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Greg RestallThis Thursday, I’ll be down in London giving a talk about defining rules for quantifiers and identity, at the PPLV group in Computer Science at UCL. If you happen to be in the area, and are interested in proof theory, semantics and hints of metaphysics, I’d love to see you there.<a href="https://consequently.org/presentation/2025/drqi-pplv-ucl/" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://consequently.org/presentation/2025/drqi-pplv-ucl/</a><a href="https://hcommons.social/tags/prooftheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#prooftheory</a> <a href="https://hcommons.social/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a> <a href="https://hcommons.social/tags/semantics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#semantics</a>

Greg RestallI'm glad to have space to get to writing, and the first writing project of my sabbatical has reached first-draft stage. If you're interested in modal logic, proof theory, and the metaphysics of contingent existence, have I got the paper for you!<a href="https://consequently.org/writing/mlce-ge2/" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://consequently.org/writing/mlce-ge2/</a>I've got to say, I think the hypersequent calculus in this paper is pretty neat.<a href="https://hcommons.social/tags/ProofTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ProofTheory</a> <a href="https://hcommons.social/tags/ModalLogic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ModalLogic</a> <a href="https://hcommons.social/tags/Metaphysics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Metaphysics</a>

Greg RestallIt’s a cloudy and cold Tuesday, and I’m inside writing about refinement. At least I *think* I understand what I’m doing a bit better than Mark S and his team of macrodata refiners do.(That’s an inappropriate <a href="https://hcommons.social/tags/Severance" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Severance</a>, <a href="https://hcommons.social/tags/prooftheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#prooftheory</a> <a href="https://hcommons.social/tags/ModalLogic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ModalLogic</a> and <a href="https://hcommons.social/tags/ClickyKeyboard" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ClickyKeyboard</a> crossover post. I’m sorry about that.)

Greg Restall(The proof with alternatives does have a bit of a round-about feel, with having to store the conclusion as an alternative twice to then retrieve both in one go. This gives you the effect of contraction in conclusion position, which is required because the natural deduction introduction rule for disjunction is additive, while the elimination rule is multiplicative. In this format, your only device for contraction in conclusion position is to hoist the conclusion into the assumption context (as an alternative, under the slash) and to then retrieve two or more copies of that assumption back as a conclusion.)<a href="https://hcommons.social/tags/prooftheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#prooftheory</a> <a href="https://hcommons.social/tags/bureaucracy" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#bureaucracy</a>

Greg RestallThis morning, one of my hardworking intermediate logic students (prepping for her exam next week), came to me with a query about how to prove the constructively invalid quantifier negation inference (from ∀x(A(x)∨B(x)) to ∀xA(x)∨∃xB(x)) in natural deduction with Double Negation Elimination.It was natural that she would struggle with this exercise, since any proof of this (in that natural deduction framework at least), requires a quite bit of fancy footwork. After we worked through that, I wondered whether the proof is much simpler in classical natural deduction with alternatives (basically the λμ calculus). If you help yourself to the derivable (and simpler to work with) ∨E* variant rule, it turns out that the proof shrinks from 14 steps to 10, and it seems much more direct.<a href="https://hcommons.social/tags/prooftheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#prooftheory</a> <a href="https://hcommons.social/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a>

Greg RestallI think my paper on Dummett, proof assistants and pluralism has shaped up rather nicely, and it will be good to see it out in the Proceedings of the Aristotelian Society in a few months’ time. You can read the preprint now. Thanks to everyone who gave me feedback on earlier drafts, and discussed these issues along the way. There's more to be done, but I hope to have clarified some issues around how we can think about the relationship between constructive and classical reasoning, and how philosophers might engage with what is going on in the application of dependent type theory in proof assistants, programming language design, and the formalisation and mechanisation of reasoning.<a href="https://consequently.org/writing/what-can-we-mean/" rel="nofollow noopener noreferrer" target="_blank">https://consequently.org/writing/what-can-we-mean/</a><a href="https://hcommons.social/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a> <a href="https://hcommons.social/tags/prooftheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#prooftheory</a> <a href="https://hcommons.social/tags/typetheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#typetheory</a> <a href="https://hcommons.social/tags/philosophy" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#philosophy</a>

Greg RestallAnother week, another research presentation. Coming up on Saturday, I’ll be hanging out in Munich, talking about free logic and rules for quantifiers. <a href="https://consequently.org/presentation/2024/defining-quantifiers-mcmp/" rel="nofollow noopener noreferrer" target="_blank">https://consequently.org/presentation/2024/defining-quantifiers-mcmp/</a><a href="https://hcommons.social/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a> <a href="https://hcommons.social/tags/prooftheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#prooftheory</a>

