20 research outputs found
A Note on R-Mingle and the Danger of Safety
Dunn has recently argued that the logic R-Mingle (or RM) is a good, and good enough, choice for many purposes in relevant and paraconsistent logic. This includes an argument that the validity of Safety principle, according to which one may infer an arbitrary instance of the law of excluded middle from an arbitrary contradiction, in RM is not a problem because it doesn’t allow one to infer anything new from a contradiction. In this paper, I argue that while Dunn’s claim holds for the logic, there is a good reason to think that it’s not the case for (prime) theories closed under the logic, and that this should give relevantists, and some paraconsistentists, pause when considering whether RM is adequate for their purposes
A Note on R-Mingle and the Danger of Safety
Dunn has recently argued that the logic R-Mingle (or RM) is a good, and good enough, choice for many purposes in relevant and paraconsistent logic. This includes an argument that the validity of Safety principle, according to which one may infer an arbitrary instance of the law of excluded middle from an arbitrary contradiction, in RM is not a problem because it doesn’t allow one to infer anything new from a contradiction. In this paper, I argue that while Dunn’s claim holds for the logic, there is a good reason to think that it’s not the case for (prime) theories closed under the logic, and that this should give relevantists, and some paraconsistentists, pause when considering whether RM is adequate for their purposes
Recent Work in Relevant Logic
This paper surveys important work done in relevant logic in the past 10 years
Relevance via decomposition: A project, some results, an open question
We report on progress and an unsolved problem in our attempt to obtain a clear rationale for relevance logic via semantic decomposition trees. Suitable decomposition rules, constrained by a natural parity condition, generate a set of directly acceptable formulae that contains all axioms of the well-known system R, is closed under substitution and conjunction, satisfies the letter-sharing condition, but is not closed under detachment. To extend it, a natural recursion is built into the ocedure for constructing decomposition trees. The resulting set of acceptable formulae has many attractive features, but it remains an open question whether it continues to satisfy the crucial letter-sharing condition
Relevance via decomposition
We report on progress and an unsolved problem in our attempt to obtain a clear rationale for relevance logic via semantic decomposition trees. Suitable decomposition rules, constrained by a natural parity condition, generate a set of directly acceptable formulae that contains all axioms of the well-known system R, is closed under substitution and conjunction, satisfies the letter-sharing condition, but is not closed under detachment. To extend it, a natural recursion is built into the procedure for constructing decomposition trees. The resulting set of acceptable formulae has many attractive features, but it remains an open question whether it continues to satisfy the crucial letter-sharing condition
Complexity Hierarchies Beyond Elementary
We introduce a hierarchy of fast-growing complexity classes and show its
suitability for completeness statements of many non elementary problems. This
hierarchy allows the classification of many decision problems with a
non-elementary complexity, which occur naturally in logic, combinatorics,
formal languages, verification, etc., with complexities ranging from simple
towers of exponentials to Ackermannian and beyond.Comment: Version 3 is the published version in TOCT 8(1:3), 2016. I will keep
updating the catalogue of problems from Section 6 in future revision
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
Relevance via decomposition
We report on progress and an unsolved problem in our attempt to obtain a clear rationale for relevance logic via semantic decomposition trees. Suitable decomposition rules, constrained by a natural parity condition, generate a set of directly acceptable formulae that contains all axioms of the well-known system R, is closed under substitution and conjunction, satisfies the letter-sharing condition, but is not closed under detachment. To extend it, a natural recursion is built into the procedure for constructing decomposition trees. The resulting set of acceptable formulae has many attractive features, but it remains an open question whether it continues to satisfy the crucial letter-sharing condition