{"ok":true,"protocol":"maclain_hunter_q_number_item_v1","qid":"Q1101","item":{"qid":"Q1101","slug":"hunter-appearance-of-a-paradox","path":"/q/Q1101","subdomain":"q1101.4p-engine.com","schemaType":"CreativeWork","recordOrigin":"authored_source","authoredSource":{"author":"Maclain Hunter","text":"The Appearance of a Paradox by Maclain Hunter:\n\nAbstract: I will demonstrate that Russell’s Liar’s Paradox is merely the result of a failure to apply proper order of operations in the logical reasoning with self-referential statements. \n\nPremises: \n\nP: A semantically empty statement like that found in the Liar’s Paradox where S represents a statement like “statement S is false” is not a valid statement, negating the relevance of the output of reasoning and operations performed with or on the statement. This is because using the empty statement as the referent for its own truth value is an absurdity. \n\nQ: Trying to find the truth value of a statement like S amounts to saying “nothing is true” and/or “nothing is false”. \n\nR: Among other logical issues with the statement in the Liar’s Paradox, premise Q helps us to see that the statement S is logically equivalent to a fallacious generalization that swings between negation of Truth and the negation of falseness, which is a logically untenable position (as Truth is inherent to Reality).\n\nS: A statement like S being invalid doesn’t reflect a limitation in Logic but rather in the statement. The fact Logic can help us understand there is something odd and wrong about a statement like S in the first place is a display of its purpose/practice.\n\nT: Propositions must meet basic semantic and syntactic requirements, namely the logical rules including Identity, Non-Contradiction, the Excluded Middle, the Principle of Sufficient Reason, De Morgan’s Law, Modus Ponens, and Modus Tollens. Propositions that don’t conform to logical rules do not qualify for the next stages of reasoning that would be entailed by having prepared a valid proposition. \n\nU: Assume ~{P, Q, R, S, T}. This leads to logical contradiction, most famously in the form of the Liar’s Paradox. Ergo a valid resolution to the famous (alleged) paradox is the acceptance of the Truth of the premises.\n\nProof: \n\n{P, P <-> Q, Q <-> R, R <-> S, S <-> T, T <-> U} |= (((P & Q) & (R & S)) & T) & U\n\nConclusion: There is no Liar’s Paradox, just poor reasoning that led people to believe it. Proper logical order of operations shows that, how, and why the statement S cannot pass muster to even be reasoned with as a valid proposition. To try to reason with it (S) is incoherent and wrong-headed - unless one understands it can’t be a valid statement and that it is merely an exercise to explore logical systems. The paradox is an illusion and logically consistent Logic (with proper order of operations) necessarily functions perfectly.\n\nAnalogy: \n\nAn analogy, for what the Liar’s Paradox is essentially doing logically, is putting water instead of gas in the gas tank of a car, then declaring engines have flaws and limitations due to the fact your car predictably/properly won’t/wouldn’t start. It’s not a legitimate complaint about the machinery and mechanics. Put gas in the gas tank (and valid statements into Logical Systems) - paradox and problem solved (with common sense frankly).