The present book is the third in a series of collaborations, its two predecessors being:

• • Beyond Sets: A Venture in Collection-Theoretic Revisionism (Frankfurt: Ontos Verlag, 2010).

• • Reflexivity: From Paradox to Consciousness (Frankfurt: Ontos Verlag, 2012).

In this book, as in the others, a key philosophical concept (collectivity/totality, reflexivity/self-orientation, and in this case categoricity/sortalization) is subject to critical scrutiny and innovative exploration.

A great deal of water has flowed under the philosophical bridge since the tract on The Conception of Types in the Light of the Modern Logic (Der Typusbegriff im Lichte der Neuen Logik) by Carl G. Hempel and Paul Oppenheim (Leiden, 1936). This classic venture at renovating categorization in the heyday of logical positivism was predicated on now-outdated perspectives and requires updating in the wake of the wider horizons that subsequent philosophizing has opened up. The present deliberations represent an attempt to accommodate these broadened perspectives.

The authors are grateful to Estelle Burris for aid in preparing their text for publication.

A number of the lessons regarding categories that we have emphasized throughout also play a role in classical puzzles and paradoxes. We have attempted to counter a range of tempting philosophical conceptions of categories, arguing among other things that:

• • Categories are not set-like entities.

• • Categories typically carry the intentionality of their membership conditions rather than being defined purely extentionally in terms of membership.

• • Both categories and the similarity relations that travel with them can be essentially vague.

• • Categories function pragmatically in terms of salient purposes and interests, and their application can thus vary with different purposes and interests.

• • Categorical classifications need not form tree-like structures of exclusive and exhaustive sub-categories.

Each of these points appears again in a consideration of category mistakes and a range of paradoxes, in either classical or contemporary form:

• • The Sorites and its ancient kin: the Phalakros and the Millet Seed.

• • The Ship of Theseus, the Statue and the Clay, Dion and Theon and the Problem of 1,001 Cats.

• • The Sancho Panza Hanging Paradox and The Contract of Protagoras.

• • The Liar.

Paradox is often presented in terms of a set up followed with a final “gotcha” question:

• • Does a trio of sand grains constitute a heap or not?

• • Once we have replaced each plank, do we have the ship of Theseus or not?

• • Is this lump of clay a statue or not?

• • Is the Liar sentence true or not?

But the appropriate response is sometimes not an attempted “solution” in the terms in which the “gotcha” question is phrased but a rejection of the question and perhaps the entire setup. For the paradoxes we will survey, the proper is often “The question, and perhaps the entire set up, is based on a misunderstanding of the nature of the categories in terms of which it is phrased.” For a range of classical paradoxes, such an approach allows us to characterize fairly precisely what that categorical misunderstanding amounts to, even if it doesn't always offer the kind of solution that the “gotcha” question seems to demand.

Our aim is to point out the crucial role that categories play – and that our assumptions regarding the nature of the categories play – in creating these puzzles in the first place. Although our analysis may at times implicitly suggest a, what we have to offer need not stand or fall with so ambitious a goal.

An Arc of History

Our concern is with categories in general, including chemical categories, biological categories, color categories, social and cultural categories, even mystery categories and categories of fictional entities. But much of the philosophical history regarding the nature of categories in general is buried beneath a history focused on a very specific set of categories: the categories, envisaged in a universal classification as the highest kinds or genera. A range of alternative and conf licting proposals as to which are the categories in this sense stretches from Aristotle through Kant and Husserl to the present.

Which are the highest categories is a topic of controversy, both historical and contemporary. We have also noted the ancient debate (as old at least as Porphyry) as to whether the categories are (mere) human contrivances or reflect objective differences in the nature of things via natural kinds. The philosophical history reveals a range of different approaches to the nature of categories – envisaged alternatively as metaphysical, epistemic, linguistic, or pragmatic. But it also reveals an intriguing trail of development through these approaches. In broad strokes, the history of categorytheorizing exhibits an increasing naturalization. We will trace that trail of changing interpretations of the categories through exemplars in Aristotle, Avicenna, the Ramist Revolt, Locke, Kant, Peirce, Frege, Husserl, and Ryle to contemporary debates.

The two central areas of debate, both historically and in contemporary guise, are debates as to which the categories are and to what they are. The lesson we draw from history is that both areas of controversy reflect deep philosophical mistakes. One mistake is the presupposition that there is some unique and ultimate set that are the categories. Another is the mistaken assumption that categories must fall to one side or the other of a false metaphysical/epistemic divide.

