{"id":21581539,"url":"https://github.com/thoughtscript/truth-eliminability-sorting-algorithm","last_synced_at":"2025-03-18T08:14:35.984Z","repository":{"id":252552104,"uuid":"840763694","full_name":"Thoughtscript/truth-eliminability-sorting-algorithm","owner":"Thoughtscript","description":"Truth Sorting Algo for T-Scheme","archived":false,"fork":false,"pushed_at":"2024-12-29T22:35:04.000Z","size":37,"stargazers_count":1,"open_issues_count":1,"forks_count":0,"subscribers_count":1,"default_branch":"main","last_synced_at":"2025-01-24T14:44:58.596Z","etag":null,"topics":["alethic-paradox","liar-paradox","logic"],"latest_commit_sha":null,"homepage":"https://thoughtscript.github.io/truth-eliminability-sorting-algorithm/","language":"HTML","has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":null,"status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/Thoughtscript.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":null,"code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null}},"created_at":"2024-08-10T16:00:04.000Z","updated_at":"2024-12-29T22:35:07.000Z","dependencies_parsed_at":"2024-12-29T23:20:10.499Z","dependency_job_id":"3cdce5e8-a23e-4d02-8aa5-a52f6914f26d","html_url":"https://github.com/Thoughtscript/truth-eliminability-sorting-algorithm","commit_stats":null,"previous_names":["thoughtscript/truth-eliminability-sorting-algorithm"],"tags_count":0,"template":false,"template_full_name":null,"repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/Thoughtscript%2Ftruth-eliminability-sorting-algorithm","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/Thoughtscript%2Ftruth-eliminability-sorting-algorithm/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/Thoughtscript%2Ftruth-eliminability-sorting-algorithm/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/Thoughtscript%2Ftruth-eliminability-sorting-algorithm/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/Thoughtscript","download_url":"https://codeload.github.com/Thoughtscript/truth-eliminability-sorting-algorithm/tar.gz/refs/heads/main","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":244181391,"owners_count":20411605,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":["alethic-paradox","liar-paradox","logic"],"created_at":"2024-11-24T14:12:51.717Z","updated_at":"2025-03-18T08:14:35.975Z","avatar_url":"https://github.com/Thoughtscript.png","language":"HTML","funding_links":[],"categories":[],"sub_categories":[],"readme":"# Truth-Eliminability-Sorting-Algorithm\r\n\r\n*WIP* (*Future Thesis and/or Dissertation.*)\r\n\r\nThe 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.\r\n\r\n\u003e Original draft hosted here: https://www.thoughtscript.io/papers/000000000002\r\n\r\nFor the more recent: ***Classical Extensions of Kripke-Feferman: Constraint Satisfaction and Alethic Paradox*** (summarizing key points)\r\n\r\n\u003e https://www.thoughtscript.io/papers/000000000013\r\n\r\n\u003e Mirrored: https://thoughtscript-io.onrender.com/papers/000000000013.html\r\n\r\n## Context\r\n\r\nThere's widespread consensus that no offered solution to the **Liar Paradox** gets all of the following:\r\n1. All the other **Alethic Paradoxes**: *Yablo-Visser*, *Liar Cycles*, *Revenge Sentences*, *Boolean Compounds*, *Curry Compounds*, *McGee Sentences*, etc.\r\n2. 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**.)\r\n3. 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.\r\n4. No solution offers both a \"philosophical explanation\" and formal proof of its correctness.\r\n5. Additionally, no formal solution is known to overcome all of the existing concerns: **Bacon 2015**, **Revenge Paradox**, etc.