Summarizing, linking, and uniting several topics.

For the DRAFT

Constraint Satisfaction and Classical Extensions of KF.

Tarski 1944 defines each **Meta Language** `Mₐ`

as containing a target **Object Language** `Oₐ`

such that a **Truth Predicate** for `a`

(`Tₐ`

) is present in `Mₐ`

about *WFF* in `Oₐ`

(but not present in `Oₐ`

).

Tarski's approach appears to commit one to the following:

- An infinite ascending hierarchy of languages.
**Profligate Truth Predicate Pluralism**- an*infinite*ascending hierarchy of**Truth-Predicates**. Each**Truth-Predicate**is immanent in each linguistic ascent.- In the absence of
**Bridge Laws**or other subsumption laws, each ascending**Truth-Predicate**will express more Truths at each ascent than those below it (where`n = a + 1`

,`Tₙ`

doesn't exist in`Mₐ`

and so*WFF*with`Tₙ`

don't exist at all below`n`

). So, each**Truth Predicate**fails to express the totality of truths (e.g. - all`> a`

) at each level`a`

. It's a*Partial***Truth Definition**at each level. - Since it ascends infinitely, there's no top-level
**Truth Predicate**capable of expressing all Truths. - While at each level
`a`

, the**Liar Sentence**is blocked for`a`

, it can be recovered at`> a`

for`a`

. And since each*WFF*(including**T-Scheme**at level`a`

) is contained in each ascent, the**Liar Sentence**re-emerges for`< n`

at`n`

.

How then are we justified in taking **T-Scheme** through such an approach:

- As a correct, universally quantified expression, schema, or law (even within the
**Object-**and**Meta Langauge**hierarchy since we cannot find a top-level**Truth Predicate**)? - How do we identify which
**Truth-Predicate**we're to use given that each**Truth-Predicate**is*Partial*and there are at least`n-many`

such predicates for each ascent`n`

? - And why not move to
`n+1`

since the ascent`n`

is contained in`n+1`

? (Why one ascent level over another?)

We might distinguish between a more conservative approach and the sketch outlined above:

- On such a conservative approach, only
**Object Langauges**one "level" (ordinal) lower are included in each**Meta Language**so that no**Truth Predicates**are imported by way of including lower ordinal**Meta Langauges**. (It's "conservative" precisely in that sense, only one language ascent is included within each**Meta Language**and not*all of them*.) - But then we run afoul of
**Logical Skepticism**since we are forced to ascend indefinitely to prove the justification of any particular language level. (And here such worries probably do matter even if we are followers of Crispin Wright since we are reasoning about Deductive systems of absolute certainty afterall.) - And again we would be committed to
**Profligate Truth Predicate Pluralism**but with the additional limitation that there's no single top-level**Truth-Predicate**able to talk about all Truths below the current Object Language "level" (ordinal). Truth is then strictly immanent (within and bound to) specific language ascents. - How then do we explain translations between languages (Artificial or otherwise), representation theorems, and the like? Through isomorphism? It's not at all clear that all Natural or Artificial languages of relevant comparison share all the relevant features sufficient and necessary for isomorphism.
- How then we explain comparing language systems (like we presently are) since Truth would appear to be bound to a specific language (per the above) and no lower ordinal
**Truth Predicates**would be imported into such a context of comparison?

- 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: 09/11/2024

##### update: 10/01/2024