The role of logic can hardly be overestimated. On one hand, logical rules determine the basic canons of correct thinking. On the other, it is well known that very large portions of mathematics can be reconstructed only using logic and definitions. But what is logic? Our aim is to offer a unified account of both the nature of logic and of our knowledge of it. Our starting point is the twofold idea that the meaning of a logical expression is determined by the rules for its correct deductive use, and that to know the meaning of a logical expression is to know how to use it correctly. This very popular idea among philosophers has never been systematically explored. We aim to fill this gap in the literature, first by clarifying the position, and then by exploring its ramifications. We believe that it has the resources to provide a very intuitive account of logic and logical knowledge. Moreover, we argue that the all-important notion of harmony – the fact that basic logical rules display a certain balance – lays the basis for a compelling approach for demarcating ‘the province of logic’.
Inference and logic: some background
We all seem to have an intuitive grasp of the notion of validity: we reject arguments as logically invalid, on the grounds that a purported conclusion does not logically follow from the premises. Similarly, we feel compelled to accept the conclusion of an argument on the grounds that we accept its premises and we regard its conclusion as a logical consequence of them. But what is validity? And, if logically valid arguments are valid in virtue of the meaning of the logical expressions, how to account for the meaning of the logical vocabulary?
Our starting point for answering these questions is the simple observation that the basic rules for the use of the logical expressions exhibit a unique kind of symmetry: the conditions under which it is correct to assert a proposition involving a logical operation precisely match the consequences that can be drawn from it. Conditions of correct assertion and instructions for drawing consequences from logically complex propositions are captured by introduction and elimination rules respectively. For instance, the I-rule for a proposition of the form ‘If A, then B’ states that, if one can derive B from A, then one may conclude ‘If A, then B’. The E-rule is the well-known rule of modus ponens: ‘If A, then B’ and A jointly entail B. But this is to say that, from ‘If A, then B’ one gets precisely what was required to introduce ‘If A, then B’ in the first place, viz. a derivation of B from A. Similarly for the rules for all the other logical expressions: their I- and E-rules are perfectly harmonious.
We think this is no coincidence. On the contrary, the characteristic harmony of logical rules reflects something distinctive about the logical vocabulary, viz. that, so to speak, logic alone doesn’t give us knowledge of the world. This conception of logic traditionally animates the twofold inferentialist idea that (i) the meanings of the logical expressions are fully determined by their introduction rules and that (ii) to understand a logical expression is to be disposed to use it according to such rules. We find such an approach to logic – logical inferentialism – attractive. For one thing, it has a built-in epistemology: to understand a logical expression is to know that some inferences involving it are, in some sense, correct. For another, it provides us with a conception of what logic is: only expressions whose I- and Erules are harmonious may count as logical.
The view was first sketched in the writings of Gerhard Gentzen, and subsequently developed, somewhat unsystematically, in the writings of Michael Dummett, Dag Prawitz and Neil Tennant. In our project, we aim to provide the first book-length investigation of the inferentialist approach to logic. The book will not only critically introduce and discuss the existing literature: it will also develop several original ideas and arguments, and contain a wealth of philosophically fecund technical results. In particular, it will provide a novel account of harmony, with the help of which we develop a novel approach to the logical demarcation problem. Furthermore, we show that logical inferentialism implies no commitment to intuitionistic logic, nor to verificationism, contrary to what Dummett, Prawitz and Tennant have influentially argued. It will draw precise boundaries between logic and mathematics, and develop a novel inferentialist epistemology of logic – one that is immune to objections recently advanced by, among others, Timothy Williamson.