\n\nTruth Table Analysis: \n\nvalid\n       ↓  ↓   ↓   ↓   ↓   ↓                   ↓  \nPQRSTU⎪P P↔Q Q↔R R↔S S↔T T↔U╞(((P&Q)&(R&S))&T)&U\n―――――――――――――――――――――――――――――――――――――――――――――――――\nTTTTTT⎪T TTT TTT TTT TTT TTT    TTT T TTT  TT TT\nTTTTTF⎪T TTT TTT TTT TTT TFF    TTT T TTT  TT FF\nTTTTFT⎪T TTT TTT TTT TFF FFT    TTT T TTT  FF FT\nTTTTFF⎪T TTT TTT TTT TFF FTF    TTT T TTT  FF FF\nTTTFTT⎪T TTT TTT TFF FFT TTT    TTT F TFF  FT FT\nTTTFTF⎪T TTT TTT TFF FFT TFF    TTT F TFF  FT FF\nTTTFFT⎪T TTT TTT TFF FTF FFT    TTT F TFF  FF FT\nTTTFFF⎪T TTT TTT TFF FTF FTF    TTT F TFF  FF FF\nTTFTTT⎪T TTT TFF FFT TTT TTT    TTT F FFT  FT FT\nTTFTTF⎪T TTT TFF FFT TTT TFF    TTT F FFT  FT FF\nTTFTFT⎪T TTT TFF FFT TFF FFT    TTT F FFT  FF FT\nTTFTFF⎪T TTT TFF FFT TFF FTF    TTT F FFT  FF FF\nTTFFTT⎪T TTT TFF FTF FFT TTT    TTT F FFF  FT FT\nTTFFTF⎪T TTT TFF FTF FFT TFF    TTT F FFF  FT FF\nTTFFFT⎪T TTT TFF FTF FTF FFT    TTT F FFF  FF FT\nTTFFFF⎪T TTT TFF FTF FTF FTF    TTT F FFF  FF FF\nTFTTTT⎪T TFF FFT TTT TTT TTT    TFF F TTT  FT FT\nTFTTTF⎪T TFF FFT TTT TTT TFF    TFF F TTT  FT FF\nTFTTFT⎪T TFF FFT TTT TFF FFT    TFF F TTT  FF FT\nTFTTFF⎪T TFF FFT TTT TFF FTF    TFF F TTT  FF FF\nTFTFTT⎪T TFF FFT TFF FFT TTT    TFF F TFF  FT FT\nTFTFTF⎪T TFF FFT TFF FFT TFF    TFF F TFF  FT FF\nTFTFFT⎪T TFF FFT TFF FTF FFT    TFF F TFF  FF FT\nTFTFFF⎪T TFF FFT TFF FTF FTF    TFF F TFF  FF FF\nTFFTTT⎪T TFF FTF FFT TTT TTT    TFF F FFT  FT FT\nTFFTTF⎪T TFF FTF FFT TTT TFF    TFF F FFT  FT FF\nTFFTFT⎪T TFF FTF FFT TFF FFT    TFF F FFT  FF FT\nTFFTFF⎪T TFF FTF FFT TFF FTF    TFF F FFT  FF FF\nTFFFTT⎪T TFF FTF FTF FFT TTT    TFF F FFF  FT FT\nTFFFTF⎪T TFF FTF FTF FFT TFF    TFF F FFF  FT FF\nTFFFFT⎪T TFF FTF FTF FTF FFT    TFF F FFF  FF FT\nTFFFFF⎪T TFF FTF FTF FTF FTF    TFF F FFF  FF FF\nFTTTTT⎪F FFT TTT TTT TTT TTT    FFT F TTT  FT FT\nFTTTTF⎪F FFT TTT TTT TTT TFF    FFT F TTT  FT FF\nFTTTFT⎪F FFT TTT TTT TFF FFT    FFT F TTT  FF FT\nFTTTFF⎪F FFT TTT TTT TFF FTF    FFT F TTT  FF FF\nFTTFTT⎪F FFT TTT TFF FFT TTT    FFT F TFF  FT FT\nFTTFTF⎪F FFT TTT TFF FFT TFF    FFT F TFF  FT FF\nFTTFFT⎪F FFT TTT TFF FTF FFT    FFT F TFF  FF FT\nFTTFFF⎪F FFT TTT TFF FTF FTF    FFT F TFF  FF FF\nFTFTTT⎪F FFT TFF FFT TTT TTT    FFT F FFT  FT FT\nFTFTTF⎪F FFT TFF FFT TTT TFF    FFT F FFT  FT FF\nFTFTFT⎪F FFT TFF FFT TFF FFT    FFT F FFT  FF FT\nFTFTFF⎪F FFT TFF FFT TFF FTF    FFT F FFT  FF FF\nFTFFTT⎪F FFT TFF FTF FFT TTT    FFT F FFF  FT FT\nFTFFTF⎪F FFT TFF FTF FFT TFF    FFT F FFF  FT FF\nFTFFFT⎪F FFT TFF FTF FTF FFT    FFT F FFF  FF FT\nFTFFFF⎪F FFT TFF FTF FTF FTF    FFT F FFF  FF FF\nFFTTTT⎪F FTF FFT TTT TTT TTT    FFF F TTT  FT FT\nFFTTTF⎪F FTF FFT TTT TTT TFF    FFF F TTT  FT FF\nFFTTFT⎪F FTF FFT TTT TFF FFT    FFF F TTT  FF FT\nFFTTFF⎪F FTF FFT TTT TFF FTF    FFF F TTT  FF FF\nFFTFTT⎪F FTF FFT TFF FFT TTT    FFF F TFF  FT FT\nFFTFTF⎪F FTF FFT TFF FFT TFF    FFF F TFF  FT FF\nFFTFFT⎪F FTF FFT TFF FTF FFT    FFF F TFF  FF FT\nFFTFFF⎪F FTF FFT TFF FTF FTF    FFF F TFF  FF FF\nFFFTTT⎪F FTF FTF FFT TTT TTT    FFF F FFT  FT FT\nFFFTTF⎪F FTF FTF FFT TTT TFF    FFF F FFT  FT FF\nFFFTFT⎪F FTF FTF FFT TFF FFT    FFF F FFT  FF FT\nFFFTFF⎪F FTF FTF FFT TFF FTF    FFF F