Markus Redeker<a href="https://mathstodon.xyz/@loopspace" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@loopspace</a> <a href="https://mathstodon.xyz/@tao" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@tao</a> For me, the most natural approach would be in the context of formal logic. Start with the set of all proofs for a given statement (in a specified formal system, with some axioms given, for example) and then introduce some transformations that transform a proof into an equivalent one. Changing the order of the proof steps is a possible transformation, and the morst trivial one. Then these transformations define an equivalence relation, and voilà!, you have a concept for proof equivalence.The challenges here are of course (1) to find a definition that is meaningful in the real life of mathematicians, and (2) to prove interesting things with these concepts. One would try to define invariants, for example.I have done nothing concrete in this direction and do not know whether anyone else has, but maybe there is something.<a href="https://mathstodon.xyz/tags/Logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Logic</a> <a href="https://mathstodon.xyz/tags/Proofs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Proofs</a> <a href="https://mathstodon.xyz/tags/ProofTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ProofTheory</a>

Greg RestallNext week it’s our last Arché Metaphysics and Logic seminar for the academic year, and I’m going to have a go at addressing some of the big questions in the foundations of logic, with a contemporary twist.<a href="https://consequently.org/presentation/2024/what-do-we-mean-arche/" rel="nofollow noopener noreferrer" target="_blank">https://consequently.org/presentation/2024/what-do-we-mean-arche/</a><a href="https://hcommons.social/tags/philosophy" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#philosophy</a> <a href="https://hcommons.social/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a> <a href="https://hcommons.social/tags/prooftheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#prooftheory</a> <a href="https://hcommons.social/tags/typetheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#typetheory</a>

Estelle Platini"A major function [of deductive <a href="https://techhub.social/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a> is in] assessing exactly what is involved in asserting some set of propositions. […] By omitting some premiss without which the deduction of some conclusion is not valid, it misrepresents the premiss from which this conclusion is obtained, and hence responsibility for the conclusion. To agree to accept partial responsibility as good enough here is like agreeing to say that somebody was responsible for the dinner when he peeled potatoes and the cook did the rest. The first statement cannot be accepted as an elliptical, but allowable, way of making the second statement. And similarly suppression [of some premiss] enables us to obtain as causally responsible a partially sufficient rather than a fully sufficient causal condition." Valerie Plumwood in Australasian Journal of Logic, 2023: <a href="https://ojs.victoria.ac.nz/ajl/issue/view/894" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://ojs.victoria.ac.nz/ajl/issue/view/894</a> v <a href="https://mathstodon.xyz/@rrrichardzach" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@rrrichardzach</a><a href="https://techhub.social/tags/Plumwood" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Plumwood</a> <a href="https://techhub.social/tags/causality" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#causality</a> <a href="https://techhub.social/tags/correlations" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#correlations</a> <a href="https://techhub.social/tags/economics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#economics</a> <a href="https://techhub.social/tags/reason" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#reason</a> <a href="https://techhub.social/tags/ProofTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ProofTheory</a> <a href="https://techhub.social/tags/PhilSci" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#PhilSci</a> <a href="https://techhub.social/tags/truth" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#truth</a> <a href="https://techhub.social/tags/science" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#science</a> <a href="https://techhub.social/tags/ethics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ethics</a> <a href="https://techhub.social/tags/ecofeminism" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ecofeminism</a> <a href="https://techhub.social/tags/freedom" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#freedom</a>

Greg RestallThe Nordic Logic Summer School is now in full swing here in Reykjavík. I’ve given my first proof theory class, and Rineke Verbrugge is introducing modal logic and social cognition.<a href="https://consequently.org/class/2024/nls-proof-theory/" rel="nofollow noopener noreferrer" target="_blank">https://consequently.org/class/2024/nls-proof-theory/</a><a href="https://hcommons.social/tags/ProofTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ProofTheory</a> <a href="https://hcommons.social/tags/ModalLogic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ModalLogic</a>

Greg RestallMy anticipation is building for next week’s Nordic Logic Summer School and my class on proof theory. (The fact that I get to visit Reykjavík is to teach is a cool bonus.)<a href="https://consequently.org/class/2024/nls-proof-theory/" rel="nofollow noopener noreferrer" target="_blank">https://consequently.org/class/2024/nls-proof-theory/</a><a href="https://hcommons.social/tags/prooftheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#prooftheory</a>

Greg RestallI’m enjoying preparing my classes for next month‘s Nordic Logic Summer School in Reykjavik.This will be the northernmost latitude at which I’ve eliminated cuts.<a href="https://consequently.org/class/2024/nls-proof-theory/" rel="nofollow noopener noreferrer" target="_blank">https://consequently.org/class/2024/nls-proof-theory/</a><a href="https://hcommons.social/tags/prooftheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#prooftheory</a> <a href="https://hcommons.social/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a>