\r\n\r\n(Optionally) Adding to the above: \r\n\r\n6. Is **Classical** (preserves **Bivalence** and the other **Classical** validities).\r\n\r\n## Definitions\r\n\r\nFormal definitions.\r\n\r\n\u003e Consult **Section 3.2** for an overview of **Truth as a Metalinguistic Predicate**.\r\n\r\n### Sentences Names, Name-Forming Operators, and Diagonalization\r\n\r\nHere and below I’ll use the convention `⟨`,`⟩` to denote the familiar **Gödel Numbering** technique:\r\n\r\n1. **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`.\r\n2. **Name-Forming Operator** - `⟨S⟩` represents the mapping of some **Sentence**, **Proposition**, or **Expression** `S` to its **Name**. `⟨S⟩ ≡ P := S` returning `P`.\r\n3. **Diagonalization** - a **Technique** that associates the **Fixed Point** of a **Sentence** containing `S` as a sub-expression so that `S` is its own name.\r\n4. (Below, if `S` is the name of a sentence containing `S` as sub-**Expression**, both `⟨S⟩` and `S` will be used interchangeably as **Names**.)\r\n\r\n\u003e 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.\r\n\r\n### T-Scheme\r\n\r\n1. *Tarski’s 1933* **Definition of Truth**  - `For all x, True(x) if and only if ϕ(x)`.\r\n2. **Modern Formulation** - (For all `S`) `T(⟨S⟩) ↔ S`:\r\n   * **Capture** (or **T-Intro**) - conditional subrule of the **T-Scheme** biconditional. The rule going from `S` to `T(⟨S⟩)`.\r\n   * **Release** - conditional subrule of the **T-Scheme** biconditional. The rule going from `T(⟨S⟩)` to `S`.\r\n\r\n### Truth Tellers\r\n\r\n```\r\nS := T(⟨S⟩)\r\n```\r\n\r\n1. **Truth Teller** - like the **Liar Sentence** but expressing **Truth** of itself. Constructed via **Fixed-Point Diagonalization** like the **Liar Sentence**.\r\n\r\n### Alethic Paradox\r\n\r\n\u003e From [Dictionary.com](https://www.dictionary.com/browse/alethic): **Alethic** \"of or relating to such philosophical concepts as truth, necessity, possibility, contingency, etc\".\r\n1. **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\r\n**Truth Predicate**).\r\n\r\n### Properties of Truth and T-Schema\r\n\r\n1. **Truth Transparency** - the principle that `S` and `T(⟨S⟩)` are always and everywhere intersubstitutable.\r\n   * *(Introductory Wikipedia article on [Referential Transparency](https://en.wikipedia.org/wiki/Referential_transparency) in Philosophy and Computer Science.)*\r\n2. **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**.\r\n   * A stronger criterion on (or reading of) **Truth Transparency** (and **T-Scheme**).\r\n   * **Truth Transparency** requires that `T(⟨S⟩)` can be rewritten in a form where no `T` appears (where **Truth** is not predicated).\r\n3. **Truth Opacity** - when a **Sentence** `S` cannot be rewritten (via **Truth Transparency**) without a `T` appearing (where **Truth** is not predicated). Such a **Sentence** is **Truth Opaque**.\r\n\r\n## Truth Eliminability Algorithm and KFG\r\n\r\nA Finite, **Sorting**, Algorithm used to determine whether a **Sentence** is **Truth Opaque** or not.\r\n\r\nMay circumvent general concerns stemming from [**Bacon 2015**](https://andrew-bacon.github.io/papers/Indeterminacy%20and%20Revenge.pdf) (my original paper was never published):\r\n* Appears to be a new \"species\" of **Restrictionist** approaches that also doesn't require every **Sentence** or **Theorem** to be \"Cleaned\", \"Healthy\", or \"Debuggered\".\r\n  * E.g. - those approaches that follow the Classical Axiom, Theorem, and Tautology: **Weakening** `P → (Q → P)`.\r\n  * The **T-Scheme** is actually a **Material Conditional** with some **Constraint**, \"Checkpoint\", or condition that must be met / a \"Restriction\" on it.\r\n* Invalidates the move from **P1** to **P2** (by Substitution or Diagonalization).