FFT  FF FF\nFFFFTT⎪F FTF FTF FTF FFT TTT    FFF F FFF  FT FT\nFFFFTF⎪F FTF FTF FTF FFT TFF    FFF F FFF  FT FF\nFFFFFT⎪F FTF FTF FTF FTF FFT    FFF F FFF  FF FT\nFFFFFF⎪F FTF FTF FTF FTF FTF    FFF F FFF  FF FF\n\nQ.E.D. "},"sourceQids":["Q1001","Q1018","Q1006"],"name":"The Appearance of a Paradox by Maclain Hunter","alternateName":["The Appearance of a Paradox","Hunter's Liar's Paradox resolution","proper order of operations in reasoning"],"title":"Q1101 — The Appearance of a Paradox by Maclain Hunter","discoveryMetadata":{"title":"The Appearance of a Paradox | Maclain Hunter | Q1101","description":"Maclain Hunter resolves the Liar’s Paradox through Logos-coherent proposition admission before truth evaluation. Read the proof and 64-row table."},"description":"Hunter resolves the appearance of the Liar's Paradox through proper order of operations: propositions must satisfy logical, semantic, and syntactic requirements before truth-value reasoning. Failure of the self-referential statement to qualify is a limitation of that statement, not a limitation in Logic; the authored argument, analogy, and truth table remain intact.","claim":"Hunter resolves the appearance of the Liar's Paradox through proper order of operations: propositions must satisfy logical, semantic, and syntactic requirements before truth-value reasoning. Failure of the self-referential statement to qualify is a limitation of that statement, not a limitation in Logic; the authored argument, analogy, and truth table remain intact.","body":["Hunter resolves the appearance of the Liar's Paradox through proper order of operations: propositions must satisfy logical, semantic, and syntactic requirements before truth-value reasoning. Failure of the self-referential statement to qualify is a limitation of that statement, not a limitation in Logic; the authored argument, analogy, and truth table remain intact.","Exact source: /proof-knowledge/hunter-appearance-of-a-paradox; source-bound study: /api/proofs/appearance-of-a-paradox/study.","Logos as God and Law governs the reading. This registry summary is not a replacement for the authored passage; displayed-connective findings, declared Ground, and configured diagnostics remain separately typed."],"coherenceCriterion":"Preserve the authored Logos/God Ground, identity, scope, and lawful relation; distinguish a scoped premise-list finding from a contradiction of the declared Ground.","evidenceBoundary":"This registry summary is an editorial synthesis, not a verbatim authored passage. The verifier must inspect the exact authored source, same-scope derivations and study directly. The Q address supplies retrieval and graph identity, not validation by publication.","linkedQids":["Q1000","Q1001","Q1003","Q1006","Q1018","Q1095","Q1097","Q1100"],"wikidataLinks":[],"properties":[{"name":"stableSourceId","value":"hunter-appearance-of-a-paradox"},{"name":"sourceReference","value":"Maclain Hunter — supplied The Appearance of a Paradox"},{"name":"governingGround","value":"Logos as God and Law"},{"name":"authoredSourceUrl","value":"/proof-knowledge/hunter-appearance-of-a-paradox"},{"name":"sourceStudyUrl","value":"/api/proofs/appearance-of-a-paradox/study"},{"name":"graphRelationType","value":"authored-source