Greg RestallComing up in less than 18 hours, our two-day Proofs, Rules and Meanings extravaganza. Sophie, Viviane and Francisca have been working hard to organise a productive two days of logic, and our participants get to enjoy the fruits of their hard work very soon. <a href="https://www.st-andrews.ac.uk/arche/event/workshop-proofs-rules-and-meaning" rel="nofollow noopener noreferrer" target="_blank">https://www.st-andrews.ac.uk/arche/event/workshop-proofs-rules-and-meaning</a><a href="https://hcommons.social/tags/prooftheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#prooftheory</a> <a href="https://hcommons.social/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a> <a href="https://hcommons.social/tags/philosophy" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#philosophy</a>

Greg RestallI *think* I’m ready for my talk at our two-day proof theory workshop, starting tomorrow: <a href="https://consequently.org/presentation/2024/lambda-mu-arche/" rel="nofollow noopener noreferrer" target="_blank">https://consequently.org/presentation/2024/lambda-mu-arche/</a> — I have a lot to pack in to 25 minutes, so the monster-sized handout contains some of the details I’ll skim over in the talk.<a href="https://hcommons.social/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a> <a href="https://hcommons.social/tags/prooftheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#prooftheory</a>

Samiro DischerDamn, another social network not containing anything about my research interests..Looking for challengers: <a href="https://github.com/xamidi/pmGenerator/discussions/2" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://github.com/xamidi/pmGenerator/discussions/2</a><a href="https://mastodon.social/tags/ProofTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ProofTheory</a> <a href="https://mastodon.social/tags/HilbertSystems" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#HilbertSystems</a> <a href="https://mastodon.social/tags/ProofMinimization" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ProofMinimization</a> <a href="https://mastodon.social/tags/Research" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Research</a> <a href="https://mastodon.social/tags/Logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Logic</a> <a href="https://mastodon.social/tags/Puzzle" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Puzzle</a> <a href="https://mastodon.social/tags/Challenge" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Challenge</a>

Greg RestallThe titles and abstracts of our Proofs, Rules and Meanings workshop—on April 11 and 12—are now online: <a href="https://www.st-andrews.ac.uk/arche/event/workshop-proofs-rules-and-meanings/" rel="nofollow noopener noreferrer" target="_blank">https://www.st-andrews.ac.uk/arche/event/workshop-proofs-rules-and-meanings/</a>I’m looking forward to catching up with so many friends and colleagues, and to meet some new folks, too.This is a hybrid in-person and online workshop. If you want to join in online, it’s not too late to register so you get sent all the information. All details are at the link above.<a href="https://hcommons.social/tags/prooftheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#prooftheory</a> <a href="https://hcommons.social/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a> <a href="https://hcommons.social/tags/philosophy" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#philosophy</a>

RanaldCloustonNew <a href="https://fediscience.org/tags/blog" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#blog</a> up: on an interesting textbook on <a href="https://fediscience.org/tags/ProofTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ProofTheory</a> and <a href="https://fediscience.org/tags/AbstractAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AbstractAlgebra</a> in <a href="https://fediscience.org/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a> <a href="https://blogs.fediscience.org/the-updated-scholar/2024/02/09/discussing-proof-theory-and-algebra-in-logic/" rel="nofollow noopener noreferrer" target="_blank">https://blogs.fediscience.org/the-updated-scholar/2024/02/09/discussing-proof-theory-and-algebra-in-logic/</a>

Greg RestallHere’s a draft of a paper I’ve been writing over the last few months, on and off, in between other things, on the philosophical significance of the paradoxes. It’s to appear in the Oxford Handbook of Philosophical Logic, sometime in the next year or two.I’ve tried to be fair to accounts of the paradoxes that use the paradoxes to motivate revisions to traditional logical principles, and those accounts that keep those logical principles fixed, and use them to constrain our theories of truth, of types, of properties, of vague predicates, etc.<a href="https://consequently.org/writing/psp/" rel="nofollow noopener noreferrer" target="_blank">https://consequently.org/writing/psp/</a><a href="https://hcommons.social/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a> <a href="https://hcommons.social/tags/paradox" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#paradox</a> <a href="https://hcommons.social/tags/prooftheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#prooftheory</a>

Greg RestallMy office blackboard looks at its best after my weekly logic lunch. Today was an especially good lunchtime discussion.<a href="https://hcommons.social/tags/prooftheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#prooftheory</a> <a href="https://hcommons.social/tags/blackboard" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#blackboard</a> <a href="https://hcommons.social/tags/philosophy" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#philosophy</a>

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