Recent content in Home on consequently.org
I recently noticed that over the last day or so I’d got increased traffic to this rarely-visited website. Quite a few folks were reaching the home page, and a surprising number were landing on an old post of mine from 2004 about Haskell. Why on earth did that happen? Well, on August 2, John MacFarlane posted a 20 year retrospective on his much-used document conversion toolkit, pandoc. In telling…
This is an intensive class on logical bilateralism, taught by Sara Ayhan and me at esslli 2026. Logical bilateralism is an approach to meaning and consequence that foregrounds a symmetry between certain notions, like assertion and denial, proof and refutation or truth and falsity. Bilateralist approaches to such dual distinctions take both sides as primitive rather than—as in conventional…
Abstract: Graham Priest’s simple three-valued logic LP has many curious properties. It has the same valid formulas as classical logic, but differs from classical logic when it comes to valid sequents. The valid sequents do not uniquely characterise the logic: it is possible to have more than one different LP-“negation”, each of which satisfies all the LP-requirements, without being equivalent.…
Abstract: In this talk I will introduce a simple hypersequent calculus for the propositional modal logic S5, explaining how this representation encodes a natural form of reasoning about possibility and necessity. I then explain how the familiar structural rules of contraction and weakening take two forms in such a hypersequent calculus: they operate inside sequents, as familiar in substructural…
Abstract: Graham Priest’s simple three-valued logic LP has many curious properties. It has the same valid formulas as classical logic, but differs from classical logic when it comes to valid sequents. The valid sequents do not uniquely characterise the logic: it is possible to have more than one different LP-“negation”, each of which satisfies all the LP-requirements, without being equivalent.…
Abstract: Graham Priest’s simple three-valued logic LP has many curious properties. It has the same valid formulas as classical logic, but differs from classical logic when it comes to valid sequents. The valid sequents do not uniquely characterise the logic: it is possible to have more than one different LP-“negation”, each of which satisfies all the LP-requirements, without being equivalent.…
Abstract: In this talk I will introduce a simple hypersequent calculus for the propositional modal logic S5, explaining how this representation encodes a natural form of reasoning about possibility and necessity. I then explain how the familiar structural rules of contraction and weakening take two forms in such a hypersequent calculus: they operate inside sequents, as familiar in substructural…
This Easter season, my church marked the occasion with an art exhibition on the theme of Betrayal, and some short reflective evening services on Maundy Thursday, Good Friday, and Holy Saturday. For the Saturday service, I was asked to give a short reflection on the gospel text: John 19:31-42. Christ in the Tomb, by Mihály Munkácsy (1881) Since I have a website and somewhere to…
This Easter season, my church marked the occasion with an art exhibition on the theme of Betrayal, and some short reflective evening services on Maundy Thursday, Good Friday, and Holy Saturday. For the Saturday service, I was asked to give a short reflection on the gospel text: John 19:31-42. Christ in the Tomb, by Mihály Munkácsy (1881) * * * I didn’t grow up in a tradition…
Abstract: Starting with the his much-discussed 2006 paper “Must Do Better,’ Timothy Williamson has offered a series of methodological reflections and recommendations for philosophers. Good philosophy, for Williamson, (1) prefers precision and rigour to depth and profundity, (2) where possible, it avails itself of formal modelling, so as to not only prescribe definitions or analyses of concepts,…
The St Andrews Union Debating Society is a long-standing institution around these parts. Founded in 1794, this student-run society hosts regular debates. Last week, the society collaborated with SUMS , the St Andrews University Mathematics Society, to host a public debate on the status of mathematics , featuring the eminent Emeritus Professor Peter Cameron and my colleague Dr Walter Pedriali on…
py4612: Advanced Logic applies the tools of formal logic to make logic itself the object of study. We will explore the power and limits of logical tools and techniques. The main goals of the module will be to come to grips with some standard ‘metatheoretical’ results about logic: (1) the Soundness and Completeness Theorems, which together show that proofs and models can be used analyse the same…
This is a longer, more relaxed and interactive version of the talk “Why Logic Matters for Philosophy, and why Philosophy Matters for Logic” that I first gave as a part of the Inaugural Lecture Showcase of the Philosophy Department. The poster for the presentation. The slides for the talk are available here.