\r\n* Sorted expressions are nevertheless given **Truth-Values** and don't entail **Untruth** (or **Falsity**).\r\n* So, both **Truth Opaque** and non-**Truth Opaque Sentences** are allowed - they are not \"banned\" or \"outlawed\".\r\n* 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 at all for **KFG**.\r\n* **Truth Opaque Expressions** aren't necessarily **Theorems** nor are the assertions of them as such.\r\n* 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**.\r\n\r\nBlocks **McGee's T-Intro** step.\r\n\r\nSatisfies **Tarski's Undefinability Theorem** for **T-Scheme** since `S` and `T(⟨S⟩)` can be **Logically Consistent** yet diverge in Truth Values within **KFG**.\r\n\r\n## Some Validations and Truth Table Proofs\r\n\r\nSome simple **Truth Table** and basic Model checking summarized succinctly below.\r\n\r\n\u003e Depicts some Models of **KFG** and proves **Consistency** of **KFG** with respect to **Liar Cycles**.\r\n\r\n\u003e Demonstrates Classically Consistent Models (the primary goal) and ways to address the ancillary goals: **KFG** global validity and embedded **Catuṣkoṭi**.\r\n\r\n### Truth Assignments \r\n\r\nThe **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.)\r\n\r\n\u003e Below, `T(S) ↔ S` refers to the specific WFF with Sentential Constant `S` substituted into **T-Scheme**.\r\n\r\n### Truth Eliminable Sentences\r\n\r\nTruth Table Semantics:\r\n\r\n| `S` | `¬S` | `C(S)` (`S ∈ C`) | `¬C(S)` | `T(S)` | `¬T(S)` | `T(S) ↔ S` | `C(S) → (T(S) ↔ S)` | \r\n| --- | --- | --- | --- | --- | --- | --- | --- |\r\n| `⊥` | `⊤` | `⊤` | `⊥` | `⊤`* | `⊥`* | `⊥`* | `⊥`* |\r\n| `⊥` | `⊤` | `⊤` | `⊥` | `⊥` | `⊤` | `⊤` | `⊤` |\r\n| `⊤` | `⊥` | `⊤` | `⊥` | `⊤` | `⊥` | `⊤` | `⊤` |\r\n| `⊤` | `⊥` | `⊤` | `⊥` | `⊥`* | `⊤`* | `⊥`* | `⊥`* |\r\n\r\nThere are two ways to read this:\r\n\r\n1. *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**.\r\n2. 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 will the above unmodified construction (whilst remaining **(Logically) Consistent** nevertheless).\r\n\r\n### Truth Opaque Sentences\r\n\r\nTruth Table Semantics:\r\n\r\n| `S` | `¬S` | `C(S)` (`S ∈ C`) | `¬C(S)` | `T(S)` | `¬T(S)` | `T(S) ↔ S` | `C(S) → (T(S) ↔ S)` |\r\n| --- | --- | --- | --- | --- | --- | --- | --- |\r\n| `⊤` | `⊥` | `⊥` | `⊤` | `⊥` | `⊤` | `⊥` | `⊤` |\r\n| `⊤` | `⊥` | `⊥` | `⊤` | `⊤` | `⊥` | `⊤` | `⊤` |\r\n| `⊥` | `⊤` | `⊥` | `⊤` | `⊥` | `⊤` | `⊤` | `⊤` |\r\n| `⊥` | `⊤` | `⊥` | `⊤` | `⊤` | `⊥` | `⊥` | `⊤` |\r\n\r\nComments:\r\n\r\n1. **Truth Teller** (and **Liar Cycle Negator**) expressions are given the second or third **Interpretations** above.\r\n2. **Liar Sentences** expressions are given the first or fourth **Interpretations** above.\r\n\r\n### Liar Cycles\r\n\r\n\u003e `S := T(Q)`, `Q := ¬T(S)`\r\n\r\nHere:\r\n1. We only need to prove that *at least one* **Consistent Interpretation** exists.\r\n2. The following pairs must share Truth-Values:\r\n   * `S`, `T(Q)` \r\n   * `¬S`, `¬T(Q)`\r\n   * `Q`, `¬T(S)` \r\n   * `¬Q`, `T(S)`\r\n\r\n| `S` | `T(Q)` | `¬S` |  `¬T(Q)` | `Q` | `¬T(S)` |  `¬Q` | `T(S)` | **Consistent** |\r\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | \r\n| `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `NO` |\r\n| `⊥` | `⊥` | `⊥` | `⊥` | `⊥` | `⊥` | `⊥` | `⊥` | `NO` |\r\n| `⊤` | `⊤` | `⊥` | `⊥` | `⊤` | `⊤` | `⊤` | `⊤` | `NO` |\r\n| `⊤` | `⊤` | `⊤` | `⊤` | `⊥` | `⊥` | `⊤` | `⊤` | `NO` |\r\n| `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `⊥` | `⊥` | `NO` |\r\n| `⊥` | `⊥` | `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `⊤` | `NO` |\r\n| `...` | `...` | `...` | `...` | `...` | `...` | `...` | `...