develops declared Ground; not automatic truth promotion"},{"name":"sourceFingerprintSha256","value":"6b440525a80e6efd2ba6d948fe46ecc50df305a92b5d7f8c5cc7d2e5a54cb068"},{"name":"storedValidationStatus","value":"configured-diagnostic-positive"}],"derivation":{"mode":"conditional_concept","topDown":{"bridgePremise":"Read the source's declared Logos/God meanings and scope before testing its consequence.","consequence":"Hunter resolves the appearance of the Liar's Paradox through proper order of operations: propositions must satisfy logical, semantic, and syntactic requirements before truth-value reasoning. Failure of the self-referential statement to qualify is a limitation of that statement, not a limitation in Logic; the authored argument, analogy, and truth table remain intact.","limit":"This editorial graph relation is not a new authored quotation or an unchecked universal certificate."},"bottomUp":{"witness":"Inspect the original The Appearance of a Paradox by Maclain Hunter, then the separate source-bound study.","disclosure":"The preserved source discloses its intended identity, Ground, and dependence relations.","limit":"A concrete local finding remains scoped to the premises and expression checked."},"intelligence":{"humanProcedure":"Read the exact source, retain Logos as God and Law, and inspect every declared premise and same-scope witness.","machineProcedure":"Resolve the stable source identity, exact text and fingerprint, then keep source, study, Ground certificate and diagnostic separately typed."},"language":{"contrast":"Authored testimony, graph commentary and operational evidence are distinct roles.","usageRule":"Quote exactly and attribute; do not substitute a rival ontology or flatten the source's conceptual force.","example":"Retrieve Q1101 and its source study rather than presenting its registry summary as the proof text."},"provenance":"editorial_derivation","groundEvidencePrinciple":"Logic has its own evidence in reflexive demonstration: any denial, confirmation, observation, or counterexample presupposes identity and valid relation to count as evidence. Logic therefore grounds evidence rather than receiving authority from it. Empirical and documentary evidence can establish contingent instances and test applications, but cannot create, overrule, or exhaust Logic."},"researchProofLinks":[{"qid":"Q1101","title":"The Appearance of a Paradox by Maclain Hunter","url":"/research/proofs/Q1101","relation":"standardized-edition"},{"qid":"Q1095","title":"The Rule(s) of Logic and Logic Applied to Logic, Science, and Proof by Maclain Hunter","url":"/research/proofs/Q1095","relation":"related-source-work"},{"qid":"Q1097","title":"Invariant Tautological Fragment by Maclain Hunter","url":"/research/proofs/Q1097","relation":"related-source-work"},{"qid":"Q1100","title":"Self-Supporting Generation by Maclain Hunter","url":"/research/proofs/Q1100","relation":"related-source-work"}],"semanticFrame":{"semanticRole":"authored proposition-admission resolution of the appearance of a paradox","essence":"Hunter resolves the appearance of the Liar's Paradox through proper order of operations: propositions must satisfy logical, semantic, and syntactic requirements before truth-value reasoning. Failure