We introduce a family of modal expansions of {\L}ukasiewicz logic that are designed to accommodate modal translations of generalized basic logic (as formulated with exchange, weakening, and falsum). We further exhibit algebraic semantics for each logic in this family, in particular showing that all of them are algebraizable in the sense of Blok and Pigozzi. Using this algebraization result and an analysis of congruences in the pertinent varieties, we establish that each of the introduced modal {\L}ukasiewicz logics has a local deduction-detachment theorem. By applying Jipsen and Montagna's poset product construction, we give two translations of generalized basic logic with exchange, weakening, and falsum in the style of the celebrated G\"odel-McKinsey-Tarski translation. The first of these interprets generalized basic logic in a modal {\L}ukasiewicz logic in the spirit of the classical modal logic S4, whereas the second interprets generalized basic logic in a temporal variant of the latter.
翻译:我们引入了一种模式扩展 {L}ukasiewicz 逻辑, 旨在适应通用基本逻辑( 以交换、 削弱和假相制成的) 的模式翻译。 我们进一步展示了这个家庭每种逻辑的代数语义, 特别是显示所有这些逻辑在布洛克 和 Pigozzi 的意义上都是可变数的。 使用这一代数分析结果和对相关品种一致性的分析, 我们确定引入的每个模式 {L}ukasiewicz 逻辑都有一种本地推分分理论。 通过应用吉普森 和蒙塔尼亚的假冒产品构造, 我们用流行的G'odel- McKinsey-Tarski 翻译的风格将通用基本逻辑与交换、 削弱和假相交译为两种通用的基本逻辑。 我们用经典模型逻辑 S4 精神中的第一个解释通用的基本逻辑 =L}ukasiewicz 逻辑, 而第二个解释则在后一种时间变法中概括基本逻辑。