We present a research program which investigates the intersection of deductive reasoning with explicit quantitative capabilities. These quantitative capabilities encompass probabilistic reasoning, counting and counting quantifiers, and similar systems.

The need to have a combined reasoning system that enables a unified way to reason with quantities has always been recognized in modern logic, as proposals of logic probabilistic reasoning are present in the work of Boole [1854]. Equally ubiquitous is the need to deal with cardinality restrictions on finite sets.

We actually show that there is a common way to deal with these several deductive quantitative capabilities, involving a framework based on Linear Algebras and Linear Programming, and the distinction between probabilistic and cardinality reasoning arising from the different family of algebras employed.

The quantitative logic systems are particularly amenable to the introduction of inconsistency measurements, which quantify the degree of inconsistency of a given quantitative logic theory, following some basic principles of inconsistency measurements.

Thus, Quantitative Reasoning is presented as a non-explosive reasoning method that provides a reasoning tool in the presence of quantitative logic inconsistencies, based on the principle that inference can be obtained by minimizing the inconsistency measurement.