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指出，广义度量空间是$[0, \infty]$（即正扩展实数的quantale）上的丰富范畴。这种丰富性是预序关系的定量类比。为了寻求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那些可在任意quantale中解释的，即有限合取和析取、张量（对于Lawvere quantale为加法）和线性蕴涵（此处为截断减法）；在此基础上，我们依次添加常数$1$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。针对每种逻辑，我们开发了一个自然演绎系统，并证明该系统相对于quantale值语义是可判定的完备的。完备性证明的核心运用了Motzkin转置定理。一致性也是可判定的；该证明利用了线性不等式的Fourier-Motzkin消去法。强完备性一般并不成立，即使（如已知的那样）对于有限多个命题变元上的理论也是如此；事实上，Pavelka或Ben Yaacov意义上的近似强完备性形式（即任意精度的可证性）也不成立。然而，对于在有限命题变元集合上由一组（未必有限）范式判定式公理化的理论，当我们限制变量不映射到$\infty$的模型时，我们可以证明强完备性；该证明使用了Farkas引理的Hurwicz广义形式。