When I did some research for the Piston project, I learned that there was a productivity technique called "meta parsing" which was used in late 60s to develop the first modern computer. This was before C. The language was Tree-Meta. Viewpoint Research Institute upgraded it to OMeta.

I thought OMeta was too complex, so I developed Piston-Meta, an alternative for Rust using a simple data structure: Start node, end node, text, bool and f64.

Asking "why do you want to be reasoning within an inconsistent system?" is like facing a dead end, because you are supposing a bias that was never there in the first place. As if, logic cares about what you want. You only get out what you put in. Bias in => bias out.

I am speculating about the following: If we don't bias ourselves in favor of consistency at the meta-level, then the correct notion of provability is HOOO EP. If we are biased, then the correct notion is Provability Logic.

In order to see HOOO EP as a provability notion, you have to interpret the axioms as a theory about provability. This requires mathematical intuition, for example, that you are able to distinguish a formal theory from its interpretation. Now, I can only suggest a formal theory, but the interpretation is up to users of that theory.

My system uses "tautological equality" and this allows me to treat them the save way for all tautological congruent operators. Ofc, you can't treat them the same if an operator is neither normal nor tautological congruent.

Even if you have a such operator `foo'(x)` you can prove `foo'(x)`, `(foo'(x) == foo'(x))^true` or `foo'(x) == foo'(x)` etc. If this is what you are talking about, then this system supports it.

The reason is that if I have a proof `((x + x) == (2 * x))^true`, then I can treat objects as if they were definitionally equal.

When definitional equality and syntactic equality are the same, one is assuming `(!(sd(a, b)) == (a == b))^true`, which obtains a proof `!sd(a, a)` for all `a`. This destroys the property of reasoning hypothetically about exponential propositions in Path Semantics. For example, I might want to reason about what if I had `sd(a, a)`, does this imply `a ~~ a` by `q_lift : sd(a, b) & (a == b) -> (a ~~ b)`? Yes!

However, if I already have `!sd(a, a)` for all `a`, then the above reasoning would be the same absurd reasoning.

I can always assume this in order to reason about it, but I don't have to make this assumption a built-in axiom of my theory.

When assuming tautological congruence e.g. `(a == b)^true` in a context, one is not enforcing observational equality. So, it is not the same as requiring the type system to be decidable. I can also make up new notions of equivalences and guard them, e.g. not making them tautological congruent.

Not only are we going to treat mathematics as subjective, but also having formal theories that reason about different notions of subjectivity. https://crates.io/crates/joker_calculus

> Could our conception of paradox be itself primal, and perhaps, in some plane, could it be something ranking higher, of first-class?

Yes! Paradoxes are statements of the form `false^a` in exponential propositions. https://crates.io/crates/hooo

> Also, I’ve been thinking, recently, on the role of time in structures. There can’t possibly be any structure whatsoever without time, or, more concretely, at least the memory of events, recollecting distinctive and contrasting entropic signatures. So, mathematics manifesting as, of, and for structure, wouldn’t it require, first and foremost, a treatment from physics? Regular or meta?

Path semantical quality models this relation, where you have different "moments" in time which each are spaces for normal logical reasoning. Between these moments, there are ways to propagate quality, which is a partial equivalence. https://github.com/advancedresearch/path_semantics

Yes. You can also think of it as a function pointer `b -> a`. Unlike lambdas/closures, a function pointer can not capture variables from the environment. So, if you have a function pointer of type `b -> a`, then you can produce an element of type `a` from an element of type `b`.

This sounds almost like the same thing as with lambdas/closures. The difference is that a lambda can capture variables from the environment, such that `b => a` is possible "at run-time" in a way that does not hold for every possible world. So, the distinction between function pointers and lambdas/closures can be thought of as different notions of provability at static compile-time and dynamic run-time.

> One more thing, what are you calling `N`?

N is the name of the axiom in Modal Logic. It is called the "Necessitation Rule". See https://en.wikipedia.org/wiki/Modal_logic#Axiomatic_systems

Naming things is hard. Given how constrained Propositional Language is as a language, I do not think there is much risk of misinterpreting it. I needed something to associate with "superposition" but also fit with "quality". Both "qubit" and "quality" starts with "qu", so I liked the name.

It does not bother me if people find another better name for it.

> I have looked through that section, and afaict, nowhere in it do you define an alternative notion of “provability”?

I do not want to create a controversy around Provability Logic by making too strong claims for some people's taste. What I meant is that this section is explaining HOOO EP and my interest is in communicating what it is on its own sake, without needing to compare it to Provability Logic all the time. However, since HOOO EP is so similar to Provability Logic, it requires some clarification. I hope you found the section useful even though you were not able to see how it is an alternative notion of provability from its definition.

> Why do you expect to be able to prove `□false => false` ? I.e. why do you expect to be able to prove `not □false`, i.e. prove the consistency of the system you are working in.

I think this is trying to think about logic as a peculiar way. HOOO EP was not developed to reason about consistency. It has its own motivation that makes sense. However, once you have HOOO EP, you can start discussing how it relates to consistency of theories.

It makes sense, in a sense of consistency, from the perspective where an inconsistent theory is absurd. Absurd theories can prove anything, so there is no distinction between true and false statements. Now, if you interpret `□false` as an assumption that one can prove false, then of course, one can prove false. `□false => false` is the same as `!□false`. Does this mean that it proves its own consistency? No, because you made the assumption `□false`. You have only talked about what you can prove in the context of `□false`. From this perspective, `□false => false` is trivially true.

