition makes them problematic.

Problematic should be understood as implying that what a proposition designates, what it means and how it is said are insufficient on their own, because they lead to paradoxes and difficulties that require responses. A proposition always has a sense within and independent of those three.

Sense generates problems and the demand to respond to them, because it is how propositions have effects on and are affected by all of the real, understood as changing things, values and intensities. Deleuze calls this interdependence of designation, signification, manifestation and sense the ‘circle’ of the proposition.

Circularity means that each of the features of propositions is completed by the others, when sense is added to them. Sense takes priority, though, because it explains why propositions matter and hence why they are problematic. Something is not a problem, if it doesn’t matter. If something matters, it will have its problems. It is therefore always inaccurate to say Deleuze is an idealist, since his philosophy takes practical and problematic challenges to life to be ubiquitous, inescapable and requiring responses.

This interdependence of problems and significance explains why Deleuze insists on the determining role of intensities, because they are a marker of something having a significant effect on an individual determination of the whole of reality: the real taken from the perspective of lived intensities.

For the proposition ‘You will come to regret your betrayal’, there is a designation (your future regret about your betrayal). There is also a meaning (for instance, combining the dictionary meanings of ‘regret’, ‘betrayal’, ‘come to’, ‘you’). There is a manifestation (the saying of the proposition to you by someone). But why each of these matters requires going into the effects of that saying, to what it does in reality, rather than its internal meaning. A prophesy is more than a reference, a meaning and a saying. It is all the lived effects of a foretelling, backwards and forwards in time: ‘… the untrammeled fulfillment of a death so clearly foretold.’ (Gabriel García Márquez, Chronicle of a Death Foretold)

The analysis of the truth value of propositions is possible – for instance, in terms of tense logic and the value of a prophesy that will be true – but it is also insufficient:

It is striking to note that all the work of logic is directly concerned with signification, with implications and conclusions, and only indirectly concerned with sense – precisely through the intermediary of paradoxes that signification cannot resolve, or even creates.

Gilles Deleuze, Logique du sens, Paris: Minuit, 1969, p34 [my translation]

Deleuze is not denying that propositions imply one another logically and lead to logical conclusions (if P, then Q; therefore Z). His own sentences are put together with close attention to this. He is saying that there is necessarily something more and this something plays out in all aspects of propositions.

Deleuze is a philosopher of arguments and structures of concepts, but he is also a philosopher of value, change and novelty, where each of these applies to and goes beyond things that can be pointed to, meanings that can be agreed upon, implications that hold, propositions that are true or false, or even undecided. In Logic of Sense, this beyond is also a logic.

The reason I have introduced the ideas of logical subspaces, completeness and conditions is to begin to answer the question ‘What is the logic of Deleuze’s metaphysics?’ It is an unusual question that I began considering when writing about Deleuze and Spencer-Brown. I was struck by the difficulty of describing Deleuze’s logic in order to contrast it to Spencer-Brown’s.

When Deleuze’s philosophy is systematised along lines taken by Spencer-Brown, the ensuing axiomatic doesn’t seem to lead to a practical form of construction. For Deleuze, repetition of the same leads to difference (>p< >p< = >p'<). This implies that any repetition is open to every outcome, otherwise it would be possible to reduce repetition back to identity through a limited range of possibilities (>p< >p< = one of this set of possibilities).

This reduction of the real to the possible is exactly what Deleuze seeks to avoid, because he wants the potential of radical novelty in every part of his system. When I called this series of posts ‘Deleuze’s timed logics’ it was an allusion to Arthur Prior, temporal logic and tense logic. For Prior, the difficulty of time for logic is a problem of undecided possibilities in the future. For Deleuze, the problem of time for logic is the difficulty of how propositions are both decidable and undecidable at all times and beyond any set of possibilities.

There has been much work on mathematics and physics in Deleuze’s philosophy, but almost nothing on his logic, as opposed to his critical work on other logics, through paradoxes, or approaches to his work on Stoic logic in Logic of Sense.

I’ll define the logic of a metaphysics as a formal simplification of how any event must work through the whole of its system. Such logics need not be unique, but they do need to be necessary. An event could be a proposition, a number, a material change, a new value – anything that happens in the system. The logic determines the path of how an event must play out. If an event does not work according to that logic, then either the logic is wrong, or the event has been misdescribed.

This logic is not necessarily how the metaphysics is argued for, since a philosopher can propose a new logic using an older one. It is not necessarily the logic a philosopher claims to be using. As a simplification of an open-ended and complex construction, logic is frequently worked out subsequently. So there is a difference between the way different statements of a system fit together – for instance, how a series of aphorisms or paragraphs are ordered – and the logic placing an event across the relations constructed by those texts.

Though some metaphysics set out their logic, others never do so and call for difficult and controversial interpretations to track how the metaphysics functions. This can lead to dismissals on the grounds that a philosophy is illogical or esoteric. It is risky to do this, since a philosophy can work – in the sense of dependably changing the world and thought – without fitting established versions of logic.

The logic of a metaphysics is not how major parts of the system fit together, since these relations do not necessarily hold for all events. A philosophy can treat ontology, epistemology, ethics and aesthetics separately. It can divide human freedom away from natural causality. However, if there’s to be a logic for the complete system, there will have to be connections between events across those fields, unless some parts are to retain independence from others to the point of leading to different philosophies (A’s work on music, as opposed to A’s work in epistemology).

The logic determines how any event fits coherently through the system. There are two clarifications and one important remark to make about this.

First, coherence is a practical test of reliable functioning on a formal level. Is there a formal pattern that holds for all events in the system? It might seem odd to combine formal and practical, but this pragmatism is about formal simplification and practical assessments of whether that simplification works reliably. The precise meaning of coherent can be internal to a philosophy, because coherence introduces logical constraints and an external definition might introduce a limitation counter to metaphysical innovations.

Second, coherence only determines the system as simplified. A philosophy cannot be reduced to its logic. The logic provides a frame for understanding how a system works – or ought to work – as an interconnection of events, but there are many other ways of evaluating and taking up a philosophy, such as its ideas, examples, concepts, language, style, critical points and principles.

Importantly, practical reliability at a formal level appears to be at odds with the most basic commitments of Deleuze’s philosophy. We understand reliable to mean repeatable, but his philosophy insists that repetition is about difference. If things become different when repeated, how can they be reliable? How can any formal simplification not contradict the change inherent to the system?

The answer is that for Deleuze coherence is about processes of transformation. It is not a matter of the same thing happening reliably, but rather about whether transformations can be given a reliable formal structure. The formal approach to this challenge comes from his philosophy of time.

When discussing time and identity, I contrasted Deleuze’s questions and problems about time and change with more traditional ones about time and identity. This contrast holds for the logic of his metaphysics. Instead of thinking about the simple formal work of an event in a metaphysical system as timeless or as occurring in or over time, Deleuze thinks of it as done by time.

The event is a process of time. This is not a metaphor, as when we describe a face as aged by time. It is a systematisation of the processes making events. In order to think through how events are made and how they spread, we can organise them into time processes: how the event passes away, how it becomes present and how it leads into the future. These processes lead to Deleuze’s logic. I will describe and criticise this new timed logic over the following posts.