consequently.org: Greg Restall’s website in consequently.org: Greg Restall’s website Greg Restall's publications on logic and philosophyGreg RestallGreg Restallgreg@consequently.orgPhilosophy, Logic, mathematics, pdf, research, University, Greg Restall, Melbourne, Australia, VictorianoHugo
https://consequently.org/
en-usThu, 21 Oct 2021 09:25:00 UTCconsequently.org: Greg Restall’s website
https://consequently.org/
Mon, 01 Jan 0001 00:00:00 UTChttps://consequently.org/Presentations
https://consequently.org/presentation/
Mon, 01 Jan 0001 00:00:00 UTChttps://consequently.org/presentation/Assertions, Denials, Questions, Answers, and the Common Ground
https://consequently.org/presentation/2021/assertion-denial-common-ground-bristol/
Sat, 16 Oct 2021 00:00:00 UTChttps://consequently.org/presentation/2021/assertion-denial-common-ground-bristol/<p><em>Abstract</em>: In this talk, I examine some of the interconnections between norms governing assertion, denial, questions and answers, and the common ground of a discourse. When we pay attention to the structure of norms governing polar (yes/no) questions, we can clarify the distinction between strong and weak denials, together with the parallel distinction between strong and weak assertion, and the way that these speech acts interact with the common ground.</p>
<p>With those connections established, I respond to two criticisms of the program sketched out in my 2005 paper “Multiple Conclusions”. First, that understanding the upshot of a valid sequent <em>X</em> ⊢ <em>Y</em> as enjoining us to not assert each member of <em>X</em> and deny each member of <em>Y</em> is altogether too weak to explain the inferential force of logical validity. Deriving <em>X</em> ⊢ <em>A</em> should tell us, after all, something about justifying <em>A</em> on the basis of <em>X</em>, rather than merely prohibiting <em>A</em>’s denial. Where is the force to actually conclude the conclusion of a proof? A second, related criticism is that the format of multiple conclusion sequents seems unsatisfactory, in that it has no place for distinguishing a single conclusion, and proofs, after all, seem to be proofs of individual claims.</p>
<p>I will argue that both of these concerns can be assuaged if we pay closer attention to the norms connecting assertions and denials along with <em>justification requests</em> — questions aiming at eliciting reasons for assertions or denials. Once we understand the connection between justification requests, definitionsand the common ground, we will see not only that the these two concerns can be met. A derivation of a sequent <em>X</em> ⊢ <em>A</em>,<em>Y</em> gives us an answer to a justification request “why <em>A</em>?” in any available context where each member of <em>X</em> has been ruled in and each member of <em>Y</em> has been ruled out, and a derivation of a sequent <em>X</em>,<em>B</em> ⊢ <em>Y</em>, similarly gives us an answer to the justification request “why not <em>B</em>?” in any such context. The picture that results utilises the full multiple premise, multiple conclusion sequent calculus of classical logic, and does due justice to the idea that a proof (or a refutation) proves (or refutes) <em>one thing</em> relative to background assumptions or premises. In addition, when we consider the connection between justification requests and the norms governing <em>definitions</em>, we can see more clearly what could be involved in taking the connective/quantifier rules of a logical system to <em>define</em> the concepts they introduce.</p>
<ul>
<li><p>The talk is an online presentation at the Philosophy Department at the University of Bristol.</p></li>
<li><p>The <a href="https://consequently.org/slides/assertion-denial-common-ground-slides-bristol.pdf">slides for the talk are available here</a>, and the <a href="https://consequently.org/handouts/assertion-denial-common-ground-handout-bristol.pdf">handout is here</a>.</p></li>
</ul>
Classes
https://consequently.org/class/
Mon, 01 Jan 0001 00:00:00 UTChttps://consequently.org/class/PY4612: Advanced Logic
https://consequently.org/class/2021/py4612/
Sun, 12 Sep 2021 00:00:00 UTChttps://consequently.org/class/2021/py4612/<p><strong><span class="caps">py4612</span>: Advanced Logic</strong> applies the tools of formal logic to make <em>logic itself</em> 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 consequence relation in two very different ways. (2) The Compactness Theorem and the Löwenheim–Skolem Theorems, which explore some of the limits of first-order classical predicate logic for classifying infinite structures. And most importantly (3) we will work through Gödel’s celebrated Incompleteness Theorems and come to grips with what they mean.
Along the way, there will be some preparatory discussion of elementary set theory, proof theory, model theory, and recursion theory.</p>
<figure>
<img src="https://consequently.org/images/godel.jpg" alt="Kurt Godel, seated">
<figcaption>Kurt Gödel, seated</figcaption>
</figure>
PY3100: Reading Philosophy 1—Texts in Language, Logic, Mind, Epistemology, Metaphysics and Science
https://consequently.org/class/2021/py3100/
Sun, 12 Sep 2021 00:00:00 UTChttps://consequently.org/class/2021/py3100/<p><strong><span class="caps">py3100</span>: Reading Philosophy 1–Texts in Language, Logic, Mind, Epistemology, Metaphysics and Science</strong> is designed to develop the philosophical skills students have acquired over the first two years of their philosophy study, and acquaint them with key works in core areas of philosophy. The module involves close study of philosophical texts – historical and contemporary – that address a variety of topics within metaphysics, epistemology, the philosophies of logic and language, mind and science. Students will be required to carry out close study and discussion of these texts in staff-led weekly workshops, thereby furthering their skills of critical evaluation and analysis. Students will also take turns in presenting papers to the workshop, in pair-groups, which will help them to develop important communication skills and provide an opportunity for teamwork.</p>
Comparing Rules for Identity in sequent systems and natural deduction
https://consequently.org/presentation/2021/identity-rules-tableaux-2021/
Thu, 02 Sep 2021 00:00:00 UTChttps://consequently.org/presentation/2021/identity-rules-tableaux-2021/<p><em>Abstract</em>: It is straightforward to treat the identity predicate in models for first order predicate logic. Truth conditions for identity formulas are given by a natural clause: a formula <em>s = t</em> is true (or satisfied by a variable assignment) in a model if and only if the denotations of the terms <em>s</em> and <em>t</em> (perhaps relative to the given variable assignment) are the same.</p>
<p>On the other hand, finding appropriate rules for identity in a sequent system or in a natural deduction proof setting leaves a number of questions open. Identity could be treated with introduction and elimination rules in natural deduction, or left and right rules, in a sequent calculus, as is standard for familiar logical concepts. On the other hand, since identity is a predicate and identity formulas are atomic, it is also very natural to treat identity by way of axiomatic sequents, rather than by inference rules. I will describe and discuss this phenomenon, and explore the relationships between different formulations of rules for the identity predicate, and attempt to account for some of the distinctive virtues of each different formulation.</p>
<ul>
<li><p>The talk is an invited online presentation for the <a href="http://tableaux2021.org/">Tableaux 2021</a>.</p></li>
<li><p>The <a href="https://consequently.org/slides/identity-rules-tableaux-2021.pdf">slides for the talk are available here</a>.</p></li>
</ul>
Proofs and Models in Philosophical Logic
https://consequently.org/presentation/2021/pmpl-blc/
Thu, 02 Sep 2021 00:00:00 UTChttps://consequently.org/presentation/2021/pmpl-blc/<p><em>Abstract</em>: In this talk, I will draw out three different ways that soundness and completeness—and the relationship between proofs and models—can teach us in something about classical propositional logic, the semantics of modal logic, and the metaphysics of quantified modal logic.</p>
<ul>
<li><p>The talk is an online presentation for the <a href="https://blc-logic.org/blc-meetings/">British Logic Colloqium</a>.</p></li>
<li><p>The <a href="https://consequently.org/slides/pmpl-blc.pdf">slides for the talk are available here</a>.</p></li>
</ul>
News
https://consequently.org/news/
Wed, 30 Jun 2021 09:00:00 +1000https://consequently.org/news/Leaving Melbourne
https://consequently.org/news/2021/leaving-melbourne/
Wed, 30 Jun 2021 09:00:00 +1000https://consequently.org/news/2021/leaving-melbourne/<p>As June 2021 turns to a close, this is my last official day at <a href="https://unimelb.edu.au">The University of Melbourne</a>. I’ve taught <a href="https://consequently.org/class/">my last classes</a>, the marking for the semester is all done, I’ve wound up all my committee work, I’ve supervised my last undergraduate theses, and wrapped up all the end-of-semester administration. I’m now packing up my office (which I’ve rarely seen over the last 18 months) and tying up lots of loose ends. If I weren’t <a href="https://www.st-andrews.ac.uk/philosophy/news/title-114634-en.php">starting a new position</a>, things would be falling eerily silent, with no Semester 2 subjects to prepare, no committee work to do, and no students to supervise. And with my <a href="https://www.st-andrews.ac.uk/semester-dates/2021-2022/">new academic year</a> starting in September, I do have a few moments to pause, to breathe, and to reflect on my 19 years at the University of Melbourne, before I head off on the next adventure.</p>
<p>I arrived at the University of Melbourne <a href="https://consequently.org/news/2002/07/05/arrival/">in July 2002</a>, hired in the wake of Graham Priest’s arrival as Boyce Gibson Professor of Philosophy. As I leave, in 2021, there has been an almost complete turnover in members of staff in the Philosophy program. I arrived as one of the youngest staff members, and I leave as, if not the oldest, at least firmly in the middle of the age distribution of Philosophy staff. I’ve seen the Philosophy Department move from being an independent department in the <a href="https://arts.unimelb.edu.au">Faculty of Arts</a>, to being a tiny School of Philosophy, when the other Departments were amalgamating into congolmerate Schoools, to then being a part of the School of <em>Philosophy, Anthropology and Social Inquiry</em>, and then, when that imploded, we joined forces with our colleagues in History, Classics and Anthropology, History and Philosophy of Science and the Centre for Cultural Materials Conservation, to form the <em><a href="https://arts.unimelb.edu.au/school-of-historical-and-philosophical-studies">School of Historical and Philosophical Studies</a></em>. This last partnership seems to have lasted the distance, and Philosophy is now a thriving member of a well-functioning and collegial school, with large numbers of undergraduate students, a diverse postgraduate community, and an active research culture.</p>
<p>The road from there to here was not without its twists and turns, and quite a few periods of anxiety and trauma. The higher education scene in Australia is in a continual state of crisis, with regular periods of restructuring, staff redundancies, “belt tightening” and ever increasing “competition” for a reducing pool of resources. We must compete to get research funding, even if our research doesn’t so much require money but <em>time</em>, partly so we can get funds to create temporary positions for junior academics post-PhD, partly to increase the prestige for the University and the Department (since securing research funding is an easily measured proxy for doing “quality research”), partly to look good on our own CVs, and, partly to help fund the institution as a whole. Somewhere in there is the importance of the research we’re trying to do, but many days that imperative seems further down the priority listing than it should be.</p>
<p>As a continuing member of staff with a relatively active research profile, I was more insulated from the ongoing turmoil than junior academics attempting to find secure employment, or staff members in fields viewed as marginal or secondary, or those whose research activity fell under the required threshold for counting as “research active”. Those pressures never fell on me anywhere near as hard as they fell on others. However, this tension is felt <em>everywhere</em> in the system, even in a relatively secure place such as the University of Melbourne, where, you would think, a Philosophy Department might be safe.</p>
<p>It turned out that the Philosophy Department was <em>relatively</em> safe, but there was a period late in the first decade of the 2000s and early into the second decade, when we were at risk of collapse. In the worst period of restructuring, we lost many of our colleagues, and the Department was halved as many colleagues retired or took redundancies and none were replaced. For a while, there were only six of us, attempting to keep the ship afloat, teaching all our students, doing our resarch, and making the case that the University needs a generalist Philosophy program, teaching a full major, despite efforts of some of our colleagues in the Faculty for us to specialise and become a boutique applied philosophy program. The situation was dire, and the workloads were atrocious, but those of us who remained worked hard, grew the department, both in terms of student numbers, and eventually, in <a href="https://arts.unimelb.edu.au/school-of-historical-and-philosophical-studies/discipline-areas/philosophy">new members of staff</a>. We have now regained all the positions we lost during that time, and for the moment, the future looks relatively bright, once we get through the pandemic, at least.</p>
<p>What I most value from my years at Melbourne are the close working communities. The longest running must be the <a href="https://blogs.unimelb.edu.au/logic/logic-seminar/">Melbourne Logic Seminar</a>, which I launched at my arrival, and which has been ably coordinated by <a href="https://standefer.net">Shawn Standefer</a> over the last few years. That group put up with so many half-baked thoughts of mind, and was the crucible in which many different ideas have been incubated and research careers have been launched. I’ve already <a href="https://consequently.org/news/2019/teaching-logical-methods/">waxed lyrical</a> about working with <a href="https://standefer.net">Shawn</a> in my teaching. The <a href="https://consequently.org/writing/collection-frames/">research collaboration</a> with Shawn has been a special delight, too.</p>
<p>I’ve also treasured the weekly lunchtime meetings with my graduate students. We’ve been doing this over most of my time at Melbourne, getting together, talking about our research, supporting each other as ideas take shape—and occasionally crumble into dust before our very eyes—and seeing projects from conception to fruition. This is one of the highlights of my week.</p>
<p>Other groups that deserve a shout-out was the shorter but very intense working group consisting of <a href="https://people.eng.unimelb.edu.au/davoren/">Jen Davoren</a>, <a href="http://rohan-french.github.io">Rohan French</a> and me, who managed to transform our interdisciplinary introductory logic subject into a successful but short-lived experiment on Coursera. The course ran for only two sessions before becoming unviable (because we had no funding for it to continue), but the lessons we learned in online pedagogy proved very useful when it came to dealing with <a href="https://consequently.org/news/2020/teaching-during-a-pandemic/">2020</a>. Teaching the interdiciplinary intro logic subject with Jen has been a constant delight over the last 15 years, and the support from our colleagues, especially the much-missed <a href="https://www.math.ucla.edu/~greg/">Greg</a> <a href="https://en.wikipedia.org/wiki/Greg_Hjorth">Hjorth</a>, <a href="https://findanexpert.unimelb.edu.au/profile/13664-lesley-stirling">Lesley Stirling</a> and <a href="http://www.stevenbird.net">Steven Bird</a>, who helped us cook up a crazy and wild <a href="https://consequently.org/class/2021/unib10002/">intro to logic</a> for students from all over the university, taking in aspects of digital systems, computer science, linguistics, philosophy and mathematics. It was a wild experiment, loved by those who took the subject, but never as popular as we had hoped for it to be. I’ve learned so much about how to teach logic to a diverse student body over these years.</p>
<p>I’ve also served on many School, Faculty and University Committees, with an aim to somehow support the teaching and research of colleagues. (I’ve never seen committee work or my time as Head of Department as a stepping stone to going on to greater administrative responsibilities. It’s necessary plumbing work to make sure that the important things—teaching and research—can continue, and hopefully, thrive.) For a crew to work with to do the hack work of keeping the institution going, I’d like to single out the Philosophy Department when we were at our lowest ebb: somehow, Chris, Howard, Laura, François, Karen and I managed to keep the place going, fighting for our discipline, keeping everything going as the institution restructured around us, and navigating a path forward in a time of crisis. Somehow, we managed to keep our heads together, work out how to navigate the difficulties and even stay relatively sane while doing it. We got through that time, fighting for our corner, looking out for each other, with the hope that things would turn the corner. And they did.</p>
<p>It’s been a wild ride over the last 19 years. I can’t wait to see what the years ahead will bring.</p>
Natural Deduction with Alternatives: on structural rules, and identifying assumptions
https://consequently.org/presentation/2021/natural-deduction-with-alternatives-aal/
Fri, 11 Jun 2021 00:00:00 UTChttps://consequently.org/presentation/2021/natural-deduction-with-alternatives-aal/<p><em>Abstract</em>: In this talk, I will introduce natural deduction with alternatives, explaining how this framework provides a simple, well-behaved, single conclusion natural deduction system for a range of logical systems, including classical logic, (classical) linear logic, relevant logic and affine logic, in addition to the familar intuitionistic restrictions of these systems. Each of these proof systems have identical connective rules. As we expect in substructural logics, different logical systems are given by varying the structural rules in play. The distinctly classical behaviour of these systems is given by the presence of alternatives (formulas in consequent, or positive position, other than the conclusion of the proof) in addition to assumptions (formulas in antecedent, or negative position). Unlike multiple conclusion proof systems, the proof system is single conclusion, since one formula in positive position is singled out as the conclusion. The context in which that formula is proved consists, in general, of formulas ruled in (assumptions) and formulas ruled out (alternatives).</p>
<p>In sequent systems, and in some natural deduction systems that use labels, the structural rules of contraction and weakening govern an explicitly represented structure, such as a set or multiset or sequence of formulas occurring in each sequent. In this natural deduction framework, the structural rules have their force at the point of discharge, or more generally, at any point at which it is important to determine whether two occurrences of the same formula (in positive position, or in negative position) are the same assumption or the same alternative. There is no explicit representation of any structure of assumptions or alternatives, other than the structure of the proof itself.</p>
<p>Along the way, this presentation will touch on (1) the connection between normalisation of a natural deduction proof and cut elimination in a corresponding sequent calculus; (2) the separation between the operational rules governing the connectives and the “antecedently given context of deducibility”, to borrow a phrase from Nuel Belnap’s essay, <a href="https://www.jstor.org/stable/3326862">“Tonk, Plonk and Plink” (1962)</a>; (3) the sense in which the operational rules for a connective might be understood as providing a definition of that connective; and (4) the use of the identity or difference of variables in type systems such as the typed λ calculus in keeping track of the sameness or difference of assumptions as opposed to the sameness or difference of the things assumed.</p>
<ul>
<li><p>The talk is an online presentation at the <a href="https://blogs.unimelb.edu.au/logic/aal-2021/">2021 Australasian Association for Logic Conference</a>.</p></li>
<li><p>The <a href="https://consequently.org/slides/natural-deduction-with-alternatives-aal.pdf">slides for the talk are available here</a>.</p></li>
</ul>
Writings
https://consequently.org/writing/
Mon, 01 Jan 0001 00:00:00 UTChttps://consequently.org/writing/Proofs and Models in Philosophical Logic
https://consequently.org/writing/pmpl-elements/
Tue, 01 Jun 2021 00:00:00 UTChttps://consequently.org/writing/pmpl-elements/<p>This is a short manuscript for the <a href="https://www.cambridge.org/core/what-we-publish/elements">Cambridge Elements</a> series in Philosophical Logic.</p>
<p>This is a general introduction to recent work in proof theory and model theory of non-classical logics, with a focus on the application of non-classical logic to the semantic paradoxes and (to a lesser extent), the sorites paradox. After a short introduction motivating general notions of proof and of models, I introduce and motivate a simple natural deduction system, and present the structure of the liar paradoxical argument (concerning truth) and Curry’s paradox (concerning class membership). I introduce and motivate the notion of a structural rule, in both natural deduction and the sequent calculus and I compare and contrast the different approaches to substructural treatments of the paradox, contrasting the roles that <em>contraction</em>, <em>cut</em> and <em>identity</em> play in the derivations of the paradoxes.</p>
<p>In the next section, I introduce model theoretic treatments of the paradoxes, introducing supervaluations, and three-valued treatments of vagueness, and of the semantic paradoxes. I explain the fixed-point model construction that shows how to construct three-valued models for theories of truth, which can be used to then give models for different logics: K3 (with truth-value gaps), LP (with truth-value gluts) and ST (which supports all of classical logic, at the cost of invalidating the <em>cut</em> rule). I compare and contrast these approaches, and introduce the Routley—Meyer ternary relational semantics one way to model logics without the structural rules of contraction, or of weakening.</p>
<p>In the final section, I explore the relationship between proofs and models, explaining how the soundness and completeness theorems are proved for classical logic and for ST. I then discuss the issue of whether proofs or models play the primary <em>semantic</em> role, relating this question to the broader context of inferentialist and representationalist theories of meaning. This section closes with a discussion of the different resources that proof-first and model-first accounts of semantics have for answering the general question of how a language might be generally resistant to triviality due to paradox.</p>
Proof, Rules and Meaning
https://consequently.org/writing/prm/
Sun, 03 Oct 2010 00:00:00 UTChttps://consequently.org/writing/prm/<p>This is my next book-length writing project. I am writing a book which aims to do these things:</p>
<ol>
<li> Be a useable introduction to philosophical logic, accessible to someone who’s done only an introductory course in logic, covering at least some model theory and proof theory of propositional logic, and maybe a little bit of predicate logic.
