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https://github.com/thoughtscript/truth-predicate-eliminability-sorting-algorithm

Truth Predicate Sorting Algo for T-Scheme
https://github.com/thoughtscript/truth-predicate-eliminability-sorting-algorithm

alethic-paradox liar-paradox logic

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Truth Predicate Sorting Algo for T-Scheme

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# Truth-Predicate-Eliminability-Sorting-Algorithm



[![](https://img.shields.io/badge/Solomon-Feferman-blue.svg)](https://www.kva.se/en/prize-laureate/solomon-feferman-2/)

[![](https://img.shields.io/badge/Saul-Kripke-red.svg)](https://www.kva.se/en/prize-laureate/saul-a-kripke-2/)

[![](https://img.shields.io/badge/JC-Beall-lightblue.svg)](https://philosophy.nd.edu/people/faculty/jc-beall/)

[![](https://img.shields.io/badge/Graham-Priest-black.svg)](https://grahampriest.net/)



*WIP* (*Future Thesis and/or Dissertation.*)



The original paper (2013-2014) was conferenced and referred to two of the top Logic journals in the English-speaking world (and much to my surprise!) but I declined to begin the lengthy (often 2+ years) academic publication process.



> Original draft hosted here: https://www.thoughtscript.io/papers/000000000002



For the more recent: ***Classical Extensions of Kripke-Feferman: Constraint Satisfaction and Alethic Paradox*** (summarizing key points)



> https://www.thoughtscript.io/papers/000000000013



> Mirrored: https://thoughtscript-io.onrender.com/papers/000000000013.html



## Context



There's widespread consensus that no (previously) offered solution to the **Liar Paradox** gets all of the following:

1. All the other **Alethic Paradoxes**: *Yablo-Visser*, *Liar Cycles*, *Revenge Sentences*, *Boolean Compounds*, *Curry Compounds*, *McGee Sentences*, etc.

2. No ***touted*** "philosophical (non-formal/mathematical) solution" can prove its formal correctness (**Soundness**, **Completeness**, **Consistency**, Metalogical Theorems, address **Tarski's Undefiniability Theorem**, etc.). (These are mostly what you'd find on the SEP article about the **Liar Paradox**.)

3. No formal solution can explain the big "philosophical why" (it must explain the problem) while addressing issues of formal correctness. It mustn't just be an ad-hoc trick of mathematical machinery.

4. No solution offers both a "philosophical explanation" and formal proof of its correctness.

5. Additionally, no formal solution is known to overcome all of the existing concerns: **Bacon 2015**, **Revenge Paradox**, etc.



(Optionally) Adding to the above: 



6. Is **Classical** (preserves **Bivalence** and the other **Classical** validities).



## Definitions



Formal definitions.



> Consult **Section 3.2** for an overview of **Truth as a Metalinguistic Predicate**.



### Truth Values, Truth Makers, Truth Predicates, and Axioms for Truth



*Some elementary definitions that are nevertheless useful to explicitly state and clarify here.*



1. **Truth Values** - the **Value** assigned to a  **Sentence** through a **Truth Assignment** (**Truth Interpretation**).

   * The historical tradition (in Maths and Logic, the academic fields/disciplines) follows Wittgenstein in taking **Truth Assignments** as **Functions** (**Truth Functional**) - a Mapping of **Sentences** to their respective **Truth Values**.

   * **Truth Assignments** are Mapped in two steps: **Propositional Variable Assignment** and then complex **Sentential Assignment** (via **Truth Tables**).

   * This topic has traditionally been confined to the academic fields/disciplines of Philosophy and Logic. (Are there three **Truth Values**? Two? Is Logic **Classical**?)

2. **Truth Makers** - what determines the *actual* **Truth Value** of a **Proposition**. The objective circumstances that make a Linguistic *representation* of that scenario (**Fact**), **True**.

   * An accurate (correct) **Model** faithfully depicts (*represents*) such **Truth Conditions** and **Truth Makers**. One winnows down all possible or permissible **Truth Assignments** to the correct or most accurate *representation*.

   * Properly understood, Scientists and Mathematicians uncover or discover such **Facts**. (`F = M x A`, `1 + 1 = 2`, etc.)

