Second tidbit - see propositional abstraction

First Pass

*Propositional Stability* ensures that when a proposition is *transacted* between two logics (more on this later) - it never acquires a new truth-value *beyond those it could have already acquired under the first logic under which it is evaluated*

where ⊶ ∈ ℕ

where ⋇ ∈ {a, ..., z, ...} | {a, ..., z, ...} = ℕ

Conventions

We write ML⊶⋇ to denote a semantics (model or truth-assignment M) for a language L⊶ with ⋇-many truth values.

We write VML⊶⋇(*p*) to denote a truth-evaluation of p under semantics (model or truth-assignment M) for a language L⊶ with ⋇-many truth values.

We write VML1aVML2b(*p*)* to denote any possible truth-evaluation of *p* to a truth-value *t* in semantics ML2b such that: t ∈ ML2b and t ∉ ML1a.

An *instruction set* is a finite procedure or algorithm mapping one input to one output.

Elaborated Definition

*Propositional stability*: a proposition or sentence *p* evaluated under semantics ML1a will preserve its truth-value under semantics ML2b whenever a ⊆ b and no *instruction set* exists to map VML1a(*p*) to any VML1aVML2b(*p*)*.

Initial Result

Any proposition under a Boolean logic will exhibit *propositional stability* against a (standard - thus far axiomatized) Kleene 3-Value Algebra. Proof: Obvious. No single proposition already assigned a truth-value of 'true' or 'false' can receive a truth-value of 'indeterminate' or 'true and false'. ∎