Problems with Meillassoux and Traditional Modal Logic

As I already mentioned in my last two posts, there are problems if it comes to reconstructing Meillassoux with modal logic. In this short post I want to collect the questions and problems that Meillassoux’s philosophy presents modal logic with.

Before that I want to reflect a little bit about the value of reconstructing Meillassoux with logic. Maybe my project is in some parts in vain. One of the reasons why it could be in vain is that (at least until now) I didn’t consider problems in the tenets or methods of Meillassoux (like Toscano does in his contribution here: https://www.re-press.org/book-files/OA_Version_Speculative_Turn_9780980668346.pdf).

On the other hand Meillassoux explicitly uses logic and reason to tackle the problem of correlationism. But the logic he uses is not completely the logic analytical philosophers are used to. Well aware of the logical problems tackled by his influences (especially Deleuze and Badiou), he uses terms and concepts outside the traditional conception of logic.

His writing style is that of a dialogue. There are often a lot of different positions and Meillassoux presents their arguments. He uses a lot of phrases like “The subjectivist would argue …”, “The correlationist would reply …”. Therefore I’m often reminded of Platonic Dialogues or Hume’s Dialogues Concerning Natural Religion.

While reading I find the arguments often very convincing. But he never formalizes them (which doesn’t mean that they unformalizable). If I try – like in my last post – to reconstruct them at least a little bit more formal I come to questions about the kind of logic and reasoning behind it. Meillassoux tackles problems of the event, contingency and the virtual. If criticisms like that of Toscano and Brassier (in the same reader linked above) are right and correlationism has to be attacked otherwise, we haven’t excluded a contingent world enriched with real virtualities and impossible events. So I think my project is not completely in vain because it can help to consider and get to the bottom of the problems of ontological and epistemic modalities.

I hope that explains a little bit my approach and attempts. Now I want to present a set of open questions:

  • In his article Potentiality and Virtuality (first English publication was in Collapse II; an open access version can – again – be fund in this reader: https://www.re-press.org/book-files/OA_Version_Speculative_Turn_9780980668346.pdf) Meillassoux tackles the problem of probabilistic reasoning. In short this problem consist in constructing a set of cases to which probabilities could be applied to. (A problem discussed in a wide variety of research areas: see e.g. Nassim Taleb: The Black Swan. The Impact of the Highly Improbable and the response Elie Ayache: The Blank Swan. The End of Probability.) This is linked to Meillassoux’s definition of “Tatsache” and “Archi-Tatsache” presented in the last post (I use the German translation because I’m still waiting for my copy of Pli with the English translation). How can we talk about the necessity or probability of facts and events, if we have a restricted or no knowledge of their being otherwise. This questions the very foundations of possibility and probability in modal logic and probability theory. Taleb and Ayache talk about a backward narrative of an event that constructs its own possibilities; after it happened! Therefore each reconstruction with traditional modalities has to be questioned regarding their metaphysical and/or ontological conception of probability and possibility. The relation between Existence and Possibility is more complicated than I presented it so far. I’m not sure how much better a reconstruction with Kripke works, but it will be more accurate than my previous version. [Note that another problem arises here. I blurred distinctions between the terms “probability”, “possibility” and “potentiality”. They are related, but are they all prone to the same problems? I’m not sure whether Meillassoux is clear in distinguishing them.]
  • There is a line of tradition from Bergson to Deleuze to Badiou and finally to Meillassoux where they think in a novel way about the concepts of virtuality and event (that will actually be the subject of my Master thesis). Traditional logic and probability don’t contain these concepts and are challenged by it. A big question is, if these concepts can be integrated in these systems, or if the systems can be expanded, or if we need completely new ones (DeLanda in his book Intensive Science and Virtual Philosophy tries to answer this question via Deleuze; at the moment I’m writing a paper about his contribution on social ontology and I want to use his book to reconstruct Deleuze’s conception of virtuality in my thesis). My method on this blog is an experimental one. I tried it with a simple concept of modal logic (System T) and will now try it with a more complex one (Kripke). Something I have to explore more is how these concepts are applied to epistemological (sometimes also called gnoseological) or ontological distinctions (see for example footnote 7 on p. 232f in Meillassoux’s article).
  • Something that’s already implicit in the points above are the relations (or more logical: opposites) between modal terms and also the relation between epistemological and ontological statements. What does it mean for the logic of a model that an event creates its own possibilities? About what kind of possibility do we talk in that moment? How is this related to novelty and is a novelty an actualization of something that was virtual or possible before? Is a novelty purely epistemological or is there meaningful way to talk about novelty in the world (and is it than on the level of virtuality, reality or actuality)?