I gave a short, 25 minute lecture, entitled “Why Logic Matters for Philosophy, and why Philosophy Matters for Logic” as a part of the Inaugural Lecture Showcase of the Philosophy Department. The slides for the talk are available here.
I will be giving the 2025 Wendy Huang Lectures at the invitation of the Taiwan Association for Logic, Methodology and Philosophy of Science, at National Taiwan University, in Taipei, from October 21 to 23. The third lecture will also be a keynote presentation at the Taiwan Philosophical Logic Colloquium. The three talks are: Inferentialism for Everyone Tuesday October 21 • (handout) I aim to give…
Alongside coordinating and lecturing in PY2010: Intermediate Logic this semester, I have the fun task of teaching one of the tutorial groups, together with our enthusiastic and capable graduate student tutors. My Tuesday morning cohort is a microcosm of the diverse student body at St Andrews. Of the 13 students, we have two each from England, China, and the USA, and we have one each from Botswana,…
In October, I will be giving the 2025 Wendy Huang Lectures at the invitation of the Taiwan Association for Logic, Methodology and Philosophy of Science, at National Taiwan University, in Taipei. In preparation for those lectures, I’m trying out the ideas in three sessions in the Metpahysics and Logic Group at Arché. The three talks are: Inferentialism for Everyone (handout) I aim to give an…
I’ve returned from a productive semester of research leave, and I’m enjoying being back in the classroom teaching Intermediate Logic. Now that my big book has a complete first draft, my writing energies are focused elsewhere. I’m gearing up to give the 2025 Wendy Huang Lectures in Taiwan next month, so from this coming Wednesday, I’ll be giving the material a test run with the home crowd.…
py2010: Intermediate Logic is a University of St Andrews undergraduate subject in which we cover important results in logic to philosophy students. It’s taught by Greg Restall, together with a committed crew of postgraduate students. The subject introduces the proof theory and model theory of propositional, modal and predicate logic–in that order. I’m using the new textbook Logical Methods,…
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 con- cepts that have been found useful in many substructural logics, namely intensional or multiplicative conjunction (sometimes known as…
This is my next book-length writing project. I am writing a book which aims to do these things: Be a useable introduction to philosophical logic, accessible to someone who's done only an introductory course in logic, covering at least some models and proofs for propositional logic, and maybe a little bit of predicate logic. Be a user-friendly, pedagogically useful and philosophically motivated…
Abstract: Starting with the his much-discussed 2006 paper “Must Do Better,’ Timothy Williamson has offered a series of methodological reflections and recommendations for philosophers. Good philosophy, for Williamson, (1) prefers precision and rigour to depth and profundity, (2) where possible, it avails itself of formal modelling, so as to not only prescribe definitions or analyses of concepts,…
Abstract: Contemporary type theorists share interests and concerns with philosophical logicians. This isn’t surprising, since Per Martin-Löf is (among other things) a philosopher, and type theory was born in philosophy. Although type theory has come into its own in computer science—and, more recently, in mathematics with the rise of assistants—the connections between type theory and philosophical…
Abstract: Starting with the his much-discussed 2006 paper “Must Do Better,’ Timothy Williamson has offered a series of methodological reflections and recommendations for philosophers. Good philosophy, for Williamson, (1) prefers precision and rigour to depth and profundity, (2) where possible, it avails itself of formal modelling, so as to not only prescribe definitions or analyses of concepts,…
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 21st century with the…
Abstract: In this talk, I will defend contingentism, the idea that some things exist contingently. It might be surprising that this needs defence, but natural reasoning principles concerning possibility and necessity on the one hand, and the existential and universal quantifiers on the other, have led some to necessitism, the view that everything that exists, exists necessarily. Almost all recent…
Abstract: In this talk, I will defend contingentism, the idea that some things exist contingently. It might be surprising that this needs defence, but natural reasoning principles concerning possibility and necessity on the one hand, and the existential and universal quantifiers on the other, have led some to necessitism, the view that everything that exists, exists necessarily. Almost all recent…
Abstract: In this talk, I will defend contingentism, the idea that some things exist contingently. It might be surprising that this needs defence, but natural reasoning principles concerning possibility and necessity on the one hand, and the existential and universal quantifiers on the other, have led some to necessitism, the view that everything that exists, exists necessarily. Almost all recent…
