Research Objectives

The notion of logical discrimination captures what can be “told apart” in a given logical system. We say, for instance, that intuitionist logic can discriminate between p and ~~p, whereas classical logic cannot, or that paraconsistent logics can tell different inconsistent theories apart while from a classical viewpoint there’s only one such theory—the trivial one. Still, even when it is acknowledged that non-classical logics like intuitionist and paraconsistent logic allow for finer discriminations than classical logic, considerations of that kind are not assumed to bear upon core issues in the philosophy of logic. Consider, for that matter, the fact that for intentional notions like belief or meaning the granularity-problem of finding the right way to discriminate meanings or to characterise the content of intentional states is a central concern (Barwise [1997], Stalnaker [1984]), while similar considerations about logical discrimination do not have a similar impact on the choice of a logic. The present project is motivated by the view that thinking explicitly about logical discrimination is as central to the choice of a logic as the traditional granularity-problem is for how we model intentional states.

The notion of logical discrimination can only take up a central role in our thinking about logic if it can stand on a par with the more established notions of consequence, truth, and validity. This presupposes that the notion in question is already sufficiently precise; which isn’t the case: Logical discrimination can refer to several distinct phenomena. Humberstone [2005], for instance, individuates four of them. The upshot of this project is double: A first aim is to clarify the concept or concepts of logical discrimination; a second aim is to give considerations about logical discrimination a central place in our theorising about logic. The latter is important because logical discrimination is already one of the central notions that are used to formulate the informational conception of logical consequence (Allo & Mares [forthcoming]), but also because implicit considerations about logical discrimination are already at work in the philosophy of non-classical logic and the debate between logical pluralists and logical monists.

To get a better grip on the different types of logical discrimination, we first need to focus on what logics (as opposed to presentations of logics) can discriminate, and then we need to introduce the distinction between (1) what can be captured by (interpreted) formal languages (the ability to discriminate between different structures), and (2) the discriminations that can be made by a specific logic. The former type of discriminatory power reduces to the expressiveness of a language; the latter type bears on the relations of synonymy and logical equivalence between formulae. This distinction between the expressiveness of a language and the granularity of the relations of synonymy and logical equivalence needs further clarification.

A detailed account of how expressivity and granularity are related will be developed by (a) providing a more precise formulation of the formal relation between discrimination as expressivity and as granularity; (b) spelling out a number of illustrative examples based on the two standard contenders of classical logic (intuitionist and relevant logic); and (c) showing how the conceptual primacy of logical discrimination within an informational semantics benefits from a generalised account of logical discrimination.

State of the Art Review

The formal interest of the two guises of logical discrimination that were introduced above can be described by referring to how these already surface in the relevant literature. Discrimination at the level of the language (and its interpretation) is the focus of model theory. Discrimination as granularity is relevant in view of the inverse-relationship between synonymy and deductive strength. The latter relation holds for a wide range of logics (the core topic of Humberstone [2005]), and is captured by the truism that “the more a logic proves, the fewer distinctions it registers” (ibid, 207). As such, logical discrimination as granularity is closely related to the traditional focus on valid arguments and the study of reasoning or inference, while the expressivity of a formal language is more concerned with our means for describing the world. The interest of both types of logical discrimination for the philosophy of logic is illustrated by two paradigm cases.

A first paradigm case refers to two ways of changing one’s logic: adopting an extension of a logic, or adopting a rival of that logic—a classical distinction due to Haack [1974]. In the first case, one extends a logic with some new logical vocabulary to obtain conservative extension of the original logic (e.g. by adding modal operators to classical propositional logic). In the second case, the language remains unchanged, but the set of theorems or valid deductions is altered (e.g. by moving from classical to intuitionist logic). Clearly, both changes involve a change in logical discrimination, but the change is of a different kind. One reason for thinking that the gap between a more fine-grained logic, and a more expressive logic isn’t absolute is that the distinction between extensions and rivals of classical logic isn’t absolute either. This is best illustrated by the well-known fact that intuitionist logic can be faithfully embedded in the classical modal logic S4 (but see Aberdein & Read [2009: 2.1.2] for a discussion of its significance). Thus, if the contrast between extended and rivalling logics can itself be retraced to a more general contrast between two guises of logical discrimination, a better understanding of the latter contrast could arguably also lead to new insights about how rivals and extensions of a logic are related. To the best of my knowledge, this line of research hasn’t been systematically pursued in the past.