All these questions and problems lead me to the possibilities of this blog. In a homework, paper or thesis I have the problem that I want to present something finished or thought through to a certain degree. I would never have submitted a paper where I use System T to reconstruct Meillassoux because I see the problems. On this blog I can publish impasses, directions of thought and experiments. Even if the results are only to see that the impasses, it is a possibility to write something, practise writing (especially writing in English and not my native language) and have feedback (at the moment only a handfull of people, but maybe that will change).

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Some Thoughts about Modal Logic

Thinking about Meillassoux’s argument from facticity and how to reconstruct it or at least know how to proof or disproof it, I had to think about modalities. (More on this argument in another post soon)

So I got back to my superficial knowledge about modal logic (sadly my philosophy department never offered a course on modal logic – at least as far as I know). Primarily I wanted a tool to think about the relations between modal propositions. Because my books about logic only deal with modal logic either superficially or simply as a marginal note, I need something else. Books and scripts about modal logic I could find on the internet or as eBooks in my libraries (physical editions in time of this crisis are unsurprisingly not available) need a lot of time to work through. At the moment I should focus and not spend too much time with this sort of thing (even if I really want to do so in the future). Nonetheless with a little internet recherche and my knowledge about traditional logic systems, I tried to develop a tool on my own. To understand Meillassoux’s argument a lot is done by understanding the relations between modal propositions.

Therefore I started with the square of oppositions. In one of my logic books this was already applied to modal logic (Zoglauer: Einführung in die formale Lokig für Philosophen). A little bit of browsing showed me that there is also a logical hexagon and a logical octagon. The result of my thoughts is a logical octagon with the oppositions between eight different modal propositions.

This logical octagon is based on an ordinary Propositional Calculus. Ordinary in this context means that it is two-valued (a propositions is either false or true). Propositional Calculus means that I’m working with propositional logic. In propositional logic the main elements that have truth-values are propositions which are not divided into smaller units (something done by syllogistic or predicate logic). Calculus may be a little bit misleading, because I don’t consider axiomatization and (even if I have superficial knowledge about Kripke worlds) I don’t consider semantics. Most of the things I take intuitively true (Np -> p; Np -> Mp) are true in a logical system that Zoglauer calls System T. With Kripke that is not so clear. This probably leads to a lot of problems. But (again) I want to think about oppositions intuitively and develop more detailed variants in the future. Because of a missing concrete calculus I cannot make real proofs here. When I write proof, I work with simple logic and not strong mathematical proof.

The modal propositions

There are eight modal propositions I considered:

Existence:

Formally: A

This is just the proposition. (You can also say: A is true.)

Non-Existence:

Formally: ¬A

This is the negation of the proposition.

Possibility

Formally: PA

This means: A is possible.

Necessity

Formally: NA

This means: A is necessary.

Impossibility

Formally: ¬MA (= N(¬A))

This means: A is not possible.

Non-Necessity

Formally: ¬NA (= M(¬A))

This means: A is not necessary.

Contingency

Formally: (MA)∧(¬NA)

This means: A is possible and not necessary.

Non-Contingency

Formally: (NA)∨(¬MA)

This means: A is either necessary or impossible.

Constructing the logical octagon

The logic book mentioned above presents a square of oppositions for the following four propositions: Necessity, Impossibility, Possibility and Non-Necessity (see figure 1).

Figure 1
Figure 1 (Zoglauer uses “p” instead of “A”)

This was the base for my extension. I ordered the other four propositions around it as it is shown in figure 2.

Figure 2
Figure 2

Now I started to ask the question in which opposition these propositions are. In classical logic since Aristotle there are four oppositions:

Contrary

Meaning the two propositions in opposition can both be false, but not both true. (Contrary means also that it is possible that one is true and one false.) [In the figures as normal arrow in both directions]

Subcontrary

This means that both proposition can be true, but they can’t be false at the same time. (Also meaning that one can be true and one false.) [In the figures as dotted arrow in both directions]

Contradictory

In a contradictory opposition one of the propositions is true and one false. No other combinations of truth-values are possible. [as thick arrow in both directions]

Subaltern

Meaning the truth of one proposition implies the truth of the other. Important is that this opposition is unidirectional. [as arrow from the implying to the implied]

More formally these oppositions can be analyzed with truth tables. Taking the modal propositions in the following logical connections:

¬(A ∧ B) for contrary
A ∨ B for subcontrary
A -> B for subaltern
A XOR B for contradictory

(Note that the A here stands for any modal proposition and not for the A as Existence.)

Figure 1 can now be enriched by filling it with arrows for the oppositions. This is shown in figure 3.

Figure 3
Figure 3

Let’s explain and proof some oppositions.

