Logicism is, roughly speaking, the doctrine that mathematics is fancy logic. So getting clear about the nature of logic is a necessary step in an assessment of logicism. Logic is the study of logical concepts, how they are expressed in languages, their semantic values, and the relationships between these things and the rest of our concepts, linguistic expressions, and their semantic values. A logical concept is what can be expressed by a logical constant in a language. So the question “What (...) is logic?” drives us to the question “What is a logical constant?” Though what follows contains some argument, limitations of space constrain me in large part to express my Credo on this topic with the broad brush of bold assertion and some promissory gestures. (shrink)

In my Steinberger 2009 I argued that Neil Tennant’s Harmony requirement is untenable because of its failure to account for the standard quantifier rules.1 Instead of justifying the customary rules for the existential and universal quantifiers, Tennant’s account appears to sanction only wholly unrestricted – and so patently disharmonious – quantifier rules. In his characteristically thoughtful response Tennant 2010, Tennant offers a sequent calculus version of his Harmony requirement that rules out such pathological would-be quantifiers. While I agree with Tennant (...) that recasting his Harmony requirement in the sequent format as he proposes blocks the said disharmonious quantifier rules, I submit that Tennant’s revamped Harmony requirement nevertheless misses the mark. I present two objections to substantiate my claim. First, I show that the crucial additional assumption underlying Tennant’s sequent calculus-based account – what I call the admissibility assumption – is excessively strong: so strong, in fact, that it renders otiose the core of Tennant’s original account. Second, I argue that the admissibility assumption, as a global requirement on deductive systems, is ill-suited for the purposes of codifying the intuitive notion of harmony. Fortunately, though, as I will demonstrate, we can dispense with the admissibility assumption altogether. (shrink)

According to Michael Dummett, we may think of the meaning of an expression as given by the principles governing the use we make of it. The principles regulating our linguistic practices can then be grouped into two broad categories (Dummett 1973: 396, 1991: 211). We might state them as follows: I-principles: state the circumstances under which an assertion of a sentence containing the expression in question is warranted. E-principles: state the consequences of asserting a sentence containing the expression. In the (...) case of the logical constants, we may associate a constant's I-principles with the set of its introduction rules and its E-principles with the set of its elimination rules (Dummett 1973: 454).1. (shrink)

The notion of harmony has played a pivotal role in a number of debates in the philosophy of logic. Yet there is little agreement as to how the requirement of harmony should be spelled out in detail or even what purpose it is to serve. Most, if not all, conceptions of harmony can already be found in Michael Dummett's seminal discussion of the matter in The Logical Basis of Metaphysics. Hence, if we wish to gain a better understanding of the (...) notion of harmony, we do well to start here. Unfortunately, however, Dummett's discussion is not always easy to follow. The following is an attempt to disentangle the main strands of Dummett's treatment of harmony. The different variants of harmony as well as their interrelations are clarified and their individual shortcomings qua interpretations of harmony are demonstrated. Though no attempt is made to give a detailed alternative account of harmony here, it is hoped that our discussion will lay the ground for an adequate rigorous treatment of this central notion. (shrink)