Provability Logic does not allow you to think of `□false` as meaning "I can prove `false`". Instead, it is interpreted as "this theory is inconsistent" but without assigning this statement a particular meaning. This means, there is a gap between "I can prove `false`" and "this theory is inconsistent". Now, if you ignore the gap, then you are just making an error of interpretation. You have to respect the formal sense, where Provability Logic can have two different notions of absurdity while naturally you would think of them as one. However, if you want to have one instead of two, then you need HOOO EP.

> Also, if you want to do reasoning from within an inconsistent theory, then I’d hope it is at least paraconsistent, as otherwise you aren’t going to get much of value?

It sounds like you are assuming HOOO EP is inconsistent? Why are you speculating about my motivations in a such context?

Yes. One way to put it: Provability Logic is balancing on a knife-edge. It works, but just barely. However, you can turn it around and say the new notion is balancing on a knife-edge by requiring DAG (Directed Acyclic Graph) at meta-level. They way I see it, is that both approaches have implicit assumptions and you have to trade one with another.

I am working on an implementation of the new notion of provability (https://crates.io/crates/hooo), after finding the axioms last year (it took several months):

    pow_lift : a^b -> (a^b)^c

    tauto_hooo_imply : (a => b)^c -> (a^c => b^c)^true

    tauto_hooo_or : (a | b)^c -> (a^c | b^c)^true

The major difference is that `|- p` equals `p^true` instead of implying, if you treat `|-` as internal. If you treat it as external, then you can think of it as N + T.

I needed this theory to handle reasoning about tautological congruent operators.

However, once you have this, you can perfectly reason about various modal logical theories by keeping separate modality operators, including Provability Logic, e.g. `modal_n_to : N' -> all(a^true => □a)`.

So, it is not a tradeoff that loses Provability Logic. Instead, you get a "finalized" IPL for exponential propositions. This is why I think of as a natural way of extending IPL with some notion of provability.

1. I only need normal congruence

2. I only need perfect information games

Under problems that are solvable using these two assumptions, there is little benefit in tying proofs back to an axiomatic basis. Once you drop one of these two assumptions, proofs get much harder and a solid foundation gets more important.

yes

> I viewed the page you linked, but I don't see anywhere where you describe the alternate notion of "provability" you have in mind.

See section "HOOO EP"

>Oh, I guess, if you want to use "there exists" in a specifically constructive/intuitionistic way? (so that to prove "[]P" you would have to show that you can produce a proof of P?)

This would be `|- p` implies `□p`, which is the N axiom in modal logic (used by Provability Logic).

In Provability Logic, you can't prove `□false => false`. If you can prove this, then Löb's axiom implies `false`, hence absurd. `□p => p` for all `p` is the T axiom in modal logic. In IPL, if have `true |- false`, then naturally you can prove `false`. So, you can only prove `□false => false` if you already have an inconsistent theory.

Provability Logic is biased toward languages that assume that the theory is consistent. However, if you want to treat provability from a perspective of any possible theory, then favoring consistency at meta-level is subjective.

> I think I've seen other people try to do away with the need for a meta-language / make it the same as the object-language, and, I think this generally ends up being inconsistent in the attempts I've seen?

I have given this some thought before and I think it is based on fix-point results of predicates of one argument in Provability Logic. For example, in Gödel's proof, he needs to encode a predicate without arguments in order to create the Gödel sentence. In a language without such fix-points, this might not be a problem.

> I don't know quite what you mean by this, but, _please_ do not call this a qubit, unless you literally mean something whose value is a vector in a 2d Hilbert space.

The name "qubit" comes from the classical model, where you generate a random truth table using the input bit vector as seed. So, the proposition is in super-position of all propositions and hence behaves like a "qubit" in a classical approximation of a quantum circuit.

Recently I found another notion of "provability" where Löb's axiom, required in Provability Logic, is absurd. It turns out that this notion fits better with Intuitionistic Propositional Logic than Provability Logic. This allows integrating the meta-language into object-language. This is pretty recent, so I think we still have much to learn about the foundations of mathematics.

To answer this question, one must say something about which language a foundation of mathematics is using. For example, Set Theory is formalized in First Order Logic. However, First Order Logic uses Propositional Logic, so to build an alternative foundation to Set Theory, you might consider starting with Propositional Logic and extending it another way.

First Order Logic extends Propositional Logic with predicates. This might seem like a good design at first, until you try to reason about uniqueness. In Set Theory, one requires an equality operator in addition to set membership, in order to be able to reason about uniqueness, at all. This equality operator is ugly, because you have to rebuild objects that are isomorphic but using different encodings.

Predicates causes problems because they are unconstrained. For easier formalizing of advanced theories, I suggested Avatar Logic, which replaces predicates with binary relations, avatars and roles. You can try it here: https://crates.io/crates/avalog

Also, most theories assume congruence for all predicates, which is bad for e.g. foundations of randomness.

The next crisis in "the foundations of mathematics" will be "tautological congruence". Luckily, this is already being worked on, by extending Intuitionistic Propositional with exponential propositions. This theory is known as "HOOO EP" and is demonstrated here: https://crates.io/crates/hooo

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