<li> Be a user-friendly, pedagogically useful and philosophically motivated presentation of cut-elimination, normalisation and conservative extension, both (a) why they’re important to semantics and (b) how to actually <em>prove</em> them. (I don’t think there are any books like this currently available, but I’d be happy to be shown wrong.)
<li> Present the duality between model theory and proof theory in a philosophically illuminating and clear fashion.
<li> And then apply these results to issues concerning meaning, epistemology and metaphysics, including issues of logical consequence and rationality, the problem of absolute generality, and the status of modality.
</ol></p>
<p>Here is an outline of the manuscript, showing how the parts hold together. At least so far — I'm still writing the third part.</p>
<p><figure>
<img src="https://consequently.org/images/ptrm-map.png" alt="Outline of the book Proof Theory, Rules and Meaning">
<figcaption>How <em>Proof Theory, Rules and Meaning</em> hangs together.</figcaption>
</figure>
The book is in three parts.
<ol>
<li> <em>Tools</em>: in which core concepts from proof theory are introduced.
<li> <em>The Core Argument</em>: in which I motivate <em>defining rules</em>, and show how they answer Arthur Prior’s challenge concerning when an inference rule defines a logical concept.
<li> <em>Insights</em>: in which we see consequences for logic and language, epistemology and metaphysics, etc.
</ol></p>
<p>The book draft was discussed at <a href="http://ba-logic.com/workshops/symposium-restall/">this symposium</a> in Buenos Aires in July 2018. I’m currently finalising the last section of the book, and updating the whole manuscript based on the feedback I got there from colleagues.</p>
Geometric Models for Relevant Logics
https://consequently.org/writing/geometric-models/
Tue, 01 Jun 2021 00:00:00 UTChttps://consequently.org/writing/geometric-models/<p>Alasdair Urquhart’s work on models for relevant logics is distinctive in a number of different ways. One key theme, present in both his undecidability proof for the relevant logic R, and his proof of the failure of interpolation in R, is the use of techniques from geometry. In this paper, inspired by Urquhart’s work, I explore ways to generate natural models of R from geometries, and different constraints that an accessibility relation in such a model might satisfy. I end by showing that a set of natural conditions on an accessibility relation, motivated by geometric considerations, is jointly unsatisfiable.</p>
Platonism, Nominalism, Realism, Anti-Realism, Reprentationalism, Inferentialism and all that
https://consequently.org/presentation/2021/platonism-nominalism-etc/
Wed, 13 May 2020 00:00:00 UTChttps://consequently.org/presentation/2021/platonism-nominalism-etc/<figure>
<img src="https://consequently.org/images/platonism-nominalism-usual-talk.jpg" alt="A close-up view of the Old Quad and Arts West at the University of Melbourne">
<figcaption>My usual talk (a close-up view of the Old Quad and Arts West at the University of Melbourne).</figcaption>
</figure>
<p><em>Abstract</em>: In this talk, I will place contemporary research in philosophical logic in a wider historical and philosophical context, showing how recent work in logic connects to the rivalry between <em>Platonism</em> and <em>Nominalism</em>, or <em>realism</em> and <em>anti-realism</em> in metaphysics, and between <em>representationalism</em> and <em>inferentialism</em> in the the philosophy of language. Along the way, I will touch on the contemporary resurgence of interest in Carnap’s logical positivism, and Robert Brandom’s turn toward Hegel.</p>
<figure>
<img src="https://consequently.org/images/platonism-nominalism-this-talk.jpg" alt="A view of the globe, with eastern Australia at dusk">
<figcaption>This talk (a view of the globe, with eastern Australia at dusk).</figcaption>
</figure>
<ul>
<li><a href="https://philevents.org/event/show/90022">Here</a> is the PhilEvents link to the event.</li>
</ul>
Comparing Rules for Identity in Sequent Systems and Natural Deduction
https://consequently.org/presentation/2021/comparing-identity-rules/
Sat, 17 Apr 2021 00:00:00 UTChttps://consequently.org/presentation/2021/comparing-identity-rules/<p><em>Abstract</em>: It is straightforward to treat the identity predicate in models for first order predicate logic. Truth conditions for identity formulas are straightforward. On the other hand, finding appropriate rules for identity in a sequent system or in natural deduction leaves many questions open. Identity could be treated with introduction and elimination rules in natural deduction, or left and right rules, in a sequent calculus, as is standard for familiar logical concepts. On the other hand, since identity is a predicate and identity formulas are atomic, it is possible to treat identity by way of axiomatic sequents, rather than inference rules. In this talk, I will describe this phenomenon, and explore the relationships between different formulations of rules for the identity predicate, and attempt to account for some of the distinctive virtues of each different formulation.</p>
<ul>
<li><p>The talk was an online presentation for the <a href="https://www.proofsociety.org/proof-theory-seminar/index.html">Proof Theory Virtual Seminar</a></p></li>
<li><p>The <a href="https://consequently.org/slides/comparing-identity-rules-ptvs.pdf">slides for the talk are available here</a>.</p></li>
<li><p>Here is the <a href="https://www.youtube.com/watch?v=BKZaihmMUo8">recording of the talk</a>: <iframe width="560" height="315" src="https://www.youtube-nocookie.com/embed/BKZaihmMUo8" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe></iframe></p></li>
</ul>
UNIB10002: Logic, Language and Information
https://consequently.org/class/2021/unib10002/
Mon, 01 Mar 2021 00:00:00 UTChttps://consequently.org/class/2021/unib10002/<p><strong><span class="caps">UNIB10002</span>: Logic, Language and Information</strong> is a <a href="http://unimelb.edu.au">University of Melbourne</a> undergraduate breadth subject, introducing logic and its applications to students from a wide range of disciplines in the Arts, Sciences and Engineering. I coordinate this subject with my colleague Dr. Jen Davoren, with help from Prof. Lesley Stirling (Linguistics), Dr. Peter Schachte (Computer Science) and Dr. Daniel Murfet (Mathematics).</p>
<p>The subject is taught to University of Melbourne undergraduate students. Details for enrolment are <a href="https://handbook.unimelb.edu.au/view/2021/UNIB10002">here</a>.</p>
PHIL30043: The Power and Limits of Logic
https://consequently.org/class/2021/phil30043/
Mon, 01 Mar 2021 00:00:00 UTChttps://consequently.org/class/2021/phil30043/
<p><strong><span class="caps">PHIL30043</span>: The Power and Limits of Logic</strong> (or, as I like to call it, <em>Kurt Gödel’s Greatest Hits</em>) is a <a href="https://handbook.unimelb.edu.au/view/2021/PHIL30043">University of Melbourne undergraduate subject</a>. It covers the metatheory of classical first order predicate logic, beginning at the <em>Soundness</em> and <em>Completeness</em> Theorems, <em>Compactness</em>, <em>Cantor’s Theorem</em>, the <em>Downward Löwenheim–Skolem Theorem</em>, <em>Recursive Functions</em>, <em>Register Machines</em>, <em>Representability</em>, the <em>Indefinability of Truth</em> and the <em>Undecidability of Predicate Logic</em>, and ending up at <em>Gödel’s Incompleteness Theorems</em> and <em>Löb’s Theorem</em>.</p>
<figure>
<img src="https://consequently.org/images/godel.jpg" alt="Kurt Godel, seated">
<figcaption>Kurt Gödel, seated</figcaption>
</figure>
<p>The subject is taught to University of Melbourne undergraduate students (for Arts students as a part of the Philosophy major, for non-Arts students, as a breadth subject). Details for enrolment are <a href="https://handbook.unimelb.edu.au/view/2021/PHIL30043">here</a>. I make use of video lectures I have made <a href="https://www.youtube.com/playlist?list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n">freely available on YouTube</a>.</p>
<h3 id="outline">Outline</h3>
<p>The course is divided into four major sections, taught over 12 weeks. Here is a list of all of the videos, in case you’d like to follow along with the content.</p>
<h4 id="soundness-and-completeness">Soundness and Completeness</h4>
<ul>
<li><a href="https://www.youtube.com/watch?v=qTAcRSnObdw&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=2&t=0s">The Language of Predicate Logic</a></li>
<li><a href="https://www.youtube.com/watch?v=JMOTYyHnkt8&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=3&t=0s">Proofs for Predicate Logic</a></li>
<li><a href="https://www.youtube.com/watch?v=1p0oTY6I-Yw&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=4&t=0s">Models for Predicate Logic and Soundness</a></li>
<li><a href="https://www.youtube.com/watch?v=58k6dNrvBoU&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=5&t=0s">Completeness for Predicate Logic</a></li>
</ul>
<h4 id="countability-and-compactness">Countability and Compactness</h4>
<ul>
<li><a href="https://www.youtube.com/watch?v=eI-i8jpcd_o&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=6&t=0s">Identity and Functions</a></li>
<li><a href="https://www.youtube.com/watch?v=NJgmRvKFGDc&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=7&t=0s">Countability and Diagonalisation</a></li>
<li><a href="https://www.youtube.com/watch?v=f-iTL4wWa8k&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=8&t=0s">Compactness and Countable Models</a></li>
</ul>
<h4 id="computability">Computability</h4>
<ul>
<li><a href="https://www.youtube.com/watch?v=83zBUpKm0GM&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=9&t=0s">Recursive Functions</a></li>
<li><a href="https://www.youtube.com/watch?v=04ol2ZkZuUk&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=10&t=0s">Register Machines</a></li>
<li><a href="https://www.youtube.com/watch?v=8rUGLIubpl0&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=11&t=0s">Register Machines compute the Recursive Functions</a></li>
<li><a href="https://www.youtube.com/watch?v=olAiZHi4ra0&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=12&t=0s">Non-Recursive Functions</a></li>
</ul>
<h4 id="undecidability-and-incompleteness">Undecidability and Incompleteness</h4>
<ul>
<li><a href="https://www.youtube.com/watch?v=btRjkOm0HFA&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=13&t=0s">Theories of Arithmetic</a></li>
<li><a href="https://www.youtube.com/watch?v=gxZgiEllHJE&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=14&t=0s">Diagonalisation and its Consequences</a></li>
<li><a href="https://www.youtube.com/watch?v=3gQWRaXhPWE&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=15&t=0s">Provability Predicates, and Beyond</a></li>
</ul>
An Inferentialist Account of Identity and Modality
https://consequently.org/presentation/2021/an-inferentialist-account-of-identity-and-modality/
Thu, 11 Feb 2021 00:00:00 UTChttps://consequently.org/presentation/2021/an-inferentialist-account-of-identity-and-modality/<p><em>Abstract</em>: In this talk I will show how <em>defining rules</em> in a hypersequent setting can give a uniform proof-theoretic semantics of identity and modality, allowing – equally naturally – for (1) modal operators for which identity statements are <em>necessary</em> (if true), and (2) modal operators for which identity statements can be <em>contingently</em> true.</p>
<ul>
<li><p>The talk is an online presentation for the <a href="https://inferentialexpressivism.com/seminar/">ERC EXPRESS Project</a> in Amsterdam, and the <a href="https://www.ihpst.cnrs.fr/en/activites/seminaires/seminaire-philmath-2019-2020-0">PHILMATH Seminar</a> in Paris.</p></li>
<li><p>The <a href="https://consequently.org/slides/an-inferentialist-account-of-identity-and-modality-express-philmath.pdf">slides for the talk are available here</a>.</p></li>
<li><p>Here is the <a href="https://www.youtube.com/watch?v=jZuMf5NL7Us">recording of the talk</a>: <iframe width="560" height="315" src="https://www.youtube-nocookie.com/embed/jZuMf5NL7Us" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe></p></li>
</ul>
Natural Deduction with Alternatives
https://consequently.org/presentation/2020/natural-deduction-with-alternatives/
Wed, 04 Nov 2020 00:00:00 UTChttps://consequently.org/presentation/2020/natural-deduction-with-alternatives/<p><em>Abstract</em>: In this talk, I will introduce natural deduction with alternatives, explaining how this framework can provide a simple well-behaved single conclusion natural deduction system for a range of logical systems, including classical logic, (classical) linear logic, relevant logic and affine logic, by varying the policy for managing discharging of assumptions and retrieval of alternatives. Along the way, the talk will touch on (1) the connection between normalisation of a natural deduction proof and cut elimination in a corresponding sequent calculus; (2) the separation between the operational rules governing the connectives and the “antecedently given context of deducibility”, to borrow a phrase from Nuel Belnap’s essay, “Tonk, Plonk and Plink” (1962); and (3) the sense in which the operational rules for a connective might be understood as providing a definition of that connective.</p>
<p><div style="padding:56.25% 0 0 0;position:relative;"><iframe src="https://player.vimeo.com/video/476143886" style="position:absolute;top:0;left:0;width:100%;height:100%;" frameborder="0" allow="autoplay; fullscreen" allowfullscreen></iframe></div><script src="https://player.vimeo.com/api/player.js"></script>
<p><a href="https://vimeo.com/476143886">Greg Restall—Natural Deduction with Alternatives</a> from <a href="https://vimeo.com/logicmelb">logicmelb</a> on <a href="https://vimeo.com">Vimeo</a>.</p></p>
<ul>
<li><p>The talk is an online presentation at our <a href="https://blogs.unimelb.edu.au/logic/applied-proof-theory-workshop/">Applied Logic Workshop</a>.</p></li>
<li><p>The <a href="https://consequently.org/slides/natural-deduction-with-alternatives-alw.pdf">slides for the talk are available here</a>.</p></li>
</ul>
Speech Acts & the Quest for a Natural Account of Classical Proof
https://consequently.org/presentation/2020/speech-acts-for-classical-natural-deduction-berkeley/
Tue, 15 Sep 2020 00:00:00 UTChttps://consequently.org/presentation/2020/speech-acts-for-classical-natural-deduction-berkeley/<p><em>Abstract</em>: It is tempting to take the logical connectives, such as conjunction, disjunction, negation and the material conditional to be defined by the basic inference rules in which they feature. Systems of “natural deduction” provide the basic framework for studying these inference rules. In natural deduction proof systems, well-behaved rules for the connectives give rise to <em>intuitionistic logic</em>, rather than classical logic. Some, like Michael Dummett, take this to show that intuitionistic logic is on a sounder theoretical footing than classical logic. Defenders of classical logic have argued that some other proof-theoretical framework, such as Gentzen’s sequent calculus, or a <em>bilateralist</em> system of signed natural deduction, can provide a proof-theoretic justification of classical logic. Such defences of classical logic have significant shortcomings, in that the systems of proof offered are much less natural than existing systems of natural deduction. Neither sequent derivations nor signed natural deduction proofs are good matches for representing the inferential structure of everyday proofs.</p>
<p>In this paper I clarify the shortcomings of existing bilateralist defences of classical proof, and, making use of recent results in the proof theory for classical logic from theoretical computer science, I show that the bilateralist can give an account of natural deduction proof that models our everyday practice of proof as well as intuitionist natural deduction, if not better.</p>
<ul>
<li><p>The talk is an online presentation at the <a href="http://logic.berkeley.edu/events.html">Berkeley Logic Group Seminar</a>.</p></li>
<li><p>The <a href="https://consequently.org/slides/speech-acts-for-classical-natural-deduction-berkeley.pdf">slides for the talk are available here</a>.</p></li>
<li><p>A <a href="https://consequently.org/papers/speech-acts-for-classical-natural-deduction.pdf">draft of the paper on which this talk is based is available here</a>.</p></li>
</ul>
Speech Acts & the Quest for a Natural Account of Classical Proof
https://consequently.org/writing/speech-acts-for-classical-proofs/
Tue, 15 Sep 2020 00:00:00 UTChttps://consequently.org/writing/speech-acts-for-classical-proofs/<p>It is tempting to take the logical connectives, such as conjunction, disjunction, negation and the material conditional to be defined by the basic inference rules in which they feature. Systems of “natural deduction” provide the basic framework for studying these inference rules. In natural deduction proof systems, well-behaved rules for the connectives give rise to <em>intuitionistic logic</em>, rather than classical logic. Some, like Michael Dummett, take this to show that intuitionistic logic is on a sounder theoretical footing than classical logic. Defenders of classical logic have argued that some other proof-theoretical framework, such as Gentzen’s sequent calculus, or a <em>bilateralist</em> system of signed natural deduction, can provide a proof-theoretic justification of classical logic. Such defences of classical logic have significant shortcomings, in that the systems of proof offered are much less natural than existing systems of natural deduction. Neither sequent derivations nor signed natural deduction proofs are good matches for representing the inferential structure of everyday proofs.</p>