Aristotle

Aristotle was equivocal as to whether categories are about language or about reality. But this is understandable, given that our informative talk about things is an endeavor to make manifest what things actually are, adequatio ad rem: “approximation to reality” usually taken as “correspondence to fact.” All of the Aristotelian categories indicate different aspects of what something is or does.

Aristotle's interpreters are not unanimous with regard to the nature and function of his Categories. But his main Anglo-Saxon expositors have approached the matter from an epistemic point of view.

Preliminaries

Categories, categorization, and the organization of information are crucial topics throughout the broad realm of cognitive studies, not merely in philosophy but in both theoretical and empirical work in linguistics, psychology, and the brain sciences. Each of these disciplines offers intriguing findings regarding the ways we categorize—findings of importance for the understanding of human knowledge.

The central philosophical point that categories are crucial for cognition, in general, is fully recognized in the empirical work of other disciplines. As the linguist George Lakoff emphasizes,

A point that is clear philosophically, and essentially a priori, is that categorization and discrimination are linked. To recognize things as belonging to different categories is to be able to discriminate, at least conceptually, between items in those different categories. The “at least conceptually” allows us to say that we can discriminate between prime numbers greater than a googol and non-primes greater than a googol, though actually being able to name any of the former may be beyond our abilities. With that proviso, indeed, to recognize things as belonging to different categories is to discriminate between them.

This in no way forces us to say that all discrimination entails categorization: that if we recognize that this thing is different than that, it must be because we have assigned the two things to different categories. Categories are kinds, with the kinship of kinds determined by the pragmatic context of our purposes. Two things x and y will always belong to different sets or different arbitrary collections: we need to merely think of (1) the set to which the coins in my pocket and x belong as members and (2) the set to which the coins in my pocket and y belong as members.

In its most general form, logical pluralism is the view that there is more than one correct logic. I call this generic claim "the plurality thesis". Different versions of logical pluralism emerge with different implementations of that thesis and, most notably, of its key components logic and correctness. On some readings of the plurality thesis, logical pluralism is completely uncontroversial, on others it may turn out to be a rather exciting position. In this opening chapter, I identify an interesting, revisionist reading of the plurality thesis that is inconsistent with both logical monism and logical nihilism. Logical pluralism, so understood, claims that there are at least two correct theories of logical consequence. The chapter sketches historical developments of the view and gives an outline of the arguments defended in the book.

The previous chapter highlighted the difficulties of combining logical pluralism with a semantic account of rivalry between correct logics. This chapter discusses the weaker conception of applicational rivalry and its relation to the idea that logical consequence has a certain kind of normative force. I argue that all variants of logical pluralism that meet the following three conditions are susceptible to what has been called the collapse problem for logical pluralism: (i) that there are at least two correct logical systems characterized in terms of different consequence relations, (ii) that there is applicational rivalry among the correct logics, and (iii) that logical consequence is normative. I argue that if a position satisfies all these conditions, then that position is unstable in the sense that it collapses into competing positions. In a final step, I show how the collapse problem persists even without an explicitly logical normativity constraint, leaving only conditions (i) and (ii). The problem can therefore be viewed as a result of two core assumptions: plurality and a very weak sense of rivalry that is endorsed by virtually all logical pluralists.

The chapter is concerned with a commitment of the logical pluralist: if there are at least two correct logics, then these logics will either involve different logical vocabulary or they will assign different meanings to the shared vocabulary. A central question is how this plurality in meaning can be implemented within a pluralist framework. Pluralists typically endorse claims to the effect that (i) the connectives of the logics have different meanings or (ii) that the notion of validity employed by one logic may be relevantly different from the one employed by the other. A further important question is whether the plurality of meanings is confined to the theoretical level only or whether a corresponding plurality is postulated regarding the extra-systematic counterparts of the logical vocabulary of correct logics. I argue that both connectives pluralism and consequence pluralism are implausible when construed as pluralistic theories. Meta-contextualism–the view that the question of whether the meaning of the logical terminology is the same or different in different contexts is itself subject to semantic variability–is shown to have exceedingly radical implications.

