Lawvere showed that generalised metric spaces are categories enriched over $[0, \infty]$, the quantale of the positive extended reals. The statement of enrichment is a quantitative analogue of being a preorder. Towards seeking a logic for quantitative metric reasoning, we investigate three $[0,\infty]$-valued propositional logics over the Lawvere quantale. The basic logical connectives shared by all three logics are those that can be interpreted in any quantale, viz finite conjunctions and disjunctions, tensor (addition for the Lawvere quantale) and linear implication (here a truncated subtraction); to these we add, in turn, the constant $1$ to express integer values, and scalar multiplication by a non-negative real to express general affine combinations. Quantitative equational logic can be interpreted in the third logic if we allow inference systems instead of axiomatic systems. For each of these logics we develop a natural deduction system which we prove to be decidably complete w.r.t. the quantale-valued semantics. The heart of the completeness proof makes use of the Motzkin transposition theorem. Consistency is also decidable; the proof makes use of Fourier-Motzkin elimination of linear inequalities. Strong completeness does not hold in general, even (as is known) for theories over finitely-many propositional variables; indeed even an approximate form of strong completeness in the sense of Pavelka or Ben Yaacov -- provability up to arbitrary precision -- does not hold. However, we can show it for theories axiomatized by a (not necessarily finite) set of judgements in normal form over a finite set of propositional variables when we restrict to models that do not map variables to $\infty$; the proof uses Hurwicz's general form of the Farkas' Lemma.

翻译：Lawvere曾指出，广义度量空间是在$[0, \infty]$（正扩展实数的量词）上富集的范畴，而富集性陈述是预序关系的定量类比。为探索定量度量推理的逻辑，我们研究了Lawvere quantale上的三种$[0,\infty]$值命题逻辑。这三种逻辑共有的基本逻辑连接词是可在任意量词中解释的那些：有限合取与析取、张量（对Lawvere quantale而言为加法）及线性蕴涵（此处为截断减法）；在此基础上，我们分别添加常量$1$以表达整数值，以及非负实数的标量乘法以表达一般仿射组合。若允许使用推理系统而非公理系统，则定量等式逻辑可在第三种逻辑中解释。我们为每种逻辑建立了自然演绎系统，并证明其相对于量词值语义是可判定的完备的；完备性证明的核心借助了Motzkin转置定理。一致性也是可判定的，其证明利用了Fourier-Motzkin消去法处理线性不等式。一般情形下强完备性不成立，即使对于有限多个命题变元上的理论亦是如此（此为主流结论）；事实上，Pavelka或Ben Yaacov意义上的近似强完备性——即任意精度下的可证性——亦不成立。然而，当模型限制为不将变元映射至$\infty$，且理论由有限命题变元集上的（未必有限的）范式判定集公理化时，我们可证明强完备性成立；该证明使用了Farkas引理的Hurwicz一般形式。