<p>In this paper I clarify the shortcomings of existing bilateralist defences of classical proof, and, making use of recent results in the proof theory for classical logic from theoretical computer science, I show that the bilateralist can give an account of natural deduction proof that models our everyday practice of proof as well as intuitionist natural deduction, if not better.</p>
Teaching During a Pandemic
https://consequently.org/news/2020/teaching-during-a-pandemic/
Thu, 06 Aug 2020 22:10:00 +1000https://consequently.org/news/2020/teaching-during-a-pandemic/<p>As I write this, the first week of the second semester of 2020 is nearing its end, and I’ve taught my first two seminars in <a href="https://consequently.org/class/2020/phil20030/">Logical Methods</a>, my main undergraduate teaching responsibility for this semester. Melbourne has just entered Stage 4 of its lockdown, as we attempt to deal with the ongoing community transmission of COVID-19. The streets are quiet, it has been over four months since I’ve been on campus, and all my teaching is done from the chair at my desk in my study, peering into the 15 inch screen of my laptop, with the green cyclops dot in the middle of the top screen bezel showing that yet again, my image is being transmitted across the internet, to students scattered across Melbourne, across Australia, and across the world.</p>
<p>I’ve started teaching online afresh this semester, taking into account some of the things I learned in the <a href="https://consequently.org/news/2020/notes-from-a-pandemic/">mad rush to online teaching in Semester 1</a>. Then, I had to juggle <a href="https://consequently.org/class/2020/unib10002">three</a> <a href="https://consequently.org/class/2020/phil30043">very different</a> <a href="https://consequently.org/class/2020/phil40013">subjects</a>. This semester, I have the chance to put into practice, from the beginning, what I learned in the experience, concentrating my energy into one unit: the Philosophy program’s second year introduction to logic, <a href="https://consequently.org/class/2020/phil20030/">Logical Methods</a>. <a href="https://shawn-standefer.github.io">Shawn Standefer</a> and I put a lot of effort <a href="https://consequently.org/news/2019/teaching-logical-methods/">developing this unit last year</a>, so the work for this semester involves figuring out how to teach it <em>online</em>.</p>
<p>Thankfully, we were set up pretty well to make the transition, at least in part. I’d already recorded hours of video lectures, designed to supplement the active seminars we used to teach face-to-face. Shawn had already prepared practice and graded quizzes the students could use to test their skills. All this still works in a pandemic, provided the student has a decent internet connection. What is less easy to replace is the <em>seminar</em>. When we taught this in person in 2019, we’d have a class of around 30 students, clustered around large tables, working on logic problems, talking to each other, as Shawn and I went from table to table, helping out, talking to each student, and then, after a reasonable amount of time, one or other of us would feed back to the whole group some thoughts about how they were going, or give some advice, or general feedback, or we’d raise for the class different questions that came up in one of the clusters of students we’d talked with. We’d have this regular rhythm, of working through questions in small groups, and feeding back to the larger group, putting things into practice, and reflecting on what we’d learned.</p>
<p>This can be <em>approximated</em> online, but it there is no way that it can be <em>replicated</em>.</p>
<p>In a classroom, I have peripheral vision. I can get a sense of how other groups are going, while I’m focussing on one group. People elsewhere in the room can catch my eye. I can sense when the groups are restless, wanting to go on, or when they’re having a lively conversation or a full-on argument, or when they’re stuck, spinning their wheels and needing help. Using <a href="https://zoom.us">Zoom</a>, a powerful video conferencing setup that allows for breakout groups that can approximate those small groups of discussion in a larger group, I can get into things with one group, but while I’m in <em>that</em> group, we’re hermetically sealed from the rest of the class. I have no idea what the other groups are doing. In this position, you either losing track of time as you deeply engage with one group of people, or you’re constantly looking at your watch, half thinking about how you should get out of there to see which of the other groups are on fire. It takes at least twice the energy to go through the same material as you could cover face-to-face, and to do that, you need more time. It’s hard work.</p>
<p>This is not a complaint. Without something like Zoom, we wouldn’t be able to even <em>approximate</em> what we do in face-to-face teaching. With videoconferincing sofware like Zoom, we can at least give the students the opportunity to work together in groups, to spend some time working independently, while also having regular feedback from a teacher, and they can have the experience of being in a larger cohort, working <em>with</em> others, of not being alone, of some of the joy of being in this <em>together</em>. That, alone, is a valuable thing in a time of physical distancing.</p>
<p>Having one semester’s experience of the transition to off-campus teaching, we noticed that it was easy for students to become disconnected. The web of informal connections that’s sustained on campus, with students bumping into each other, of chance encounters in the library, or waiting in line for a coffee, or of just hanging out with other students, all these things are <em>gone</em> in the move to working online. None of those informal connections are vital in and of themselves, but the loss of the whole campus experience is experienced as just that: it’s a loss. Some students are able to manage, but some have so much going on (it’s a pandemic, after all!) that they’re unable to sustain the effort required to complete a semester’s course of study, because it takes significant energy to recreate all the habits and routines of on campus study, when uprooted to the new, online, context.</p>
<p>So, this semester I have tried to recreate in conscious, explicit ways, what was habitual and implicit in life on campus. Instead of a student chatting with me as I walked back to my office after a class, I’ve made the decision to let the whole class know that I’ll stay around online in the Zoom session for 15 to 30 minutes after class, for anyone to talk informally about whatever is on their mind—whether the content of the subject, or anything else. It’s just one way to allow space for informal conversations, to allow things to arise naturally, without having to book an appointment, or send an email, but to just hang out and chat, like we might if we’d bumped into each other on campus.</p>
<p>That’s one difference this year, but I decided to do more help provide other means of connection. Since we started revising <em>Logical Methods</em> last year, we’ve been concerned to make logic teaching in the Philosophy program much more accessible and better integrated with the rest of the curriculum. The work we’ve done has been paying off, but by its nature, <em>any</em> logic subject is going to stand out in a humanities program. When you look at what we do, it’s formal, it’s technical, and it looks mathematical. Students approach this material with a range of expectations. Some love it, some fear it. Since logic is connected to many disciplines, the pool of students is broad, too. This year about 65% of the students aim to complete a philosophy major, but that leaves a third of my students as coming from all over the university. We have the usual mix of mathematics, computer science, and engineering students, but we have a good supply of media, politics, commerce and economics students, and more. With a student body like this, students are coming with a diverse range of needs and expectations. And in this isolated time, students have much less access to each other, and less access to their teachers. It’s easy for them to flounder in the initial weeks, unsure of how to work with this new material, and with little idea of what to do. I wanted to do something to help each student start off the semester well.</p>
<p>So this time around, when the course website launched, in the first announcement I sent out (10 days before the semester started), I let students know that I’d be available for a short 5 to 10 minute one-on-one consultation, where (1) I’d ask them what they wanted to get out of the subject (to help orient <em>me</em> as their teacher) and (2) they could ask me anything I wanted. That way, I’d get to know some students just a little bit, and deepen the connection beyond me seeing them restricted to small rectangles on my screen, and as names on email addresses or assignment submissions. But at the same time, I’d break the ice for them, and hopefully establish the semester as one where we’re available to each other, and the subject is much more interactive than what might amount to a bunch of videos on the internet.</p>
<p>I thought that maybe 50% of the students would take me up on the offer. And at the time I made the offer, our enrolment was at around 70 students for the subject (already up on the 65 who completed the unit last year). So I thought I could manage that many short interviews in the first week of classes.</p>
<p>Well, about 95% of the students have signed up for these interviews, and our enrolment is now on the other side of 110. So I’ve been busy, getting to know my students. They’re a bright, engaged, bunch, and to a person they’ve been grateful at the opportunity to talk one-on-one with their lecturer. Some have been disappointed with the move to online teaching, keenly feeling the loss of the campus experience. Others have found that they’ve managed well so far. But in each case, I got to hear from students, about what they want to learn, some of their hopes or fears, I’ve heard their excitement and their nervousness. For some, it’s continuing an already established passion for learning philosophy or learning logic. For others, it is the first dip of a toe into the water. Some are doing it because they want to do well in logic questions in the LSAT, and for others, they’ve just heard that it’s fun. In each case I’ve been able to help set expectations, to explain how the subject fits together, and how it might scratch where they’re itching, or how it might stretch them beyond what they’re looking for. These conversations have helped me as I present material in class, to orient things <em>toward</em> the students I have, and to help take the interests and passions they have, to show how they can pursue them while they study logical methods with me.</p>
<p>This is a bittersweet time, as we start our second semester of online teaching, while our community struggles with a pandemic and with living in a lockdown and all that entails. But one of its highlights, so far, is getting to know over 100 students, who are each, in their own way, keen to make connections, and to make the most of the opportunity to learn some logic. Being a part of that is enough to keep me going.</p>
PHIL20030: Logical Methods
https://consequently.org/class/2020/phil20030/
Sun, 01 Mar 2020 00:00:00 UTChttps://consequently.org/class/2020/phil20030/
<p><strong><span class="caps">PHIL20030</span>: Logical Methods</strong> is a <a href="http://unimelb.edu.au">University of Melbourne</a> undergraduate subject introducing logic to philosophy students. It’s taught by <a href="http://consequently.org">Greg Restall</a>.</p>
<p>The subject introduces the proof theory and model theory of propositional, modal and predicate logic–in that order. I’m using an introductory text <em>Logical Methods</em>, written with my colleague <a href="https://shawn-standefer.github.io">Shawn Standefer</a> for this course.</p>
<p>Here’s the outline of the subject.</p>
<h3 id="preliminaries">Preliminaries</h3>
<ul>
<li>Introduction
<ul>
<li>Arguments and Trees</li>
<li>Sentences and Formulas</li>
</ul></li>
</ul>
<h3 id="propositional-logic">Propositional Logic</h3>
<ul>
<li>Connectives: and & if
<ul>
<li>Conjunction</li>
<li>Conditional</li>
<li>Biconditional</li>
</ul></li>
<li>More connectives: not & or
<ul>
<li>Negation and falsum</li>
<li>Disjunction</li>
<li>Our System of Proofs</li>
</ul></li>
<li>Facts about proofs & provability
<ul>
<li>Facts about provability</li>
<li>Normalisation</li>
<li>The Subformula Property</li>
<li>Consequences of Normalisation</li>
</ul></li>
<li>Models & counterexamples
<ul>
<li>Models and truth tables</li>
<li>Counterexamples and validity</li>
<li>Model-theoretic validity</li>
</ul></li>
<li>Soundness & completeness
<ul>
<li>Soundness</li>
<li>Completeness</li>
<li>Proofs first or models first?</li>
<li>Heyting algebras</li>
</ul></li>
</ul>
<h3 id="modal-logic">Modal Logic</h3>
<ul>
<li>Necessity & possibility
<ul>
<li>Possible worlds models</li>
<li>Validity</li>
<li>Strict conditionals and ambiguities</li>
<li>Propositions</li>
<li>Another notion of necessity</li>
<li>Equivalence relations and epistemic logic</li>
</ul></li>
<li>Actuality & two-dimensional logic
<ul>
<li>Actuality models and double indexing</li>
<li>Validity</li>
<li>Fixity and diagonal propositions</li>
<li>Real world validity</li>
</ul></li>
<li>Natural deduction for modal logics
<ul>
<li>Natural deduction for S4</li>
<li>Natural deduction for S5</li>
<li>Features of S5</li>
</ul></li>
</ul>
<h3 id="predicate-logic">Predicate Logic</h3>
<ul>
<li>Quantifiers
<ul>
<li>Syntax</li>
<li>Natural deduction for CQ</li>
<li>What is provable?</li>
<li>Generality and eliminating detours</li>
</ul></li>
<li>Models for first-order logic
<ul>
<li>Models and assignments of values</li>
<li>Substitution</li>
<li>Counterexamples and validity</li>
<li>Compactness and what this means</li>
</ul></li>
</ul>
<p>One novelty in our approach to the subject is the balance between proof theory and model theory. We introduce propositional logic by way of Gentzen/Prawitz-style natural deduction—for intuitionistic logic—and along the way, each time we introduce the rules for a connective, we show that they are in harmony. So, it’s not too hard to show that proofs in the whole system can be normalised and we get the subformula property for normal proofs. (So, we can gesture in the direction of provability being <em>analytic</em> in a strong sense, since a normal proof literally <em>analyses</em> the premises and conclusion into components and connects them using the fundamental rules governing the concepts involved.)</p>
<p>Once that’s done, we then introduce Boolean valuations (and truth tables), and we can show that the proof system is sound but not complete for validity defined as the absence of a Boolean counterexample. Approaching things this way means we have an interesting discussion about soundness and completeness, and about intuitionistic and classical logic, and whether we should be happy with the gap between proofs and models or not, and if not, whether we should close that gap by adding to our proof system (that way lies <em>classical</em> natural deduction), or whether we should close the gap by enriching our class of models to serve as counterexamples (here we sketch Heyting algebras, as generalisations of Boolean valuations, but we point to Kripke models, too). There’s also scope for a discussion of whether we should understand logic in a proof-first way or a model-first way (or both, or neither), and how proofs and models relate to however it is that words and concepts get their meanings.</p>
<p>With that done, we’re halfway through the subject. Having arrived at Boolean valuations, it’s a short hop, skip and jump to Carnap’s models for modality, and their generalisation, universal models for the modal logic S5. So, we look at these models for possibility and necessity, and show how these possible worlds models can be used to analyse modality, strict conditionality, and similar notions.</p>
<p>Then with models like these we can be of service to our colleagues by introducing double-indexing and two-dimensional modal logic, and the analysis of fixedly diagonal propositions, and the relationship between analyticity, necessity and <em>a priority</em>.</p>
<p>With these model-theoretic considerations in hand, we turn to the question of what it might be to <em>derive</em> a modal claim, and we turn to the natural deduction rules for modals, which introduce constraints on assumptions. One way to prove that \(A\) is necessary, after all, is to prove \(A\) from claims of the form \(\Box B\), for those claims hold not only <em>here</em>, but also in any alternate circumstances, too. So, we get natural deduction systems for S4 and S5 rather straightforwardly.</p>
<p>Proving something more <em>general</em> than \(A\) by proving \(A\) from premises satisfying certain conditions sounds familiar if you’ve dealt with <em>quantifiers</em> before. How to you show that <em>everything</em> is an \(F\)? By proving that \(Fa\) when we have assumed <em>nothing about \(a\)</em>. Then our proof applies <em>no matter what \(a\) is</em>. So, we can generalise the conditions for modal proof to proofs with <em>quantifiers</em> too. So, we introduce the logic of first-order quantifiers with natural deduction first, and once we’ve done that, we turn back to models at last.</p>
