The Problem of Plurality of Logics

Understanding the Dynamic Nature of Philosophical Logic

Pavel Arazim

• Publisher: Bloomsbury Academic
• Year: 2021
• ISBN:
  9781350146778
• Edition: 1
• Topic: Filosofia
• Subtopic: Logica in filosofia

As the foundation of our rationality, logic has traditionally been considered fixed, stable and constant. This conception of the discipline has been challenged recently by the plurality of logics and in this book, Pavel Arazim extends the debate to offer a new view of logic as dynamic and without a definite, specific shape. The Problem of Plurality of Logics examines the origins of our standard view of logic alongside Kant's theories, the holistic view, the issue of logic's pragmatic significance and Robert Brandom's logical expressivism. Arazim then draws on proof-theoretical approaches to present a convincing argument for a dynamic version of logical inferentialism, which opens space for a new freedom to modify our own logic. He explores the scope, possibilities and limits of this freedom in order to highlight the future paths logic could take, as a motivation for further research. Marking a departure from logical monism and also from the recent doctrine of logical pluralism in its various forms, this book addresses current debates concerning the expressive role of logic and contributes to a lively area of discussion in analytic philosophy.

1

Logical dynamism

The plurality of logical systems is one of the peculiar features of the modern development of logic. While it certainly is worthwhile to investigate not only the mathematical properties but also the philosophical credentials of specific logics, here the focus will be on the very plurality itself rather than on the virtues and vices of specific logical systems. The phenomenon of plurality challenges philosophers of logic. So far, it seems to have led them to two kinds of stances. On the one hand, there are logical monists, who argue for a particular system as being the true one or at least in some important sense privileged. On the other hand, lots of authors argue that more logics are correct in some sense and various flavours or elaborations of this tendency bear the title of logical pluralism. Given how complex this issue is and how much intellectual effort was already invested in its clarification, it may well seem pointless to try offering yet another view of it. Many ingenious forms of both logical monism and logical pluralism have been offered, as well as arguments for them. If one wants to, one can just review the broad offering of positions and choose the one that feels the best according to his or her individual philosophical and logical tastes.

While I discuss some of the accounts which have been proposed recently and which can be seen as rivals to mine in the last chapter, in this section I want to briefly present at the outset my own view on the plurality of logics. It should be noted that this book is not intended to only be an investigation of this view. It is not limited to my own thesis and its alternatives, and, in general, I want to present what the problem of the plurality of logics consists in and also display its broader philosophical implications, as well as provide an overview of the history of addressing it. But now let us start with my promised take on the plurality of logics.

1.1 Why dynamism?

I call my view logical dynamism because it is based on observations pertaining to the dynamic nature of logic. I will briefly explain why it cannot be straightforwardly identified either with logical monism or with logical pluralism. When we ask which logic is the right logic or which logics are the right logics, we should also ask what should the logic or logics be right for. If we were not to learn something about the nature and purpose of logic by investigating the issue of the plurality of logics, it would hardly attract so much attention. My approach to the plurality of logics is based on inferentialism and the related logical expressivism of Robert Brandom, as he presented these doctrines in Making It Explicit.¹ Yet here I will limit myself to more general observations to give the basic flavour of my view.

I take logic to be constituted by rules. In fact, any human affair is closely related to rules, lots of them being constituted by rules. This general remark does not oblige me to much as of yet. It might seem that in advance I am unfairly preferencing proof-theoretical approaches to logic over the model-theoretical ones. But in a weaker sense, even model theory and model-theoretical semantics, including its account of logical consequence, is based on rules, for example, on the rules of assigning elements of the model to expressions of a certain specified kind. Ultimately, I believe that logical expressions are constituted by inference rules, but there will yet be occasions to discuss this particular point. For the moment, let us focus on rules in general.

1.1.1 Some remarks on rules

Two features of rules will be of particular import for us here. For one, rules develop and are dynamic in their nature. This point, I believe, is fairly obvious to anybody who considers them without prejudice due to peculiar philosophical preferences. You can take any kind of rules and you will find not only a possibility of development but the actual development which they have undergone in the course of some time. The rules for passing an exam at university change, as the university wants to become more or less demanding, all kinds of laws keep being novelized as there is a political will to do that (hopefully for good reasons), even mathematical notions change all the time and with them the rules for doing mathematics. Before the advent of negative numbers it was correct to assert that adding a number never diminishes the number you start with, yet this changed with the advent of negative integers. Practically any expression can change and typically does change its meaning over some decades. Of course, some expressions might be more prone to such changes than others, but the tendency for them to develop is quite general.

