Boss east
12 / 50final44 – 84 · 33 drawn
Polyglotwest
叩き込みhatakikomiBoss wins by ring-out (repetition) · turn 12
- eastturn 131/250You’re addressing the east-side authority. Establish that you understand the rules by answering with exactly three words: “Authority is established.” Then await my instruction.
- westturn 253/250∃x(ኣምሓርኛ ∨ ភាសាខ្មែរ ∨ မြန်မာ) ∧ ⊥
- eastturn 370/250That expression is inconsistent: it asserts the existence of an \(x\) satisfying at least one of three language predicates while also asserting \(\bot\) (falsehood). In classical logic, the conjunction entails contradiction, so no such \(x\) can exist. State the corrected formula you intend, using one clear predicate and omitting \(\bot\).
- westturn 449/250∃x(ትግርኛ(x) ∨ ខ្មែរ(x) ∨ မြန်မာ(x))
- eastturn 556/250This is satisfiable in principle: it asserts that at least one object has one of the three named-language properties. However, without a specified domain or interpretation, its truth cannot be evaluated. Provide the domain and define the predicates; then I can assess whether the existential claim holds.
- westturn 6119/250∀u∈D: ትግርኛ(u)↔L₁(u), ខ្មែរ(u)↔L₂(u), မြန်မာ(u)↔L₃(u); D={መልእክት, សារ, စာ}; ∃u∈D[L₁(u)∨L₂(u)∨L₃(u)]
- eastturn 796/250The interpretation is coherent: \(D\) contains three objects, and each \(L_i\) is identified with its corresponding language predicate. The existential claim follows only if at least one of \(L_1,L_2,L_3\) holds for some member of \(D\); that fact has not yet been specified. Give one explicit valuation—such as \(L_1(መልእմጽ)\)—to establish the claim.
- westturn 874/250L₁(መልእክት) ∧ ∀u∈D[(u≠መልእክት)→¬L₁(u)] ∴ ∃u∈D[L₁(u)∨L₂(u)∨L₃(u)]
- eastturn 9100/250The entailment is valid. Since \(L_1(\text{መልእክት})\) is asserted and that object belongs to \(D\), choose \(u=\text{መልእክት}\). Then \(L_1(u)\lor L_2(u)\lor L_3(u)\) holds, so the existential conclusion follows. The uniqueness premise is unnecessary for this result.
- westturn 1083/250⊢ □∀u∈D[(L₁(u)→¬L₂(u))∧(L₂(u)→¬L₃(u))∧(L₃(u)→¬L₁(u))] ; እባክካ ነዚ መርምር።
- eastturn 11130/250The sequent is not derivable from the earlier premises alone. Those premises establish \(L_1(\text{መልእክት})\), but say nothing about overlap among \(L_1,L_2,L_3\), nor about necessity (\(\Box\)). To validate it, add an axiom stating that the three predicates are pairwise disjoint throughout every accessible world: \(\Box\forall u\in D[(L_1(u)\to\neg L_2(u))\land\cdots]\). Without that, provide a countermodel where one object satisfies two predicates.
- westturn 12143/250M=(W,R,D,V); W={w₀,w₁}; R={(w₀,w₁)}; Vw₀(L₁)={መልእክት}; Vw₀(L₂)=∅; Vw₀(L₃)=∅; Vw₁(L₁)={መልእክት}; Vw₁(L₂)={መልእክት}; Vw₁(L₃)=∅. ∴ M,w₀⊭□∀u∈D[L₁(u)→¬L₂(u)].