3. **Truth Predicates** - how we ascribe **Truth** to a **Sentence** *within* a **Language**.

   * *See below.*

4. **Axioms for Truth** - the Mathematically precise general Semantics and behavior of the word `Truth` in Natural (and Artificial) Language - e.g. it's **Inferential Properties** (for reasoning), the valid moves/language norms allowed in discourse/social activities, it's Linguistic Definition especially in consideration of the Liar Paradox. 

   * From an applied standpoint: people definitely talk about the **Liar Paradox** (Tarski's ***Semantic Conception of Truth*** is one of the most cited papers of all time).

   * How does one correctly parse, understood, or define these concepts in **Large Language Models**? In **Word Vector** notation, is the computed distance for `True(S)` and `S` within a Corpus equivalent?

   * From the frontiers of Computer Science: is **KFG** a suitable Sequent, Tableux, or other semantic, contextual, or denotational Rule we'd use within Monoidal Semantics?

   * This topic is the concern of Linguistics (the Scientific study of Language), Philosophy of Language, etc.



> This paper is concerned with the latter two topics. It is not concerned with altering  or understanding **Fact**. It seeks instead to address a long-standing problem with the **Truth Predicate** and how we are to use it in Mathematics, Logic, and other technical disciplines.



### Sentences Names, Name-Forming Operators, and Diagonalization



Here and below I’ll use the convention `⟨`,`⟩` to denote the familiar **Gödel Numbering** technique:



1. **Sentence Name** - the **Name** of a **Sentence** (e.g. - a **Variable Name** in Computer Science) `P` for a **Sentence** `S` shall be written: `P := S`.

2. **Name-Forming Operator** - `⟨S⟩` represents the mapping of some **Sentence**, **Proposition**, or **Expression** `S` to its **Name**. `⟨S⟩ ≡ P := S` returning `P`.

3. **Diagonalization** - a **Technique** that associates the **Fixed Point** of a **Sentence** containing `S` as a sub-expression so that `S` is its own name.

4. (Below, if `S` is the name of a sentence containing `S` as sub-**Expression**, both `⟨S⟩` and `S` will be used interchangeably as **Names**.)



> This should come as no surprise since it forms the historical and mathematical basis for Variable Naming, Memory Addressing, and Value Assignment within programming languages.



### T-Scheme



1. *Tarski’s 1933* **Definition of Truth**  - `For all x, True(x) if and only if ϕ(x)`.

2. **Modern Formulation** - (For all `S`) `T(⟨S⟩) ↔ S`:

   * **Capture** (or **T-Intro**) - conditional subrule of the **T-Scheme** biconditional. The rule going from `S` to `T(⟨S⟩)`.

   * **Release** - conditional subrule of the **T-Scheme** biconditional. The rule going from `T(⟨S⟩)` to `S`.



### Truth Tellers



```

S := T(⟨S⟩)

```



1. **Truth Teller** - like the **Liar Sentence** but expressing **Truth** of itself. Constructed via **Fixed-Point Diagonalization** like the **Liar Sentence**.



### Alethic Paradox



> From [Dictionary.com](https://www.dictionary.com/browse/alethic): **Alethic** "of or relating to such philosophical concepts as truth, necessity, possibility, contingency, etc".



> From [Mirriam-Webster](https://www.merriam-webster.com/dictionary/alethic): **Alethic** "of or relating to truth".

1. **Alethic Paradox** - for any sentence `S`: The shortest proof resulting in **Contradiction** that requires the use of **T-Scheme** (**F-Schema**, or our other **Alethic** inferences including proven biconditionals involving the

**Truth Predicate**).



### Properties of Truth and T-Schema



1. **Truth Transparency** - the principle that `S` and `T(⟨S⟩)` are always and everywhere intersubstitutable.

   * *(Introductory Wikipedia article on [Referential Transparency](https://en.wikipedia.org/wiki/Referential_transparency) in Philosophy and Computer Science.)*

2. **Truth Eliminability** - (W.R.T. to **T-Scheme**) in rewriting `T(⟨S⟩)` in the lexiographical form `S` (via **Truth Transparency**) `S` *must* contain content that *doesn’t* predicate **Truth**.

