Preservation of Interpolation Features by Fibring

Fibring is a metalogical constructor that permits to combinedifferent logics by operating on their deductive systems undercertain natural restrictions, as for example that the two givenlogics are presented by deductive systems of the same type.Under such circumstances, fibring will produce a new deductivesystem by means of the free use of inference rules from bothdeductive systems, provided the rules are schematic, in thesense of using variables that are open for application to formulaswith new linguistic symbols (from the point of view of eachlogic component). Fibring is a generalization of fusion, a lessgeneral but wider developed mechanism which permits resultsof the following kind: if each logic component is decidable(or sound, or complete with respect to a certain semantics)then the resulting logic heirs such a property. The interestfor such preservation results for combining logics is evident,and they have been achieved in the more general setting of fibringin several cases. The Craig interpolation property and the Maeharainterpolation have a special significance when combining logics,being related to certain problems of complexity theory, someproperties of model theory and to the usual (global) metatheoremof deduction. When the peculiarities of the distinction betweenlocal and global deduction interfere, justifying what we callcareful reasoning, the question of preservation of interpolationbecomes more subtle and other forms of interpolation can bedistinguished. These questions are investigated and several(global and local) preservation results for interpolation areobtained for fibring logics that fulfill mild requirements.     

Fibring Non-Truth-Functional Logics: Completeness Preservation

Fibring has been shown to be useful for combining logics endowed withtruth-functional semantics. However, the techniques used so far are unableto cope with fibring of logics endowed with non-truth-functional semanticsas, for example, paraconsistent logics. The first main contribution of thepaper is the development of a suitable abstract notion of logic, that mayalso encompass systems with non-truth-functional connectives, and wherefibring can still be dealt with. Furthermore, it is shown that thisextended notion of fibring preserves completeness under certain reasonableconditions. This completeness transfer result, the second main contributionof the paper, generalizes the one established in Zanardo et al. (2001) butis obtained using new techniques that explore the properties of a suitablemeta-logic (conditional equational logic) where the (possibly)non-truth-functional valuations are specified. The modal paraconsistentlogic of da Costa and Carnielli (1988) is studied in the context of this novel notionof fibring and its completeness is so established.