A second paradigm case concerns what one believes to be the core subject-matter of logic. Orthodoxy has it that this is the concept of logical consequence (see, for instance, the second chapter of Beall & Restall [2006]). As such, the traditional view ties logical theorising to the study of the deductive strength of specific logics. If, by contrast, one looks at some of the most important results in twentieth-century logic, it is clear that the notion of definability is equally important (van Benthem [2008]). The connection with expressivity is plain in the latter case, but also on the more traditional view there’s a connection with logical discrimination via the inverse-relation between deductive strength and granularity. Consequently, getting a better grip on how granularity and expressivity are related, might help us to resolve the tension between the two competing views about what the primary subject-matter of logic might be.

Research Project

Connecting the two notions

The purpose of connecting two guises of logical discrimination isn’t to reduce one to the other, but rather to show, first, how they are formally related, and, secondly, how considerations about both granularity and expressivity function in the process of logical modelling. Expressivity is in the first place a property of formal languages and their intended interpretation, while granularity is a property of specific logics. By making this clear, we can already show that the adoption of an extended logic isn’t merely about enhancing the expressive means of our logic, but actually involves two choices of logical discrimination: one aimed at the language, and the other aimed at the logic itself.

In rough outline, we can thus characterise the distinction between expressivity and granularity as follows. When we decide to use a certain formal language, we decide on the in-principle available distinctions. When, furthermore, we settle on a given consequence-relation over that language, we decide which of the in-principle available distinctions really matter. That is, we decide which distinctions are retained, and which distinctions are collapsed. This type of interaction is nicely illustrated by the well-known fact that while we can embed classical logic (the more coarse grained logic) into intuitionist logic (a more fine-grained logic), we can only embed intuitionist logic (a more fine-grained logic) in a modal extension of classical logic (a more expressive logic with a more coarse grained propositional fragment).

When we think of logic as a modelling tool (Shapiro [2006: Chapt. 2]), we can relate considerations about granularity and expressivity to the criteria we use when we try to construct a good model. Basically, finding a good model means choosing the right degree of logical discrimination. The expressivity of the language we use is closely tied to one purpose of models: the description of the world. The granularity of a logic is also related to this descriptive aim, but is more intimately tied to another purpose: deductive inference. Again, we encounter the two already mentioned purposes of logical theorising, but since each aim can be furthered by choosing the right degree of logical discrimination, there’s no insurmountable gap between them. As illustrated above with respect to embedding one logic in another, the two processes of deciding on a set of in-principle available distinctions and the decision on which of these distinctions can be collapses do not occur independently of each other, but need to be balanced. Furthermore, there is no reason to presuppose that one type of logical discrimination is more basic than the other. Our means for describing the world influence what we can reason about, but the opposite is equally true. A pre-existing deductive practice also limits the distinctions that can be usefully made by a language. This two-way interaction seems closer to the practice of logical modelling, and needs to be made more precise to show how the descriptive and deductive aim of logic are related.

Applications

The main reason for the study of logical discrimination as an independent notion is its intended role within the informational conception of logic described in Allo & Mares [forthcoming]. The informational conception is meant as a contender for the traditional truth-conditional and inferential conceptions of logic. In particular, it proposes a double inversion of the usual order of explanation: Information comes before meaning, and information comes before possibility (Barwise [1997]). It is based on the assumption that the way we individuate informational content should be understood relative to how we access and use information. This approach presupposes that information is itself a relational concept: It is there to be accessed, but how it can be accessed depends as much on the informee as on the world. More exactly, it depends on what the world is like (i.e. how information is distributed), but also on the distinctions that are already available to the informee and on the distinctions the informee chooses to ignore. These features of the world and of the informee give rise to so-called global constraints on the logical space, which determine the possibilities, and thus yield a consequence relation.

This description of informational semantics implicitly favours the granularity account of logical discrimination, but thinking about the distinctions that are available to an informee also forces us the integrate the expressivity account of logical discrimination. A more elaborate description and defence of the informational conception of logic will therefore have to rely on a better understanding of the two main guises of logical discrimination.

Two further topics in the philosophy of logic that benefit from a better insight in the phenomenon of logical discrimination are described below.

Discussions about how logic and information are related pop up every now and then in the literature. A recent disagreement concerns the so-called modal and categorical information-theories (van Benthem [2010], Sequoiah-Grayson [2010]). The former is customarily known as the information-as-range paradigm (Stalnaker [1984]). The latter is related to the study of substructural logics, and the view that the Lambek calculus is a kind of basic logic for information flow and in particular inference. One aim of my “The Many Faces of Closure and Introspection” was to show that we could remain within the framework of modal information to model inferential processes, and thus avoid categorical information to explain inference as a dynamic process. The upshot of that proposal is, however, not to show that categorical information theory should be reduced to, or even be replaced with, modal information theory.

