Arnold Koslow

.  On a structuralist account of logic, the logical operators, as well as modal operators are defined by the specific ways that they interact with respect to implication. As a consequence, the same logical operator (conjunction, negation etc.) can appear to be very different with a variation in the implication relation of a structure. We illustrate this idea by showing that certain operators that are usually regarded as extra-logical concepts (Tarskian algebraic operations on theories, mereological sum, products and negates of individuals, intuitionistic operations on mathematical problems, epistemic operations on certain belief states) are simply the logical operators that are deployed in different implication structures. That makes certain logical notions more omnipresent than one would think.

The original motivation of D. Gabbay’s concept of Fibring concerned the combination of logics, and initially it involved the syntactic introduction of modals into formulations of intuitionistic logic in which modals are syntactically absent. We show, using the notion of structural modals that there are many modals of intuitionism, and logics for subjunctive and epistemic conditionals which are not syntactically evident in our best formulations of them. We discuss some cases when the attempt to make them syntactically evident can have undesirable consequences.

Carnap in the 1930s discovered that there were non-normal interpretations of classical logic - ones for which negation and conjunction are not truth-functional so that a statement and its negation could have the same truth value, and a disjunction of two false sentences could be true. Church ar-gued that this did not call for a revision of classical logic. More recent writers seem to disa-gree. We provide a definition of "non-normal interpretation" and argue that Church was right, and in fact, the existence of non-normal interpretations tells us something important about the condi-tions of extensionality of the classical logical operators