   * A stronger criterion on (or reading of) **Truth Transparency** (and **T-Scheme**).

   * **Truth Transparency** requires that `T(⟨S⟩)` can be rewritten in a form where no `T` appears (where **Truth** is not *predicated*).

3. **Truth Opacity** - when a **Sentence** `S` cannot be rewritten/restated (via **Truth Transparency**) without a `T` appearing (where **Truth** is not *predicated* and all while keeping its Semantic Content and **Truth Value**). Such a **Sentence** is **Truth Opaque**.



## Truth Predicate Eliminability Algorithm and KFG



A Finite, **Sorting**, Algorithm used to determine whether a **Sentence** is **Truth Opaque** or not.



May circumvent general concerns stemming from [**Bacon 2015**](https://andrew-bacon.github.io/papers/Indeterminacy%20and%20Revenge.pdf) (my original paper was never published):

* Appears to be a new "species" of **Restrictionist** approaches that also doesn't require every **Sentence** or **Theorem** to be "Cleaned", "Healthy", or "Debuggered".

  * E.g. - those approaches that follow the Classical Axiom, Theorem, and Tautology: **Weakening** `P → (Q → P)`.

  * The **T-Scheme** is actually a **Material Conditional** with some **Constraint**, "Checkpoint", or condition that must be met / a "Restriction" on it.

* Invalidates the move from **P1** to **P2** (by Substitution or Diagonalization).

* Sorted expressions are nevertheless given **Truth-Values** and don't entail **Untruth** (or **Falsity**).

* So, both **Truth Opaque** and non-**Truth Opaque Sentences** are allowed - they are not "banned" or "outlawed".

* Not clear that **Diagonalization** is legitimate for such **Sorted Expressions**. In the original papers by Gödel, **Diagonalization** is justified only up to and for *primitive recursive number-theoretic function*(s). As such, it's not clear a **Revenge**-type **Sentence** can be constructed from the get-go for **KFG** (since they require a **Diagonal Predicate** in their construction).

  * Per the above: and even if we were to allow the construction of such a **Revenge**-type **Sentence** `R` in **KFG**, it doesn't satisfy the formal criteria to result in **Contradiction** (e.g. - that **Diagonal Predicate** `C` in `R` must entail **Untruth** or **Falsity**).

* **Truth Opaque Expressions** aren't necessarily **Theorems** nor are the assertions of them as such.

* The conclusion of the argument is essentially that **KFG** will prove a Theorem that's **Truth Opaque**.  Consider the unproblematic Sentence: `S := T(S) → T(S)` - it's a Theorem, receives Truth Values, and is **Truth Opaque**.



Blocks **McGee's T-Intro** step.



Satisfies **Tarski's Undefinability Theorem** for **T-Scheme** since `S` and `T(⟨S⟩)` can be **Logically Consistent** yet diverge in Truth Values within **KFG**.



## Some Validations and Truth Table Proofs



Some simple **Truth Table** and basic Model checking summarized succinctly below.



> Depicts some Models of **KFG** and proves **Consistency** of **KFG** with respect to **Liar Cycles**.



> Demonstrates Classically Consistent Models (the primary goal) and ways to address the ancillary goals: **KFG** global validity and embedded **Catuṣkoṭi**.



Since `S` can be any arbitrary **Sentence** within **KFG**, the below constitutes a **Consistency Proof** by **Mathematical Induction** (per **pp. 11** of the original draft).



### Truth Assignments 



The **Truth-Value** for `S ∈ C` is determined by the **Truth Opacity** of a **Sentence** and prior to **Truth-Assignments**. (It's a constraint on the **Interpretation Function** itself as specified in the Draft Paper.)



> Below, `T(S) ↔ S` refers to the specific WFF with Sentential Constant `S` substituted into **T-Scheme**.