This final chapter summarizes the main arguments given in the book. The central aim has been to defend logical monism–the view that there is only one correct answer to the question of whether or not a given argument is valid–against the challenges raised by the logical pluralist. The first task was to get clear on what, exactly, those challenges amount to. It turned out that pluralism, understood as the thesis that there is more than one correct logic, is not necessarily a controversial view. In some readings, it is obviously true. Crucially, logical monism, properly understood, needs no defense against those readings. But there are other versions of logical pluralism that do conflict with logical monism. Those are the readings I call revisionist. The account offered in this book allows for the obviously true readings of logical pluralism while resisting the revisionist approaches pursued by some pluralists. The basic tenets of this account are (i) that there is exactly one notion of extra-systematic logical consequence and (ii) that there is exactly one logical theory that provides the best account of this notion.

This chapter explores a number of ways to understand the key notions of the plurality thesis. First, I disambiguate three readings of the term logic: (i) purely formal systems, (ii) interpreted logical theories, and (iii) the subject matter of logical theories. I argue that this distinction is relatively lightweight and should be acceptable on all prominent views about the nature of logical consequence. Building on those readings of logic, I then explore different conceptions of what it means for a logic to be correct. In particular, I present a generic view of correctness of logical theories which is broad enough not to exclude pluralists who claim that the plurality thesis should better be put in terms of the legitimacy or the usefulness of a logic. I propose different ways to strengthen the generic view by means of a weak or a strong version of the correspondence view or the logic-as-modeling view. Finally, I introduce different implementations of the plurality thesis resulting from the different readings of logic and of correctness and identify the interesting version of the thesis which will be the subject of the rest of the book.

This chapter offers a detailed discussion of domain-based pluralism. In line with observations of previous chapters, the main focus is on the claim that logic in its canonical application to logical consequence is domain-dependent. I first review arguments brought forward in support of the domain-dependence of logic understood in that sense. I argue that none of them is conclusive. I then discuss two indirect arguments for domain-dependence in the form of arguments against universal applicability and argue that both can be resisted. I then highlight some open problems for domain-based logical pluralism. Combining the insights of these discussions, I argue that, as things stand, there is no good reason to assume that logical theories are domain-dependent.

Logical pluralism is sometimes motivated by the claim that it affords a more charitable interpretation of important debates in philosophical logic than monism does. This chapter argues that this claim is false. Pluralists are unable to account for important parts of logical and mathematical practice since they are in no position to account for potential rivalry between different logics. This is a problem for the charity-based project because most non-classical logicians formulate their theories of logical consequence as rivals to classical logic. I introduce three approaches to rivalry that focus on semantics, metasemantics, and applications, respectively. I argue in this chapter that neither the semantic nor the metasemantic approach offer a sense of rivalry that is plausible from a pluralist perspective, leaving the applicational approach for the following chapter. I further argue that this supports the conclusion that, at least as far as issues concerning the meanings of logical vocabulary are concerned, the correct logics are ultimately compatible. If we are interested in potential rivalry between the logics, we will have to look elsewhere.

In previous chapters, I construed logical pluralism as the view that there are multiple correct theories of extra-systematic logical consequence. Against this background, it may be tempting to think that logical pluralists are committed to the postulation of a plurality of extra-systematic logical consequence relations. In this chapter I argue that further options are available. I first show that, depending on the underlying notion of correctness, logical pluralism is compatible with any account of the cardinality of extra-systematic logical consequence. I then identify readings of the plurality thesis that give rise to the revisionist reading of logical pluralism that is the target of this book. The most obvious one is genuine plurality—the view that there is more than one extra-systematic consequence relation. A less obvious one acknowledges monism about extra-systematic consequence but argues that there cannot be a single precise theory that captures this relation. I propose a monist approach to logic in both the theory sense and the subject of investigation sense that rejects revisionist logical pluralism.

This chapter explores three dimensions on which logical plurality may arise. The first is concerned with the application of logic. Traditionally, logic was taken to be universally applicable in the sense that a deductively valid argument can be applied in any discourse or inquiry whatsoever. Some pluralists oppose that view by arguing that there are arguments which, though deductively valid, cannot be applied across the board. Deductive validity, on that view, is domain-dependent. The second dimension concerns semantics. Typically, if logics differ in their logical vocabulary, then they will draw the line between valid and invalid arguments in different ways. Even if the logical vocabulary of two logics is superficially the same, the sets of arguments the logics classify as valid may differ due to differences in the meaning of the logical vocabulary. The third dimension concerns the nature of validity. The most substantial kind of pluralism amounts to claiming that there is more than one extra-systematic relation that qualifies as a relation of logical consequence. The chapter outlines both the pluralist and the monist positions on those dimensions and identifies some core commitments.