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# Fun Math Stuff and the Philosopher's Stone #3

1. I previously discussed inter-axiom-predication whereby two approaches (techniques, theories, specific axiomatizations) have the same models. Apparently, there are known parallels that have been studied.

`1a.` We can obtain something like the same results as the Restriction Technique (which caveats or conditionalizes the T-Schema) with Quantification Restriction/Typing. I alluded to Quantification Typing in my draft as a reply to context instability woes surrounding the T-Schema (contra Read).

`1b.` Strengthening my argument further: we can also think of a consistent T-Schema as being expressed in two ways: (The Pure Classical Form) `T𝑓(s) ↔ s` (albeit with Quantification Typing - e.g. more precisely expressed as: `∀x(T𝑓(x) ↔ x)` such that `x` ranges over Truth-Based/Truth-Grounded sentences).

`1c.` Or, the Restricted form `p → (T𝑓(s) ↔ s)` - more precisely `∀y∀x(y → (T𝑓(x) ↔ x))` (or `∀y(y → ∀x(T𝑓(x) ↔ x))`) where `x`,`y` is allowed to range over any sentence type.

`1d.` I think the observation above (in tandem with the basic fact that if `p is a tautology` so too is `q implies p`) makes my view far more palatable. Proponents of the T-Schema get to keep an analytically true axiom (if the T-Schema is as they insist since they also insist it's Analytic) that's aesthetically the same (lexicographically isomorphic but Quantifier Type Restricted).

1. Curry–Howard Correspondence which establishes a link between Formulas and Constructive Proof.

`2a.` Curry–Howard Correspondence can be used to frame my formal definition of Alethic Paradox (the very first to my knowledge): an Alethic Paradox is any proof to contradiction whose shortest proof requires the use of T-Schema or F-Schema.

`2b.` Apparently, Curry–Howard Correspondence is often used in tandem with topological (or tableaux) notions (Paths) and Normalization Techniques that simplify or shorten a "route" from an expression to some ending expression (preventing "meandering" or "wandering off the trail"). In this way, we can establish the "sameness" of techniques through a variety of ways (not just via "sameness" in computable models).