Abstract We consider a family of non-classical three-valued logics proposed to model indicative conditionals in natural language. Among these, systems introduced by B. De Finetti, W.S. Cooper, J. Cantwell and R.J. Farrell, as well as some variants that have not appeared in the literature, but seem nevertheless to be natural objects of interest from a formal point of view. Most of these logics are not easily treatable with the standard techniques of algebraic logic. We therefore resort to non-deterministic structures and multiple-conclusion calculi to provide alternative semantical characterizations and axiomatizations. In the best cases—logics given by a finite monadic matrix—this can be done directly, in a modular way, through a procedure due to Shoesmith and Smiley. In the more involved ones—logics preserving degrees of truth—some ingenuity and more sophisticated techniques are required. We characterize these logics by a partial non-deterministic matrix, and show how to produce analytic (and effective) calculi that are complete with respect to this generalized semantics. In all cases, the calculi thus obtained can be straightforwardly converted, by a uniform procedure, into traditional single-conclusion Hilbert-style axiomatizations.

Abstract We obtain, for the first time, a modular many-valued semantics for combined logics, which is built directly from many-valued semantics for the logics being combined, by means of suitable universal operations over partial non-deterministic logical matrices. Our constructions preserve finite-valuedness in the context of multiple-conclusion logics, whereas, unsurprisingly, it may be lost in the context of single-conclusion logics. Besides illustrating our constructions over a wide range of examples, we also develop concrete applications of our semantic characterizations, namely regarding the semantics of strengthening a given many-valued logic with additional axioms, the study of conditions under which a given logic may be seen as a combination of simpler syntactically defined fragments whose calculi can be obtained independently and put together to form a calculus for the whole logic, and also general conditions for decidability to be preserved by the combination mechanism.

Abstract Perfect paradefinite algebras are De Morgan algebras expanded with an operation that allows for the full behavior of classical negation to be restored. They form a variety that is term-equivalent to the variety of involutive Stone algebras. Their associated multiple-conclusion (Set-Set) and single-conclusion ( ) order-preserving logics are non-algebraizable self-extensional logics of formal inconsistency and undeterminedness determined by a six-valued matrix. We studied these logics extensively in Gomes et al. ((2022). Electronic Proceedings in Theoretical Computer Science357 56–76.) from both the algebraic and the proof-theoretical perspectives. In the present paper, we continue that study by investigating directions for conservatively expanding these logics with an implication connective (essentially, one that admits the deduction-detachment theorem). We first consider logics given by very simple and manageable non-deterministic semantics whose implication (in isolation) is classical. These, nevertheless, fail to be self-extensional. We then consider the implication realized by the relative pseudo-complement over the six-valued perfect paradefinite algebra. Our strategy is to expand the language of the latter algebra with this connective and study the (self-extensional) Set-Set and order-preserving and $\top$ -assertional logics of the variety induced by the resulting algebra. We provide axiomatizations for such new variety and for such logics, drawing parallels with the class of symmetric Heyting algebras and with Moisil’s “symmetric modal logic.” For the order-preserving Set-Set logic, in particular, we obtain a Set-Set axiomatization that is analytic. We close by studying interpolation properties for these logics and concluding that the new variety has the Maehara amalgamation property.

Abstract Recent compositionality results in logic have highlighted the advantages of enlarging the traditional notion of logical matrix semantics, namely by incorporating non-determinism and partiality. Still, several important properties which are known to be computable for finite logical matrices have not been studied in the wider context of partial non-deterministic matrices (PNmatrices). In this paper, we study how incorporating non-determinism and/or partiality in logical matrices impacts on the computational properties of some natural problems regarding their induced logics and concretely their sets of theorems. We show that, while for some of these problems there is no relevant computational impact, there are problems whose computational complexity increases and still other problems that simply become undecidable. In particular, we show that the problem of checking whether the logics characterized by two finite PNmatrices have the same set of theorems is not decidable. This undecidability result explores the connection between PNmatrices and term-DAG-automata, where the universality problem is known to be undecidable. This link also motivates a final contribution, in the form of a pumping-like lemma, which can be used, in some cases, to show that a given logic cannot be characterized by a finite PNmatrix.

Fibring is a general mechanism for combining logics that provides valuable insight on designing and understanding complex logical systems. To date, most research on fibring has focused on its model and proof-theoretic aspects, and on transference results for relevant metalogical properties. But we are still far from understanding in full the way mixed reasoning emerges from the logics being combined, which is preventing us from having a fully satisfactory semantics for fibred logics and, consequently, limiting the usability of the general results obtained. In previous work, assuming no shared connectives, we have presented an effective characterization of mixed reasoning in terms of the component logics, taking only variables as hypotheses. Despite these restrictions, the result immediately proved to have very interesting applications. In this article, we extend our previous characterization of mixed reasoning for disjoint fibring to arbitrary non-mixed hypotheses. While still not completely satisfactory, as the characterization still cannot cover reasoning from mixed hypotheses, and even less fibred logics with shared connectives, the result again proves to be extremely useful. We illustrate its power by exploring two meaningful applications. To start with, we provide the first full characterization of conservativity for logics obtained by disjoint fibring, extending the partial results of Schechter (2011). Then, we take a semantic detour and use our characterization of mixed reasoning to show that (disjoint) fibring does not preserve finite (N)valuedness.

Abstract We show the first examples of recursively enumerable (even decidable) two-dimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linearly ordered first components must be infinite in two senses: It should contain infinitely many propositional variables, and formulas of arbitrarily large modal nesting-depth.