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This chapter distinguishes between various senses in which one might be a pluralist about logic. In several of these, pluralism is uncontentiously correct. The difficult issue is whether there are different accounts of validity that are equally correct. It argues for logical monism in this regard.

This book presents and defends what it calls logical pluralism, the view that there is more than one genuine deductive consequence relation, and that this plurality arises not merely because there are different languages, but rather arises even within the kinds of claims expressed in any one language. First, it sets logical consequence in its historical context, explains why it is important, and clarifies what is assumed to be the settled core of this notion. It defines logical pluralism and indicates the type of arguments that it takes to support the position. Concrete examples of the various consequence relations endorsed by qua pluralists are given.

By way of further clarifying logical pluralism, this chapter turns to some general objections. One objection is to argue for a pluralism about logical consequence which is wide enough to encompass classical, relevant, and intuitionistic logic. In doing so, this chapter proposes a view which the adherents of relevant and intuitionistic logic by and large would find repugnant. In order to endorse classical logic all instances of the law of the excluded middle should be taken to be necessary. But intuitionists take the law of excluded middle to have counterexamples. The premises of this objection are correct, but the conclusion, that pluralism is uninteresting, should be resisted. Other objections discussed in this chapter relate to cases, transitivity and reflexivity, warrant and entitlement, warrant and disjunctive syllogism, meaning theory, logical truth, Carnapian pluralism, and logical form.

This chapter continues elaborating and defending alethic functionalism. Particular attention is paid to how the functionalist can counter the problems of mixed inferences and mixed compounds that plague pluralist theories of truth. The second half of the chapter discusses the question of whether alethic functionalism entails a form of logical pluralism — the idea that there is more than one form of logical consequence.

This chapter first describes older approaches to the unity of science: strong reductionism and supervenience, and relates their drawbacks to the widespread practice—in the sciences—of referring to entities that don’t exist. Then it is shown how actual confirmation and deduction relations are established between discourses containing very different kinds of vocabulary by the use of gross correlational regularities, and it is shown how this is compatible with a minimal form of physicalism. Two extended illustrations, one from biology and the other from neurophysiology, are characterized. The chapter then turns to explaining how the forgoing analysis bears negatively on two forms of pluralism: logical pluralism and truth pluralism. The latter is undercut both in a form that takes our various discourses to have different truth predicates, and in the more innocuous form of taking there to be various truth-properties that different statements have.

Logical pluralism is a pluralism about logical consequence. Crudely put, a pluralist maintains that there is more than one relation of logical consequence. By way of illustrating the kind of claim involved in logical pluralism, a few analogies are considered in this chapter. According to the Generalised Tarski Thesis (GTT), an argument is valid if and only if, in every case in which the premises are true, so is the conclusion. Logical pluralism is the claim that at least two different instances of GTT provide admissible precisifications of logical consequence. Unlike the restricted Tarski Thesis, which admits only one instance of case (Tarski's models), the pluralist endorses at least two instances, giving rise to two different accounts of deductive logical consequence (for the same language), two different senses of ‘follows from’. There are at least two ways to not be a logical pluralist: reject the GTT or endorse exactly one instance.

The highly general notion of structure lets us raise metaphysical questions about logic. Is it conjunction or disjunction that is most fundamental? (An uncomfortable question! ‐ fortunately we may answer “both”; the joint‐carving notions can contain some redundancy.) Do logical notions carve at the joints? (Surely yes, since they are indispensable in fundamental theories.) Do metalogical notions ‐ such as logical truth and logical consequence ‐ carve at the joints? (Probably not; unlike logical notions they are not needed in fundamental theories.) Is logic classical? (At the fundamental level, yes; but in nonfundamental languages perhaps we should say not, because of vagueness and the liar paradox.)

This chapter looks at just a few of the ways in which a pluralistic view of logical consequence could be developed. It considers the upshot of recognising the plurality of consequence relations; how the plurality can be exploited; and ways that pluralism might be further developed in the theory of proofs. The moral of logical pluralism is that not every consequence relation needs to do everything that must be done by consequence relations. Failure to learn that (pluralistic) moral unnecessarily hinders logical theorising. It is one thing to recognise plurality in the development of a logical system, but something more interesting to let pluralism go somewhere. We can exploit the fact that we have more than one consequence relation, and allow the presence of a number of distinct consequence relations to do some work for us.

According to an increasingly popular view, the source of logical necessity is to be found in the essences of logical entities. One might be tempted to extend the view further in using it to tackle fundamental questions surrounding logical consequence. This chapter enquires: how does a view according to which the facts about logical consequence are determined by the essences of logical entities look in detail? Are there any more or less obvious problems arising for such a view? The chapter uncovers a prima facie result in favour of logical pluralism. However, it then goes on to raise some concerns for this result. It argues that, considered generally, it is difficult to see how essence could do all of the requisite work alone. The chapter also shows how considering things from the perspective of disputes between particular rival logics makes an interesting and important difference to the picture of things presented by the essentialist account.

This chapter explores the concept of logical consequence and defends logical pluralism. Logic, in the core tradition, involves the study of formal languages. However, the primary aim is to consider such languages as interpreted: languages which may be used either directly to make assertions and denials, or to analyse natural languages. Logic, whatever it is, must be a tool useful for the analysis of the inferential relationships between premises and conclusions expressed in arguments actually employed. If a discipline does not manage this, it cannot claim to be logic in its traditional sense. Many different basic units of evaluation have been taken to be the components of arguments: assertions, judgements, claims, utterances, interpreted sentences, sentences in contexts-of-use and propositions have all been singled out as the right kinds of things for logic to take as its raw material. Three features of logical consequence are necessity, normativity, and formality.

