HEREDITARILY STRUCTURALLY COMPLETE POSITIVE LOGICS

Positive logics are $\{ \wedge , \vee , \to \}$-fragments of intermediate logics. It is clear that the positive fragment of $Int$ is not structurally complete. We give a description of all hereditarily structurally complete positive logics, while the question whether there is a structurally complete positive logic which is not hereditarily structurally complete, remains open.

The Grammar of Capital

Karl Marx’s Capital does not start with the commodity, it starts with wealth. This has enormous consequences, both theoretically and politically. To start with the commodity leads into the analysis of capitalism as a system of domination. To start with wealth and its movement in-against-and-beyond the commodity form takes the reader into a world of struggle. In Capital there are two antagonistic categorial series. The familiar series of the forms of domination: commodity, value, abstract labor, money, person, capital, profit, interest, rent. These forms have their own grammar: a positive logic that imposes itself on the flow of life. But there is also a series of subversive categories rebelling against the logical chain of derivation: wealth, use value, concrete labor or doing, anti-person, and so on. Here is a negative, defetishizing, detotalizing grammar that moves against the rigid cohesion of the first series.

POSITIVE LOGIC WITH ADJOINT MODALITIES: PROOF THEORY, SEMANTICS, AND REASONING ABOUT INFORMATION

We consider a simple modal logic whose nonmodal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of (e.g.) T, S4, and S5, such logics are useful, as shown in previous work by Baltag, Coecke, and the first author, for encoding and reasoning about information and misinformation in multiagent systems. For the propositional-only fragment of such a dynamic epistemic logic, we present an algebraic semantics, using lattices with agent-indexed families of adjoint pairs of operators, and a cut-free sequent calculus. The calculus exploits operators on sequents, in the style of “nested” or “tree-sequent” calculi; cut-admissibility is shown by constructive syntactic methods. The applicability of the logic is illustrated by reasoning about the muddy children puzzle, for which the calculus is augmented with extra rules to express the facts of the muddy children scenario.