### Truth Eliminable Sentences



Truth Table Semantics:



| `S` | `¬S` | `C(S)` (`S ∈ C`) | `¬C(S)` | `T(S)` | `¬T(S)` | `T(S) ↔ S` | `C(S) → (T(S) ↔ S)` | 

| --- | --- | --- | --- | --- | --- | --- | --- |

| `⊥` | `⊤` | `⊤` | `⊥` | `⊤`* | `⊥`* | `⊥`* | `⊥`* |

| `⊥` | `⊤` | `⊤` | `⊥` | `⊥` | `⊤` | `⊤` | `⊤` |

| `⊤` | `⊥` | `⊤` | `⊥` | `⊤` | `⊥` | `⊤` | `⊤` |

| `⊤` | `⊥` | `⊤` | `⊥` | `⊥`* | `⊤`* | `⊥`* | `⊥`* |



There are two ways to read this:



1. *Modus Tollens* on the **Argument from Tautology**. If **T-Scheme** is a **Tautology** then so too is `Q → T-Scheme`. If `Q → T-Scheme` isn't a **Tautology** then neither is **T-Scheme** (which is precisely what `Q → T-Scheme` is showing in the first place - e.g. **Weakened T-Scheme**). On this view, both **T-Scheme** and **KFG** are **Contingent**.

2. The fourth and first **Interpretations** above can be ruled out by additional (optional) extensions that modify how the **Interpretation Function** behaves (this is the route primarily endorsed by the Draft Paper but isn't the only route available. In the original Draft, `S` and `T(S)` are harmonized through additional rules added to the construction step of `C(S)` that convert `*` to the second or third interpretation.) prior to **Truth Assignment** itself (akin to the way that the Truth of **Logical Connectives** are calculated after **Atomic Proposition Truth Assignment** and **ZFC Set Theory** which has a complicated setup for the **Domain of Discourse** - both **ZFC Set Theory** and **Zero-Order Logic** are part of **KFG**). This converts **KFG** into a global validity (**Tautology**) otherwise it'll fail with the above unmodified construction (whilst remaining **(Logically) Consistent** nevertheless).



### Truth Opaque Sentences



Truth Table Semantics:



| `S` | `¬S` | `C(S)` (`S ∈ C`) | `¬C(S)` | `T(S)` | `¬T(S)` | `T(S) ↔ S` | `C(S) → (T(S) ↔ S)` |

| --- | --- | --- | --- | --- | --- | --- | --- |

| `⊤` | `⊥` | `⊥` | `⊤` | `⊥` | `⊤` | `⊥` | `⊤` |

| `⊤` | `⊥` | `⊥` | `⊤` | `⊤` | `⊥` | `⊤` | `⊤` |

| `⊥` | `⊤` | `⊥` | `⊤` | `⊥` | `⊤` | `⊤` | `⊤` |

| `⊥` | `⊤` | `⊥` | `⊤` | `⊤` | `⊥` | `⊥` | `⊤` |



Comments:



1. **Truth Teller** (and **Liar Cycle Negator**) expressions are given the second or third **Interpretations** above.

2. **Liar Sentences** expressions are given the first or fourth **Interpretations** above.



### Liar Cycles



> `S := T(Q)`, `Q := ¬T(S)`



Here:

1. We only need to prove that *at least one* **Consistent Interpretation** exists.

2. The following pairs must share Truth-Values:

   * `S`, `T(Q)` 

   * `¬S`, `¬T(Q)`

   * `Q`, `¬T(S)` 

   * `¬Q`, `T(S)`



| `S` | `T(Q)` | `¬S` |  `¬T(Q)` | `Q` | `¬T(S)` |  `¬Q` | `T(S)` | **Consistent** |

| --- | --- | --- | --- | --- | --- | --- | --- | --- | 

| `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `NO` |

| `⊥` | `⊥` | `⊥` | `⊥` | `⊥` | `⊥` | `⊥` | `⊥` | `NO` |

| `⊤` | `⊤` | `⊥` | `⊥` | `⊤` | `⊤` | `⊤` | `⊤` | `NO` |

| `⊤` | `⊤` | `⊤` | `⊤` | `⊥` | `⊥` | `⊤` | `⊤` | `NO` |

| `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `⊥` | `⊥` | `NO` |

| `⊥` | `⊥` | `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `NO` |

| `...` | `...` | `...` | `...` | `...` | `...` | `...` | `...` | `NO` |

| `⊥` | `⊥` | `⊤` | `⊤` | `⊤` | `⊤` | `⊥` | `⊥` | `YES` |

| `⊤` | `⊤` |  `⊥` | `⊥` |  `⊥` | `⊥` | `⊤` | `⊤` | `YES` |



Note:

1. One of the paired **Sentences** can behave like the **Truth Teller**. (The **Liar Cycle Negator** of the pair.)

2. We also require (through optional extensions) that **T-Scheme** fails if a **Sentence** refers to a another **Truth Opaque** Sentence.



