“distributivity”的概念、定义、翻译、参考文献-科学参考

distributivity

Logic

1. The property of two connectives $\circ_{0}$ and $\circ_{1}$ when one may be distributed over the other, that is, when the formulae $φ \circ_{1} (ψ \circ_{0} ξ)$ is interderivable with $(φ \circ_{0} ψ) \circ_{1} (φ \circ_{0} ξ)$ . In this case one says that $\circ_{0}$ distributes over $\circ_{1}$ .
2. A rule of inference that corresponds to the distributivity of one connective over another. For example, classical logic admits inferences corresponding to the distributivity of conjunction and disjunction over one another:
$\frac{φ \land (ψ \lor ξ)}{(φ \lor ψ) \land (φ \lor ξ)} \frac{φ \lor (ψ \land ξ)}{(φ \land ψ) \lor (φ \land ξ)}$
and over themselves:
$\frac{φ \land (ψ \land ξ)}{(φ \land ψ) \land (φ \land ξ)} \frac{φ \lor (ψ \lor ξ)}{(φ \lor ψ) \lor (φ \lor ξ)}$
Distribution rules hold of other connectives as well.
3. Rules of inference relating the universal and particular quantifiers to conjunction and disjunction. In classical logic, for example, one has the distribution of the universal quantifier over conjunction and the existential quantifier over disjunction:
$\frac{\forall x (φ \land ψ)}{\forall x φ \land ψ} \frac{\exists x (φ \lor ψ)}{\exists x φ \lor \exists x ψ}$

单词	distributivity
释义	distributivity Logic 1. The property of two connectives $\circ_{0}$ and $\circ_{1}$ when one may be distributed over the other, that is, when the formulae $φ \circ_{1} (ψ \circ_{0} ξ)$ is interderivable with $(φ \circ_{0} ψ) \circ_{1} (φ \circ_{0} ξ)$ . In this case one says that $\circ_{0}$ distributes over $\circ_{1}$ . 2. A rule of inference that corresponds to the distributivity of one connective over another. For example, classical logic admits inferences corresponding to the distributivity of conjunction and disjunction over one another: $\frac{φ \land (ψ \lor ξ)}{(φ \lor ψ) \land (φ \lor ξ)} \frac{φ \lor (ψ \land ξ)}{(φ \land ψ) \lor (φ \land ξ)}$ and over themselves: $\frac{φ \land (ψ \land ξ)}{(φ \land ψ) \land (φ \land ξ)} \frac{φ \lor (ψ \lor ξ)}{(φ \lor ψ) \lor (φ \lor ξ)}$ Distribution rules hold of other connectives as well. 3. Rules of inference relating the universal and particular quantifiers to conjunction and disjunction. In classical logic, for example, one has the distribution of the universal quantifier over conjunction and the existential quantifier over disjunction: $\frac{\forall x (φ \land ψ)}{\forall x φ \land ψ} \frac{\exists x (φ \lor ψ)}{\exists x φ \lor \exists x ψ}$
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