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.

The Boolean Pythagorean Triples problem asks: does there exist a binary coloring of the natural numbers such that every Pythagorean triple contains an element of each color? This problem was first solved in 2016, when Heule, Kullmann and Marek encoded a finite restriction of this problem as a propositional formula and showed its unsatisfiability. In this work we formalize their development in the theorem prover Coq. We state the Boolean Pythagorean Triples problem in Coq, define its encoding as a propositional formula and establish the relation between solutions to the problem and satisfying assignments to the formula. We verify Heule et al.'s proof by showing that the symmetry breaks they introduced to simplify the propositional formula are sound, and by implementing a correct-by-construction checker for proofs of unsatisfiability based on reverse unit propagation.

Joint work with João Marques-Silva and Peter Schneider-Kamp.

We give a first definitive step towards endowing the general mechanism for combining logics known as fibring with a meaningful and useful semantics given by non-deterministic logical matrices (Nmatrices). We present and study the properties of two semantical operations: a unary operation of ω-power of a given Nmatrix, and a binary operation of strict product of Nmatrices with disjoint similarity types (signatures). We show that, together, these operations can be used to characterize the disjoint fibring of propositional logics, when each of these logics is presented by a single Nmatrix. As an outcome, we also provide a decidability and complexity result about the resulting fibred logic. We illustrate the constructions with a few meaningful examples.

Joint work with Carlos Caleiro.

Possibility and necessity theories rival with probability in representing uncertain knowledge, while offering a more qualitative view of uncertainty. Moreover, necessity and possibility measures constitute, respectively, lower and upper bounds for probability measures, with the advantage of avoiding the complications of the notion of probabilistic independence.

On the other hand, paraconsistent formal systems, especially the Logics of Formal Inconsistency, are capable of quite carefully expressing the circumstances of reasoning with contradictions. The aim of this talk is to merge these ideas, by precisely defining new notions of possibility and necessity theories involving the concept of consistency (generalizing the proposal by (Besnard & Lang 1994)) based on connecting them to the notion of partial and conclusive evidence. This combination permits a whole treatment of contradictions, both local and global, including a gradual handling of the notion of contradiction, thus obtaining a really useful tool for AI and machine learning, with potential applications in logic programming via appropriate resolution rules.