Necessity and Impossibility are in a contrary opposition. If it is the case that A is contingent, then both are false. If A is necessary or impossible, then one of the propositions is true and the other false. Therefore the option that both are true is excluded.

Necessity and Non-Necessity are in a contradictory opposition. If A is necessary, it is by definition not possible at the same time to be non-necessary. The same is true for Impossibility and Possibility. Either something is possible or it is not.

Necessity implies that it is possible. But Possibility doesn’t imply necessity. Therefore they are in subcontrary opposition. The same can be said about Impossibility and Non-Necessity. Impossibility implies Non-Necessity, but Non-Necessity doesn’t imply Impossibility. (Like many other sentences in this post you have to proof this in more detailed versions of modal logic.)

Possibility and Non-Necessity are in a subcontrary opposition. If A is contingent, then both are true. If A is possible, A can be non-necessary (as the previous sentence already said), but A can also be actually necessary which means that A is not non-necessary. They cannot be both false, because one of the following cases has to be true, implying either Possibility, Non-Necessity or both: Existence, Necessity, Impossibility or Non-Existence.

The next step is add the four remaining propositions. Melting figure 2 and 3 resulting in figure 4:

Figure 4
Figure 4

By checking the oppositions I could now fill into figure 4 the arrows for all the oppositions. The result is shown in figure 5:

Figure 5.3
Figure 5

Now I want to proof some of these new arrows and explain why there are four possible connections missing.

Contingency implies Possibility and Non-Necessity. If A is contingent, it cannot be necessary, impossible or non-contingent.
Necessity implies Existence and Existence implies Possibility (but not the other way around). The same can be said about the other side of the octagon: Impossibility implies Non-Existence and Non-Existence implies Non-Necessity.

Impossibility and Existence are in a contray opposition. If A is impossible, it doesn’t exist (cannot both be true). If A is not impossible (=possible), it can exist as well as not exist. Therefore A can be not impossible and not existent (both false). Necessity and Non-Existence are also in a contrary opposition. Either A is necessary and must exist (cannot both be true) or A is possibly existent and it is neither necessary not non-existent (both false.

Existence and Non-Necessity are in subcontrary opposition. They can be both true, if A exists, but not necessarily does so. One of them can be true and the other false, if A is necessary A or if A is non-existent. But they cannot both be false, because if A is not not necessary, it is necessary and therefore A has to exist.

Possibility and Non-Existence are in subcontrary opposition. If A is possible, A can be existent or non-existent. If A is not possible, it is impossible and the Non-Existence has to be true. Therefore they can both be true, but not both false and are in subcontrary opposition.

There are four oppositions that are missing in figure 5. These are the oppositions between contingency and existence, contingency and non-existence, non-contingency and existence, non-contingency and non-existence. The reason there are no arrows is that this is a new case of opposition. As I will show, the truth-value of the one side of the opposition doesn’t imply anything for the other side of the opposition. Therefore I will now call them independent.

Let’s proof this independence. I will only consider one of the four cases, but this proof can be made for the others as well. We examine contingency and existence. If existence is true, the contingency can be true as well as false (e.g. if A is NA which contradicts contingency). If existence is false, contingency can be true as well as false (e.g. if A is impossible which is contradictory to contingency). We can consider the other side of the opposition. If contingency is true, a can be true as well as false (via definition). If A is not contingent, there are three possibilities. From figure 5 we see there are three contradictory propositions to contingency: necessity, impossibility and non-contingency. Necessity implies A, impossibility implies ¬A and non-contingency means that A is either impossible or necessary. Therefore if A is contingent, A can be existent as well as non-existent. The conclusion is existence and contingency are independent.

Conclusion

Hopefully this construction of a logical octagon is not completely in vain. I haven’t considered the dependence on the different versions of modal logical calculus. At the moment my order of modal logical oppositions seems coherent and conclusive to me. I hope that I can use them as a tool to analyze Meillassoux’s argument from facticity with it.

I would really be delighted, if someone can either support this construction or show me at which points I made mistakes.

A look ahead

Even if I want to use this octagon as a tool to analyze Meillassoux, I think I cannot go too far with it. I think the octagon can show why Meillassoux’s argument from facticity seems to be conclusive. But a closer look will reveal some problems: the modalities Meillassoux uses operate on different levels. One of these levels can be called epistemic, because it considers the questions of knowledge; another one can be called ontological, because it considers how things are. Therefore the model has to be expanded and maybe some of the modal propositions are only useful on some, but not on all levels. There will also be questions about the notion of virtuality and how it interacts with modalities and the different levels they appear on.

[Figure 1 is a photography of my edition of Zoglauer, Thomas: Einführung in die formale Logik für Philosophen 4th edition. Figures 2 – 5 are made with a program called yEd. I’m very new to this program and these figures are my first results.]