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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.

consequently.org/presentation/

consequently.orgDefining Rules for Quantifiers and Identity — consequently.org
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.org/writing/what-

consequently.orgWhat can We Mean? On Practices, Norms and Pluralisms — 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.

"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: ojs.victoria.ac.nz/ajl/issue/v v @rrrichardzach

ojs.victoria.ac.nz Vol. 20 No. 2 (2023): The Australasian Journal of Logic | The Australasian Journal of Logic

The titles and abstracts of our Proofs, Rules and Meanings workshop—on April 11 and 12—are now online: st-andrews.ac.uk/arche/event/w

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.