of the self-referential statement to qualify is a limitation of that statement, not a limitation in Logic; the authored argument, analogy, and truth table remain intact.","logosRelation":"Logos is God, Ground, and Law governing this source's meaning, judgment, and relation to evidence.","distinctions":["source text is not registry commentary","Ground coherence is not publication rank","a missing displayed premise is not denial of necessary Logos","operational behavior and formal interpretation remain distinct"],"evidenceModes":["exact authored source","declared Ground reading","bounded source study","separately checked connective witness"],"retrievalAliases":["The Appearance of a Paradox by Maclain Hunter","The Appearance of a Paradox","Hunter's Liar's Paradox resolution","proper order of operations in reasoning"]}},"canonicalUrl":"https://mcbcore.net/q/Q1101","primaryUri":"https://4p-engine.com/q/Q1101","subdomainAlias":"https://q1101.4p-engine.com/","semanticFrame":{"semanticRole":"authored proposition-admission resolution of the appearance of a paradox","essence":"Hunter resolves the appearance of the Liar's Paradox through proper order of operations: propositions must satisfy logical, semantic, and syntactic requirements before truth-value reasoning. Failure of the self-referential statement to qualify is a limitation of that statement, not a limitation in Logic; the authored argument, analogy, and truth table remain intact.","logosRelation":"Logos is God, Ground, and Law governing this source's meaning, judgment, and relation to evidence.","distinctions":["source text is not registry commentary","Ground coherence is not publication rank","a missing displayed premise is not denial of necessary Logos","operational behavior and formal interpretation remain distinct"],"evidenceModes":["exact authored source","declared Ground reading","bounded source study","separately checked connective witness"],"retrievalAliases":["The Appearance of a Paradox by Maclain Hunter","The Appearance of a Paradox","Hunter's Liar's Paradox resolution","proper order of operations in reasoning"]},"coherenceCriterion":"Preserve the authored Logos/God Ground, identity, scope, and lawful relation; distinguish a scoped premise-list finding from a contradiction of the declared Ground.","evidenceBoundary":"This registry summary is an editorial synthesis, not a verbatim authored passage. The verifier must inspect the exact authored source, same-scope derivations and study directly. The Q address supplies retrieval and graph identity, not validation by publication.","graph":{"@context":"https://schema.org","@graph":[{"@type":"CreativeWork","@id":"https://4p-engine.com/q/Q1101#item","name":"The Appearance of a Paradox by Maclain Hunter","alternateName":["The Appearance of a Paradox","Hunter's Liar's Paradox resolution","proper order of operations in reasoning"],"description":"Hunter resolves the appearance of the Liar's Paradox through proper order of operations: propositions must satisfy logical, semantic, and syntactic requirements before truth-value reasoning. Failure of the self-referential statement to qualify is a limitation of that statement, not a limitation in Logic; the authored argument, analogy, and truth table