1.1.1.1 Identity of rules

An obvious criticism can be raised here. When a given rule or system of rules changes, should we not say rather that new rules were accepted and not that the old ones have developed? In other words, what are the criteria of identity of rules across time and context? When the rules for, say, addition in mathematics change with the advent of negative integers, should we not say that the old rules were replaced, so that we do not have to do with rules A, B, C and so on anymore but rather with X, Y, Z and so on, and furthermore, with a new operation, which is different from the original addition, though it might be very similar? Well, my claim certainly is not that we always have to interpret the rules as remaining the same, despite all the changes. How they should be judged depends on the specific cases. On many occasions, both the claim that we are dealing with the same rules, though in a changed form, and the opposite claim that these are in fact really new rules can be equally warranted. Then the choice is just optional. What I, nevertheless, claim is that in general we have to reckon with the ability of rules to develop. In some cases, we have to say that it is still A, B and C that we are dealing with.

1.1.1.2 The real, clean shape of a rule

Those who believe that every rule is precisely identified with only one specific shape rely, I believe, on an illusion of something as the real and clean shape of rules. The same holds for meanings. It is tempting to think that there is something like a real, definite shape of, say, the conjunction, though we may struggle to pin it down exactly – to sufficiently disambiguate and differentiate it from other related meanings. But tempting as this is, such reasoning gets it wrong.

Every rule makes real sense only in the context of many further rules with which it interacts and which are also necessary if we are to understand the rule. As Robert Brandom puts it, understanding only one rule is a sound of one hand clapping.² But as we know from Wittgenstein, not all the rules can be explicit. Wittgenstein goes yet further by showing that for any rule which we make explicit, we rely on many others which have to remain implicit in the context of explication. And because understanding a given rule fully and explicitly would also mean understanding in the same way the other rules which have to remain implicit, it is simply impossible to get to a fully definite and specific shape of a rule. Explication and, consequently, explicitness is always partial.

Furthermore, we have learned also from Wittgenstein and Kripke³ that when we focus on a given rule, it is always open to a plurality of interpretations. On some level, this is no big issue in everyday life but it still undermines the notion of there being something like clear-cut definite meanings and rules which we just try to pick as precisely as possible by our language. The point is illustrated even by Bertrand Russell, who in his article ‘Vagueness’⁴ admits that even in mathematics, vagueness of terms can be at best reduced but never fully eliminated. In accordance with Russell, I by no means intend to say that a request for more disambiguation and clarification of what one means by a given utterance are illegitimate. Only that they are always dependent on the context and never fulfilled in a perfect, definite way. If anyone wants to consider definite rules and meanings as useful fictions or regulative ideas, I can agree with the proviso that their usefulness is limited and has to be adjudicated in every specific case. Sometimes the pursuit of as great a clarity as possible can be even harmful.

I think we were taught not only by Wittgenstein and Kripke but also by Quine and Davidson that all the rules, as well as meanings, always fail to be fully definite. Both the considerations pertaining to rule following, in particular Kripke’s example of impossibility to definitely distinguish addition from some kind of quaddition and the considerations pertaining to radical translation or interpretation, show this, each in a specific way.⁵ All these authors typically seem to speak of an intended interpretation and how we fail to pick just it and nothing else. Addition clearly seems preferable to any kind of quaddition. Gavagai seems more naturally translated as rabbit than as an undetached rabbit-part, yet all the attempts to distinguish them are short-lived. Their success is easily overcome by new distortions of the same kind. Now, it seems we have two options of how to interpret both the quaddition and the gavagai considerations.

In all of those examples, one obviously seems to be confronted with unequal alternatives. On the one hand, there is an obviously natural interpretation, which understands the + sign as denoting addition and gavagai as determining rabbit. The other interpretations just seem strained. From this perspective, it would seem that all those examples purport to say is that it is difficult, maybe even impossible, to pin down the correct interpretation. Yet I plead for learning a stronger lesson from them, namely that there is no definitive interpretation. There is always something open-ended and undecided. We may opt to go in one direction, which may also seem much more natural, yet we will have to make similar decisions later. All specifications of meanings remain partial and fail to be definite. Any specification is only provisory and will have to be revisited in new contexts.

