An early draft of mine on *Propositional Stability*:

Read the full draft.

Refining the verbiage a tad:

Propositional stabilityis a two-place predicate (dyadicorarity two)P(x,y):xexhibits (or has)propositional stabilitywith respect toy.

A great paper on Cohen Forcing.

For any two (*classical*, *deductive*) models M, M* of the same theory T such that:

- M* is an extension of M (by
*Cohen Forcing*). - Both M, M* satisfy proposition P.

P exhibits:

*Truth stability*under M* with respect to M. (*Truth instability*otherwise.)*Strong propositional stability*.

**Proof.** Obvious. Trivially, any *classically-consistent*, *deductive* proof system, model, and logic that's a model-theoretic *extension* of another (of the same theory T) has, by definition, the same *instruction set* mapping propositions to truth values (e.g. the semantics for the logical connectives). What diverges between the two models, is that the background universe assumed in M* is larger than M.

**Note:** a *model* is a background universe (roughly, a "semantics for a logic") that makes all the sentences of the *theory* (the grammatically valid sentences of a language) and *logic* (the language and proofs system) evaluate to `true`

. By the *Principle of Explosion*, any *contradiction* in a *logic* entails that anything can be proven within it (e.g. - that every sentence is `true`

). So, the above holds even when the *models* are *inconsistent*.

So, trivially, the cases of interest in many

Cohen Forcingscenarios are described byFact 2(with models also taken into consideration) within the draft. Many of the most interesting results are those in which theories differ (Axiom of Choice versus no Axiom of Choice, etc.).

For example, cases where a background universe of sets (and say, the axioms of ZFC) has added to it some set for which the axioms of ZFC set theory fails (self-referential sets). We can, of course, define more or less rigorous notions of a model so that consistency is required of them (which further delimits the cases alluded to).

Edit: I've significantly updated and revised this remark to clarify and remove some typographical errors.

The original paper

Also: Kremer and notes: http://web.mit.edu/~24.118/www/handouts/KripkeTruth.pdf and http://web.mit.edu/~24.118/www/handouts/KripkeTruth.pdf

Kripke outline several techniques in his landmark paper:

- The general use of
*fixed points*from elsewhere in mathematics to sentences of paradox.*Fixed points*are commonly used in Number Theory.

`i.`

A *fixed point* of a function *F* is some *x* such that: F(x) = x.

`ii.`

See: the Wolfram MathWorld entry.

- A closed-off construction (with two-valued semantics).
- A three-valued construction (using Kleene 3-valued semantics and models thereof) with truth values:
`true`

,`false`

,`neither true nor false`

.

I'm curious about any intersections between my nascent

Propositional Stabilityformulation andfixed pointsemantics.

Consider a two-valued formulation with predicate `T`

added to it along with T-Schema constraints:

- Using Kripke's technique, we define an extension of
`T`

(`ext(T)`

) corresponding to true sentences and an anti-extension of`T`

(`anti(T)`

) corresponding to those sentences that we don't assign truth to. We also use the same convention that coded sentences are formulated and placed into respective extensions (representing truth statements about other truth statements and attributions of truth to names of sentences or their untruth):`code(T)`

and`code(anti(T))`

. - A
*fixed point*here is then any model in which`code(T)`

=`ext(T)`

and`code(anti(T))`

=`anti(T)`

(which corresponds to the T-Schema constraint in model format). - If our interpretation function,
`I`

assigns primitive truth in the standard fashion (lacking any alterations) to the*Liar Sentence*, the*Liar Sentence*gets assigned to both`ext(T)`

and`anti(T)`

(as well as both coded extensions). Consequently:

`i.`

The *Liar Sentence* exhibits *truth stability* between any two fixed points (and at the same fixed point).

`ii.`

The *Liar Sentence* exhibits *truth instability* between any two fixed points (and at the same fixed point).

`iii.`

The *Liar Sentence* exhibits *strong propositional stability* between any two fixed points.

A *proof* trivially exists at every fixed point *h*:

- The
*Liar Sentence*is*true*(not*false*). - The
*Liar Sentence*is*false*(not*true*). - The
*instruction set*remains unchanged.

So, trivially, (by the *Principle of Explosion* - proof following a derived contradiction - or otherwise through direct proof):

- One can prove the same set of truth values at each fixed point,
*h*, relative to every other fixed point*h**:`true`

,`false`

- One can prove a different truth value at each fixed point
*h*(but not at the exclusion of the others).

Note:sentences of paradox are the only class of sentences that exhibit all the properties above.

What are differences between models of paradoxical sentences in terms of propositional stability? How do sentences of paradox differ from others in terms of propositional stability? How do consistent models differ in terms of propositional stability? Now, we have some answers.

Regarding *Kripke's Three-Valued Fixed Point Construction*.

If we take a weaker reading about *truth-values* and disallow *n* to count as such (`{t,f,n}`

is the set of values that our propositions can be mapped to but count only `{t,f}`

as truth-values proper):

- The
*Liar Sentence*exhibits*strong propositional stability*between any two fixed points.

**Proof.** Obvious. Following the definitions from Kremer:

Say that a sentence

Aisρ-grounded iff lfp(ρ) = t or f, andρ-intrinsic iff gifp(ρ) = t or f.

The

Liar Sentenceis neitherκ-groundednorκ-intrinsicsince it gets the valuenat everyfixed point h.

- The
*Liar Sentence*receives the value*n*at every*fixed point h*. - The
*instruction set*remains unchanged.

If we take a stronger reading about *truth-values* and disallow *n* to count as such (since the spirit of the original proposal was to allow truth-value gaps):

- The
*Liar Sentence*only exhibits*weak propositional stability*between any two fixed points (since it would lack a truth-value). - The
*Liar Sentence*could be said to exhibit*truth stability*between any two fixed points (since it would lack a truth-value at any fixed point). - The
*Liar Sentence*could also be said to exhibit*truth instability*between any two fixed points (since it would lack any truth-value at any fixed point).

This turns on how we understand a truth-value. I think the more natural reading is the one above where "truth-value gap" just means "not being true or false".

There's more to be explored here with similar systems that just don't define a total truth predicate - e.g. the Liar Sentence receives no truth-value at all (not even

n) in a two-valued but non-bivalent logic. The latter scenario above would likely describe that scenario.

*Propositional stability* is a logical interface.

Relevant to Cross Logics (acting as "shared intermediaries" between logics) or across permanent changes in logics.

One additional insight that's relevant to Truth Grounding and Liar:

- Kripke's Fixed Point construction identifies the smallest fixed point.
- This corresponds to identifying the most minimal set of sentences to restrict the T-Schema by (when comparing the primary machinery between the two proposals).
- I opt to restrict the axiom itself (its still a tautology afterwards and particularly if one also accepts the analyticity of the T-Schema itself) rather than a model.

A great read about some of these observations and the applications to programming languages.

- Non-Eternalist Logic
- Transactional Logic
- Logical Module
- Dimensional and Hyper-Dimensional Logics
- Dimensional and Hyper-Dimensional Logics #2
- Language of God
- Propositional Stability and Cohen Forcing

##### post: 7/11/2021

##### update: 9/29/2021