186 research outputs found
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
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
Strong Completeness and the Finite Model Property for Bi-Intuitionistic Stable Tense Logics
Bi-Intuitionistic Stable Tense Logics (BIST Logics) are tense logics with a Kripke semantics where worlds in a frame are equipped with a pre-order as well as with an accessibility relation which is ‘stable’ with respect to this pre-order. BIST logics are extensions of a logic, BiSKt, which arose in the semantic context of hypergraphs, since a special case of the pre-order can represent the incidence structure of a hypergraph. In this paper we provide, for the first time, a Hilbert-style axiomatisation of BISKt and prove the strong completeness of BiSKt. We go on to prove strong completeness of a class of BIST logics obtained by extending BiSKt by formulas of a certain form. Moreover we show that the finite model property and the decidability hold for a class of BIST logics
Fixed-point elimination in the Intuitionistic Propositional Calculus (extended version)
It is a consequence of existing literature that least and greatest
fixed-points of monotone polynomials on Heyting algebras-that is, the alge-
braic models of the Intuitionistic Propositional Calculus-always exist, even
when these algebras are not complete as lattices. The reason is that these
extremal fixed-points are definable by formulas of the IPC. Consequently, the
-calculus based on intuitionistic logic is trivial, every -formula
being equiv- alent to a fixed-point free formula. We give in this paper an
axiomatization of least and greatest fixed-points of formulas, and an algorithm
to compute a fixed-point free formula equivalent to a given -formula. The
axiomatization of the greatest fixed-point is simple. The axiomatization of the
least fixed- point is more complex, in particular every monotone formula
converges to its least fixed-point by Kleene's iteration in a finite number of
steps, but there is no uniform upper bound on the number of iterations. We
extract, out of the algorithm, upper bounds for such n, depending on the size
of the formula. For some formulas, we show that these upper bounds are
polynomial and optimal.Comment: extended version of arXiv:1601.0040
Syntactic completeness of proper display calculi
A recent strand of research in structural proof theory aims at exploring the
notion of analytic calculi (i.e. those calculi that support general and modular
proof-strategies for cut elimination), and at identifying classes of logics
that can be captured in terms of these calculi. In this context, Wansing
introduced the notion of proper display calculi as one possible design
framework for proof calculi in which the analiticity desiderata are realized in
a particularly transparent way. Recently, the theory of properly displayable
logics (i.e. those logics that can be equivalently presented with some proper
display calculus) has been developed in connection with generalized Sahlqvist
theory (aka unified correspondence). Specifically, properly displayable logics
have been syntactically characterized as those axiomatized by analytic
inductive axioms, which can be equivalently and algorithmically transformed
into analytic structural rules so that the resulting proper display calculi
enjoy a set of basic properties: soundness, completeness, conservativity, cut
elimination and subformula property. In this context, the proof that the given
calculus is complete w.r.t. the original logic is usually carried out
syntactically, i.e. by showing that a (cut free) derivation exists of each
given axiom of the logic in the basic system to which the analytic structural
rules algorithmically generated from the given axiom have been added. However,
so far this proof strategy for syntactic completeness has been implemented on a
case-by-case base, and not in general. In this paper, we address this gap by
proving syntactic completeness for properly displayable logics in any normal
(distributive) lattice expansion signature. Specifically, we show that for
every analytic inductive axiom a cut free derivation can be effectively
generated which has a specific shape, referred to as pre-normal form.Comment: arXiv admin note: text overlap with arXiv:1604.08822 by other author
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