<p>So, the introduction to logic has a rhythm, taking us from proofs to models of propositional logic, through models and then proofs for modal logic, and then to proofs and models for predicate logic. Along the way we look at issues in the philosophy of logic and the applications of logic to different issues in philosophy.</p>
<p>Although this curriculum and the course material is all ours, we are indebted to our colleagues for many discussions concerning the pedagogy of logic. I’ll single out two here. Allen Hazen talked to GR for many years about the pedagogical virtues of introducing modal logic before predicate logic to philosophy students. And <a href="http://davewripley.rocks">Dave Ripley</a> has, for the last couple of years, introduced logic using intuitionistic natural deduction and classical truth tables, making a virtue out of the soundness and <em>in</em>completeness of the pairing between the proof theory and the model theory. Neither Allen nor Dave would teach things how we have, but we’ve valued talking over the pedagogy with them over the years.</p>
<p>If you’d like to compare your mastery of logic, in comparison to what our students are learning, you can try your own hand at our <a href="https://consequently.org/resources/PHIL20030-2019-class-tasks-1-6.pdf">in-class tasks for weeks 1 to 6</a>.</p>
Notes from a Pandemic
https://consequently.org/news/2020/notes-from-a-pandemic/
Wed, 27 May 2020 21:19:47 +1000https://consequently.org/news/2020/notes-from-a-pandemic/<p>I’ve been up to a few things during the pandemic. Quite a few things, it seems. Here are links to some of the traces you can find elsewhere on the internet.</p>
<p>I wouldn’t say that I’ve become <em>good</em> at using Zoom, but I have been doing a heck of a lot of it. <a href="http://consequently.org/class/2020/unib10002/">My</a> <a href="http://consequently.org/class/2020/phil30043/">three</a> <a href="http://consequently.org/class/2020/phil40013/">subjects</a> for this semester moved online, and running seminars, workshops, classes over Zoom has become a part (only a part) of keeping the ship going. I don’t record those classes (for obvious reasons!) but I have recorded a couple of research seminars presentations I’ve made over the last few weeks.</p>
<p>First up, as the shift to off-campus online teaching was just starting, in late March, I decided to try giving a research seminar presentation over Zoom, for the <a href="https://blogs.unimelb.edu.au/logic/logic-seminar/">Melbourne Logic Group Seminar</a>. That was fun, because I got to see an audience from all over the world. You can <a href="https://consequently.org/presentation/2020/geometric-models-logicmelb/">see the recording here</a>.</p>
<p>Then, months later, in early May, I got to do another research presentation, after having had over 50 Zoom classes/seminars/meetings as practice in the mean time. Maybe you can see some difference in <a href="https://consequently.org/presentation/2020/assertion-denial-common-ground-acu/">the second talk, which you can see here</a>. That was a talk “at” the new <a href="https://www.acu.edu.au/research/our-research-institutes/dianoia-institute-of-philosophy">Dianoia Institute of Philosophy</a> at the Australian Catholic University.</p>
<p>One of the upsides of the pandemic’s forced move to online work has been the internationalisation of our research seminars. My colleague <a href="https://shawn-standefer.github.io">Shawn Standefer</a> was instrumental in establishing the <a href="http://ba-logic.com/logic-supergroup/">Logic Supergroup</a>, an international collaboration between research groups in logic all over the globe, and we’ve been sharing seminars and enjoying catching up with each other during this time of isolation. I wonder what the “new normal” will be, after the pandemic. I hope we keep some of this different way of working after we are able to resume face-to-face meetings.</p>
<p>Shawn and I have also somehow been able to keep writing during this time. I’m most proud of this <a href="https://consequently.org/writing/collection-frames/">joint paper</a>, which brings together ideas we’ve had over the last 18 months or so. I think it’s a genuine advance in our understanding of the frame semantics for relevant logics, and it’s the paper I’m most proud of having written, in recent years.</p>
<p>Finally, last week I recorded a wide-ranging conversation with Aleks Hammo, for his podcast, <em><a href="https://aleks.co">Aleks Listens</a></em>. We talked about logic, philosophy, belief, and the state of the world. If you like that sort of thing, you might like to <a href="https://alekslistens.podbean.com/e/43-prof-greg-restall-logic-society-belief-and-the-self/">listen along, here</a>.</p>
Collection Frames for Substructural Logics
https://consequently.org/writing/collection-frames/
Thu, 14 May 2020 00:00:00 UTChttps://consequently.org/writing/collection-frames/<p>We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalization of Routley–Meyer ternary frames and a simplification of them. The key innovation of this semantics is the use of a single accessibility relation to relate collections of points to points. Different logics are modeled by varying the kinds of collections used: they can be sets, multisets, lists or trees. We show that collection frames on trees are sound and complete for the basic positive substructural logic B<sup>+</sup>, that collection frames on multisets are sound and complete for RW<sup>+</sup> (the relevant logic R<sup>+</sup>, without contraction, or equivalently, positive multiplicative and additive linear logic with distribution for the additive connectives), and that collection frames on sets are sound for the positive relevant logic R<sup>+</sup>. (The completeness of set frames for R<sup>+</sup> is, currently, an open question.)</p>
Assertions, Denials, Questions, Answers, and the Common Ground
https://consequently.org/presentation/2020/assertion-denial-common-ground-acu/
Sat, 25 Apr 2020 00:00:00 UTChttps://consequently.org/presentation/2020/assertion-denial-common-ground-acu/<p><em>Abstract</em>: In this talk, I examine some of the interconnections between norms governing assertion, denial, questions and answers, and the common ground of a discourse. When we pay attention to the structure of norms governing polar (yes/no) questions, we can clarify the distinction between strong and weak denials, together with the parallel distinction between strong and weak assertion, and the way that these speech acts interact with the common ground.</p>
<p>With those connections established, I respond to two criticisms of the program sketched out in my 2005 paper “Multiple Conclusions”. First, that understanding the upshot of a valid sequent <em>X</em> ⊢ <em>Y</em> as enjoining us to not assert each member of <em>X</em> and deny each member of <em>Y</em> is altogether too weak to explain the inferential force of logical validity. Deriving <em>X</em> ⊢ <em>A</em> should tell us, after all, something about justifying <em>A</em> on the basis of <em>X</em>, rather than merely prohibiting <em>A</em>’s denial. Where is the force to actually conclude the conclusion of a proof? A second, related criticism is that the format of multiple conclusion sequents seems unsatisfactory, in that it has no place for distinguishing a single conclusion, and proofs, after all, seem to be proofs of individual claims.</p>
<p>I will argue that both of these concerns can be assuaged if we pay closer attention to the norms connecting assertions and denials along with <em>justification requests</em> — questions aiming at eliciting reasons for assertions or denials. Once we understand the connection between justification requests, definitionsand the common ground, we will see not only that the these two concerns can be met. A derivation of a sequent <em>X</em> ⊢ <em>A</em>,<em>Y</em> gives us an answer to a justification request “why <em>A</em>?” in any available context where each member of <em>X</em> has been ruled in and each member of <em>Y</em> has been ruled out, and a derivation of a sequent <em>X</em>,<em>B</em> ⊢ <em>Y</em>, similarly gives us an answer to the justification request “why not <em>B</em>?” in any such context. The picture that results utilises the full multiple premise, multiple conclusion sequent calculus of classical logic, and does due justice to the idea that a proof (or a refutation) proves (or refutes) <em>one thing</em> relative to background assumptions or premises. In addition, when we consider the connection between justification requests and the norms governing <em>definitions</em>, we can see more clearly what could be involved in taking the connective/quantifier rules of a logical system to <em>define</em> the concepts they introduce.</p>
<ul>
<li><p>The talk is an online presentation at the Dianoia Institute of Philosophy Seminar at the Australian Catholic University, in Melbourne.</p></li>
<li><p>The <a href="https://consequently.org/slides/assertion-denial-common-ground-slides-acu.pdf">slides for the talk are available here</a>, and the <a href="https://consequently.org/handouts/assertion-denial-common-ground-handout-acu.pdf">handout is here</a>.</p></li>
<li><p>There is a (low resolution) recording of the talk <a href="https://youtu.be/9ZqcTEX4v9E">on YouTube</a> if you’d like to see what you missed.</p></li>
</ul>
<iframe width="560" height="315" src="https://www.youtube.com/embed/9ZqcTEX4v9E" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
Proofs and Models in Naive Property Theory: A Response to Hartry Field's “Properties, Propositions and Conditionals”
https://consequently.org/writing/proofs-and-models-in-npt/
Thu, 02 Apr 2020 00:00:00 UTChttps://consequently.org/writing/proofs-and-models-in-npt/<p>In our response Field’s “Properties, Propositions and Conditionals”, we explore the methodology of Field’s program. We begin by contrasting it with a proof-theoretic approach and then commenting on some of the particular choices made in the development of Field’s theory. Then, we look at issues of property identity in connection with different notions of equivalence. We close with some comments relating our discussion to Field’s response to Restall’s “<a href="https://consequently.org/writing/stp/">What are we to accept, and what are we to reject, when saving truth from paradox?</a>”.</p>
Geometric Models for Relevant Logics
https://consequently.org/presentation/2020/geometric-models-logicmelb/
Thu, 19 Mar 2020 00:00:00 UTChttps://consequently.org/presentation/2020/geometric-models-logicmelb/<p><em>Abstract</em>: Alasdair Urquhart’s work on models for relevant logics is distinctive in a number of different ways. One key theme, present in both his undecidability proof for the relevant logic R, and his proof of the failure of interpolation in R, is the use of techniques from geometry. In this talk, inspired by Urquhart’s work, I explore ways to generate natural models of R from geometries, and different constraints that an accessibility relation in such a model might satisfy. I end by showing that a set of natural conditions on an accessibility relation, motivated by geometric considerations, is jointly unsatisfiable</p>
<ul>
<li><p>This talk is an online presentation for the Melbourne Logic Group. The <a href="https://consequently.org/slides/geometric-models-talk-logicmelb.pdf">slides for the talk are here</a>.</p></li>
<li><p>There is a (low resolution) recording of the talk <a href="https://youtu.be/0XRF4VD1Qno">on YouTube</a> if you’d like to see what you missed.</p></li>
</ul>
<iframe width="560" height="315" src="https://www.youtube.com/embed/0XRF4VD1Qno" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
UNIB10002: Logic, Language and Information
https://consequently.org/class/2020/unib10002/
Sun, 01 Mar 2020 00:00:00 UTChttps://consequently.org/class/2020/unib10002/<p><strong><span class="caps">UNIB10002</span>: Logic, Language and Information</strong> is a <a href="http://unimelb.edu.au">University of Melbourne</a> undergraduate breadth subject, introducing logic and its applications to students from a wide range of disciplines in the Arts, Sciences and Engineering. I coordinate this subject with my colleague Dr. Jen Davoren, with help from Prof. Lesley Stirling (Linguistics), Dr. Peter Schachte (Computer Science) and Dr. Daniel Murfet (Mathematics).</p>
<p>The subject is taught to University of Melbourne undergraduate students. Details for enrolment are <a href="https://handbook.unimelb.edu.au/view/2020/UNIB10002">here</a>.</p>
PHIL40013: Uncertainty, Vagueness and Disagreement
https://consequently.org/class/2020/phil40013/
Sun, 01 Mar 2020 00:00:00 UTChttps://consequently.org/class/2020/phil40013/<p><strong><span class="caps">PHIL40013</span>: Uncertainty, Vagueness and Disagreement</strong> is a <a href="http://unimelb.edu.au">University of Melbourne</a> honours seminar subject for fourth-year students. Our aim in the Honours program is to introduce students to current work in research in philosophy of logic and language.</p>
<p>In 2020, we’re covering the connections between speech acts, epistemology and normative theory.</p>
<ol>
<li><strong>Introduction and overview, background</strong></li>
<li><strong>Speech acts: what are they?</strong>
<ul>
<li>J. L. Austin, <em>How to Do things with Words</em>, Clarendon Press,
Oxford, 1962. [<strong><em>Read Lecture 9</em></strong>]</li>
<li>H. P. Grice, “Logic and Conversation,” pages 41–58 in <em>Syntax and
Semantics</em>: <em>Speech Acts</em>, edited by P. Cole and J. L. Morgan,
Academic Press, New York, 1975.</li>
<li>Sarah E. Murray and William B. Starr, “<a href="http://dx.doi.org/10.1093/oso/9780198738831.003.0009">Force and Conversational States</a>,” pages 202–236 in <em>New Work on Speech Acts</em>, edited by Daniel Fogal, Daniel Harris and Matthew Moss, Oxford University Press, 2018. [<strong><em>Read Sections 9.1 and 9.2</em></strong>]</li>
<li>Nuel Belnap “<a href="http://dx.doi.org/10.1007/BF00368389">Declaratives are not Enough</a>”, <em>Philosophical Studies</em> 59:1 (1990) 1–30.</li>
<li>Mark Lance and Rebecca Kukla “<a href="http://dx.doi.org/10.1086/669565">Leave the Gun; Take the Cannoli! The Pragmatic Topography of Second-Person Calls</a>” <em>Ethics</em> 123:3 (2013) 456–478.</li>
<li>Mark Lance and Rebecca Kukla <em>Yo! and Lo! The Pragmatic Topography of the Space of Reasons,</em> Harvard University Press, 2009. [<strong><em>Read Chapter 1</em></strong>]</li>
<li>Craige Roberts “<a href="http://dx.doi.org/10.1093/oso/9780198738831.001.0001">Speech Acts in Discourse Context</a>”, pages 317–359 in <em>New Work on Speech Acts</em>, edited by Daniel Fogal, Daniel Harris and Matthew Moss, Oxford University Press, 2018.</li>
</ul></li>
<li><strong>Assertion</strong>
<ul>
<li>John Macfarlane, “<a href="http://dx.doi.org/10.1093/acprof:oso/9780199573004.001.0001">What is Assertion?</a>” pages 79–96 in <em>Assertion</em>:
<em>New Philosophical Essays</em>, edited by Jessica Brown and Herman
Cappelen, Oxford University Press, 2011.</li>
<li>Ishani Maitra, “<a href="http://dx.doi.org/10.1093/acprof:oso/9780199573004.001.0001">Assertion, Norms, and Games</a>” pages 277–296 in
<em>Assertion</em>: <em>New Philosophical Essays</em>, edited by Jessica Brown and
Herman Cappelen, Oxford University Press, 2011.</li>
<li>Jennifer Lackey, “<a href="http://dx.doi.org/10.1111/j.1468-0068.2007.00664.x">Norms of Assertion</a>,” <em>Noûs</em> 41:4 (2007) 594–626.</li>
<li>Rachel Mckinnon, <em>The Norms of Assertion</em>: <em>Truth, Lies, and Warrant,</em> Palgrave, 2015.</li>
<li>Peter Pagin, “<a href="http://plato.stanford.edu/archives/spr2015/entries/assertion/">Assertion</a>”, <em>The Stanford Encyclopedia of Philosophy,</em> 2015.</li>
</ul></li>
<li><strong>Common Ground and Accommodation</strong>
<ul>
<li>Robert Stalnaker, “<a href="http://dx.doi.org/10.1023/A:1020867916902">Common Ground</a>,” <em>Linguistics and Philosophy</em> 25:5–6 (2002) 701–721.</li>
<li>Mandy Simons, “<a href="http://dx.doi.org/10.1023/A:1023004203043">Presupposition and Accommodation: Understanding the Stalnakerian Picture</a>,” <em>Philosophical Studies</em> 112:3 (2003) 251–278.</li>
<li>Craige Roberts, “<a href="https://onlinelibrary-wiley-com/doi/pdf/10.1002/9781118398593.ch22">Accommodation in a Language Game</a>”, pages 345–366 in <em>A Companion to David Lewis</em>, edited by Barry Loewer and Jonathan Schaffer, John Wiley & Sons, Ltd., 2015.</li>
<li>David Lewis, “<a href="http://dx.doi.org/10.1007/BF00258436">Scorekeeping in a Language Game</a>”, <em>Journal of Philosophical Logic</em> 8:1 (1979) 339–359.</li>
<li>Paal Antonsen, “<a href="http://dx.doi.org/10.1093/analys/anx145">Scorekeeping</a>”, <em>Analysis</em> 78:4 (2018) 589–595.</li>
</ul></li>
<li><strong>Cooperation, Convention and Norms</strong>
<ul>
<li>Sarah E. Murray and William B. Starr, “<a href="http://dx.doi.org/10.1093/oso/9780198738831.003.0009">Force and Conversational States</a>,” pages 202–236 in <em>New Work on Speech Acts</em>, edited by Daniel Fogal, Daniel Harris and Matthew Moss, Oxford University Press, 2018. [<strong><em>Read Sections 9.3 to 9.5</em></strong>]</li>
<li>Cristina Bicchieri, <em>The Grammar of Society</em>: <em>the nature and dynamics of social norms</em>, Cambridge University Press, 2006. [<strong><em>Read Chapter 1</em></strong>]</li>
<li>Cristina Bicchieri, <a href="http://dx.doi.org/10.1093/acprof:oso/9780190622046.001.0001"><em>Norms in the Wild</em>: <em>how to diagnose, measure, and change social norms</em></a>, Oxford University Press, 2017.</li>
