RanaldClouston<p>I am going to make an attempt to <a href="https://fediscience.org/tags/blog" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>blog</span></a> a bit again, reading and writing about papers and books, old and new, that are cited by recent work in my area. This week, we look at a <a href="https://fediscience.org/tags/proofTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>proofTheory</span></a> <a href="https://fediscience.org/tags/logic" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>logic</span></a> textbook. <a href="https://blogs.fediscience.org/the-updated-scholar/2025/08/08/discussing-basic-proof-theory/" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">blogs.fediscience.org/the-upda</span><span class="invisible">ted-scholar/2025/08/08/discussing-basic-proof-theory/</span></a></p>
#prooftheory
1 post1 participant0 posts today
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.)
Continued thread
(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.
consequently.orgWhat can We Mean? On Practices, Norms and Pluralisms — consequently.org
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/
consequently.orgDefining Quantifiers — consequently.org
Replied in thread
@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/
consequently.orgWhat Do We Mean? Semantics, Practices and Pluralism — consequently.org
"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
ojs.victoria.ac.nz
Vol. 20 No. 2 (2023): The Australasian Journal of Logic
| The Australasian Journal of Logic
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.)
consequently.orgProof Theory — consequently.org
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.
consequently.orgProof Theory — consequently.org
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/