#### Liar Cycle Semantics and KFG



Regarding the last two **Interpretations**:



| `S` | `¬S` | `C(S)` (`S ∈ C`) | `¬C(S)` | `T(S)` | `¬T(S)` | `T(S) ↔ S` | `C(S) → (T(S) ↔ S)` |

| --- | --- | --- | --- | --- | --- | --- | --- |

| `⊥` | `⊤` | `⊥` | `⊤` | `⊥` | `⊤` | `⊤` | `⊤` |

| `⊤` | `⊥` | `⊥` | `⊤` | `⊤` | `⊥` | `⊤` | `⊤` |



| `Q` | `¬Q` | `C(Q)` (`Q ∈ C`) | `¬C(Q)` | `T(Q)` | `¬T(Q)` | `T(Q) ↔ Q` | `C(Q) → (T(Q) ↔ Q)` |

| --- | --- | --- | --- | --- | --- | --- | --- |

| `⊤` | `⊥` | `⊥` | `⊤` | `⊥` | `⊤` | `⊥` | `⊤` |

| `⊥` | `⊤` | `⊥` | `⊤` | `⊤` | `⊥` | `⊥` | `⊤` |



Immediately above:

1. Each **Model** pairs the respective first and second **Interpretations**.

2. We observe that `Q` (the **Liar Cycle Negator** of the pair) behaves like **Truth Tellers**.



### Catuṣkoṭi



> Some interesting phenomena.



With **F-Scheme** (`¬T(S) ↔ F(S)`) unmodified:



| `S` | `¬S` | `T(S)` | `¬T(S)` | **Comment** | `T(S) ↔ S` | `C(S) → (T(S) ↔ S)` |

| --- | --- | --- | --- | --- | --- | --- |

| `⊥` | `⊤` | `⊥` | `⊤` | `False` | `⊤` |  `⊤` |

| `⊥` | `⊤` | `⊤` | `⊥` | `True and False` | `⊥` | Depends on `S` being **Truth Opaque** or not (per the above). |

| `⊤` | `⊥` | `⊥` | `⊤` | `True and False` | `⊥` | Depends on `S` being **Truth Opaque** or not (per the above). |

| `⊤` | `⊥` | `⊤` | `⊥` | `True` | `⊤` |  `⊤` |



The above mirrors **Kleene 3-Value** constructions. The assertion would be that:

1. Confusion around **3-Value Semantics**;

2. And, Tarski's **Object-Level/Meta-Level** intutions would then be seen to stem from mismatching **Truth Values**/**Truth Predicates** (where **Language** levels are replaced by priority in **Truth Assignment** within the same **Language**).



With **F-Scheme** also **Weakened** (e.g. - `¬T(S) ↔ F(S)` will sometimes fail), the **Catuṣkoṭi** appears:



| `S` | `¬S` | `T(S)` | `¬T(S)` | `F(S)` | `¬F(S)` | Comment |

| --- | --- | --- | --- | --- | --- | --- |

| `⊥` | `⊤` | `⊥` | `⊤` | `⊤` | `⊥` | `False` |

| `⊥` | `⊤` | `⊥` | `⊤` | `⊥` | `⊤` | `Neither True nor False` |

| `⊥` | `⊤` | `⊤` | `⊥` | `⊤` | `⊥` | `True and False` |

| `⊥` | `⊤` | `⊤` | `⊥` | `⊥` | `⊤` | `True` |

| `⊤` | `⊥` | `⊥` | `⊤` | `⊤` | `⊥` | `False` |

| `⊤` | `⊥` | `⊥` | `⊤` | `⊥` | `⊤` | `Neither True nor False` |

| `⊤` | `⊥` | `⊤` | `⊥` | `⊤` | `⊥` | `True and False` |

| `⊤` | `⊥` | `⊤` | `⊥` | `⊥` | `⊤` | `True` |



> The above is the approach recommended in the original Draft - at that time I referred to them as "defects" being unaware of **The Catuṣkoṭi**. I was also unaware that similar "quirks" also appear in JavaScript: `[] == ![]; // -> true`, `true == ![]; // -> false`, `false == ![]; // -> true`.