Logical pluralism addresses the following conditions. First, the settled core of consequence is given in the Generalised Tarski Thesis (GTT). Second, an instance of GTT is obtained by a specification of the cases in GTT, and a specification of the relation is true in a case. Such a specification can be seen as a way of spelling out truth conditions. Third, an instance of GTT is admissible if it satisfies the settled role of consequence, and if its judgements about consequence are necessary, normative, and formal. Fourth, a logic is given by an admissible instance of GTT. Lastly, there are at least two different admissible instances of GTT. Logic is a matter of preservation of truth in all cases, which lies at the heart of logical consequence, the settled core of follows from. This chapter considers two well-known specifications of cases _x : possible worlds and models for classical predicate logic. It notes the extent to which the canvassed accounts of consequence are admissible: the extent to which their respective judgements are necessary, normative, and formal.

This chapter presents variations on logical consequence and logical pluralism. If the arguments presented so far are sound, then there are different ways to settle the notion of logical consequence, each equally deserving the title ‘logic’. In this chapter, some other degrees of variation possible for the logical pluralist are considered. It examines an issue raised by such variations, as well as by variations considered in earlier chapters: whether plurality in logic comes only by way of plurality of languages. The chapter concludes by tying up some loose ends, especially with respect to paraconsistency, dialetheism, and the notion of endorsing a logic. Free logic, second-order logic, and higher-order logic are also discussed.

This introduction takes notes of the fact that we have many different kinds of discourses—moral discourse, mathematical discourse, scientific discourse, fictional discourse, and so on—and that one response to this has been various pluralist doctrines: pluralism about logic, about metaphysics, and about truth. The external discourse demand and the rejection of Quine’s criterion, and a companion doctrine about singular idioms can be used against the various pluralist views. Crucial in the attack is the fact that statements from a discourse-domain can be brought to bear evidentially and deductively to establish and refute statements in quite different discourse-domains. It is a puzzle how this is possible, one which it is promised will be resolved in part II, and specifically in chapter 4.

This chapter turns to specific objections against logical pluralism. There are a number of objections to classical, intuitionistic, and relevant logic which are specific to those logics, and not to the scheme of pluralism as a whole. Since it makes sense not only to be a pluralist about logical consequence, but to be a pluralist who endorses each of classical, constructive, and relevant reasoning, it becomes obligatory to say something about such logic-specific objections. These objections relate to coherence and relevant models, relevance and explosion, intensionality and the slingshot, and constructivity and semantics.

Logical conventionalism leads to logical pluralism. The chapter discusses various arguments for pluralism, based on more and less demanding principles of translation. The crucial problem case of a tonk language is discussed in detail and related to various philosophical points and distinctions from the previous chapters. The chapter also provides a general account of logical and conceptual pluralism in terms of structural inferential role or semantic counterparts. This machinery is then applied to give a conventionalist-friendly account of equivalence between logics. The chapter closes by distinguishing between different types of disagreements in the philosophy of logic – descriptive disputes, normative disputes, and metaphysical disputes. Together chapters 3, 4, and 5 constitute a full development of an inferentialist-conventionalist theory of logic.

This essay introduces the reader to the central issues in philosophical logic. The chapter rehearses the main proposals and problems driving contemporary debate in the field, and describes how the individual chapters in the present volume contribute to these debates.

This chapter investigates what we disagree about when we disagree about logic, on the assumption that judgments of logical validity are normative. If logic is normative, then the popular anti-realist thesis that there are no normative facts or properties generalizes—it entails that there are no logical facts or properties. When faced with this anti-realism, it is tempting to endorse a pluralist thesis, according to which two people who disagree about the validity of an argument can both say something true. This chapter explores the limitations of three prominent forms of pluralism: contextualism, relativism, and expressivism. It argues that none of these forms of pluralism gives an adequate account of what we disagree about when we disagree about logic.

This chapter explores the ramifications of a relativist view of logic with respect to the meanings of logical terms. It is a twist on the old question of whether classicists and intuitionists have any substantial disagreement, or whether they are merely talking past each other as they attach different meanings to the crucial logical terminology. The author argues that the very question of whether the meanings are the same or different is itself a context sensitive matter: it depends on what aspects of the situation are salient.

This chapter develops an interpretation of Wittgenstein’s method of language-games as a method of logic that exhibits continuities with Frege’s, Russell’s, and the early Wittgenstein’s conceptions of logic and logical analysis as the method of philosophy. On the proposed interpretation, the method of language-games is a method for isolating and modelling aspects of the uses of linguistic expressions embedded in human activities that enables one to clarify complex uses of expressions by gradually building up the complexity of clarificatory models. Wittgenstein’s introduction of the language-game method constitutes an attempt to overcome certain limitations of calculus-based logical methods, and to extend the reach of logical methods, so as to make accessible areas of language use and thought that are not accessible by means of calculus-based methods. Rather than rejected, calculus-based methods are retained as a special case of logical methodology. The method of language-games is thus compatible with the employment of calculus-based methods in logic and philosophy, and the method of language-games involves no exclusive claim to being the correct method.