</ul></li>
<li><strong>Stereotypes and Generics</strong>
<ul>
<li>Sarah-Jane Leslie, “<a href="http://dx.doi.org/10.1111/j.1520-8583.2007.00138.x">Generics and the Structure of the Mind</a>,” <em>Philosophical Perspectives</em> 21:1 (2007) 375–403.</li>
<li>Sally Haslanger, “<a href="http://dx.doi.org/10.1007/978-90-481-3783-1_11">Ideology, Generics, and Common Ground</a>,” pages 179–207 in <em>Feminist Metaphysics</em>: <em>Explorations in the Ontology of Sex, Gender and the Self</em>, edited by Charlotte Witt, Springer, Dordrecht, 2011.</li>
<li>Rachel Katharine Sterken, “<a href="http://dx.doi.org/10.1111/phc3.12431">The Meaning of Generics</a>” <em>Philosophy Compass,</em> 12:8 (2017) e12431.</li>
<li>Jennifer Saul, “<a href="http://dx.doi.org/10.1080/0020174x.2017.1285995">Are Generics Especially Pernicious?</a>” <em>Inquiry,</em> advance access (2019), 1–18.</li>
</ul></li>
<li><strong>Authority and Epistemic Territory</strong>
<ul>
<li>Jennifer Nagel, “<a href="http://dx.doi.org/10.1017/epi.2015.4">The Social Value of Reasoning in Epistemic
Justification</a>,” <em>Episteme</em> 12:2 (2015) 297–308.</li>
<li>John Heritage, “<a href="http://dx.doi.org/10.1080/08351813.2012.646684">Epistemics in Action: Action Formation and Territories of Knowledge</a>,” <em>Research on Language and Social Interaction</em> 45:1 (2012) 1–29.</li>
<li>Akio Kamio, <em>Territory of Information,</em> John Benjamins, 1997.</li>
<li>Hugo Mercier and Dan Sperber, “<a href="http://dx.doi.org/10.1017/s0140525x10000968">Why do Humans Reason? Arguments for an argumentative theory</a>,” <em>Behavioral and Brain Sciences</em> 34:2 (2011) 57–74.</li>
</ul></li>
<li><strong>Illocutionary Silencing</strong>
<ul>
<li>Rae Langton, “<a href="https://www-jstor-org/stable/2265469">Speech Acts and Unspeakable Acts</a>,” <em>Philosophy</em> & <em>Public Affairs</em> 22:4 (1993) 293–330.</li>
<li>Ishani Maitra, “<a href="https://www-jstor-org/stable/27822050">Silencing Speech</a>,” <em>Canadian Journal of Philosophy</em> 39:2 (2009) 309–338.</li>
<li>Alessandra Tanesini, “<a href="http://aristoteliansupp.oxfordjournals.org/content/90/1/71">“Calm Down, Dear”: Intellectual Arrogance,
Silencing and Ignorance</a>,” <em>Aristotelian Society Supplementary Volume</em> 90:1 (2016) 71–92.</li>
<li>Alexander Bird, “<a href="https://doi-org/10.1111/1468-0114.00137">Illocutionary Silencing</a>,” <em>Pacific Philosophical Quarterly</em> 83:1 (2002) 1–15.</li>
<li>Mari Mikkola, “<a href="http://dx.doi.org/10.1111/j.1468-0114.2011.01404.x">Illocution, Silencing and the Act of Refusal</a>,” <em>Pacific Philosophical Quarterly</em> 92:3 (2011) 415–437.</li>
<li>Kristie Dotson, “<a href="http://dx.doi.org/10.1111/j.1527-2001.2011.01177.x">Tracking Epistemic Violence, Tracking Practices of Silencing</a>,” <em>Hypatia</em> 26:2 (2011) 236–257.</li>
</ul></li>
<li><strong>Gaslighting</strong>
<ul>
<li>Kate Abramson, “<a href="http://dx.doi.org/10.1111/phpe.12046">Turning up the Lights on Gaslighting</a>,” <em>Philosophical Perspectives</em> 28:1 (2014) 1–30.</li>
<li>Kate Manne, <a href="http://dx.doi.org/10.1093/oso/9780190604981.001.0001"><em>Down Girl</em>: <em>the logic of misogyny</em></a>, Oxford
Univeristy Press, 2018. [<strong><em>Read Chapter 1</em></strong>]</li>
<li>Andrew D. Spear, “<a href="http://dx.doi.org/10.1007/s11245-018-9611-z">Gaslighting, Confabulation, and Epistemic
Innocence</a>,” <em>Topoi</em> early access (2018).</li>
<li>Cynthia A. Stark, “<a href="http://dx.doi.org/10.1093/monist/onz007">Gaslighting, Misogyny, and Psychological
Oppression</a>,” <em>The Monist</em> 102:2 (2019) 221–235.</li>
</ul></li>
</ol>
<p>For further information, contact me. To participate, check <a href="https://handbook.unimelb.edu.au/view/2019/PHIL40013">the handbook</a>.</p>
PHIL30043: The Power and Limits of Logic
https://consequently.org/class/2020/phil30043/
Sun, 01 Mar 2020 00:00:00 UTChttps://consequently.org/class/2020/phil30043/
<p><strong><span class="caps">PHIL30043</span>: The Power and Limits of Logic</strong> (or, as I like to call it, <em>Kurt Gödel’s Greatest Hits</em>) is a <a href="https://handbook.unimelb.edu.au/view/2020/PHIL30043">University of Melbourne undergraduate subject</a>. It covers the metatheory of classical first order predicate logic, beginning at the <em>Soundness</em> and <em>Completeness</em> Theorems, <em>Compactness</em>, <em>Cantor’s Theorem</em>, the <em>Downward Löwenheim–Skolem Theorem</em>, <em>Recursive Functions</em>, <em>Register Machines</em>, <em>Representability</em>, the <em>Indefinability of Truth</em> and the <em>Undecidability of Predicate Logic</em>, and ending up at <em>Gödel’s Incompleteness Theorems</em> and <em>Löb’s Theorem</em>.</p>
<figure>
<img src="https://consequently.org/images/godel.jpg" alt="Kurt Godel, seated">
<figcaption>Kurt Gödel, seated</figcaption>
</figure>
<p>The subject is taught to University of Melbourne undergraduate students (for Arts students as a part of the Philosophy major, for non-Arts students, as a breadth subject). Details for enrolment are <a href="https://handbook.unimelb.edu.au/view/2020/PHIL30043">here</a>.</p>
<p>The subject is taught to University of Melbourne undergraduate students (for Arts students as a part of the Philosophy major, for non-Arts students, as a breadth subject). Details for enrolment are <a href="https://handbook.unimelb.edu.au/view/2020/PHIL30043">here</a>. I make use of video lectures I have made <a href="https://www.youtube.com/playlist?list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n">freely available on YouTube</a>.</p>
<h3 id="outline">Outline</h3>
<p>The course is divided into four major sections, taught over 12 weeks. Here is a list of all of the videos, in case you’d like to follow along with the content.</p>
<h4 id="soundness-and-completeness">Soundness and Completeness</h4>
<ul>
<li><a href="https://www.youtube.com/watch?v=qTAcRSnObdw&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=2&t=0s">The Language of Predicate Logic</a></li>
<li><a href="https://www.youtube.com/watch?v=JMOTYyHnkt8&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=3&t=0s">Proofs for Predicate Logic</a></li>
<li><a href="https://www.youtube.com/watch?v=1p0oTY6I-Yw&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=4&t=0s">Models for Predicate Logic and Soundness</a></li>
<li><a href="https://www.youtube.com/watch?v=58k6dNrvBoU&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=5&t=0s">Completeness for Predicate Logic</a></li>
</ul>
<h4 id="countability-and-compactness">Countability and Compactness</h4>
<ul>
<li><a href="https://www.youtube.com/watch?v=eI-i8jpcd_o&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=6&t=0s">Identity and Functions</a></li>
<li><a href="https://www.youtube.com/watch?v=NJgmRvKFGDc&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=7&t=0s">Countability and Diagonalisation</a></li>
<li><a href="https://www.youtube.com/watch?v=f-iTL4wWa8k&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=8&t=0s">Compactness and Countable Models</a></li>
</ul>
<h4 id="computability">Computability</h4>
<ul>
<li><a href="https://www.youtube.com/watch?v=83zBUpKm0GM&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=9&t=0s">Recursive Functions</a></li>
<li><a href="https://www.youtube.com/watch?v=04ol2ZkZuUk&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=10&t=0s">Register Machines</a></li>
<li><a href="https://www.youtube.com/watch?v=8rUGLIubpl0&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=11&t=0s">Register Machines compute the Recursive Functions</a></li>
<li><a href="https://www.youtube.com/watch?v=olAiZHi4ra0&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=12&t=0s">Non-Recursive Functions</a></li>
</ul>
<h4 id="undecidability-and-incompleteness">Undecidability and Incompleteness</h4>
<ul>
<li><a href="https://www.youtube.com/watch?v=btRjkOm0HFA&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=13&t=0s">Theories of Arithmetic</a></li>
<li><a href="https://www.youtube.com/watch?v=gxZgiEllHJE&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=14&t=0s">Diagonalisation and its Consequences</a></li>
<li><a href="https://www.youtube.com/watch?v=3gQWRaXhPWE&list=PLvROQ_RldgC8KYmkQsF_zKqAXD_Xphr9n&index=15&t=0s">Provability Predicates, and Beyond</a></li>
</ul>
A Place for Logic in the Humanities?
https://consequently.org/presentation/2020/a-place-for-logic-ballarat/
Fri, 21 Feb 2020 00:00:00 UTChttps://consequently.org/presentation/2020/a-place-for-logic-ballarat/<p><em>Abstract</em>: Logic has been an important part of philosophy in the western tradition since the work of Aristotle in the 4th Century BCE. Developments of the 19th and the 20th Century have seen an incredible flowering of mathematical techniques in logic, and the discipline transformed beyond recognition into something that can seem forbiddingly technical and formal. The discipline of logic plays a vital role in mathematics, linguistics, computer science and electrical engineering, and it may seem that it no longer has a place within the humanities. In this short talk, I’ll argue that this perception is misplaced and dangerous, and that in a time of increasing specialisation and differentiation between the cultures of the humanities, the sciences, and of engineering, logic not only has much to give to the humanities, it also has much to learn from them.</p>
<ul>
<li>The talk is an presentation at <a href="https://www.facebook.com/PhilosophyBallarat/photos/a.2201699763185207/2912554252099751/?type=3&__xts__%5B0%5D=68.ARAUQL5e5CWr3eRtb2U-cmHriKCZV61-JE0xvtbuBzQQkwwxzLLnhuyU8orEMXUy1-53DNMtC2VdDVR84dKrwcKD1HPl0-UMpEYpr_KmR0gOMH5q0wZzesi-ga40uAuWZ8cHdjbIXzCP_D4-8EPVk6luxXZcFTi_F9VPEaORiq63QmyuZ6k1MGgz8gGC4lStyGicz14v_oxKMqzehNjVTBa29TeKgNYpE7UrjMV6nEM6mK-uASfuiFX12zFOb4uajox4zthWhvfa4r2ZkQlmAJW59PlmPieeO3qT1zMzkLfB6sWxhrbOoi6a7UyWiRhZdXd2REcI8Dxa9HXrCjoaJSwjzOY4">The Ballarat Philosophy Symposium 2020</a>, at Federation University, Ballarat.</li>
</ul>
Two Negations are More than One
https://consequently.org/writing/two-negations/
Thu, 12 Dec 2019 00:00:00 UTChttps://consequently.org/writing/two-negations/<p>In models for paraconsistent logics, the semantic values of sentences and their negations are less tightly connected than in classical logic. In “American Plan” logics for negation, truth and falsity are, to some degree, independent. The truth of \({\mathord\sim}p\) is given by the falsity of \(p\), and the falsity of \({\mathord\sim}p\) is given by the truth of \(p\). Since truth and falsity are only loosely connected, \(p\) and \({\mathord\sim}p\) can both hold, or both fail to hold. In “Australian Plan” logics for negation, negation is treated rather like a modal operator, where the truth of \({\mathord\sim}p\) in a situation amounts to \(p\) failing in <em>certain other situations</em>. Since those situations can be different from this one, \(p\) and \({\mathord\sim}p\) might both hold here, or might both fail here.</p>
<p>So much is well known in the semantics for paraconsistent logics, and for first degree entailment and logics like it, it is relatively easy to translate between the American Plan and the Australian Plan. It seems that the choice between them seems to be a matter of taste, or of preference for one kind of semantic treatment or another. This paper explores some of the differences between the American Plan and the Australian Plan by exploring the tools they have for modelling a language in which we have <em>two</em> negations.</p>
<p>This paper is dedicated to my friend and mentor, Professor Graham Priest.</p>
Generics: Inference & Accommodation
https://consequently.org/presentation/2019/generics-inference-accommodation-mit/
Thu, 05 Dec 2019 00:00:00 UTChttps://consequently.org/presentation/2019/generics-inference-accommodation-mit/<p>In this talk, I aim to give an account of norms governing our uses of <em>generic judgements</em> (like “kangaroos have long tails”, “birds lay eggs”, or “logic talks are boring”), norms governing <em>inference</em>, and the relationship between generics and inference. This connection goes some way to explain why generics exhibit some very strange behaviour: Why is it, for example, that “birds lay eggs” seems true, while “birds are female” seems false, despite the fact that only female birds lay eggs? Generics exhibit this behaviour because they make inferences and explanations explicit, and inferences and explanations have exactly the same sort of behaviour as generics.</p>
<p>Given the connection between generics and inference, we will be able to see how inference is involved in the process of <em>accommodation</em>, which plays a significant role in how we manage dialogue and conversation. A generic of the form *F*s are *G*s can enter the common ground when we allow the inference from <em>Fx</em> to <em>Gx</em> to pass without question in conversation. With this connection in hand, I will begin to explore what this means for social kind generics and how we use them.</p>
<ul>
<li><p>This is a talk for the <a href="https://blogs.unimelb.edu.au/social-hierarchy/events/mit-workshop-constructing-social-hierarchy-ii/">Constructing Social Hierarchy 2</a> Workshop at MIT in December 6, 2019.</p></li>
<li><p>The <a href="https://consequently.org/slides/accommodation-mit-workshop.pdf">slides for the talk are available here</a>, and the <a href="https://consequently.org/handouts/accommodation-mit-workshop-handout.pdf">handout is here</a>.</p></li>
</ul>
What's So Special About Logic? Practices, Rules and Definitions
https://consequently.org/presentation/2019/whats-so-special-about-logic-smith/
Wed, 04 Dec 2019 00:00:00 UTChttps://consequently.org/presentation/2019/whats-so-special-about-logic-smith/<p><em>Abstract</em>: Over the last century or so, the discipline of logic has grown and transformed into a powerful set of tools and techniques that find their use in fields as far apart as philosophy, mathematics, computer science, electrical engineering and linguistics. Is there anything distinctive about logic and its results, or is it just another kind of abstract mathematics, or another kind of empirical scientific theory? In this talk I’ll explain why the distinctive subject matter of logical theory means that the tools of logic (<em>proofs</em> and <em>models</em>) can play a special role in our thought and in our talk. This explanation will turn crucially on our practices of assertion and denial, and how it can constrain those practices by using rules and definitions.</p>
<ul>
<li><p>The talk is the 21st Annual Alice Ambrose Lazerowitz/Thomas Tymocko Logic Lecture at Smith College.</p></li>
<li><p>The <a href="https://consequently.org/slides/whats-so-special-about-logic-smith.pdf">slides for the talk are available here</a>.</p></li>
</ul>
Negation on the Australian Plan
https://consequently.org/writing/nap/
Tue, 18 Sep 2018 00:00:00 UTChttps://consequently.org/writing/nap/<p>We present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompat<em>ibility</em> is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilities between such points. We defuse a number of objections to this Plan, raised by supporters of the American Plan for negation, in which negation is handled via a many-valued semantics. We show that the Australian Plan has substantial advantages over the American Plan.</p>
Teaching Logical Methods
https://consequently.org/news/2019/teaching-logical-methods/
Thu, 14 Nov 2019 20:01:45 +1100https://consequently.org/news/2019/teaching-logical-methods/<p>It’s been a <em>big year</em>. At the start of 2019, <a href="https://shawn-standefer.github.io">Shawn Standefer</a> and I decided to throw all our cards in the air and upend the curriculum for the <a href="https://consequently.org/class/2019/PHIL20030">Level 2 logic unit</a> in the philosophy program here at Melbourne. We wrote 200 pages of a draft textbook (while I really should have been finishing my <a href="http://consequently.org/writing/ptrm">other book</a>). Shawn designed and implemented a whole raft of multiple choice practice questions, and we worked on a range of class activities to help our class of 60 students grapple with the material. I recorded hours of video lectures covering the content. We stuffed all of this into the <a href="https://lms.unimelb.edu.au">LMS</a>. And we spent hours in the classroom teaching 60 students the ins and outs of proof theory and model theory for propositional logic, modal logic (including two-dimensional modal logic), and first-order predicate logic. Like I said, it was a <em>big year</em> putting all of this together. Now we’ve wrapped up our first semester teaching the new unit, so we can sit back, breathe and reflect on how things went.</p>
<p>While I was taking that breath and thinking about how things went, Liam Kofi Bright posted <a href="https://sootyempiric.blogspot.com/2019/11/just-humble-philosopher.html">a thoughtful reflection</a> on what he hopes to achieve when he teaches formal methods to his students at the LSE. Our aims for our logic class were similar to Liam’s, but we come at things from a slightly different angle. Since I found reading Liam’s reflections helpful, it makes sense to put my thoughts down in public, in case others might benefit in some way.</p>