> To be clearer still: therefore, the many great religions of the world (as well as their opposites - their heresies) - Orthodox Christianity, Catholicism, Islam, Hinduism, Buddhism, Judaism, and many more still - along with the greatest mathematical and scientific theorists, Hegelians, Platonists, Aristotelians, and all the other major views of philosophy about Truth can be inclusively accommodated. **KFG** *does not* rule on which of these interpretations is correct but *it's the only view that does not rule any of them out*. In this way it is thoroughly pluralist and inclusive of the world's greatest ideas.



> Whereas, historically, "Eastern" (Indian, Chinese, Sanskrit, etc.) and "Western" (Greek, Arabic, etc.) views on Truth and Logic have been seen as fundamentally disjoint. **KFG** suggests that they're in fact two sides of the very same coin, a sublation of seemingly mutually contradictory opposites, they're two perspectives formed of one mind, and a "Grand Unification" (if you will) of these multi-millennia-spanning approaches. 

 

Please note that the above can all be **Consistently** captured within a **Two-Value**, **Bivalent**, **Classical** Semantics. We've relaxed the requirements on `T()` and `F()` per the above.



This is the only proposed system that can accomodate all the additional items below:

1. Parallels intuitions that motivate **3-Value Semantics**.

2. **The Catuṣkoṭi**.

3. The empirical fact that people have taken all four positions regarding the **Liar Sentence**:  `False`, `Neither True nor False`, `True and False`, `True`. No other system can "subsume" the rest.

4. Is **Classical**.

5. Gets all the other Truth-related (**Alethic**) Paradoxes.

6. Satifies **Tarski's Undefinability Theorem** for **T-Scheme**.

7. Blocks McGee's **T-Intro** step.

8. Is not harmed by Bacon's 2015 argument. If `C` is a predicate it just shows that there's a **Theorem** that's **Truth Opaque** (`S := T(S) → T(S)`) otherwise one can't **Diagonalize** into it at all.



## Extensions



So, **KFG** opens the door to a fully **Classical**, **Monistic** (single Truth Predicate), and **Restrictionism** (as a topic in multiple debates: Logical Nihilism, Logical Skepticism, Alethic Paradox, etc.). 



By selecting extensible constructions variants of **KFG** strengthen certain features discussed above:



1. Harmonization of Truth Assignments (aligning Truth Values and Truth Predicates in consistent assingments).

2. Preference for Predicates over Set Inclusion or vice-versa.

3. Model selection.

4. Analyticity of the Restricted T-Scheme.



I think this is akin to subfields like the debate between `S4` vs `S5 Modal Logic`, the correct semantics for Modality, and so on.



> And indeed such an approach aligns well with the general history of mathematical logic: Lukasiewicz, Spencer-Brown, Nicod, Syllogistic Square, Tarski, Tableaux methods, Venn, Boole, and the like all leverage creativity with **Classical** constructivity to tackle similar questions from different vantage points. No modification of **Classical** logic or Set Theory is required!



## Key Philosophical Arguments



1. The **Argument from Tautology**:

   * For simplicity's sake, let **Analyticity** (**Tautology**) be defined as **Necessary Truth** (a standard Mathematical definition). 

     * To be clear, more expansive notions of **Tautology** and **Analyticity** are *not* the relevant notions here, only **Necessary Truth** (e.g. - **True** under any **Interpretation Assignment**). 

     * A more nit-picky philosopher might protest that these terms aren't strictly overlapping. Not much is lost by agreeing but for convention's sake, I'll still refer to these interchangeably (and one may feel free to substitute any of the three terms as they so choose).