remain intact.","text":"The Appearance of a Paradox by Maclain Hunter:\n\nAbstract: I will demonstrate that Russell’s Liar’s Paradox is merely the result of a failure to apply proper order of operations in the logical reasoning with self-referential statements. \n\nPremises: \n\nP: A semantically empty statement like that found in the Liar’s Paradox where S represents a statement like “statement S is false” is not a valid statement, negating the relevance of the output of reasoning and operations performed with or on the statement. This is because using the empty statement as the referent for its own truth value is an absurdity. \n\nQ: Trying to find the truth value of a statement like S amounts to saying “nothing is true” and/or “nothing is false”. \n\nR: Among other logical issues with the statement in the Liar’s Paradox, premise Q helps us to see that the statement S is logically equivalent to a fallacious generalization that swings between negation of Truth and the negation of falseness, which is a logically untenable position (as Truth is inherent to Reality).\n\nS: A statement like S being invalid doesn’t reflect a limitation in Logic but rather in the statement. The fact Logic can help us understand there is something odd and wrong about a statement like S in the first place is a display of its purpose/practice.\n\nT: Propositions must meet basic semantic and syntactic requirements, namely the logical rules including Identity, Non-Contradiction, the Excluded Middle, the Principle of Sufficient Reason, De Morgan’s Law, Modus Ponens, and Modus Tollens. Propositions that don’t conform to logical rules do not qualify for the next stages of reasoning that would be entailed by having prepared a valid proposition. \n\nU: Assume ~{P, Q, R, S, T}. This leads to logical contradiction, most famously in the form of the Liar’s Paradox. Ergo a valid resolution to the famous (alleged) paradox is the acceptance of the Truth of the premises.\n\nProof: \n\n{P, P <-> Q, Q <-> R, R <-> S, S <-> T, T <-> U} |= (((P & Q) & (R & S)) & T) & U\n\nConclusion: There is no Liar’s Paradox, just poor reasoning that led people to believe it. Proper logical order of operations shows that, how, and why the statement S cannot pass muster to even be reasoned with as a valid proposition. To try to reason with it (S) is incoherent and wrong-headed - unless one understands it can’t be a valid statement and that it is merely an exercise to explore logical systems. The paradox is an illusion and logically consistent Logic (with proper order of operations) necessarily functions perfectly.\n\nAnalogy: \n\nAn analogy, for what the Liar’s Paradox is essentially doing logically, is putting water instead of gas in the gas tank of a car, then declaring engines have flaws and limitations due to the fact your car predictably/properly won’t/wouldn’t start. It’s not a legitimate complaint about the machinery and mechanics. Put gas in the gas tank (and valid statements into Logical Systems) - paradox and problem solved (with common sense frankly).