Abstract: In this talk, I will defend contingentism, the idea that some things exist contingently. It might be surprising that this needs defence, but natural reasoning principles concerning possibility and necessity on the one hand, and the existential and universal quantifiers on the other, have led some to necessitism, the view that everything that exists, exists necessarily. Almost all recent…
Abstract: In this talk, I will defend contingentism, the idea that some things exist contingently. It might be surprising that this needs defence, but natural reasoning principles concerning possibility and necessity on the one hand, and the existential and universal quantifiers on the other, have led some to necessitism, the view that everything that exists, exists necessarily. Almost all recent…
Abstract: Twenty-first century mathematics has seen the rise of the proof assistant, and mathematical practice has changed significantly with the rise of these new tools. One consequence of this change, for the philosopher of mathematics, is the wider adoption in mathematics of intuitionistic logic, since many mathematicians now rush to formalise results in the language of constructive type theory…
Abstract: In this talk, I will defend contingentism, the idea that some things exist contingently. It might be surprising that this needs defence, but natural reasoning principles concerning possibility and necessity on the one hand, and the existential and universal quantifiers on the other, have led some to necessitism, the view that everything that exists, exists necessarily. Almost all recent…
In a few days, I’m heading off to North America, for a brief trip to give a few talks, at Chapman University in California, Calgary University in Alberta, and finally, a brief stop at CUNY in New York. I’m looking forward to meeting new friends and catching up with old ones, as well as the chance to talk with smart people about my research. Meanwhile, I’ve managed to do a bit of reading during…
In this paper, I show how to incorporate insights from the model-theoretic semantics for negation (insights due the late J. Michael Dunn, among others, in his paper “Star and Perp: Two Treatments of Negation”), into a proof-first understanding of the semantics of negation. I then discuss the how a logical pluralist may understand the underlying accounts of proofs and their…
Abstract: Suppose we have a language involving non-denoting singular terms. (The language of everyday mathematics provides one example. Terms like \(\frac{n}{m}\) and \(\lim_{x\to\infty} f(x)\) do not denote, for appropriate choices of \(m\) and of \(f\).) It is not too difficult to define inference rules for an appropriately free logic that incorporates non-denoting terms. If \(t\) does not…
This month’s reading was mostly nonfiction, featuring Kevin Hart’s Contemplation (The Movements of the Soul), Haruki Murakami’s essay collection Novelist as a Vocation, the harrowing Chasing Shadows: The Life and Death of Peter Roebuck by Tim Lane and Elliot Cartledge, Drew Neil’s Practical Vim (I’m trying to brush up my text editing skills), and finally, Timothy Larsen’s George MacDonald in the…
In this paper, I defend contingentism, the natural idea that some things exist contingently. Had my parents not met, I would not have existed. It is perhaps surprising that an everyday idea like contingentism needs defence, but natural reasoning principles concerning possibility and necessity on the one hand, and the existential and universal quantifiers on the other, have led some to necessitism,…
Happy New Year, everyone! As 2024 draws to a close, I’ve finished another month of reading, so let’s close out my log of books read over 2024 with a short description of December’s reading. First up, I enjoyed reading the second entry in Dan Moren ’s Galactic Cold War series: The Bayern Agenda . As with the previous entry , this was a fun spy thriller, with a cold war…
I’ve completed the moderation of the exam for Intermediate Logic, and with that, the final administrative responsibilities for this semester are complete. Now it’s time to take a short break over Christmas and the New Year, and then to start a semester of research leave. I’m looking forward to time set aside to think, to write, and to talk to colleagues, near and far. I have some…
Abstract: What does the semantically anti-realist revisionary programme of Michael Dummett have to do with contemporary work on proof assistants? What are mathematicians doing when they encode their proofs in these proof assistants, based on constructive type theory? What does all this have to do with the(?) norms of assertion? (And are these norms of assertion relative or absolute, anyway?) Can I…
My last PhD student at the University of Melbourne has completed his project, and is now Dr John Cleary. Congratulations, John! It was so much fun to help supervise your project. I’ve learned a lot about Albert Lautman, and his account of the development of mathematics and the dialectic of ideas, problems and mathematical progress.