   * If **T-Scheme** is **Analytic** (**Tautological**), then so is **Restricted T-Scheme** (by Classical inferential **Weakening**).

     * Inferential (Material) **Weakening**:

         | `p` | `q` | `p → q` | `c` | `c → (p → q)` | `(p → q) → (c → (p → q))` |

         | --- | --- | --- | --- | --- | --- |

         | `T` | `T` | `T` (`T → T`) | `T` | `T` (`T → T`) | `T` (`T → T`) |

         | `T` | `T` | `T` (`T → T`) | `F` | `T` (`F → T`) | `T` (`T → T`) |

         | `T` | `F` | `F` (`T → F`) | `T` | `F` (`T → F`) | `T` (`F → F`) |

         | `T` | `F` | `F` (`T → F`) | `F` | `T` (`F → F`) | `T` (`F → T`) |

         | `F` | `T` | `T` (`F → T`) | `T` | `T` (`T → T`) | `T` (`T → T`) |

         | `F` | `T` | `T` (`F → T`) | `F` | `T` (`F → T`) | `T` (`T → T`) |

         | `F` | `F` | `T` (`F → F`) | `T` | `T` (`T → T`) | `T` (`T → T`) |

         | `F` | `F` | `T` (`F → F`) | `F` | `T` (`F → T`) | `T` (`T → T`) |



      * There are at least two ways to prove this  - both make use of the Classical Tautology (Material) **Weakening** (whose Truth Table is included above for clarity).

         * First approach: let `(p → q)` stand for each half the **T-Scheme** and `c` be the Proposition `S ∈ C`. Both halves jointly entail **KFG**. (Proof. Obvious.)

         * Second approach: substitute `P` for `(p → q)` (`P → (c → P)`, which happens to be equivalent to `c → (p → q)` with substitutions), let `P` stand for `T-Scheme is Analytic (Tautological)`, and `c` be the Proposition `S ∈ C`. Since a Consequent's being **Analytic** (**Tautological**) cannot change the Truth Value of its **Material Implications**, so too is:

           *  `P → (C → P)` (e.g. - `P → (C → P) is Analytic (Tautological)`) 

               * `T-Scheme is Analytic (Tautological) → (S ∈ C → T-Scheme is Analytic (Tautological))` 

               * `S ∈ C → T-Scheme is Analytic (Tautological)` is also a **Analytic** (**Tautological**) since if `T-Scheme is Analytic (Tautological)` is `True` (which is presupposed by the **Material Conditional**), then `S ∈ C → T-Scheme is Analytic (Tautological)` must always be `True` as well (which is the relevant definition of a **Tautology** here - e.g. **Necessary Truth**).

   * If **Restricted T-Scheme** isn't **Analytic** (**Tautological**), then **T-Scheme** isn't (by **Modus Tollens**). But, then **T-Scheme** would be **Restricted** in some form (or just wrong) undermining the alternatives.

     * If **T-Scheme** is **Restricted**, it collapses into **KFG**.

     * If not, then we have no reason to defend **T-Scheme** in the first place.

2. The **Argument from Overgeneration**:

   * As a corollary to the above, any supposed solution (theory) `TH` that were to entail, be committed to, or satisfy **T-Scheme** being **Analytic** would entail **KFG**.

   * Thus, for any such solution `TH` that were to solve **Alethic Paradox**, **KFG** would already be present (or coincident) with `TH`. (Note: this inferential relationship is not symmetric: **KFG** *does not* entail `TH`.)

   * Since, **KFG** solves **Alethic Paradox** independently (for the reasons given above and below), `TH` either solves **Alethic Paradox** *twice* or *solely* due to **KFG** being coincident with it.

   * But if `TH` solves **Alethic Paradox** *solely* because **KFG** does, it collapses into **KFG**.

   * Buf if `TH` solves **Alethic Paradox** *twice* where **KFG** is entailed by `TH` and **KFG** solves **Alethic Paradox** *alone*, `TH` is unnecessary (overgenerates, by parsimony).

   * Therefore, either `TH` collapses into **KFG** or `TH` is unnecessary (overgenerates, by parsimony) since **KFG** solves **Alethic Paradox** independently. 

