Per Fun Math Stuff and the Philosopher's Stone #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).

- 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).

- Remarks on Truth-Grounding and the Liar
- Remarks on Truth-Grounding and the Liar #2
- Ground Facts and Truth
- Ground Facts and Truth #2
- Addressing Metalogical Justification
- Propositional Stability and Cohen Forcing
- Truth as a Prosentential Operator
- The Liar Paradox in Programming
- Fun Math Stuff and the Philosopher's Stone
- Fun Math Stuff and the Philosopher's Stone #2
- Fun Math Stuff and the Philosopher's Stone #3
- Restrictionism and the Four Corners of Logic

##### post: 01/14/2024