A graph-theoretic account of logics

A graph-theoretic account of logics is explored based on the general notion of m-graph (that is, a graph where each edge can have a finite sequence of nodes as source). Signatures, interpretation structures and deduction systems are seen as m-graphs. After defining a category freely generated by a m-graph, formulas and expressions in general can be seen as morphisms. Moreover, derivations involving rule instantiation are also morphisms. Soundness and completeness theorems are proved. As a consequence of the generality of the approach our results apply to very different logics encompassing, among others, substructural logics as well as logics with nondeterministic semantics, and subsume all logics endowed with an algebraic semantics

Time-stamped claim logic

The main objective of this paper is to define a logic for reasoning about distributed time-stamped claims. Such a logic is interesting for theoretical reasons, i.e., as a logic per se, but also because it has a number of practical applications, in particular when one needs to reason about a huge amount of pieces of evidence collected from different sources, where some of the pieces of evidence may be contradictory and some sources are considered to be more trustworthy than others. We introduce the Time-Stamped Claim Logic including a sound and complete sequent calculus. In order to show how Time-Stamped Claim Logic can be used in practice, we consider a concrete cyber-attribution case study

Modulated fibring and the collapsing problem

Fibring is recognized as one of the main mechanisms in combining logics, with great significance in the theory and applications of mathematical logic. However, an open challenge to fibring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse to one of them, indicating that fibring imposes unwanted interconnections between the given logics. Modulated fibring allows a finer control of the combination, solving the collapsing problem both at the semantic and deductive levels. Main properties like soundness and completeness are shown to be preserved, comparison with fibring is discussed, and some important classes of examples are analyzed with respect to the collapsing problem.6741541156

Asymmetric Combination of Logics is Functorial: A Survey

Asymmetric combination of logics is a formal process that develops the characteristic features of a specific logic on top of another one. Typical examples include the development of temporal, hybrid, and probabilistic dimensions over a given base logic. These examples are surveyed in the paper under a particular perspective—that this sort of combination of logics possesses a functorial nature. Such a view gives rise to several interesting questions. They range from the problem of combining translations (between logics), to that of ensuring property preservation along the process, and the way different asymmetric combinations can be related through appropriate natural transformations

Object Inheritance Beyond Subtyping

. A categorial semantic domain for objects is presented in order to clarify both aggregation and specialization. Three kinds of specialization are discussed: (1) subtyping (specialization with no side effects and no nonmonotonic overriding); (2) monotonic specialization (possibly with side effects but still only with monotonic overriding); and (3) non-monotonic specialization (possibily with side effects and non-monotonic overriding). A sequence of three categories of objects differing only in the morphisms is presented. The first one is used to explain object aggregation (respecting locality through a frame constraint) and the strictest form of specialization (subtyping). The second category is shown to be adequate for explaining specialization with side effects (by relaxing the frame constraint). Finally, the third category supports also nonmonotonic overriding, by adopting as morphisms suitable partial morphisms of the second one. All these categories are complete and cocomplete. Co..

Fibring Logics with Topos Semantics

The concept of fibring is extended to higher-order logics with arbitrary modalities and binding operators. A general completeness theorem is established for such logics including HOL and with the meta-theorem of deduction. As a corollary, completeness is shown to be preserved when fibring such rich logics. This result is extended to weaker logics in the cases where fibring preserves conservativeness of HOL-enrichments. Soundness is shown to be preserved by fibring without any further assumptions

Combining Logic Systems: Why, how, what for?

ness is preserved by a combination mechanism . and it is known that logic system is given by . L ## , then the completeness of follows from the completeness of ## . No wonder that much theoretical e#ort has been dedicated to establishing preservation results and/or finding preservation counterexamples about di#erent combination mechanisms. For an early overview of the practical and theoretical issues see also [4]. Several forms of combination have been studied, like product [30, 21, 22, 23], fusion [38, 28, 29, 40, 19], temporalization [12, 13, 41, 14], parameterization [6], synchronization [33] and fibring [15, 16, 3, 17, 34, 42]. Fusion is the best understood combination mechanism. In short, the fusion of two modal systems leads to a bimodal system including the two original modal operators and common propositional connectives. Several interesting properties of logic systems (like soundness, weak completeness, Craig interpolation property and decidability) were show

Fibring: Completeness Preservation

A completeness theorem is established for logics with congruence endowed with general semantics (in the style of general frames). As a corollary, completeness is shown to be preserved by fibring logics with congruence provided that congruence is retained in the resulting logic. The class of logics with equivalence is shown to be closed under fibring and to be included in the class of logics with congruence. Thus, completeness is shown to be preserved by fibring logics with equivalence and general semantics. An example is provided showing that completeness is not always preserved by fibring logics endowed with standard (non general) semantics. A categorial characterization of fibring is provided using coproducts and cocartesian liftings