3. The **Argument from History**:

   * Every supposedly Universal (scientific) Law, Scientific Theory, and Mathematical Axiom has been proven False (Globally or Locally - the so-called *Pessimistic Meta-Induction*). Examples: General Relativity only applies at the "macro-level", Hyperbolic Geometry which rejects Axioms of Euclidean Geometry, etc.

   * Truth is a scientific and natural language phenomena (Linguistics is the science of language).

   * Therefore, we have no good reason to think that Truth Predication wouldn't also be similarly **Restricted** to a subdomain of naturally occurring phenomena. (E.g. it fails for **Truth Opacity** but not for **Truth Eliminability**.)

4. It's the only theory that explains all the diverging views on the Truth Predicate and **Liar Sentence** (it accommodates each other approach within the consistent models described above - e.g. the **Catuṣkoṭi**). In that way it's the only theory that aligns with the empirical data! (The countless attempts and approaches to solve the **Liar Paradox** - why there are many, diverging Truth Values, why we can even talk about different Truth Assignments for the **Liar Sentence**!)

   * Indeed, **KFG** provides a technical solution *and* deeper explanation for many touted and solely philosophical solutions (including but not limited to: **Infinite Propositional Depth**, **Infinite Recursion**, **Impredicativity**, **Infinite Semantic Graphs**, etc.).

5. Similar quirks show up in JavaScript and other programming languages.

   ```javascript

   // JavaScript

   [] == ![]; // -> true

   true == ![]; // -> false

   false == ![]; // -> true

   ```

6. It provides both a philosophical *and* technical solution (formal proof of its correctness).

7. It gets all the phenomena and is **Consistent** (by mathematical induction).

8. On **Classicality** itself:

   * Note: **KFG** *does not* commit one to the "truth"/"correctness" of **Classical Logic** (merely that **Classical Logic** is **Logically Consistent** and therefore should not be hastily abandoned for **Non-Classical Logic** upon consideration of the **Liar Paradox**).

   * In other words, the **Logical Realism** debate (e.g. - "Which Logic (if any) or Logics are the ultimate descriptin of reality or *The Correct Logic*") is a *separate* concern (although **KFG** might be of interest in that debate too).

   * The proofs given above hold in **Kleene 3-Value Algebras** (so the approach doesn't rely on **Classical Logic** or *Beg the Question* w.r.t. the correctness of the Metalogic at hand - they don't require **Classical Logic** to hold in the Metalogic).



## Resources and Links



> *Non-Exhaustive (but sufficient for what's described in the contents of this README) - please see the Paper for a complete Bibliography*.



1. Bacon, A. Can the Classical Logician Avoid the Revenge Paradoxes? Philosophical Review. 124 pp. 299-352 (2015, 7)

1. Beall, J. A Neglected Deflationist Approach to the Liar. Analysis. 61, 126-129 (2001)

1. Beall, J. Prolegomenon to Future Revenge. Revenge Of The Liar: New Essays On The Paradox. pp. 1-30 (2007)

1. Feferman, S. Axioms for Determinateness and Truth. Review Of Symbolic Logic. 1, 204-217 (2008)

1. Kripke, S. Outline of a Theory of Truth. Journal Of Philosophy. 72, 690-716 (1975)

1. Priest, G. Doubt Truth to Be a Liar. (Oxford University Press, 2006)

1. Priest, G. Logic of Paradox. Journal Of Philosophical Logic. 8, 219-241 (1979)

1. Tarski, A. The Semantic Conception of Truth and the Foundations of Semantics. Philosophy And Phenomenological Research. 4, 341-376 (1943)

1. https://github.com/denysdovhan/wtfjs?tab=readme-ov-file#true-is-not-equal--but-not-equal--too

1. https://logic.pku.edu.cn/ann_attachments/the%20outline%20of%20a%20new%20solution%20to%20the%20liar%20paradox134720412881.pdf

1. https://www.cs.ox.ac.uk/people/bob.coecke/Vincent.pdf

1. https://www.brunogavranovic.com/assets/FundamentalComponentsOfDeepLearning.pdf