<p>Our <a href="https://consequently.org/class/2019/PHIL20030">Logical Methods</a> unit is unashamedly a <em>logic</em> class. (We don’t try to teach the wide range of formal methods. There’s no probability calculus, decision theory or anything beyond <em>logic</em>.) Our subject is designed for philosophy students, though at least a third of the enrolment were students coming from other majors, and even other degree programs. Still, our aim was to give philosophy students the skills and the vocabulary from logic that they will find useful in the rest of their engagement with philosophy, but at the same time, get a sense of logic as a field of philosophical reflection all of its own. Yes, we wanted students to come away able to both construct proofs and make models in formal systems for propositional, modal and predicate logic, and to get a real feel for what we can <em>do</em> with proofs and with models, and how we can use tools from logic—from proof theory and from model theory—to understand and analyse arguments, and to explain and explore the connections between the claims we make. One of our aims was for students to acquire (or strengthen) some logical reflexes. To become familiar with basic inference principles, and to become comfortable with combining them. To get a sense of how to build a model and to construct a counterexample to an argument. To understand what kinds of moves are appropriate when reasoning about possibility or necessity, or with the quantifiers, and what kinds of mistakes you can make if you fail to pay attention to scope distinctions and ambiguity. These reflexes are useful when it comes to philosophical reflection, and it’s good to have a place to practice with them in their own right, before wielding them out in the Real World of other philosophy tutorials.</p>
<p>But in addition to those useful skills, we wanted to give the class a sense of some of the active debates in contemporary work in logic. Since Shawn and I are pluralists and not partisans by temperament, it was natural to us to introduce both intutitionistic logic (when we started venturing into propositional logic with Prawitz-style natural deduction as our account of proofs) and classical logic (when we introduced boolean valuations and truth tables as our account of models). This makes the mismatch between validity-as-given-by-proofs and validity-as-given-by-absence-of-counterexample a problem to be investigated. Do we close the gap by adding more proofs, or by adding more models, or do we live with this gap? That is a question that is worth asking, since working out an answer helps you articulate what sorts of things proofs and models are <em>for</em>. So, we tried to set up the curriculum so that students could understand proofs and models well enough to be able to both use those techniques whenever they’re doing reasoning, and then also to understand the kinds of questions that occupy those of us who work in these areas now.</p>
<p>In <a href="https://sootyempiric.blogspot.com/2019/11/just-humble-philosopher.html">his reflections</a>, Liam described the tension between the two distinct aims of helping students overcome ‘math anxiety’ on the one hand, and helping induce wonder and humility before the world’s complexities, on the other. There is something curious about the seeming opposition between these two goals, but as I thought about Liam’s piece, it struck me that I have never experienced these two goals to be in any way opposed in practice. If you think of logic as some finite collection of simple tools and techniques to be practiced and honed and mastered (like, say, learning the alphabet, or the addition and multiplication tables for the numbers from 1 to 12—or the truth tables for the boolean connectives), then once you’ve dealt with your math anxiety, there would be no humility before any awesome complexity, because the field would be all rather humdrum. But that’s not how logic has developed, and to stay at the level of multiplication tables (or boolean truth tables) is to miss out on what logic has become, and to be blind to the enormous range of the kinds of questions we are able to ask, and which we’re fortunate enough to occasionally be able to answer.</p>
<p>Liam’s reflections pointed me to a nice paper by <a href="https://www.nacadajournal.org/doi/abs/10.12930/0271-9517-10.1.47">Sheila Tobias</a> on “math anxiety”. According to Tobias’ research, the significant variables associated with students’ inability to do college level mathematics are (1) fear, (2) the belief that mathematics is a white male domain, and (3) the belief that you are either good at mathematics or good at languages and never both. These concerns ring true to my experience of dealing with student concerns about logic teaching. To those worries, I will add another concern that turns discourages many from doing logic: (4) the belief that <a href="http://web.apsanet.org/cswp/wp-content/uploads/sites/4/2015/08/bian-lin-et-al.-gender-stereotypes-abt-intell-ability-emerge-early-Science-Jan-2017.pdf">to do well in philosophy <em>or</em> mathematics you have to be <em>brilliant</em></a>. Worries like these have been on my mind for some time.</p>
<p>We’ve tried to design a subject that addresses concerns like these. We are nowhere near done with them–especially addressing the gender and race issues–but the results of our initial steps have been encouraging. Here is a little of what we’ve done so far.</p>
<ul>
<li><p><em>On the brilliance effect</em>: At every stage of learning a new technique or method or result, we try to operationalise what you need to learn, and to break these things down into practice tasks. When we teach the basics of natural deduction, there are separate tasks involving (<em>a</em>) reading proofs, (<em>b</em>) checking proofs to spot mistakes, (<em>c</em>) filling in the gaps in proofs, (<em>d</em>) making your own proofs. We don’t throw students into the deep end, hoping that they’ll “get it.” Whenever there are separable components to a skill, we break things down, bit by bit, and slowly build them up. We show what you need to learn in able to do something well, and then we give them tasks to practice. To master this stuff, you don’t need to just “get it”. There are skills to learn and practice, and you can get there if you put in the time and the work. Thanks to Shawn’s work on the Learning Management System, we have a range of practice exercises that students can hack away at, getting quick feedback on their work. Regular checkpoints show when a student has mastered reading proofs, or writing their own, checking truth in a model, or identifying facts about logical consequence.</p>
<p>This is one place where logic has it relatively easy, compared to other areas in philosophy. It’s a disciplinary norm to define things precisely and work things out piece by piece. It’s our job as educators to introduce those concepts in a manageable way. If we do that well, we <em>show</em> as well as <em>say</em> that these are tools and techniques that can be mastered by practice.</p></li>
<li><p><em>On the perceived difference between linguistic and mathematical expertise</em>: Here we try to use a range of different metaphors and explanatory strategies to explain what we’re doing when we’re doing formal logic. Thankfully, it is relatively easy to make clear that we’re not doing <em>mathematics</em> in the sense of doing a lot of calculations or heavy duty algebra. We’re up front that we teach <em>formal</em> logic, but we’re careful to explain that patterns are everywhere, and not just in mathematics classes. Here, the example of our colleagues in linguistics is helpful to us. We explain that we’re looking at human activities of speaking, thinking and representing, and identifying patterns and structures that recur in what we can <em>say</em> and <em>mean</em>, in order to activate the imagination and expertise of those who are confident with language.</p>
<p>Given that we have an interdisciplinary class (some from the Sciences, but most are Humanities students), we try to walk both sides of the street when it comes to explanatory metaphors. Instead of always talking about algorithms or programs, we describe things as recipes or routines. When we talk about model building, sometimes we emphasise the routine and the systematic, but we also describe specifying a model as an act of imagination, engaging our creativity. Given that the entire subject is built around the duality between proof theory and model theory, we take the perspective-switching nature of looking at a phenomenon from more than one side <em>seriously</em>, so it comes naturally to attempt to describe things in more than one way.</p></li>
<li><p><em>On gender and race</em>: Here we have much more to do than what we’ve done so far. We own the fact that we have taught the subject as two white males. It is a challenge for us to expand our citations beyond the usual small circle of white male experts, but we’ve expanded it just a little. We point to women logicians along the way, and our students’ final project involved the logic and metaphysics of putting together modal logic and first-order predicate logic, it focussed on Ruth Barcan Marcus and the Barcan formulas. So, we put her work front and centre as the culmination of the semester. There is much more to be done–especially on decolonising our curriculum–but the results of these tiny steps have been encouraging so far. With all the changes we made, our retention rate for non-male students was <em>significantly</em> higher than it has been in previous years. The class is still male dominated, but significantly less so than in previous occasions.</p></li>
<li><p><em>On fear</em>: We have not measured students’ anxiety or fears in any way, but we’ve attempted to address that fear in a number of ways. We are fortunate, at Melbourne, to have very good undergraduate students. They are smart, and they are engaged and they are willing to put in the work. So we explain that the subject will involve effort, but that the effort will pay off if they put in the time. We structured the assessments to be predictable and manageable. Not only were the practice tasks broken down into bite-size repeatable skills to master. We assessed them on the same basis. At four points through the semester, there was a multiple choice test, worth 15% of their final score, on just these skills. They weren’t all <em>easy</em> (they added up to some quite challenging tasks) but they were manageable, they were predictable and most importantly, they <em>weren’t an exam</em>. 60% of the assessment task could be done by grinding out effort. It is a way to manage the fear of not “getting it”.</p>
<p>The remaining 40% of the assessment was two projects. The second project was on quantified modal logic and Ruth Barcan Marcus, as I mentioned, and the first was on truth tables and proofs for three-valued logics, and the ways these could be used to model different sorts of phenomena. Both of these projects were clearly related to the work they’d done in class, but they allowed students to reach beyond what they’d practiced, and learn to apply their skills in different environments. The results have been much better than we had expected they’d be. The students loved being stretched, they took things seriously, and wrote quality work. Passing turned out to be relatively easy, if they put in the effort to grind. Then with the fear of passing dealt with, they could make the effort to excel if they wanted to. From the looks of it, almost all of them <em>really</em> wanted to excel, and many of them have.</p></li>
</ul>
<p>So, that’s been our experience of teaching logical methods in 2019. It’s been a wild ride, and it’s been such a pleasure to be on that ride with <a href="https://shawn-standefer.github.io">Shawn</a> and 60 willing students. Thanks to friends and colleagues, like <a href="http://davewripley.rocks">Dave Ripley</a>, François Schroeter, Allen Hazen, and my current and former graduate students, Kai, Timo, Lian, John, Sakinah and Adam who have helped us sort out some of our thinking about these issues.</p>
What's So Special About Logic? Practices, Rules and Definitions
https://consequently.org/presentation/2019/whats-so-special-about-logic-logicmelb/
Thu, 31 Oct 2019 00:00:00 UTChttps://consequently.org/presentation/2019/whats-so-special-about-logic-logicmelb/<p><em>Abstract</em>: Over the last century or so, the discipline of logic has grown and transformed into a powerful set of tools and techniques that find their use in fields as far apart as philosophy, mathematics, computer science, electrical engineering and linguistics. Is there anything distinctive about logic and its results, or is it just another kind of abstract mathematics, or another kind of empirical scientific theory? In this talk I’ll explain why the distinctive subject matter of logical theory means that the tools of logic (<em>proofs</em> and <em>models</em>) can play a special role in our thought and in our talk. This explanation will turn crucially on our practices of assertion and denial, and how it can constrain those practices by using rules and definitions.</p>
<ul>
<li><p>The talk is an presentation at <a href="https://philevents.org/event/show/76846">Melbourne Logic Day 2019</a>.</p></li>
<li><p>The <a href="https://consequently.org/slides/whats-so-special-about-logic-logicmelb.pdf">slides for the talk are available here</a>.</p></li>
</ul>
Assertions, Denials, Questions, Answers, and the Common Ground
https://consequently.org/presentation/2019/assertion-denial-qa-common-ground-arche/
Sun, 29 Sep 2019 00:00:00 UTChttps://consequently.org/presentation/2019/assertion-denial-qa-common-ground-arche/<p><em>Abstract</em>: In this talk, I examine some of the interconnections between norms governing assertion, denial, questions and answers, and the common ground of a discourse. When we pay attention to the structure of norms governing polar (yes/no) questions, we can clarify the distinction between strong and weak denials, together with the parallel distinction between strong and weak assertion, and the way that these speech acts interact with the common ground.</p>
<p>With those connections established, I respond to two criticisms of the program sketched out in my 2005 paper “Multiple Conclusions”. First, that understanding the upshot of a valid sequent <em>X</em> ⊢ <em>Y</em> as enjoining us to not assert each member of <em>X</em> and deny each member of <em>Y</em> is altogether too weak to explain the inferential force of logical validity. Deriving <em>X</em> ⊢ <em>A</em> should tell us, after all, something about justifying <em>A</em> on the basis of <em>X</em>, rather than merely prohibiting <em>A</em>’s denial. Where is the force to actually conclude the conclusion of a proof? A second, related criticism is that the format of multiple conclusion sequents seems unsatisfactory, in that it has no place for distinguishing a single conclusion, and proofs, after all, seem to be proofs of individual claims.</p>
<p>I will argue that both of these concerns can be assuaged if we pay closer attention to the norms connecting assertions and denials along with <em>justification requests</em> – questions aiming at eliciting reasons for assertions or denials. Once we understand the connection between justification requests, definitionsand the common ground, we will see not only that the these two concerns can be met. A derivation of a sequent <em>X</em> ⊢ <em>A</em>,<em>Y</em> gives us an answer to a justification request “why <em>A</em>?” in any available context where each member of <em>X</em> has been ruled in and each member of <em>Y</em> has been ruled out, and a derivation of a sequent <em>X</em>,<em>B</em> ⊢ <em>Y</em>, similarly gives us an answer to the justification request “why not <em>B</em>?” in any such context. The picture that results utilises the full multiple premise, multiple conclusion sequent calculus of classical logic, and does due justice to the idea that a proof (or a refutation) proves (or refutes) <em>one thing</em> relative to background assumptions or premises. In addition, when we consider the connection between justification requests and the norms governing <em>definitions</em>, we can see more clearly what could be involved in taking the connective/quantifier rules of a logical system to <em>define</em> the concepts they introduce.</p>
<ul>
<li><p>The talk is a presentation at the Metaphysics and Logic Group seminar at Arché at the University of St Andrews.</p></li>
<li><p>The <a href="https://consequently.org/slides/assertion-denial-qa-common-ground-slides-arche.pdf">slides for the talk are available here</a>, and the <a href="https://consequently.org/handouts/assertion-denial-qa-common-ground-handout-arche.pdf">handout is here</a>.</p></li>
</ul>
Collection Frames for Substructural Logics
https://consequently.org/presentation/2019/collection-frames-lancog-lisbon/
Tue, 24 Sep 2019 00:00:00 UTChttps://consequently.org/presentation/2019/collection-frames-lancog-lisbon/<p><em>Abstract</em>: In this talk I present a new frame semantics for positive substructural and relevant propositional logics. This frame semantics is both a <em>generalisation</em> of Routley–Meyer ternary frames and a <em>simplification</em> of them. The key innovation is the use of a single accessibility relation to relate collections of points to points. Different logics are modelled by varying the kinds of collections featuring in the relation: for example, they can be sets, multisets, lists or trees. In this talk I will focus on multiset frames, which are sound and complete for the logic RW+ (positive multiplicative and additive linear logic with distribution for the additive connectives, or equivalently, the relevant logic R+ without contraction).</p>
<p>This is joint work with Shawn Standefer.</p>
<ul>
