Truth-Only Logic #4

Some initial thoughts.

Truth Types, Tarski, and Liskov

Suppose we think of a Tarskian Truth Type T in terms of Liskov Substitution Principles:

Subtype Requirement: Let ϕ(x) be a property provable about objects x of type T. Then ϕ(y) should be true for objects y of type S where S is a subtype of T.

We might helpfully parse the above in the following way:

1. Given: S < T, t ∈ T, s ∈ S
2. ϕ(t) → ϕ(s)

dl.acm.org - 197320.197383

Suppose that there is a top-level Tarskian Truth Predicate T*:

1. And we'll define ϕ (from above) like so: ϕ: T*(s) ↔ s (letting ϕ be the T-Schema).
2. Now suppose T*-1 is a subtype of T* (T*-1 < T*). If Liskov Substitution holds for T*-1, T* then any paradox from ϕ would presumably appear in T*-1.
3. Then to avoid Tarskian-style Semantic Paradox either ϕ must not entail a Contradiction (which it does from the Liar Paradox), Liskov Substitution must not hold, T*-1 < T* must be False, or ϕ must be incorrect.
4. We're mostly reframing standard-fare Semantic Paradox with the above (and equipped with concepts inherited from Liskov).

I've addressed the fourth (last) option here and the first elsewhere. But what about the other/middle two? Those seem like novel ways to frame the problem and consider the relation of Truth and its Properties.

The original formulation (sketch) of a Truth-Only Logic uses no Type Theory (only some Ordering) and in the light of the above brief considerations, that seems like a good starting point.

1. Truth is identified with a top-level Entity which exhibits an internal structure (Ordering of Truths) such that Truth is one singular Entity (TRUTH) and many composing and ordered Truths: TRUTH :df {T0, T1, T2, ..., Tn}. Falsehood is trivially mapped to T0 and Boolean True to Tn.
2. While something like the prohibition that no Type T can be its own (proper) subtype seems to be usually implied in contexts of Liskov Substitution (noting that Liskov Substitution appears to be compatible with the rule that for every T: T < T), we observe TRUTH and Tm | Tm ∈ TRUTH are distinctive Entities and Types (if were to force a Typology or Typing scheme).
3. This is a trivial but subtle distinction in these kinds of debates which are inundated with conceptions of Truth hierarchies (of Types, Predicates [which are Types], and the like).

The Structure of Truth

Previously, we considered Truth as a rather one-dimensional kind of thing - ordered along one dimension of consideration.

1. Consider a scenario where the Single Truth Value is a (complex) Lattice, not a single-dimensioned ordering. (Kripke explored Fixed-Point Semantics for Predicates and concluded that Truth is the triad T, F, NTF. It's also common practice to consider the relationship between distinctive Truth, Falsity, and any third Truth Values as Lattices.) On such an approach Truth is identified with the Lattice (or Graph for that matter) not a Vertex, Node, Edge, etc.
2. Or a Topological Space.
3. Or a Field - (e.g. a Function distributed across a Space).