` | `NO` |\r\n| `⊥` | `⊥` | `⊤` | `⊤` | `⊤` | `⊤` | `⊥` | `⊥` | `YES` |\r\n| `⊤` | `⊤` |  `⊥` | `⊥` |  `⊥` | `⊥` | `⊤` | `⊤` | `YES` |\r\n\r\nNote:\r\n1. One of the paired **Sentences** can behave like the **Truth Teller**. (The **Liar Cycle Negator** of the pair.)\r\n2. We also require (through optional extensions) that **T-Scheme** fails if a **Sentence** refers to a another **Truth Opaque** Sentence.\r\n\r\n#### Liar Cycle Semantics and KFG\r\n\r\nRegarding the last two **Interpretations**:\r\n\r\n| `S` | `¬S` | `C(S)` (`S ∈ C`) | `¬C(S)` | `T(S)` | `¬T(S)` | `T(S) ↔ S` | `C(S) → (T(S) ↔ S)` |\r\n| --- | --- | --- | --- | --- | --- | --- | --- |\r\n| `⊥` | `⊤` | `⊥` | `⊤` | `⊥` | `⊤` | `⊤` | `⊤` |\r\n| `⊤` | `⊥` | `⊥` | `⊤` | `⊤` | `⊥` | `⊤` | `⊤` |\r\n\r\n| `Q` | `¬Q` | `C(Q)` (`Q ∈ C`) | `¬C(Q)` | `T(Q)` | `¬T(Q)` | `T(Q) ↔ Q` | `C(Q) → (T(Q) ↔ Q)` |\r\n| --- | --- | --- | --- | --- | --- | --- | --- |\r\n| `⊤` | `⊥` | `⊥` | `⊤` | `⊥` | `⊤` | `⊥` | `⊤` |\r\n| `⊥` | `⊤` | `⊥` | `⊤` | `⊤` | `⊥` | `⊥` | `⊤` |\r\n\r\nImmediately above:\r\n1. Each **Model** pairs the respective first and second **Interpretations**.\r\n2. We observe that `Q` (the **Liar Cycle Negator** of the pair) behaves like **Truth Tellers**.\r\n\r\n### Catuṣkoṭi\r\n\r\n\u003e Some interesting phenomena.\r\n\r\nWith **F-Scheme** (`¬T(S) ↔ F(S)`) unmodified:\r\n\r\n| `S` | `¬S` | `T(S)` | `¬T(S)` | **Comment** | `T(S) ↔ S` | `C(S) → (T(S) ↔ S)` |\r\n| --- | --- | --- | --- | --- | --- | --- |\r\n| `⊥` | `⊤` | `⊥` | `⊤` | `False` | `⊤` |  `⊤` |\r\n| `⊥` | `⊤` | `⊤` | `⊥` | `True and False` | `⊥` | Depends on `S` being **Truth Opaque** or not (per the above). |\r\n| `⊤` | `⊥` | `⊥` | `⊤` | `True and False` | `⊥` | Depends on `S` being **Truth Opaque** or not (per the above). |\r\n| `⊤` | `⊥` | `⊤` | `⊥` | `True` | `⊤` |  `⊤` |\r\n\r\nThe above mirrors **Kleene 3-Value** constructions. The assertion would be that:\r\n1. Confusion around **3-Value Semantics**;\r\n2. 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**).\r\n\r\nWith **F-Scheme** also **Weakened** (e.g. - `¬T(S) ↔ F(S)` will sometimes fail), the **Catuṣkoṭi** appears:\r\n\r\n| `S` | `¬S` | `T(S)` | `¬T(S)` | `F(S)` | `¬F(S)` | Comment |\r\n| --- | --- | --- | --- | --- | --- | --- |\r\n| `⊥` | `⊤` | `⊥` | `⊤` | `⊤` | `⊥` | `False` |\r\n| `⊥` | `⊤` | `⊥` | `⊤` | `⊥` | `⊤` | `Neither True nor False` |\r\n| `⊥` | `⊤` | `⊤` | `⊥` | `⊤` | `⊥` | `True and False` |\r\n| `⊥` | `⊤` | `⊤` | `⊥` | `⊥` | `⊤` | `True` |\r\n| `⊤` | `⊥` | `⊥` | `⊤` | `⊤` | `⊥` | `False` |\r\n| `⊤` | `⊥` | `⊥` | `⊤` | `⊥` | `⊤` | `Neither True nor False` |\r\n| `⊤` | `⊥` | `⊤` | `⊥` | `⊤` | `⊥` | `True and False` |\r\n| `⊤` | `⊥` | `⊤` | `⊥` | `⊥` | `⊤` | `True` |\r\n\r\n\u003e 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: `[] == ![]; // -\u003e true`, `true == ![]; // -\u003e false`, `false == ![]; // -\u003e true`.\r\n\r\n\u003e 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.\r\n\r\nPlease note that the above can all be **Consistently** captured within a **Two-Value**, **Bivalent**, **Classical** Semantics. We've relaxed the required on `T()` and `F()` per the above.\r\n\r\nThis is the only proposed system that can accomodate all the additional items below:\r\n1. Parallels intuitions that motivate **3-Value Semantics**.\r\n2. **The Catuṣkoṭi**.\r\n3. 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.\r\n4. Is **Classical**.\r\n5. Gets all the other Truth-related (**Alethic**) Paradoxes.\r\n6. Satifies **Tarski's Undefinability Theorem** for **T-Scheme**.\r\n7. Blocks McGee's **T-Intro** step.\r\n8. 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.\r\n\r\n## Extensions\r\n\r\nSo, **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.). \r\n\r\nBy selecting extensible constructions variants of **KFG** strengthen certain features discussed above:\r\n\r\n1. Harmonization of Truth Assignments (aligning Truth Values and Truth Predicates in consistent assingments).