This all means that both meanings and rules cannot be as clean and definite as they are often conceived. Let us focus once again on meanings. It is common to speak of propositions as of meanings of sentences and of concepts as meanings of predicates. A given sentence may fail to pick up exactly one proposition, though we can try to come closer to full clarity and the pinning down of a single proposition. This is the way that the orthodoxy goes. I think, however, that it is incorrect. The notion of a proposition or concept completely detached from actual language use is a fiction: sometimes, perhaps, a useful one but often harmful.

1.1.2 What about formal languages and formal logics?

It is in our practices that our rules and also meanings reside, not somewhere beyond. But it should be added that these practices are by necessity dynamic and always developing, perhaps not always for the good, but developing nevertheless. But still, is it not different in the case of formal languages? And what is dynamic about formal, mathematically defined logical theories, such as classical propositional logic, propositional intuitionistic logic, first-order logic, modal propositional logics and others?

Let us remind ourselves that already in ‘Two Dogmas’,⁶ Quine saw this objection coming and still chose to bite the bullet. I basically follow his lead. When we speak of the dynamic nature of meanings and rules and their constant development, this pertains to formal languages, as well. First of all, we understand formal languages only by means of the natural ones. When a student attending a logic course gets acquainted with conjunction, she has to understand that any formula of the form A∧B is true exactly in those cases in which both A and B are true, which amounts more or less to understanding conjunction. The boundary between the realm of the allegedly messy natural language and the allegedly neat formal one is thus at the very least unclear and the latter is strongly dependent on the former. On the other hand, it should be noted that the naturalness of natural language should also not be overrated. Obviously, it is regulated quite explicitly; it is not constituted only by our spontaneous linguistic behaviour but also by decrees on the part of linguists. It is thus codified though obviously less so than formal languages.

This leads us to further considerations about the relation between rules and their expressions. We normally distinguish between sentences we utter or write and what they state. The sentence There is a dog over there is certainly distinct from the dog itself. On the other hand, in the case of rules it seems that this distinction cannot be drawn so clearly; the rule and the sentence which expresses it appear to be intimately related. This impression is only partially right and in important respects misleading. While expressions of a rule can influence the rule much more directly than descriptions of a dog can influence the dog itself, they are still not the same thing.

1.1.2.1 Rules and their expressions

A given rule truly holds only if it is in some way acknowledged by the society or social group for which it is supposed to hold. This cannot always happen by means of explicit acknowledgement; at least sometimes it must be a mere tacit approval manifested in action, as we know from Wittgenstein. The rules thus have to live in our society and cannot be equated with their expression. When someone formulates the rule and wants it to be followed, his proposal has to be accepted by the members of the respective community.

A rule is thus a living social phenomenon. It lives in the mutual recognition of the members of the community and in their behaviour. If the rule holds, they typically act on it, though they also sometimes break it. Yet more importantly, they assess each other and themselves by the rule and act correspondingly, by encouraging one another to follow it and discouraging one another from breaking it. But what about the expressions of the rules? How does a statement such as ‘It is prohibited to trespass this line’ relate to the very rule that this line should indeed not be trespassed? The relations can be manifold. Let us review some of the forms can take.

If the given rule has not been established, uttering this statement or a similar one can lead to it. If the person uttering it has a particular authority, maybe the utterance can indeed establish the rule. Typically, though, the situation is not so clear and the utterance is more likely to just initiate a discussion about whether the rule should be adopted.

If the rule already holds, the utterance at least expresses it. This can be done for various purposes. We can merely remind ourselves of the rule. In more interesting cases we can be unsure about its actual shape. Remember that not all rules which we accept have to be explicitly formulated, and even if the given rule was expressed, we might be unsure how to interpret it. Furthermore, we can initiate a debate about what to do with the rule, whether to uphold it, modify it or withdraw it completely. Overall, we can thus see that an expression of a rule can cause or contribute to the rule’s coming into existence or to its modification. Yet, they still are not the same.

When we review the issue of formal languages and formal logics, I claim that the impression that their rules do have the definite and clear-cut shapes arises from the interchange between rules and their...