New: pmGenerator, since version 1.2.2, can
- compress Hilbert-style proofs via exhaustive search on user-provided proof data
- convert Fitch-style natural deduction proofs into any sufficiently explored Hilbert system
#Logic #HilbertSystems #NaturalDeduction #FormalMethods #ProofTheory #Mathematics
I’m in Amsterdam, about to give a talk about proof theory for modal predicate logic at the ILLC, the home base of the modal industrial complex. I have no idea how this is going to go over, but it should be a fun ride, however it turns out.
This 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.
I'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!
https://consequently.org/writing/mlce-ge2/
I've got to say, I think the hypersequent calculus in this paper is pretty neat.
It’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 #Severance, #prooftheory #ModalLogic and #ClickyKeyboard crossover post. I’m sorry about that.)
(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.)
This 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.
I 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.
Another week, another research presentation. Coming up on Saturday, I’ll be hanging out in Munich, talking about free logic and rules for quantifiers. https://consequently.org/presentation/2024/defining-quantifiers-mcmp/
@loopspace @tao 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.
Next 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.
https://consequently.org/presentation/2024/what-do-we-mean-arche/
"A major function [of deductive #logic 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: https://ojs.victoria.ac.nz/ajl/issue/view/894 v @rrrichardzach
#Plumwood #causality #correlations #economics #reason #ProofTheory #PhilSci #truth #science #ethics #ecofeminism #freedom
The 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.
My 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.)
I’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.
Catching up on my #blog writing with my second post in two days! I look at how a variation on the standard notion of sequent calculus led to new insights into structural #proofTheory for intuitionistic #logic. https://blogs.fediscience.org/the-updated-scholar/2024/04/24/discussing-nested-sequents-for-intuitionistic-logics/
The Proof Society - https://www.proofsociety.org/
The Proof Theory Blog - https://prooftheory.blog/
Coming 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.
https://www.st-andrews.ac.uk/arche/event/workshop-proofs-rules-and-meaning
I *think* I’m ready for my talk at our two-day proof theory workshop, starting tomorrow: https://consequently.org/presentation/2024/lambda-mu-arche/ — 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.
An explanation of what axioms and mathematical proofs really are. With a reference to my tool that helps exploring some of them.
https://math.codidact.com/posts/290943/291261#answer-291261
#Logic #Axioms #Mathematics #ProofTheory #HilbertSystems #ModalLogic #Research #Software
