646 research outputs found
Cut-elimination for the mu-calculus with one variable
We establish syntactic cut-elimination for the one-variable fragment of the
modal mu-calculus. Our method is based on a recent cut-elimination technique by
Mints that makes use of Buchholz' Omega-rule.Comment: In Proceedings FICS 2012, arXiv:1202.317
On an Intuitionistic Logic for Pragmatics
We reconsider the pragmatic interpretation of intuitionistic logic [21]
regarded as a logic of assertions and their justications and its relations with classical
logic. We recall an extension of this approach to a logic dealing with assertions
and obligations, related by a notion of causal implication [14, 45]. We focus on
the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on
polarized bi-intuitionistic logic as a logic of assertions and conjectures: looking at the
S4 modal translation, we give a denition of a system AHL of bi-intuitionistic logic
that correctly represents the duality between intuitionistic and co-intuitionistic logic,
correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism
as a distributed calculus of coroutines is then used to give an operational
interpretation of subtraction.Work on linear co-intuitionism is then recalled, a linear
calculus of co-intuitionistic coroutines is dened and a probabilistic interpretation
of linear co-intuitionism is given as in [9]. Also we remark that by extending the
language of intuitionistic logic we can express the notion of expectation, an assertion
that in all situations the truth of p is possible and that in a logic of expectations
the law of double negation holds. Similarly, extending co-intuitionistic logic, we can
express the notion of conjecture that p, dened as a hypothesis that in some situation
the truth of p is epistemically necessary
Finitary proof systems for Kozen’s μ.
We present three finitary cut-free sequent calculi for the modal [my]-calculus.
Two of these derive annotated sequents in the style of Stirling’s ‘tableau proof
system with names’ (4236) and feature special inferences that discharge open
assumptions. The third system is a variant of Kozen’s axiomatisation in which
cut is replaced by a strengthening of the v-induction inference rule. Soundness
and completeness for the three systems is proved by establishing a sequence
of embeddings between the calculi, starting at Stirling’s tableau-proofs and
ending at the original axiomatisation of the [my]-calculus due to Kozen. As a
corollary we obtain a completeness proof for Kozen’s axiomatisation which
avoids the usual detour through automata or games
Proofs and Refutations for Intuitionistic and Second-Order Logic
The ?^{PRK}-calculus is a typed ?-calculus that exploits the duality between the notions of proof and refutation to provide a computational interpretation for classical propositional logic. In this work, we extend ?^{PRK} to encompass classical second-order logic, by incorporating parametric polymorphism and existential types. The system is shown to enjoy good computational properties, such as type preservation, confluence, and strong normalization, which is established by means of a reducibility argument. We identify a syntactic restriction on proofs that characterizes exactly the intuitionistic fragment of second-order ?^{PRK}, and we study canonicity results
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
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