I'm studying the foundations of mathematics, and I've encountered a problem with the dependency structure of the usual developments.

Seemingly every rigorous development of first-order logic presupposes set theory, while every development of axiomatic set theory presupposes first-order logic. For example, Hinman (which I have not read beyond the introductory material) presupposes set theory, and Jech (which I have not read beyond the introductory material) presupposes first-order logic.

These are hard dependencies. You cannot do first-order logic without set theory, and you cannot do axiomatic set theory without first-order logic.

In order to break this "vicious" dependency cycle, I reasoned that I should first develop a very rigorous informal set theory, upon which first-order logic could depend. Then, axiomatic set theory could be developed without the aforementioned cycle.

However, despite my best efforts looking, I cannot find a sufficiently rigorous and systematic development of this preliminary set theory.

For example, Hrbacek and Jech (which I have not read beyond the introductory material) relies on considerable informal metatheory at the lowest level. This is precisely the material I want to develop with the greatest rigor.

To be precise, I want definitions, inferences, etc., specified in a sufficiently rigorous manner that an algorithm could, in principle, systematically reproduce the development. I am not demanding that mathematics be developed without primitive assumptions. I am demanding that those assumptions be explicit, and that subsequent reasoning actually follow from them. Metatheoretic notions just won't cut it; I am desperately trying to build a computational mechanism.

I understand that the set theory used in the metatheory of FOL is not the same thing as axiomatic set theory formulated in FOL.

I'm looking for a serious textbook that develops a robust symbolic set theory prior to FOL, or develops the two together in a coherent, noncircular manner.

I'm not looking for an elementary introduction to proofs or informal set-theoretic reasoning, such as Hammack. I want a very rigorous development that actually establishes the machinery it uses, even if that requires substantial technical machinery.

I've been thinking a lot about the current state of AI in academia. We've seen incredible breakthroughs in biology (like AlphaFold mapping proteins) and formal mathematics (like AlphaProof and Lean solving Olympiad-level problems).

But I keep wondering: why hasn't this translated into philosophy? When I use AI for my own work, it's great at checking routine math, formatting, or finding logical inconsistencies. But it seems to struggle with actually holding a philosophical position or defending it against counterarguments.

Is it just a matter of training data, or is there something fundamentally different about philosophical reasoning that current LLMs can't capture? I'd love to hear what people working in formal epistemology or logic think about this. Do you see AI as a useful tool for philosophy right now, or is it still too early?

Hello, this is my first substack blog post. It is kind of short and weirdly written, but I still would appreciate any thoughts.

Edit: the article's title is: "If AND is like Multiplication and (X)OR is like Addition Then What is Division?" And it is basically about finding an operator or function that would be analogical to division but for propositional logic (supposing AND is multiplication)

Hi. I'm 13 and I study analytic philosophy. I'm building a multiparameter method for evaluating philosophical schools: Φ⁺ for macro-philosophy (ontology, modality, fundamentality) and Φ⁻ for micro-philosophy (epistemology, ethics). A prototype of Φ⁺ already exists (LaTeX document, working formula, pilot on Platonism/nominalism/structuralism).

I'm looking for:

Logicians — to formalize consistency conditions for Φ⁺ and Φ⁻.

Philosophers — to independently score 3–5 schools (inter-rater reliability).

Programmers — to build a small calculator for weights and sensitivity analysis.

Critics — to stress-test the method with counterexamples.

Systematizers — to organize parameters, scales, and documentation.

This is not "philosophy is fully formalizable." It's "evaluation can be structured" if we separate levels, declare normative assumptions, and calibrate on classic schools.

If interested — DM me. Age 13–22, but what matters is willingness to argue and check.

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Episode 1

Let's start with three completely meaningless symbols: p, q, and - (a dash). In Chapter 2 of Gödel, Escher, Bach, Hofstadter introduces a formal system called the pq-system, which does nothing but blindly shuffle these characters based on a few basic typographical rules. Imagine a machine completely devoid of understanding, just churning out strings like --p---q-----.

This is exactly where the magic of human cognition—and the core concept of isomorphism—kicks in. Our brains detect a pattern. If we map p to "plus," q to "equals," and treat the dashes as numbers, that sterile string suddenly snaps into a universal truth: 2+3=5. We project form and meaning onto lifeless symbols.

Hofstadter uses this to demonstrate that meaning isn't an intrinsic property of symbols. It emerges from discovering the structural correspondence between formal systems and reality. Meanwhile, the system's rules couldn't care less about our interpretations; the mechanism grinds on exactly the same way whether we see it as profound mathematics or just a tally of apples and oranges.

What are your thoughts on this? Don't these striking isomorphisms suggest that math actually resides in its own independent, abstract realm, making our formal systems nothing more than the nets we weave to catch those pre-existing truths?