This week, Aaron Cotnoir’s Instruments of Unity project and I are hosting a short visit from our friend (and my PhD supervisor), Professor Graham Priest . It’s always enjoyable to spend time with him, and tomorrow, we’re going to teach a the second-last lecture class for my Intermediate Logic cohort together, on the liar paradox and non-classical logic. Today, he gave a talk on nothing…
I mentioned yesterday that this month I’ve enjoyed rereading Kim Stanley Robinson’s Mars Trilogy . This time around, after completing my re-read, I’ve enjoyed listening to Marooned on Mars , a podcast devoted to Kim Stanley Robinson’s fiction. The initial conceit of the podcast was that the hosts, Matt Hauske and Hilary Strang (two humanities academics, based in Chicago)…
This teaching semester has been keeping me so busy that I have not kept up with my monthly reading logs. I’ve had enough time to read , but I haven’t found the time to keep you, my reader, up with what I’ve been reading. I’ll attempt to remedy this now, by giving a very brisk run-down of my reading over the last three months. This last three months has been dominated by fiction…
Abstract: Suppose we have a language involving non-denoting singular terms. (The language of everyday mathematics provides one example. Terms like \(\frac{n}{m}\) and \(\lim_{x\to\infty} f(x)\) do not denote, for appropriate choices of \(m\) and of \(f\).) It is not too difficult to define inference rules for an appropriately free logic that incorporates non-denoting terms. If \(t\) does not…
Abstract: Michael Dummett, in a presentation to the Aristotelian Society 65 years ago (in 1959) inaugurated a long-running debate over semantic realism and anti-realism, and the issue of settling an appropriate logic as a necessary prolegomenon to fruitful discussion metaphysics. Dummett argued that the principles of intuitionistic logic are semantically neutral, but that essentially classical…
Generic claims, such as Birds fly, Men are violent, and Mosquitos carry Ross River Fever, seem pervasive across human thought and talk. We use generic claims to express our understanding of the world around us and our place in it. These generic claims are useful even though they admit exceptions. We can agree that birds fly, even though emus don’t. Mosquitos carry Ross River Fever, but not…
Abstract: When I use a calculator to tell me that 245 × 46 = 11,270, I learn something that I didn’t know before, even though calculators don’t have any beliefs or knowledge. Even small children know how to count things, and it is through our own capacity to enumerate and count things that we learn basic arithmetic. Calculators do not count things in any sense like we do, yet we can use them to…
py2010: Intermediate Logic is a University of St Andrews undergraduate subject in which we cover important results in logic to philosophy students. It’s taught by Greg Restall, together with a committed crew of postgraduate students. The subject introduces the proof theory and model theory of propositional, modal and predicate logic–in that order. I’m using the new textbook Logical Methods,…
Here’s August’s book haul: This month I enjoyed three novels. The most experimental of which was Olga Ravn ’s The Employees: A workplace novel of the 22nd Century , which has the form of a series of witness statements from the crew of a ship, now far away from earth. The workers, both human and artificial, have been tending a number of exotic objects from the planet New Discovery, and they find…