The difference between the modal and the categorical paradigm can be usefully compared to the tension between rivals and extensions of classical logic. Categorical information-theory, through its use of a substructural logic, proposes a rival of classical logic. Modal information-theory, through its use of modal logic, proposes an extension of classical logic. This insight situates the debate in the broader context of the integration of two guises of logical discrimination, and suggests that this is an area where a better understanding of how discrimination as granularity and as expressivity are related can be put to good use.

A from a formal point of view even more challenging application of insights about how granularity and expressivity are related, arises in the context of modal reconstructions of nonmonotonic logic (e.g. my “Adaptive Logic as a Modal Logic”). As with the previously discussed examples, this is a case where a weaker than classical logic is embedded in an extension of classical logic. The main difference is that here it is a nonmonotonic logic that is embedded in a monotonic logic. Unlike the former examples, it is not all that obvious how this can be understood in terms of two types of logical discrimination. For sure, the modal logic used to reformulate the adaptive consequence relation is more expressive, but it is much less clear whether we can understand the standard presentation of adaptive logics in terms of more fine-grained synonymy and logical equivalence relations. Yet, if this can be achieved, it promises to integrate the topic of defeasible inference within the informational account of logical consequence; which would count as a major advantage of the informational conception vis-à-vis the traditional truth-conditional and inferential conceptions.

Work Plan (based on full-time research position)

Because the completion of the proposed project requires the study of several formal notions, but also the development of a broader philosophical framework, the work plan described below develops two parallel paths that come together in the final part.

1. Preliminary work (year 1)

Formal path: Study the notions of synonymy and logical equivalence as they are used in, specifically, Humberstone [2005], and, more broadly, in the literature on algebraic semantics.

Philosophical path: Expand on previous work on “informational semantics,” with particular attention to the contrast between logical and non-logical forms of discrimination.

2. In-depth investigation (year 1–3)

Formal path: (i) Investigate how logical discrimination surfaces in algebraic and Kripke-style semantics. Use these two formalisms to analyse and compare the granularity and expressivity of modal and substructural logics. (ii) Investigate how logical discrimination surfaces in the Kripke-style semantics for adaptive logics.

Philosophical path: Further develop the logic-as-modelling view, and show how, for a given application, considerations about granularity and expressivity interact, and jointly determine how we choose a logic for that application.

3. Application (year 4–5)

Informational semantics: Describe how considerations about logical discrimination and expressivity function within an informational conception of logic. Explain what this means for the philosophy of logic (topics: logical pluralism, subject-matter of logic, contrast between rival and extended logics).

Modal and categorical information: See to what extent the contrast between modal and categorical information is analogous to the contrast between rival and extended logics.

Informational semantic for adaptive logic: Use the insights on how logical discrimination surface in Kripke-style models for adaptive logics to integrate adaptive logics within the informational conception of logic. Investigate what this means for the philosophy of logic.

References

Barwise, J.: 1997, ‘Information and Impossibilities.’ Notre Dame Journal of Formal Logic 38, 488-515.

Stalnaker, R.: 1984, Inquiry. MIT Press, Cambridge Ma.

Humberstone, I. L.: 2005. ‘Logical Discrimination’, in J.-Y. Béziau, (ed.), Logica Universalis, Birkhäuser Verlag, Basel, pp. 207–228.

Allo, P. and Mares, E.: 2011, ‘Informational Semantics as a Third Alternative?’, Erkenntnis, (forthcoming).

Haack, S.: 1974, Deviant Logic. Some Philosophical Issues. Cambridge University Press, Cambridge.

Aberdein, A. and S. Read: 2009. 'The Philosophy of Alternative Logics', in L. Haaparanta, (ed.), The Development of Modern Logic, Oxford University Press, Oxford, pp. 613–723.

Beall, Jc & G. Restall: 2006. Logical Pluralism. Oxford University Press, Oxford.

van Benthem, J.: 2008, ‘Logical dynamics meets logical pluralism?’ Australasian Journal of Logic 6, 182–209.

Shapiro, S.: 2006, Vagueness in Context. Oxford University Press, Oxford.

van Benthem, J.: 2011, ‘Categorical versus Modal Information Theory.’ Linguistic Analysis 36, 533–540.

Sequoiah-Grayson, S.: 2010, ‘Epistemic closure and commutative, nonassociative residuated structures.’ Synthese, 1-16. (Online First)