Exploring the relationship between a restricted diagonal predicate and a restricted axiom (conditionalizing the T-Schema:
Tf(S) ⟷ S becomes
C(S) → (Tf(S) ⟷ S) per the tautology
(P → Q) → (R → (P → Q))):
P✦ is a diagonal predicate for which no fixed-point is defined for any x (a fixed-point gap).
P✦ for any (sentence) x.
i. It's interesting to think about this basic relationship: how the use of a specific kind of predicate might accomplish the same work in a language as an axiom. The newly defined (and restricted) predicate plays the same role as a conditionalization rule (specifically,
(P → Q) → (R → (P → Q))).
ii. Intriguing how a grammar change can play the same functional role as an axiom/inference change. (Interestingly, the Sheffer Stroke/NAND operator allows one to compress the axioms of classical logic into a single rule: http://fitelson.org/ar.html per Nicod and Lukasiewicz.)
More precisely, that the same set of wff and models both satisfy or are consistent with the
P✦ theory and the truth-restriction theory (one can swap out the same model and set of wff and get the same result - e.g. similar to the notion of empirical adequacy in the philosophy of science). This should come as no surprise since these basic notions drive most intuitions about the isomorphism of theories.
The two theories can also be seen as rotations, transformations, or translations of each other with the invariant being the wff and models. This supplements standard lattice-theoretic treatments about the relationships between axioms, logics, and sets of sentences.
Call this inter-axiom-predication - trading a predicate(s) for a axiom (or vice-versa). Here, perhaps, the problem lies not in the T-Schema nor Classical Logic but in the definition of the name-forming predicate (Gödel Numbering following Tarski).
Note: that merely altering the diagonal predicate
P✦does not by itself preclude Liar Cycles or Yablo Sequences from wrecking logical catastrophe although it also prevents Curry Compounds from imploding the logic.
One grammatical change (rather than an axiomatic or semantic one) has allowed us to knock out 2 of the 4 Alethic Paradoxes. Are there similar grammatical changes that might get the others?
In other words:
P✦₁-technique ∪ P✦₂-technique ∪ ... ∪ P✦ᵢ-technique
Are there other kinds of entity equivalences in this way?
Per the Sheffer Stroke/NAND operator, we know that logical connectives and other symbolic items can be used to compress the number of rules within axiom systems. So, if there's no further way to do this with
P✦, it might be possible to find some C-technique for some logical connective C.
Some possibly relevant parallels between Univalent Foundations and Homotopy Type Theory vs. Thinking Notation and Connection Theory:
Univalent Foundations and Homotopy Type Theory is a model of constructive logic (intuitionist logic) and involves topological concepts like homotopy retracts and path deformation.
Thinking Notation and Connection Theory can also be a model constructive logic (intuitionist logic)
i. Thinking Notation also models classical logic but not as an internal logic.
ii. Thinking Notation also involves topological notions like homotopy retracts and path deformation.
I've been puzzling about some of the ideas I had earlier about Thinking Notation, Connection Theory, and how the two interact/are related.
Per the first section above, it might be the case that such a relationship is similar to or of the same kind as inter-axiom-predication. Can certain algebraic systems be transformed into certain logical systems, generally? (Observing that Classical Sentential Logic is isomorphic to Boolean Algebra.) If not generally, what are the properties at play? How do essentially symbolic reasoning systems "become" predominantly structured systems (of sets or some equivalently functional underlying entity)?
Reminds me of the "Great Work" - the so-called Philosopher's Stone which enables one to transmute anything into anything else.