3,981 research outputs found
Co-constructive logic for proofs and refutations
This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to equate falsity with the non-existence of a proof since this would render a statement such as âpi is transcendentalâ false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorovâs logic of problems
Co-constructive logics for proofs and refutations
This paper considers logics which are formally dual to intuition- istic logic in order to investigate a co-constructive logic for proofs and refu- tations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely- held assumptions, to a justification of bivalence. For example, we do not want to equate falsity with the non-existence of a proof since this would render a statement such as âpi is transcendentalâ false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for state- ments for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorovâs logic of problems
Kripke Semantics and Proof Systems for Combining Intuitionistic Logic and Classical Logic
International audienceWe combine intuitionistic logic and classical logic into a new, first-order logic called Polarized Intuitionistic Logic. This logic is based on a distinction between two dual polarities which we call red and green to distinguish them from other forms of polarization. The meaning of these polarities is defined model-theoretically by a Kripke-style semantics for the logic. Two proof systems are also formulated. The first system extends Gentzen's intuitionistic sequent calculus LJ. In addition, this system also bears essential similarities to Girard's LC proof system for classical logic. The second proof system is based on a semantic tableau and extends Dragalin's multiple-conclusion version of intuitionistic sequent calculus. We show that soundness and completeness hold for these notions of semantics and proofs, from which it follows that cut is admissible in the proof systems and that the propositional fragment of the logic is decidable
Proof search and counter-model construction for bi-intuitionistic propositional logic with labelled sequents
Bi-intuitionistic logic is a conservative extension of
intuitionistic logic with a connective dual to implication, called
exclusion. We present a sound and complete cut-free labelled sequent
calculus for bi-intuitionistic propositional logic, BiInt,
following S. Negri's general method for devising sequent calculi
for normal modal logics. Although it arises as a natural
formalization of the Kripke semantics, it is does not directly
support proof search. To describe a proof search procedure, we
develop a more algorithmic version that also allows for
counter-model extraction from a failed proof attempt.Estonian Science Foundation - grants no. 5567; 6940Fundação para a CiĂȘncia e a Tecnologia (FCT)RESCUE - no. PTDC/EIA/65862/2006TYPES - FP6 ISTCentro de matemĂĄtica da Universidade do Minh
Complement-Topoi and Dual Intuitionistic Logic
Mortensen studies dual intuitionistic logic by dualizing topos internal logic, but he did not study a sequent calculus. In this paper I present a sequent calculus for complement-topos logic, which throws some light on the problem of giving a dualization for LJ
Relating Sequent Calculi for Bi-intuitionistic Propositional Logic
Bi-intuitionistic logic is the conservative extension of intuitionistic logic
with a connective dual to implication. It is sometimes presented as a symmetric
constructive subsystem of classical logic.
In this paper, we compare three sequent calculi for bi-intuitionistic
propositional logic: (1) a basic standard-style sequent calculus that restricts
the premises of implication-right and exclusion-left inferences to be
single-conclusion resp. single-assumption and is incomplete without the cut
rule, (2) the calculus with nested sequents by Gore et al., where a complete
class of cuts is encapsulated into special "unnest" rules and (3) a cut-free
labelled sequent calculus derived from the Kripke semantics of the logic. We
show that these calculi can be translated into each other and discuss the
ineliminable cuts of the standard-style sequent calculus.Comment: In Proceedings CL&C 2010, arXiv:1101.520
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
Focusing and Polarization in Intuitionistic Logic
A focused proof system provides a normal form to cut-free proofs that
structures the application of invertible and non-invertible inference rules.
The focused proof system of Andreoli for linear logic has been applied to both
the proof search and the proof normalization approaches to computation. Various
proof systems in literature exhibit characteristics of focusing to one degree
or another. We present a new, focused proof system for intuitionistic logic,
called LJF, and show how other proof systems can be mapped into the new system
by inserting logical connectives that prematurely stop focusing. We also use
LJF to design a focused proof system for classical logic. Our approach to the
design and analysis of these systems is based on the completeness of focusing
in linear logic and on the notion of polarity that appears in Girard's LC and
LU proof systems
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