— If_P_Then_Panic

A=A or "Hesperus is Hesperus” simply mean the same thing; therefore, it is analytical and a priori.

But if we say,

"(This) Odyssey is (the) Odyssey."

Can we count that as a priori too? Or is it both analytical and synthetic?

It was intended jokingly, but how would you analyze the sentence: "Special pleading is a fallacy except in the case of this sentence"?

Hello, I’ve recently started studying mathematical logic on my own and came across a problem I had with a textbook I was reading (introduction to mathematical logic Mendelson second edition) the excersize in question asked me to write (C v ((B ^ (~D)) ⊃ C )) in Polish notation. I came up with the answer vC ^ B ~D ⊃ C.
However according to the book the answer is v C ⊃ ^ B ~DC. I’m very confused how it could come to that answer and the book itself is pretty old so I haven’t been able to google a proper answer. Would someone be willing to explain to me both how I got it wrong and the steps that it takes to annotate the problem properly? Thank you in advance.

If we claim that logic has limits when speaking about God, what logical grounds justify that claim?

TLDR: On empirical grounds we can reasonably justify that human use of logic is restricted and has limits when talking about Allah, Blessed Be He and may He bless and greet His Final Messenger Muhammad ﷺ.

I am analyzing a parameterized dynamic scaling function designed to model state-space transitions. The heuristic function is formulated as f(n) = 2^(n / ((p(n))^2 * (3n+1))), where p(n) is a strictly increasing prime-product (primorial n#) and (3n+1) is a linear dynamics operator. Asymptotically, as n approaches infinity, the exponent collapses to 0, forcing f(n) to achieve absolute convergence to 1.0 at a super-exponential decay rate of O(e^(-2n)). I am trying to formalize its implications on a Nondeterministic Turing Machine (NDTM) configuration tree under a proposed Dynamic Branching Law, B(n) = 2^(f(n) - 1), where the compression drives the effective branching factor to exactly 1.0 in the limit. I would highly appreciate structural feedback from the logic community on these points: 1. Is the first-order asymptotic decay bound of O(e^(-2n)) derived via Taylor expansion robust enough to mathematically bound non-deterministic expansion? 2. Does an asymptotic compression of a branching factor to 1.0 formally imply a structural reduction of the search space, or is it isolated as a non-constructive mathematical artifact? 3. Which intermediate automata-theoretic or resource-bounded information frameworks are best suited to formalize these structural bottlenecks? Thank you for your time and guidance!

Suppose a hardest logically solvable question H.

Asking if H is the hardest question could then become H

But then we can repeat the process again, since "Asking if H (original) is the hardest question" is now our new H we can just do the same to it, then the same to the new one and over and over again.

Thanks!

I'm doing some practice exercises in natural deduction, but the exercises don't come with an answer key, so I really have no clue how to figure this out:

A → (B → C). Therefore (A ∧ B) → C

I get as far as assuming A in a subproof, assuming B in a further subproof, and connecting A and B via conjunction introduction, but then I have no clue how to get back to showing that (A ∧ B) → C. What am I missing here? My thinking is that I need to prove that A and prove that B, so that if I assume their conjunction I can show that their conjunction implies C. But I have no idea how to do this in a way that allows me to cite the subproofs.

Any help is appreciated!

additionally most undergrad math theorems like lagrange's subgroup can be written in this form?
Could I find atleast one theorem presented in such format?
And where all could the group axioms go? gamma or leafs?

If somone said this "Cant believe you wrote a dissertation on my one paragraph", in order to undermine my point or get a laugh from thier audience or to not put in effort to combat my point.

'Dissertation' would be very much hyperbolic.

Would i be gish galloping? I believe my points were succint to the best of my ability.

Hi everyone,

I have a B.S. in Computer Science and really enjoyed the discrete mathematics course in my first three semesters. But I started working as a software engineer early, my focus drifted to IT-specific topics, and I never made my logic knowledge solid. Now I want to relearn it properly from scratch.

So I put together a self-study book for myself, drafted with an AI assistant, and I’m sharing it in case it helps someone else:
https://swicherwich.github.io/cogito-ergo-sum/

What’s in it: a “from absolute zero” part (reasoning, math language, sets and functions, a symbol atlas), propositional logic and natural deduction, predicate logic with careful treatment of the ∀/∃ rules, proof technique and induction, an overview of soundness, completeness and Gödel’s theorems, and a small Python proof checker and resolution prover with tests. It’s free (CC BY-SA) and works on phones, with dark mode and a dyslexia-friendly font.

Honest caveats: it hasn’t been reviewed by a logician, so there are probably mistakes, and the metatheory chapters are condensed rather than fully rigorous. If you spot an error or an unclear explanation, please open an issue: https://github.com/swicherwich/cogito-ergo-sum/issues

Again, hope it help somebody, and thank you for the feedback!