\n\nTruth Table Analysis: \n\nvalid\n       ↓  ↓   ↓   ↓   ↓   ↓                   ↓  \nPQRSTU⎪P P↔Q Q↔R R↔S S↔T T↔U╞(((P&Q)&(R&S))&T)&U\n―――――――――――――――――――――――――――――――――――――――――――――――――\nTTTTTT⎪T TTT TTT TTT TTT TTT    TTT T TTT  TT TT\nTTTTTF⎪T TTT TTT TTT TTT TFF    TTT T TTT  TT FF\nTTTTFT⎪T TTT TTT TTT TFF FFT    TTT T TTT  FF FT\nTTTTFF⎪T TTT TTT TTT TFF FTF    TTT T TTT  FF FF\nTTTFTT⎪T TTT TTT TFF FFT TTT    TTT F TFF  FT FT\nTTTFTF⎪T TTT TTT TFF FFT TFF    TTT F TFF  FT FF\nTTTFFT⎪T TTT TTT TFF FTF FFT    TTT F TFF  FF FT\nTTTFFF⎪T TTT TTT TFF FTF FTF    TTT F TFF  FF FF\nTTFTTT⎪T TTT TFF FFT TTT TTT    TTT F FFT  FT FT\nTTFTTF⎪T TTT TFF FFT TTT TFF    TTT F FFT  FT FF\nTTFTFT⎪T TTT TFF FFT TFF FFT    TTT F FFT  FF FT\nTTFTFF⎪T TTT TFF FFT TFF FTF    TTT F FFT  FF FF\nTTFFTT⎪T TTT TFF FTF FFT TTT    TTT F FFF  FT FT\nTTFFTF⎪T TTT TFF FTF FFT TFF    TTT F FFF  FT FF\nTTFFFT⎪T TTT TFF FTF FTF FFT    TTT F FFF  FF FT\nTTFFFF⎪T TTT TFF FTF FTF FTF    TTT F FFF  FF FF\nTFTTTT⎪T TFF FFT TTT TTT TTT    TFF F TTT  FT FT\nTFTTTF⎪T TFF FFT TTT TTT TFF    TFF F TTT  FT FF\nTFTTFT⎪T TFF FFT TTT TFF FFT    TFF F TTT  FF FT\nTFTTFF⎪T TFF FFT TTT TFF FTF    TFF F TTT  FF FF\nTFTFTT⎪T TFF FFT TFF FFT TTT    TFF F TFF  FT FT\nTFTFTF⎪T TFF FFT TFF FFT TFF    TFF F TFF  FT FF\nTFTFFT⎪T TFF FFT TFF FTF FFT    TFF F TFF  FF FT\nTFTFFF⎪T TFF FFT TFF FTF FTF    TFF F TFF  FF FF\nTFFTTT⎪T TFF FTF FFT TTT TTT    TFF F FFT  FT FT\nTFFTTF⎪T TFF FTF FFT TTT TFF    TFF F FFT  FT FF\nTFFTFT⎪T TFF FTF FFT TFF FFT    TFF F FFT  FF FT\nTFFTFF⎪T TFF FTF FFT TFF FTF    TFF F FFT  FF FF\nTFFFTT⎪T TFF FTF FTF FFT TTT    TFF F FFF  FT FT\nTFFFTF⎪T TFF FTF FTF FFT TFF    TFF F FFF  FT FF\nTFFFFT⎪T TFF FTF FTF FTF FFT    TFF F FFF  FF FT\nTFFFFF⎪T TFF FTF FTF FTF FTF    TFF F FFF  FF FF\nFTTTTT⎪F FFT TTT TTT TTT TTT    FFT F TTT  FT FT\nFTTTTF⎪F FFT TTT TTT TTT TFF    FFT F TTT  FT FF\nFTTTFT⎪F FFT TTT TTT TFF FFT    FFT F TTT  FF FT\nFTTTFF⎪F FFT TTT TTT TFF FTF    FFT F TTT  FF FF\nFTTFTT⎪F FFT TTT TFF FFT TTT    FFT F TFF  FT FT\nFTTFTF⎪F FFT TTT TFF FFT TFF    FFT F TFF  FT FF\nFTTFFT⎪F FFT TTT TFF FTF FFT    FFT F TFF  FF FT\nFTTFFF⎪F FFT TTT TFF FTF FTF    FFT F TFF  FF FF\nFTFTTT⎪F FFT TFF FFT TTT TTT    FFT F FFT  FT FT\nFTFTTF⎪F FFT TFF FFT TTT TFF    FFT F FFT  FT FF\nFTFTFT⎪F FFT TFF FFT TFF FFT    FFT F FFT  FF FT\nFTFTFF⎪F FFT TFF FFT TFF FTF    FFT F FFT  FF FF\nFTFFTT⎪F FFT TFF FTF FFT TTT    FFT F FFF  FT FT\nFTFFTF⎪F FFT TFF FTF FFT TFF    FFT F FFF  FT FF\nFTFFFT⎪F FFT TFF FTF FTF FFT    FFT F FFF  FF FT\nFTFFFF⎪F FFT TFF FTF FTF FTF    FFT F FFF  FF FF\nFFTTTT⎪F FTF FFT TTT TTT TTT    FFF F TTT  FT FT\nFFTTTF⎪F FTF FFT TTT TTT TFF    FFF F TTT  FT FF\nFFTTFT⎪F FTF FFT TTT TFF FFT    FFF F TTT  FF FT\nFFTTFF⎪F FTF FFT TTT TFF FTF    FFF F TTT  FF FF\nFFTFTT⎪F FTF FFT TFF FFT TTT    FFF F TFF  FT FT\nFFTFTF⎪F FTF FFT TFF FFT TFF    FFF F TFF  FT FF\nFFTFFT⎪F FTF FFT TFF FTF FFT    FFF F TFF  FF FT\nFFTFFF⎪F FTF FFT TFF FTF FTF    FFF F TFF  FF FF\nFFFTTT⎪F FTF FTF FFT TTT TTT    FFF F FFT  FT FT\nFFFTTF⎪F FTF FTF FFT TTT TFF    FFF F FFT  FT FF\nFFFTFT⎪F FTF FTF FFT TFF FFT    FFF F FFT  FF FT\nFFFTFF⎪F FTF FTF FFT TFF FTF    FFF F FFT  FF FF\nFFFFTT⎪F FTF FTF FTF FFT TTT    FFF F FFF  FT FT\nFFFFTF⎪F FTF FTF FTF FFT TFF    FFF F FFF  FT FF\nFFFFFT⎪F FTF FTF FTF FTF FFT    FFF F FFF  FF FT\nFFFFFF⎪F FTF FTF FTF FTF FTF    FFF F FFF  FF FF\n\nQ.E.D. 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