\r\n2. Preference for Predicates over Set Inclusion or vice-versa.\r\n3. Model selection.\r\n4. Analyticity of the Restricted T-Scheme.\r\n\r\nI think this is akin to subfields like the debate between `S4` vs `S5 Modal Logic`, the correct semantics for Modality, and so on.\r\n\r\n\u003e 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!\r\n\r\n## Key Philosophical Arguments\r\n\r\n1. The **Argument from Tautology**:\r\n   * If **T-Scheme** is **Analytic**, then so is **Restricted T-Scheme** (by Classical inferential **Weakening**)\r\n   * If **Restricted T-Scheme** isn't **Analytic**, then **T-Scheme** isn't (**Modus Tollens**). But, then **T-Scheme** would be **Restricted** in some form (or just wrong) undermining the alternatives.\r\n     * If **T-Scheme** is **Restricted**, it collapses into **KFG**.\r\n     * If not, then we have no reason to defend **T-Scheme** in the first place.\r\n2. The **Argument from History**:\r\n   * Every supposedly Universal (scientific) Law, Scientific Theory, and Mathematical Axiom has been proven False (Globally or Locally - the so-called *Pessimistic Meta-Induction*). Examples: Relativity only applies at the \"macro-level\", Hyperbolic Geometry which rejects Axioms of Euclidean Geometry, etc.\r\n   * Truth is a scientific and natural language phenomena (Linguistics is the science of language).\r\n   * 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**.)\r\n3. It's the only theory that explains all the diverging views on the Truth Predicate and **Liar Sentence** (it accommodate 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**!)\r\n4. Similar quirks show up in JavaScript and other programming languages.\r\n   ```javascript\r\n   // JavaScript\r\n   [] == ![]; // -\u003e true\r\n   true == ![]; // -\u003e false\r\n   false == ![]; // -\u003e true\r\n   ```\r\n5. It provides a philosophical and technical solution (formal proof of correctness).\r\n6. It gets all the phenomena and is **Consistent**.\r\n\r\n## Resources and Links\r\n\r\n\u003e *Non-Exhaustive (but sufficient for what's described in the contents of this README) - please see the Paper for a complete Bibliography*.\r\n\r\n1. Bacon, A. Can the Classical Logician Avoid the Revenge Paradoxes? Philosophical Review. 124 pp. 299-352 (2015, 7)\r\n1. Beall, J. A Neglected Deflationist Approach to the Liar. Analysis. 61, 126-129 (2001)\r\n1. Beall, J. Prolegomenon to Future Revenge. Revenge Of The Liar: New Essays On The Paradox. pp. 1-30 (2007)\r\n1. Feferman, S. Axioms for Determinateness and Truth. Review Of Symbolic Logic. 1, 204-217 (2008)\r\n1. Kripke, S. Outline of a Theory of Truth. Journal Of Philosophy. 72, 690-716 (1975)\r\n1. Priest, G. Doubt Truth to Be a Liar. (Oxford University Press, 2006)\r\n1. Priest, G. Logic of Paradox. Journal Of Philosophical Logic. 8, 219-241 (1979)\r\n1. Tarski, A. The Semantic Conception of Truth and the Foundations of Semantics. Philosophy And Phenomenological Research. 4, 341-376 (1943)\r\n1. https://github.com/denysdovhan/wtfjs?tab=readme-ov-file#true-is-not-equal--but-not-equal--too\r\n1. https://logic.pku.edu.cn/ann_attachments/the%20outline%20of%20a%20new%20solution%20to%20the%20liar%20paradox134720412881.pdf\r\n\r\n\r\n\r\n\r\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fthoughtscript%2Ftruth-eliminability-sorting-algorithm","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fthoughtscript%2Ftruth-eliminability-sorting-algorithm","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fthoughtscript%2Ftruth-eliminability-sorting-algorithm/lists"}