Also see the related (and equally as wonderful) technical report www.cl.cam.ac.uk/techreports/...

I can't begin to describe what joy this technical report ("On using Edinburgh LCF to prove the correctness of a parsing algorithm") brings me, since it's a major sticking point in this "FS0/Lisp fusion" project I'm working on.

www.cl.cam.ac.uk/techreports/...

Otten and Kreitz's "T-String-Unification: Unifying Prefixes in Non-Classical Proof Methods" seems like a suitable proxy, though...

(Well, I hope, because I don't have access to Wallen's book, so I cannot say if it is adequate or not as a substitute.)

Intuitionistic tableaux just seems to be "Classical tableaux with pre- and post-processing to make sure it's Intuitionistically provable".

I just need to track down a copy of Wallen's "Automated deduction in nonclassical logic" to appreciate this madness...

Well, the kernel has 320 lines of code, so I think that qualifies as "small".

If I include the term and formula code, it's 675 lines of code, which is getting a little heavy (almost certainly there's a bug in the unimportant parts of the code).

Hugh from the Armando Iannucci Show

“Of course in the old days Bluesky was only on for three hours a day. We used to get dressed up in our Sunday best to log on to it. Then when it shut down again we’d all stand up and sing the national anthem”

I just have 7 more theorems to prove about quantifiers in Intuitionistic logic, then I can get started with tactics and whatnot.

I'm actually doubtful that FS0 works with Intuitionistic logic without modifying a few of the axioms.

On purpose? Or in a "the surgeon dropped his wristwatch into you during surgery" way?

Hey, is there some big book consisting of theorems in first-order logic, just enumerated out Principia style?

I vaguely recall checking out such a book at university, but I cannot remember its title for the life of me.

I still haven't filed my taxes yet 😅

I'm sad to hear this.

I do like the funeral instructions: "Don't buy flowers. Buy books."

I still haven't started. Every year feels like, well...

youtu.be/ZnJcZ-5P8hE?...

Thanks, that's good to know. Now I just need to wrestle with adapting this to Intuitionistic natural deduction...

de Groote's λ^λ (from his paper "Defining λ-typed λ-calculi by axiomatizing the typing relation") is a bit tidier presentation of ΔΛ

members.loria.fr/PdeGroote/pa...

I highly recommend that everyone follow artists. If you do, your feed is filled with art every day. Real, original art. It's bonkers that you can even have this. Amazing. Undeniably cool.

And then support those artists, when you can, because that helps the delight continue.

Hmm...is there a resolution or tableaux method for Intuitionistic first-order logic? I might have painted myself into a corner 🫣

Because you can clearly tell if you are looking at Lisp-like code or ML-like code, almost immediately you know if it's the object logic or not (whether you're proving theorems with FS0 as a logical framework, or whether you're proving metatheorems about the object logic).

I say "Lisp" because it has S-expression syntax, macros, and so on, but not arbitrary recursion (only primitive recursion). Why lean into the "Lispiness" of FS0?

Ultimately, FS0 will act as a "primitive recursive Lisp REPL", and the user can "implement" their preferred foundations of Mathematics in it.

In this sense, the user just needs to build a (hierarchical/nested) finite state machine using this "Lisp" to specify the object logic's mechanisms.

I abstracted away the FS0 primitive notions into their own module. Well, the constructors for the AST.

Then I just hardcoded the FS0 axioms into the kernel.

I am bouncing between (a) implementing derived inference rules, and (b) implementing the definitional mechanism.

Student: I'm an ultrafinitist. I believe there's a largest natural number.

Me: Ah, yes, 17.

Sequent calculus would treat this as `or(not(and()), or()) == or(False, False) == False`.

But we could implement (in Standard ML) this as `variadic_implies xs = List.foldr implies true xs`...which suggests the default value should be "true".

But this would suggest `variadic_implies(A) != variadic_implies(A, variadic_implies()) != variadic_implies(and(), A)`.

Perhaps it's best to default to `variadic_implies() = True`...

It seems obvious that `variadic_implies(A) = A`, but `variadic_implies()` is...well...uh...

If we take `variadic_implies(A, B, C)` as `or(not(and(A,B)), C)`, then it makes sense to use "False" as the default value since it'd be just an empty `or` (which is False).

Variadic implies is sensible for n > 1 arguments, if you have `variadic_implies(A, B, C...) == variadic_implies(A, variadic_implies(B, C...))` (right associativity).

What should `variadic_implies(A)` be? And `variadic_implies()`?

I have just simplified 450 lines of Standard ML to a mere 44 lines.

(I suspect I'll make up that deficit, though, and soon too, because they implemented the primitives of FS0 in the Term module. I'm just implementing them in the FS0 module instead.)

The Mysteries of Lisp -- I: The Way to S-expression Lisp Despite its old age, Lisp remains mysterious to many of its admirers. The mysteries on one hand fascinate the language, on the other hand also obscure it. Following Stoyan but paying attention to what...

A fun little review (that I suspect could easily be expanded into a book): arxiv.org/abs/1505.07375

Should I cite a source for S-expressions? 🤔

It *feels* like overkill, but I also feel like there's some great review paper (which I am ignorant of) specifically on the topic of S-expressions...

"Cognitive surrender" leads AI users to abandon logical thinking, research finds Experiments show large majorities uncritically accepting "faulty" AI answers.

"bUt WhY dOn'T yOu UsE aI, bRo?"

arstechnica.com/ai/2026/04/r...

I Work Very Hard, And I Would Like To Try Cake By A Horse Hello. I am a horse. I work very hard at my job of being a horse. When humans say move the heavy thing, I move the heavy thing. When humans sit on top of me and pull on my head, I carry them where they want to go. The main food the humans give me is hay and oats. But I am thinking it would be nice to have a different food. I am thinking I would like to try cake. Yes, yes. Cake. I know all about it. When humans eat cake, it is in glad times. It is the food for a celebration, such as when a woman becomes 47. I have seen cake on the Fourth of July. When humans have a cake, they stand around it and clap hands and smile and say happy birthday at each other. Sometimes there are beautiful markings on a cake, such as balloons or a pink shape. Sometimes the top of a cake is on fire and a boy must blow on the fire with mouth wind. This is the scariest cake. I do not want this kind. But I will eat any other cake. Any cake that is not the fire cake that tries to kill the boy. Please understand: I do not get money for doing work. I do not get to go inside the house. All I am either doing my horse job or standing in my pen or eating food off the floor. I always do these things. But I have never once gotten cake and I would like it very much. I have noticed that human children get to eat cake. But I am bigger than the children. I am more helpful to the farm. Children do not move the heavy things like me or let anyone ride on them. And yet they get cake. Maybe the humans will realize this. Maybe they will say, "You know who deserves cake? That horse. That horse whose back we are always on." Every day I dream about what it will be like if I get to eat cake. Here is what will happen. First, I will walk to the cake and putt my nose at it like hrrfff to make and stomping my hooves to make sure it is not a snake. Then I will trot in a circle to show that I am a horse and I am large. After that, I will nuzzle the cake to …

The horse op-ed is an instant classic. I can't tell you how much joy this piece gives me.

It should be taught in every introductory writing class in no small part because the horse arguments are so compelling. "I have noticed that human children get to eat cake. But I am bigger than the children."