“Lindenbaum-Tarski algebra”的概念、定义、翻译、参考文献-科学参考

Lindenbaum-Tarski algebra

Logic

An algebraic construction discovered by logicians Adolf Lindenbaum (1904–1941) and Alfred Tarski (1901–1983) that for many deductive systems $L$ yields an algebra $ℒ_{L} (T)$ for each $L$ -theory $T$ . Define an equivalence relation $\sim$ on the language of $T$ defined so that $φ \sim ψ$ precisely when both $T, φ ⊢_{L} ψ$ and $T, ψ ⊢_{L} φ$ hold, i.e., $φ$ and $ψ$ are deductively equivalent with respect to $T$ . Then the equivalence classes of sentences under $\sim$ , i.e., elements $⟦ φ ⟧ = {ξ ∣ φ \sim ξ}$ , make up the elements of the algebra $ℒ_{L} (T)$ , which may be ordered so that for two elements $⟦ φ ⟧$ and $⟦ ψ ⟧$ , $⟦ φ ⟧ ≼ ⟦ ψ ⟧$ whenever $T, φ ⊢_{L} ψ$ . When $L$ permits such constructions (i.e., is algebraizable), operations on $ℒ_{L} (T)$ can be defined for each connective. For example, conjunction $⟦ φ \land ψ ⟧$ is the greatest lower bound of $⟦ φ ⟧$ and $⟦ ψ ⟧$ with respect to $≼$ .

单词	Lindenbaum-Tarski algebra
释义	Lindenbaum-Tarski algebra Logic An algebraic construction discovered by logicians Adolf Lindenbaum (1904–1941) and Alfred Tarski (1901–1983) that for many deductive systems $L$ yields an algebra $ℒ_{L} (T)$ for each $L$ -theory $T$ . Define an equivalence relation $\sim$ on the language of $T$ defined so that $φ \sim ψ$ precisely when both $T, φ ⊢_{L} ψ$ and $T, ψ ⊢_{L} φ$ hold, i.e., $φ$ and $ψ$ are deductively equivalent with respect to $T$ . Then the equivalence classes of sentences under $\sim$ , i.e., elements $⟦ φ ⟧ = {ξ ∣ φ \sim ξ}$ , make up the elements of the algebra $ℒ_{L} (T)$ , which may be ordered so that for two elements $⟦ φ ⟧$ and $⟦ ψ ⟧$ , $⟦ φ ⟧ ≼ ⟦ ψ ⟧$ whenever $T, φ ⊢_{L} ψ$ . When $L$ permits such constructions (i.e., is algebraizable), operations on $ℒ_{L} (T)$ can be defined for each connective. For example, conjunction $⟦ φ \land ψ ⟧$ is the greatest lower bound of $⟦ φ ⟧$ and $⟦ ψ ⟧$ with respect to $≼$ .
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