Greg Restall

Martin Hägglund’s This Life ( Hägglund 2019 ) is an important treatise on metaphysics, philosophical anthropology, and political philosophy. It is also a critique of religious orientations to the world. I will reflect on Hägglund’s account of value and our finitude, and on his criticism of Martin Luther King, Jr’s political theology. According to Hägglund, King’s appeal to God when calling for justice should be replaced by an appeal to our communal norms. To defer to God is at best a colourful way of depicting our own commitments. At worst, it has no determinate content, and it threatens to absolve us of making justice here and now. I will show that while Hägglund’s account is a salutary corrective to a pervasive bad faith, this criticism misses its target. Any identification of God’s justice with our communal norms is to mischaracterise concepts like these. They are ideals that direct our attention outside ourselves, in the same way that love orients us towards an other . Hägglund’s account points us to the crucial concept of dependence , and that once we understand this better, we will see that a religious orientation towards the infinite and secular faith are compatible.

Two kinds of formal semantics for negation can help us understand puzzles in the psychology of reasoning: the Australian Plan semantics , which generalises the account of negation found in frame semantics for intuitionistic logic, relevant and substructural logics; and negation-as-failure , found in logic programming and default logic. We provide a framework where the two kinds of negation coexist, mapping them to the celebrated distinction between System 1 and System 2 thinking. We use the framework to target the phenomenon of Matching Bias : specifically, to explain why reasoners fare differently in card selection tasks, depending on whether the target property is phrased positively or negatively.

This paper is a series of reflections on Ross Brady’s favourite substructural logic, the logic MC of meaning containment. In the first section, I describe some of the distinctive features of MC, including depth relevance, and its principled rejection of some concepts that have been found useful in many substructural logics, namely intensional or multiplicative conjunction (sometimes known as ‘fusion’), the Church constants (⊤ and ⊥), and the Ackermann constants (t and f). A further distinctive feature of the axiomatic formulation of MC is its meta-rule, which is a unique feature of MC Hilbert proofs. This meta-rule gives rise to one further special property of MC, in that the logic is distributive in one sense, and non-distributive in another. The distribution of additive conjunction over disjunction (the step from p∧(q∨r) to (p∧q)∨(p∧r)) holds in MC as a rule, but not as a provable conditional, and in this way, MC is distinctive among popular substructural logics. (Anderson and Belnap’s favourite logics R and E are distributive in both senses, while Girard’s linear logic is distributive in neither.) In this paper, I aim to enhance our understanding of each of these distinctive features of MC, giving an account of what it might take for a propositional logic to meet these constraints. I will start with a presentation of Hilbert proofs for MC, and then showing how Brady Lattices (a natural class of algebraic models for MC) can help us understand each of these special features of Brady’s logic of meaning containment.

Last century, Michael Dummett argued that the principles of intuitionistic logic are semantically neutral, and that classical logic involves a distinctive commitment to realism. The ensuing debate over realism and anti-realism and intuitionistic logic has now receded from view. The situation is reversed in mathematics: constructive reasoning has become more popular in the twenty-first century with the rise of proof assistants based on constructive type theory. In this paper, I revisit Dummett’s concerns in the light of these developments, arguing that both constructive and classical reasoning are recognizable and coherent assertoric and inferential practices.

In this paper, I examine connections between the speech acts of assertion, denial, polar questions and justification requests, and the common ground. When we pay attention to the structure of norms governing polar questions, we can clarify the distinction between strong and weak denial, together with the parallel distinction between strong and weak assertion, and the distinct way that these speech acts interact with the common ground. In addition, once we pay attention to the distinct norms concerning justification requests, we can give a distinctive answer to Carroll’s puzzle concerning the force of the logical must and the sense in which certain rules for logical concepts can indeed count as definitions.

Natural deduction with alternatives extends Gentzen–Prawitz-style natural deduction with a single structural addition: negatively signed assumptions, called alternatives. It is a mildly bilateralist, single-conclusion natural deduction proof system in which the connective rules are unmodi_ed from the usual Prawitz introduction and elimination rules — the extension is purely structural. This framework is general: it can be used for (1) classical logic, (2) relevant logic without distribution, (3) affine logic, and (4) linear logic, keeping the connective rules fixed, and varying purely structural rules. The key result of this paper is that the two principles that introduce kinds of irrelevance to natural deduction proofs: (a) the rule of explosion (from a contradiction, anything follows); and (b) the structural rule of vacuous discharge; are shown to be two sides of a single coin, in the same way that they correspond to the structural rule of weakening in the sequent calculus. The paper also includes a discussion of assumption classes, and how they can play a role in treating additive connectives in substructural natural deduction.

In this book Catarina Dutilh Novaes—the accomplished philosopher of logic and historian of logic—takes a wide-angle view of the field and argues that the great tree of logic can be better understood if we pay close attention to its roots. Those roots, for Dutilh Novaes, are not the traditional foundations of logic, whether they be understood in terms of metaphysics, epistemology, semantics or mathematics, but rather, its genealogy. What is the soil out of which our competence with logic grew, out of which our logical practices took shape? This book is a sustained argument for the conclusion that logic took root and flourished out of our ability to dialogue, and that once we recognise that fact, puzzles about the nature of logic are clarified and a new perspective is gained.