<li><p>The talk is a presentation at the <a href="http://cful.letras.ulisboa.pt/events/workshop-on-substructural-logics/">LanCog Workshop on Substructural Logics</a> at the Facultate de Letras at the University of Lisbon.</p></li>
<li><p>The <a href="https://consequently.org/slides/collection-frames-talk-lancog-lisbon.pdf">slides for the talk are available here</a>.</p></li>
</ul>
PHIL40013: Uncertainty, Vagueness and Disagreement
https://consequently.org/class/2019/phil40013/
Wed, 24 Jul 2019 00:00:00 UTChttps://consequently.org/class/2019/phil40013/<p><strong><span class="caps">PHIL40013</span>: Uncertainty, Vagueness and Disagreement</strong> is a <a href="http://unimelb.edu.au">University of Melbourne</a> honours seminar subject for fourth-year students. Our aim in the Honours program is to introduce students to current work in research in philosophy of logic and language.</p>
<p>In 2019, we’re covering the connections between speech acts, epistemology and normative theory.</p>
<ol>
<li><strong>Introduction and overview, background</strong></li>
<li><strong>Speech acts: what are they?</strong>
<ul>
<li>J. L. Austin, <em>How to Do things with Words</em>, Clarendon Press,
Oxford, 1962. [<strong><em>Read Lecture 9</em></strong>]</li>
<li>H. P. Grice, “Logic and Conversation,” pages 41–58 in <em>Syntax and
Semantics</em>: <em>Speech Acts</em>, edited by P. Cole and J. L. Morgan,
Academic Press, New York, 1975.</li>
<li>Sarah E. Murray and William B. Starr, “<a href="http://dx.doi.org/10.1093/oso/9780198738831.003.0009">Force and Conversational States</a>,” pages 202–236 in <em>New Work on Speech Acts</em>, edited by Daniel Fogal, Daniel Harris and Matthew Moss, Oxford University Press, 2018. [<strong><em>Read Sections 9.1 and 9.2</em></strong>]</li>
<li>Nuel Belnap “<a href="http://dx.doi.org/10.1007/BF00368389">Declaratives are not Enough</a>”, <em>Philosophical Studies</em> 59:1 (1990) 1–30.</li>
<li>Mark Lance and Rebecca Kukla “<a href="http://dx.doi.org/10.1086/669565">Leave the Gun; Take the Cannoli! The Pragmatic Topography of Second-Person Calls</a>” <em>Ethics</em> 123:3 (2013) 456–478.</li>
<li>Mark Lance and Rebecca Kukla <em>Yo! and Lo! The Pragmatic Topography of the Space of Reasons,</em> Harvard University Press, 2009. [<strong><em>Read Chapter 1</em></strong>]</li>
<li>Craige Roberts “<a href="http://dx.doi.org/10.1093/oso/9780198738831.001.0001">Speech Acts in Discourse Context</a>”, pages 317–359 in <em>New Work on Speech Acts</em>, edited by Daniel Fogal, Daniel Harris and Matthew Moss, Oxford University Press, 2018.</li>
</ul></li>
<li><strong>Assertion</strong>
<ul>
<li>John Macfarlane, “<a href="http://dx.doi.org/10.1093/acprof:oso/9780199573004.001.0001">What is Assertion?</a>” pages 79–96 in <em>Assertion</em>:
<em>New Philosophical Essays</em>, edited by Jessica Brown and Herman
Cappelen, Oxford University Press, 2011.</li>
<li>Ishani Maitra, “<a href="http://dx.doi.org/10.1093/acprof:oso/9780199573004.001.0001">Assertion, Norms, and Games</a>” pages 277–296 in
<em>Assertion</em>: <em>New Philosophical Essays</em>, edited by Jessica Brown and
Herman Cappelen, Oxford University Press, 2011.</li>
<li>Jennifer Lackey, “<a href="http://dx.doi.org/10.1111/j.1468-0068.2007.00664.x">Norms of Assertion</a>,” <em>Noûs</em> 41:4 (2007) 594–626.</li>
<li>Rachel Mckinnon, <em>The Norms of Assertion</em>: <em>Truth, Lies, and Warrant,</em> Palgrave, 2015.</li>
<li>Peter Pagin, “<a href="http://plato.stanford.edu/archives/spr2015/entries/assertion/">Assertion</a>”, <em>The Stanford Encyclopedia of Philosophy,</em> 2015.</li>
</ul></li>
<li><strong>Common Ground and Accommodation</strong>
<ul>
<li>Robert Stalnaker, “<a href="http://dx.doi.org/10.1023/A:1020867916902">Common Ground</a>,” <em>Linguistics and Philosophy</em> 25:5–6 (2002) 701–721.</li>
<li>Mandy Simons, “<a href="http://dx.doi.org/10.1023/A:1023004203043">Presupposition and Accommodation: Understanding the Stalnakerian Picture</a>,” <em>Philosophical Studies</em> 112:3 (2003) 251–278.</li>
<li>Craige Roberts, “<a href="https://onlinelibrary-wiley-com/doi/pdf/10.1002/9781118398593.ch22">Accommodation in a Language Game</a>”, pages 345–366 in <em>A Companion to David Lewis</em>, edited by Barry Loewer and Jonathan Schaffer, John Wiley & Sons, Ltd., 2015.</li>
<li>David Lewis, “<a href="http://dx.doi.org/10.1007/BF00258436">Scorekeeping in a Language Game</a>”, <em>Journal of Philosophical Logic</em> 8:1 (1979) 339–359.</li>
<li>Paal Antonsen, “<a href="http://dx.doi.org/10.1093/analys/anx145">Scorekeeping</a>”, <em>Analysis</em> 78:4 (2018) 589–595.</li>
</ul></li>
<li><strong>Cooperation, Convention and Norms</strong>
<ul>
<li>Sarah E. Murray and William B. Starr, “<a href="http://dx.doi.org/10.1093/oso/9780198738831.003.0009">Force and Conversational States</a>,” pages 202–236 in <em>New Work on Speech Acts</em>, edited by Daniel Fogal, Daniel Harris and Matthew Moss, Oxford University Press, 2018. [<strong><em>Read Sections 9.3 to 9.5</em></strong>]</li>
<li>Cristina Bicchieri, <em>The Grammar of Society</em>: <em>the nature and dynamics of social norms</em>, Cambridge University Press, 2006. [<strong><em>Read Chapter 1</em></strong>]</li>
<li>Cristina Bicchieri, <a href="http://dx.doi.org/10.1093/acprof:oso/9780190622046.001.0001"><em>Norms in the Wild</em>: <em>how to diagnose, measure, and change social norms</em></a>, Oxford University Press, 2017.</li>
</ul></li>
<li><strong>Stereotypes and Generics</strong>
<ul>
<li>Sarah-Jane Leslie, “<a href="http://dx.doi.org/10.1111/j.1520-8583.2007.00138.x">Generics and the Structure of the Mind</a>,” <em>Philosophical Perspectives</em> 21:1 (2007) 375–403.</li>
<li>Sally Haslanger, “<a href="http://dx.doi.org/10.1007/978-90-481-3783-1_11">Ideology, Generics, and Common Ground</a>,” pages 179–207 in <em>Feminist Metaphysics</em>: <em>Explorations in the Ontology of Sex, Gender and the Self</em>, edited by Charlotte Witt, Springer, Dordrecht, 2011.</li>
<li>Rachel Katharine Sterken, “<a href="http://dx.doi.org/10.1111/phc3.12431">The Meaning of Generics</a>” <em>Philosophy Compass,</em> 12:8 (2017) e12431.</li>
<li>Jennifer Saul, “<a href="http://dx.doi.org/10.1080/0020174x.2017.1285995">Are Generics Especially Pernicious?</a>” <em>Inquiry,</em> advance access (2019), 1–18.</li>
</ul></li>
<li><strong>Authority and Epistemic Territory</strong>
<ul>
<li>Jennifer Nagel, “<a href="http://dx.doi.org/10.1017/epi.2015.4">The Social Value of Reasoning in Epistemic
Justification</a>,” <em>Episteme</em> 12:2 (2015) 297–308.</li>
<li>John Heritage, “<a href="http://dx.doi.org/10.1080/08351813.2012.646684">Epistemics in Action: Action Formation and Territories of Knowledge</a>,” <em>Research on Language and Social Interaction</em> 45:1 (2012) 1–29.</li>
<li>Akio Kamio, <em>Territory of Information,</em> John Benjamins, 1997.</li>
<li>Hugo Mercier and Dan Sperber, “<a href="http://dx.doi.org/10.1017/s0140525x10000968">Why do Humans Reason? Arguments for an argumentative theory</a>,” <em>Behavioral and Brain Sciences</em> 34:2 (2011) 57–74.</li>
</ul></li>
<li><strong>Illocutionary Silencing</strong>
<ul>
<li>Rae Langton, “<a href="https://www-jstor-org/stable/2265469">Speech Acts and Unspeakable Acts</a>,” <em>Philosophy</em> & <em>Public Affairs</em> 22:4 (1993) 293–330.</li>
<li>Ishani Maitra, “<a href="https://www-jstor-org/stable/27822050">Silencing Speech</a>,” <em>Canadian Journal of Philosophy</em> 39:2 (2009) 309–338.</li>
<li>Alessandra Tanesini, “<a href="http://aristoteliansupp.oxfordjournals.org/content/90/1/71">“Calm Down, Dear”: Intellectual Arrogance,
Silencing and Ignorance</a>,” <em>Aristotelian Society Supplementary Volume</em> 90:1 (2016) 71–92.</li>
<li>Alexander Bird, “<a href="https://doi-org/10.1111/1468-0114.00137">Illocutionary Silencing</a>,” <em>Pacific Philosophical Quarterly</em> 83:1 (2002) 1–15.</li>
<li>Mari Mikkola, “<a href="http://dx.doi.org/10.1111/j.1468-0114.2011.01404.x">Illocution, Silencing and the Act of Refusal</a>,” <em>Pacific Philosophical Quarterly</em> 92:3 (2011) 415–437.</li>
<li>Kristie Dotson, “<a href="http://dx.doi.org/10.1111/j.1527-2001.2011.01177.x">Tracking Epistemic Violence, Tracking Practices of Silencing</a>,” <em>Hypatia</em> 26:2 (2011) 236–257.</li>
</ul></li>
<li><strong>Gaslighting</strong>
<ul>
<li>Kate Abramson, “<a href="http://dx.doi.org/10.1111/phpe.12046">Turning up the Lights on Gaslighting</a>,” <em>Philosophical Perspectives</em> 28:1 (2014) 1–30.</li>
<li>Kate Manne, <a href="http://dx.doi.org/10.1093/oso/9780190604981.001.0001"><em>Down Girl</em>: <em>the logic of misogyny</em></a>, Oxford
Univeristy Press, 2018. [<strong><em>Read Chapter 1</em></strong>]</li>
<li>Andrew D. Spear, “<a href="http://dx.doi.org/10.1007/s11245-018-9611-z">Gaslighting, Confabulation, and Epistemic
Innocence</a>,” <em>Topoi</em> early access (2018).</li>
<li>Cynthia A. Stark, “<a href="http://dx.doi.org/10.1093/monist/onz007">Gaslighting, Misogyny, and Psychological
Oppression</a>,” <em>The Monist</em> 102:2 (2019) 221–235.</li>
</ul></li>
</ol>
<p>For further information, contact me. To participate, check <a href="https://handbook.unimelb.edu.au/view/2019/PHIL40013">the handbook</a>.</p>
PHIL20030: Meaning, Possibility and Paradox
https://consequently.org/class/2019/phil20030/
Wed, 24 Jul 2019 00:00:00 UTChttps://consequently.org/class/2019/phil20030/
<p><strong><span class="caps">PHIL20030</span>: Meaning, Possibility and Paradox</strong> is a <a href="http://unimelb.edu.au">University of Melbourne</a> undergraduate subject introducing logic to philosophy students. It’s taught by <a href="http://consequently.org">Greg Restall</a> and <a href="https://shawn-standefer.github.io">Shawn Standefer</a>.</p>
<p>This year, we have completely revised our curriculum. Now the subject introduces the proof theory and model theory of propositional, modal and predicate logic–in that order. We’re writing an introductory text <em>Logical Methods</em>, which we’re trialling with this class, as well as producing explanatory videos to use along with the text.</p>
<p>Here’s the outline of the subject.</p>
<h3 id="preliminaries">Preliminaries</h3>
<ul>
<li>Introduction
<ul>
<li>Arguments and Trees</li>
<li>Sentences and Formulas</li>
</ul></li>
</ul>
<h3 id="propositional-logic">Propositional Logic</h3>
<ul>
<li>Connectives: and & if
<ul>
<li>Conjunction</li>
<li>Conditional</li>
<li>Biconditional</li>
</ul></li>
<li>More connectives: not & or
<ul>
<li>Negation and falsum</li>
<li>Disjunction</li>
<li>Our System of Proofs</li>
</ul></li>
<li>Facts about proofs & provability
<ul>
<li>Facts about provability</li>
<li>Normalisation</li>
<li>The Subformula Property</li>
<li>Consequences of Normalisation</li>
</ul></li>
<li>Models & counterexamples
<ul>
<li>Models and truth tables</li>
<li>Counterexamples and validity</li>
<li>Model-theoretic validity</li>
</ul></li>
<li>Soundness & completeness
<ul>
<li>Soundness</li>
<li>Completeness</li>
<li>Proofs first or models first?</li>
<li>Heyting algebras</li>
</ul></li>
</ul>
<h3 id="modal-logic">Modal Logic</h3>
<ul>
<li>Necessity & possibility
<ul>
<li>Possible worlds models</li>
<li>Validity</li>
<li>Strict conditionals and ambiguities</li>
<li>Propositions</li>
<li>Another notion of necessity</li>
<li>Equivalence relations and epistemic logic</li>
</ul></li>
<li>Actuality & two-dimensional logic
<ul>
<li>Actuality models and double indexing</li>
<li>Validity</li>
<li>Fixity and diagonal propositions</li>
<li>Real world validity</li>
</ul></li>
<li>Natural deduction for modal logics
<ul>
<li>Natural deduction for S4</li>
<li>Natural deduction for S5</li>
<li>Features of S5</li>
</ul></li>
</ul>
<h3 id="predicate-logic">Predicate Logic</h3>
<ul>
<li>Quantifiers
<ul>
<li>Syntax</li>
<li>Natural deduction for CQ</li>
<li>What is provable?</li>
<li>Generality and eliminating detours</li>
</ul></li>
<li>Models for first-order logic
<ul>
<li>Models and assignments of values</li>
<li>Substitution</li>
<li>Counterexamples and validity</li>
<li>Compactness and what this means</li>
</ul></li>
</ul>
<p>One novelty in our approach to the subject is the balance between proof theory and model theory. We introduce propositional logic by way of Gentzen/Prawitz-style natural deduction—for intuitionistic logic—and along the way, each time we introduce the rules for a connective, we show that they are in harmony. So, it’s not too hard to show that proofs in the whole system can be normalised and we get the subformula property for normal proofs. (So, we can gesture in the direction of provability being <em>analytic</em> in a strong sense, since a normal proof literally <em>analyses</em> the premises and conclusion into components and connects them using the fundamental rules governing the concepts involved.)</p>
<p>Once that’s done, we then introduce Boolean valuations (and truth tables), and we can show that the proof system is sound but not complete for validity defined as the absence of a Boolean counterexample. Approaching things this way means we have an interesting discussion about soundness and completeness, and about intuitionistic and classical logic, and whether we should be happy with the gap between proofs and models or not, and if not, whether we should close that gap by adding to our proof system (that way lies <em>classical</em> natural deduction), or whether we should close the gap by enriching our class of models to serve as counterexamples (here we sketch Heyting algebras, as generalisations of Boolean valuations, but we point to Kripke models, too). There’s also scope for a discussion of whether we should understand logic in a proof-first way or a model-first way (or both, or neither), and how proofs and models relate to however it is that words and concepts get their meanings.</p>
<p>With that done, we’re halfway through the subject. Having arrived at Boolean valuations, it’s a short hop, skip and jump to Carnap’s models for modality, and their generalisation, universal models for the modal logic S5. So, we look at these models for possibility and necessity, and show how these possible worlds models can be used to analyse modality, strict conditionality, and similar notions.</p>
<p>Then with models like these we can be of service to our colleagues by introducing double-indexing and two-dimensional modal logic, and the analysis of fixedly diagonal propositions, and the relationship between analyticity, necessity and <em>a priority</em>.</p>
<p>With these model-theoretic considerations in hand, we turn to the question of what it might be to <em>derive</em> a modal claim, and we turn to the natural deduction rules for modals, which introduce constraints on assumptions. One way to prove that \(A\) is necessary, after all, is to prove \(A\) from claims of the form \(\Box B\), for those claims hold not only <em>here</em>, but also in any alternate circumstances, too. So, we get natural deduction systems for S4 and S5 rather straightforwardly.</p>
<p>Proving something more <em>general</em> than \(A\) by proving \(A\) from premises satisfying certain conditions sounds familiar if you’ve dealt with <em>quantifiers</em> before. How to you show that <em>everything</em> is an \(F\)? By proving that \(Fa\) when we have assumed <em>nothing about \(a\)</em>. Then our proof applies <em>no matter what \(a\) is</em>. So, we can generalise the conditions for modal proof to proofs with <em>quantifiers</em> too. So, we introduce the logic of first-order quantifiers with natural deduction first, and once we’ve done that, we turn back to models at last.</p>
<p>So, the introduction to logic has a rhythm, taking us from proofs to models of propositional logic, through models and then proofs for modal logic, and then to proofs and models for predicate logic. Along the way we look at issues in the philosophy of logic and the applications of logic to different issues in philosophy.</p>
<p>Although this curriculum and the course material is all ours, we are indebted to our colleagues for many discussions concerning the pedagogy of logic. I’ll single out two here. Allen Hazen talked to GR for many years about the pedagogical virtues of introducing modal logic before predicate logic to philosophy students. And <a href="http://davewripley.rocks">Dave Ripley</a> has, for the last couple of years, introduced logic using intuitionistic natural deduction and classical truth tables, making a virtue out of the soundness and <em>in</em>completeness of the pairing between the proof theory and the model theory. Neither Allen nor Dave would teach things how we have, but we’ve valued talking over the pedagogy with them over the years.</p>
<p>If you’d like to compare your mastery of logic, in comparison to what our students are learning, you can try your own hand at our <a href="https://consequently.org/resources/PHIL20030-2019-class-tasks-1-6.pdf">in-class tasks for weeks 1 to 6</a>.</p>
Assertions, Denials, Questions, Answers, and the Common Ground
https://consequently.org/presentation/2019/assertion-denial-qa-common-ground-express/
Thu, 13 Jun 2019 00:00:00 UTChttps://consequently.org/presentation/2019/assertion-denial-qa-common-ground-express/<p><em>Abstract</em>: In this talk, I examine some of the interconnections between norms governing assertion, denial, questions and answers, and the common ground of a discourse. When we pay attention to the structure of norms governing polar (yes/no) questions, we can clarify the distinction between strong and weak denials, together with the parallel distinction between strong and weak assertion, and the way that these speech acts interact with the common ground.</p>
