337 research outputs found
Constructive Logic with Strong Negation is a Substructural Logic. II
The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL ew of the substructural logic FL ew . The main result of Part I of this series [41] shows that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL ew (namely, a certain variety of FL ew -algebras) are term equivalent. In this paper, the term equivalence result of Part I [41] is lifted to the setting of deductive systems to establish the definitional equivalence of the logics N and NFL ew . It follows from the definitional equivalence of these systems that constructive logic with strong negation is a substructural logi
Reasoning about Knowledge in Linear Logic: Modalities and Complexity
In a recent paper, Jean-Yves Girard commented that âit has been a long time since philosophy has stopped intereacting with logicâ[17]. Actually, it has no
Interpolation in Linear Logic and Related Systems
We prove that there are continuum-many axiomatic extensions of the full
Lambek calculus with exchange that have the deductive interpolation property.
Further, we extend this result to both classical and intuitionistic linear
logic as well as their multiplicative-additive fragments. None of the logics we
exhibit have the Craig interpolation property, but we show that they all enjoy
a guarded form of Craig interpolation. We also exhibit continuum-many axiomatic
extensions of each of these logics without the deductive interpolation
property
Logics for modelling collective attitudes
We introduce a number of logics to reason about collective propositional
attitudes that are defined by means of the majority rule. It is well known that majoritarian
aggregation is subject to irrationality, as the results in social choice theory and judgment
aggregation show. The proposed logics for modelling collective attitudes are based on
a substructural propositional logic that allows for circumventing inconsistent outcomes.
Individual and collective propositional attitudes, such as beliefs, desires, obligations, are
then modelled by means of minimal modalities to ensure a number of basic principles. In
this way, a viable consistent modelling of collective attitudes is obtained
Classical BI: Its Semantics and Proof Theory
We present Classical BI (CBI), a new addition to the family of bunched logics
which originates in O'Hearn and Pym's logic of bunched implications BI. CBI
differs from existing bunched logics in that its multiplicative connectives
behave classically rather than intuitionistically (including in particular a
multiplicative version of classical negation). At the semantic level,
CBI-formulas have the normal bunched logic reading as declarative statements
about resources, but its resource models necessarily feature more structure
than those for other bunched logics; principally, they satisfy the requirement
that every resource has a unique dual. At the proof-theoretic level, a very
natural formalism for CBI is provided by a display calculus \`a la Belnap,
which can be seen as a generalisation of the bunched sequent calculus for BI.
In this paper we formulate the aforementioned model theory and proof theory for
CBI, and prove some fundamental results about the logic, most notably
completeness of the proof theory with respect to the semantics.Comment: 42 pages, 8 figure
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