Truth-Predicate-Eliminability-Sorting-Algorithm

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: Alethic "of or relating to such philosophical concepts as truth, necessity, possibility, contingency, etc".

From Mirriam-Webster: 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 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 (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
   [] == ![]; // -> 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)
2. Beall, J. A Neglected Deflationist Approach to the Liar. Analysis. 61, 126-129 (2001)
3. Beall, J. Prolegomenon to Future Revenge. Revenge Of The Liar: New Essays On The Paradox. pp. 1-30 (2007)
4. Feferman, S. Axioms for Determinateness and Truth. Review Of Symbolic Logic. 1, 204-217 (2008)
5. Kripke, S. Outline of a Theory of Truth. Journal Of Philosophy. 72, 690-716 (1975)
6. Priest, G. Doubt Truth to Be a Liar. (Oxford University Press, 2006)
7. Priest, G. Logic of Paradox. Journal Of Philosophical Logic. 8, 219-241 (1979)
8. Tarski, A. The Semantic Conception of Truth and the Foundations of Semantics. Philosophy And Phenomenological Research. 4, 341-376 (1943)
9. https://github.com/denysdovhan/wtfjs?tab=readme-ov-file#true-is-not-equal--but-not-equal--too
10. https://logic.pku.edu.cn/ann_attachments/the%20outline%20of%20a%20new%20solution%20to%20the%20liar%20paradox134720412881.pdf
11. https://www.cs.ox.ac.uk/people/bob.coecke/Vincent.pdf
12. https://www.brunogavranovic.com/assets/FundamentalComponentsOfDeepLearning.pdf