<p>With those connections established, I respond to two criticisms of the program sketched out in my 2005 paper “Multiple Conclusions”. First, that understanding the upshot of a valid sequent <em>X</em> ⊢ <em>Y</em> as enjoining us to not assert each member of <em>X</em> and deny each member of <em>Y</em> is altogether too weak to explain the inferential force of logical validity. Deriving <em>X</em> ⊢ <em>A</em> should tell us, after all, something about justifying <em>A</em> on the basis of <em>X</em>, rather than merely prohibiting <em>A</em>’s denial. Where is the force to actually conclude the conclusion of a proof? A second, related criticism is that the format of multiple conclusion sequents seems unsatisfactory, in that it has no place for distinguishing a single conclusion, and proofs, after all, seem to be proofs of individual claims.</p>
<p>I will argue that both of these concerns can be assuaged if we pay closer attention to the norms connecting assertions and denials along with <em>justification requests</em> — questions aiming at eliciting reasons for assertions or denials. Once we understand the connection between justification requests, definitionsand the common ground, we will see not only that the these two concerns can be met. A derivation of a sequent <em>X</em> ⊢ <em>A</em>,<em>Y</em> gives us an answer to a justification request “why <em>A</em>?” in any available context where each member of <em>X</em> has been ruled in and each member of <em>Y</em> has been ruled out, and a derivation of a sequent <em>X</em>,<em>B</em> ⊢ <em>Y</em>, similarly gives us an answer to the justification request “why not <em>B</em>?” in any such context. The picture that results utilises the full multiple premise, multiple conclusion sequent calculus of classical logic, and does due justice to the idea that a proof (or a refutation) proves (or refutes) <em>one thing</em> relative to background assumptions or premises. In addition, when we consider the connection between justification requests and the norms governing <em>definitions</em>, we can see more clearly what could be involved in taking the connective/quantifier rules of a logical system to <em>define</em> the concepts they introduce.</p>
<ul>
<li><p>The talk is an invited address at the <a href="https://inferentialexpressivism.com/workshop/">Workshop on Bilateral Approches to Meaning</a>, at the University of Amsterdam.</p></li>
<li><p>The <a href="https://consequently.org/slides/assertion-denial-qa-common-ground-slides-express.pdf">slides for the talk are available here</a>, and the <a href="https://consequently.org/handouts/assertion-denial-qa-common-ground-handout-express.pdf">handout is here</a>.</p></li>
</ul>
Assertions, Denials, Questions, Answers, and the Common Ground
https://consequently.org/presentation/2019/assertion-denial-qa-common-ground-mcmp/
Thu, 13 Jun 2019 00:00:00 UTChttps://consequently.org/presentation/2019/assertion-denial-qa-common-ground-mcmp/<p><em>Abstract</em>: In this talk, I examine interconnections between norms governing assertion, denial, questions and answers, and the common ground of a discourse. When we pay attention to the structure of norms governing polar (yes/no) questions, we can clarify the distinction between strong and weak denials, together with the parallel distinction between strong and weak assertion, and the way that these speech acts interact with the common ground.</p>
<p>With those connections established, I respond to two criticisms of the program sketched out in my 2005 paper “Multiple Conclusions”. First, that understanding the upshot of a valid sequent <em>X</em> ⊢ <em>Y</em> as enjoining us to not assert each member of <em>X</em> and deny each member of <em>Y</em> is altogether too weak to explain the inferential force of logical validity. Deriving <em>X</em> ⊢ <em>A</em> should tell us, after all, something about justifying <em>A</em> on the basis of <em>X</em>, rather than merely prohibiting <em>A</em>’s denial. Where is the force to actually conclude the conclusion of a proof? A second, related criticism is that the format of multiple conclusion sequents seems unsatisfactory, in that it has no place for distinguishing a single conclusion, and proofs, after all, seem to be proofs of individual claims.</p>
<p>I will argue that both of these concerns can be assuaged if we pay closer attention to the norms connecting assertions and denials along with <em>justification requests</em> — questions aiming at eliciting reasons for assertions or denials. Once we understand the connection between justification requests, definitionsand the common ground, we will see not only that the these two concerns can be met. A derivation of a sequent <em>X</em> ⊢ <em>A</em>,<em>Y</em> gives us an answer to a justification request “why <em>A</em>?” in any available context where each member of <em>X</em> has been ruled in and each member of <em>Y</em> has been ruled out, and a derivation of a sequent <em>X</em>,<em>B</em> ⊢ <em>Y</em>, similarly gives us an answer to the justification request “why not <em>B</em>?” in any such context. The picture that results utilises the full multiple premise, multiple conclusion sequent calculus of classical logic, and does due justice to the idea that a proof (or a refutation) proves (or refutes) <em>one thing</em> relative to background assumptions or premises. In addition, when we consider the connection between justification requests and the norms governing <em>definitions</em>, we can see more clearly what could be involved in taking the connective/quantifier rules of a logical system to <em>define</em> the concepts they introduce.</p>
<ul>
<li><p>The talk is a <a href="https://www.mcmp.philosophie.uni-muenchen.de/events_this-_week/restall_20190618/index.html">Seminar at the Munich Centre for Mathematical Philosophy</a>.</p></li>
<li><p>The <a href="https://consequently.org/slides/assertion-denial-qa-common-ground-slides-mcmp.pdf">slides for the talk are available here</a>, and the <a href="https://consequently.org/handouts/assertion-denial-qa-common-ground-handout-mcmp.pdf">handout is here</a>.</p></li>
</ul>
Assertions, Denials, Questions, Answers, and the Common Ground
https://consequently.org/presentation/2019/assertion-denial-qa-common-ground-logicmelb/
Thu, 06 Jun 2019 00:00:00 UTChttps://consequently.org/presentation/2019/assertion-denial-qa-common-ground-logicmelb/<p><em>Abstract</em>: In this talk, I examine some of the interconnections between norms governing assertion, denial, questions and answers, and the common ground of a discourse. When we pay attention to the structure of norms governing polar (yes/no) questions, we can clarify the distinction between strong and weak denials, together with the parallel distinction between strong and weak assertion, and the way that these speech acts interact with the common ground.</p>
<p>With those connections established, I respond to two criticisms of the program sketched out in my 2005 paper “Multiple Conclusions”. First, that understanding the upshot of a valid sequent <em>X</em> ⊢ <em>Y</em> as enjoining us to not assert each member of <em>X</em> and deny each member of <em>Y</em> is altogether too weak to explain the inferential force of logical validity. Deriving <em>X</em> ⊢ <em>A</em> should tell us, after all, something about justifying <em>A</em> on the basis of <em>X</em>, rather than merely prohibiting <em>A</em>’s denial. Where is the force to actually conclude the conclusion of a proof? A second, related criticism is that the format of multiple conclusion sequents seems unsatisfactory, in that it has no place for distinguishing a single conclusion, and proofs, after all, seem to be proofs of individual claims.</p>
<p>I will argue that both of these concerns can be assuaged if we pay closer attention to the norms connecting assertions and denials along with <em>justification requests</em> — questions aiming at eliciting reasons for assertions or denials. Once we understand the connection between justification requests, definitionsand the common ground, we will see not only that the these two concerns can be met. A derivation of a sequent <em>X</em> ⊢ <em>A</em>,<em>Y</em> gives us an answer to a justification request “why <em>A</em>?” in any available context where each member of <em>X</em> has been ruled in and each member of <em>Y</em> has been ruled out, and a derivation of a sequent <em>X</em>,<em>B</em> ⊢ <em>Y</em>, similarly gives us an answer to the justification request “why not <em>B</em>?” in any such context. The picture that results utilises the full multiple premise, multiple conclusion sequent calculus of classical logic, and does due justice to the idea that a proof (or a refutation) proves (or refutes) <em>one thing</em> relative to background assumptions or premises. In addition, when we consider the connection between justification requests and the norms governing <em>definitions</em>, we can see more clearly what could be involved in taking the connective/quantifier rules of a logical system to <em>define</em> the concepts they introduce.</p>
<ul>
<li><p>The talk is a <a href="https://philevents.org/event/show/73102">Melbourne Logic Seminar</a>.</p></li>
<li><p>The <a href="https://consequently.org/slides/assertion-denial-qa-common-ground-slides-logicmelb.pdf">slides for the talk are available here</a>.</p></li>
</ul>
Isomorphisms in a Category of Proofs
https://consequently.org/presentation/2019/isomorphisms-pts3/
Thu, 07 Mar 2019 00:00:00 UTChttps://consequently.org/presentation/2019/isomorphisms-pts3/<p><em>Abstract</em>: In this talk, I show how a category of classical proofs can give rise to three different hyperintensional notions of sameness of content. One of these notions is very fine-grained, going so far as to distinguish \(p\) and \(p\land p\), while identifying other distinct pairs of formulas, such as \(p\land q\) and \(q\land p\); \(p\) and \(\neg\neg p\); or \(\neg(p\land q)\) and \(\neg p\lor\neg q\). Another relation is more coarsely grained, and gives the same account of identity of content as equivalence in Angell’s logic of analytic containment. A third notion of sameness of content is defined, which is intermediate between Angell’s and Parry’s logics of analytic containment. Along the way, we show how purely classical proof theory gives resources to define hyperintensional distinctions thought to be the domain of properly non-classical logics.</p>
<ul>
<li>This is a talk at the <a href="http://ls.informatik.uni-tuebingen.de/PTS3/overview.html">Third Tübingen Conference on Proof-Theoretic Semantics</a>, 27–30 March 2019.</li>
<li>The slides are <a href="https://consequently.org/slides/isomorphisms-talk-tubingen-2019.pdf">available here</a>, while a handout <a href="https://consequently.org/handouts/isomorphisms-handout-tubingen-2019.pdf">is here</a>.</li>
</ul>
Collection Frames for Substructural Logics
https://consequently.org/presentation/2019/collection-frames-logicmelb/
Thu, 07 Mar 2019 00:00:00 UTChttps://consequently.org/presentation/2019/collection-frames-logicmelb/<p><em>Abstract</em>: In this talk I present a new frame semantics for positive substructural and relevant propositional logics. This frame semantics is both a <em>generalisation</em> of Routley–Meyer ternary frames and a <em>simplification</em> of them. The key innovation is the use of a single accessibility relation to relate collections of points to points. Different logics are modelled by varying the kinds of collections featuring in the relation: for example, they can be sets, multisets, lists or trees. In this talk I will focus on multiset frames, which are sound and complete for the logic RW+ (positive multiplicative and additive linear logic with distribution for the additive connectives, or equivalently, the relevant logic R+ without contraction).</p>
<p>This is joint work with Shawn Standefer.</p>
<ul>
<li><p>The talk is a <a href="https://philevents.org/event/show/69618">Melbourne Logic Seminar</a>.</p></li>
<li><p>The <a href="https://consequently.org/slides/collection-frames-talk-logicmelb.pdf">slides for the talk are available here</a>.</p></li>
</ul>
PHIL30043: The Power and Limits of Logic
https://consequently.org/class/2019/phil30043/
Sat, 02 Mar 2019 00:00:00 UTChttps://consequently.org/class/2019/phil30043/
<p><strong><span class="caps">PHIL30043</span>: The Power and Limits of Logic</strong> is a <a href="https://handbook.unimelb.edu.au/view/2019/PHIL30043">University of Melbourne undergraduate subject</a>. It covers the metatheory of classical first order predicate logic, beginning at the <em>Soundness</em> and <em>Completeness</em> Theorems (proved not once but <em>twice</em>, first for a tableaux proof system for predicate logic, then a Hilbert proof system), through the <em>Deduction Theorem</em>, <em>Compactness</em>, <em>Cantor’s Theorem</em>, the <em>Downward Löwenheim–Skolem Theorem</em>, <em>Recursive Functions</em>, <em>Register Machines</em>, <em>Representability</em> and ending up at <em>Gödel’s Incompleteness Theorems</em> and <em>Löb’s Theorem</em>.</p>
<figure>
<img src="https://consequently.org/images/godel.jpg" alt="Kurt Godel, seated">
<figcaption>Kurt Gödel, seated</figcaption>
</figure>
<p>The subject is taught to University of Melbourne undergraduate students (for Arts students as a part of the Philosophy major, for non-Arts students, as a breadth subject). Details for enrolment are <a href="https://handbook.unimelb.edu.au/view/2018/PHIL30043">here</a>. I make use of video lectures I have made <a href="http://vimeo.com/album/2262409">freely available on Vimeo</a>.</p>
<h3 id="outline">Outline</h3>
<p>The course is divided into four major sections and a short prelude. Here is a list of all of the videos, in case you’d like to follow along with the content.</p>
<h4 id="prelude">Prelude</h4>
<ul>
<li><a href="http://vimeo.com/album/2262409/video/59401942">Logical Equivalence</a></li>
<li><a href="http://vimeo.com/album/2262409/video/59403292">Disjunctive Normal Form</a></li>
<li><a href="http://vimeo.com/album/2262409/video/59403535">Why DNF Works</a></li>
<li><a href="http://vimeo.com/album/2262409/video/59463569">Prenex Normal Form</a></li>
<li><a href="http://vimeo.com/album/2262409/video/59466141">Models for Predicate Logic</a></li>
<li><a href="http://vimeo.com/album/2262409/video/59880539">Trees for Predicate Logic</a></li>
</ul>
<h4 id="completeness">Completeness</h4>
<ul>
<li><a href="http://vimeo.com/album/2262409/video/59883806">Introducing Soundness and Completeness</a></li>
<li><a href="http://vimeo.com/album/2262409/video/60249309">Soundness for Tree Proofs</a></li>
<li><a href="http://vimeo.com/album/2262409/video/60250515">Completeness for Tree Proofs</a></li>
<li><a href="http://vimeo.com/album/2262409/video/61677028">Hilbert Proofs for Propositional Logic</a></li>
<li><a href="http://vimeo.com/album/2262409/video/61685762">Conditional Proof</a></li>
<li><a href="http://vimeo.com/album/2262409/video/62221512">Hilbert Proofs for Predicate Logic</a></li>
<li><a href="http://vimeo.com/album/2262409/video/103720089">Theories</a></li>
<li><a href="http://vimeo.com/album/2262409/video/103757399">Soundness and Completeness for Hilbert Proofs for Predicate Logic</a></li>
</ul>
<h4 id="compactness">Compactness</h4>
<ul>
<li><a href="http://vimeo.com/album/2262409/video/63454250">Counting Sets</a></li>
<li><a href="http://vimeo.com/album/2262409/video/63454732">Diagonalisation</a></li>
<li><a href="http://vimeo.com/album/2262409/video/63454732">Compactness</a></li>
<li><a href="http://vimeo.com/album/2262409/video/63455121">Non-Standard Models</a></li>
<li><a href="http://vimeo.com/album/2262409/video/63462354">Inexpressibility of Finitude</a></li>
<li><a href="http://vimeo.com/album/2262409/video/63462519">Downward Löwenheim–Skolem Theorem</a></li>
</ul>
<h4 id="computability">Computability</h4>
<ul>
<li><a href="http://vimeo.com/album/2262409/video/64162062">Functions</a></li>
<li><a href="http://vimeo.com/album/2262409/video/64167354">Register Machines</a></li>
<li><a href="http://vimeo.com/album/2262409/video/64207986">Recursive Functions</a></li>
<li><a href="http://vimeo.com/album/2262409/video/64435763">Register Machine computable functions are Recursive</a></li>
<li><a href="http://vimeo.com/album/2262409/video/64604717">The Uncomputable</a></li>
</ul>
<h4 id="undecidability-and-incompleteness">Undecidability and Incompleteness</h4>
<ul>
<li><a href="http://vimeo.com/album/2262409/video/65382456">Deductively Defined Theories</a></li>
<li><a href="http://vimeo.com/album/2262409/video/65392670">The Finite Model Property</a></li>
<li><a href="http://vimeo.com/album/2262409/video/65393543">Completeness</a></li>
<li><a href="http://vimeo.com/album/2262409/video/65440901">Introducing Robinson’s Arithmetic</a></li>
<li><a href="http://vimeo.com/album/2262409/video/65442289">Induction and Peano Arithmetic</a></li>
<li><a href="http://vimeo.com/album/2262409/video/65443650">Representing Functions and Sets</a></li>
<li><a href="http://vimeo.com/album/2262409/video/65483655">Gödel Numbering and Diagonalisation</a></li>
<li><a href="http://vimeo.com/album/2262409/video/65497886">Q (and any consistent extension of Q) is undecidable, and incomplete if it’s deductively defined</a></li>
<li><a href="http://vimeo.com/album/2262409/video/65498016">First Order Predicate Logic is Undecidable</a></li>
<li><a href="http://vimeo.com/album/2262409/video/65501745">True Arithmetic is not Deductively Defined</a></li>
<li><a href="http://vimeo.com/album/2262409/video/65505372">If Con(PA) then PA doesn’t prove Con(PA)</a></li>
</ul>