1,721,054 research outputs found
Craig interpolation for semilinear substructural logics
No full textThe Craig interpolation property is investigated for substructural logics whose algebraic semantics are varieties of semilinear (subdirect products of linearly ordered) pointed commutative residuated lattices. It is shown that Craig interpolation fails for certain classes of these logics with weakening if the corresponding algebras are not idempotent. A complete characterization is then given of axiomatic extensions of the "R-mingle with unit" logic (corresponding to varieties of Sugihara monoids) that have the Craig interpolation property. This latter characterization is obtained using a model-theoretic quantifier elimination strategy to determine the varieties of Sugihara monoids admitting the amalgamation property. © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Possibilistic conditioning framed in fuzzy logics
No full textAbstractThe notion of conditional possibility derived from marginal possibility measures has received different treatments. As shown by Bouchon-Meunier et al., conditional possibility can be introduced as a primitive notion generalizing simple possibility measures. In this paper, following an approach already adopted by the author w.r.t. conditional probability, we build up the fuzzy modal logic FCΠ, relying on Rational Pavelka Logic RPL, so as to reason about coherent conditional possibilities and necessities. First, we apply a modal operator ♢ over conditional events ϕ∣χ to obtain modal formulas of the type (ϕ∣χ)♢ whose reading is “ϕ∣χ is possible”. Then, we define the truth-value of the modal formulas as corresponding to a conditional possibility measure. The logic FCΠ is shown to be strongly complete for finite theories w.r.t. to the class of the introduced conditional possibility Kripke structures. Then, we show that any rational assessment of conditional possibilities is coherent iff a suitably defined theory over FCΠ is consistent. We also prove compactness for rational coherent assessments of conditional possibilities. We derive the notion of generalized conditional necessity from the notion of generalized conditional possibility, and we show and discuss how to represent those concepts introducing some logics generalizing FCΠ. Finally we show how to frame qualitative comparative relations in this logical framework
Representing upper probability Measures over rational Lukasiewicz logic
No full textUpper probability measures are measures of uncertainty that generalize
probability measures in order to deal with non-measurable events. Following
an approach that goes back to previous works by H ajek, Esteva, and Godo,
we show how to expand Rational Lukasiewicz Logic by modal operators
in
order to reason about upper probabilities of classical Boolean events
'
so that
(
'
) can be read as \the upper probability of
'
". We build the logic
U
(R L)
for representing upper probabilities and show it to be complete w.r.t. a class
of Kripke structures equipped with an upper probability measure. Finally,
we prove that the set of
U
(R L)-satis able formulas is NP-complete.Peer Reviewe
On deductive interpolation for the weak nilpotent minimum logic
No full textThe Weak Nilpotent Minimum logic WNM was introduced by Esteva and Godo in [6] and shown to be the logic of the class of weak nilpotent minimum triangular norms. In this article, we prove that WNM admits the Deductive Interpolation Property by showing through a model-theoretic argument that the corresponding variety of algebras has the Amalgamation Property
Some notes on elimination properties for the theory of Riesz MV-Chains
No full textRiesz MV-algebras are a variety of algebras strongly connected to Riesz spaces. In this short article we investigate some elimination properties of the first-order theory RMV of linearly ordered Riesz MV-algebras and show that RMV admits elimination of quantifiers and uniform elimination of imaginary elements. In the process, we also prove several other results such as modelcompleteness, o-minimality, definability of Skolem functions, and a version of the Di Nola Representation Theorem for Riesz MV-algebras
Amalgamation through quantifier elimination for varieties of commutative residuated lattices
No full textThis work presents a model-theoretic approach to the study of the amalgamation property for varieties of semilinear commutative residuated lattices. It is well-known that if a first-order theory T enjoys quantifier elimination in some language L, the class of models of the set of its universal consequences T ∀ has the amalgamation property. Let Th(K) be the theory of an elementary subclass K of the linearly ordered members of a variety V of semilinear commutative residuated lattices. We show that whenever Th(K) has elimination of quantifiers, and every linearly ordered structure in V is a model of Th ∀(K), then V has the amalgamation property. We exploit this fact to provide a purely model-theoretic proof of amalgamation for particular varieties of semilinear commutative residuated lattices
Ordered fields and ŁΠ1/2-algebras
No full textIn this work we further explore the connection between ŁΠ1/ 2-algebras and ordered fields. We show that any two ŁΠ1/ 2-chains generate the same variety if and only if they are related to ordered fields that have the same universal theory. This will yield that any ŁΠ1/2-chain generates the whole variety if and only if it contains a subalgebra isomorphic to the ŁΠ1/2-chain of real algebraic numbers, that consequently is the smallest ŁΠ1/ 2-chain generating the whole variety. We also show that any two different subalgebras of the ŁΠ1/2-chain over the real algebraic numbers generate different varieties. This will be exploited in order to prove that the lattice of subvarieties of ŁΠ1/ 2-algebras has the cardinality of the continuum. Finally, we will also briefly deal with some model-theoretic properties of ŁΠ1/ 2-chains related to real closed fields, proving quantifier-elimination and related results
On computational complexity of semilinear varieties
No full textWe propose a method for characterizing the complexity of satisfiability and tautologicity of equational theories of varieties of algebras by relying on their representability in the theory of the ordered additive group of reals R with rational constants. We call semilinear those varieties which are generated by a subclass of algebras in which the operations are representable as semilinear functions with rational coefficients. Those functions are definable in the theory of R which admits quantifier elimination and whose existential theory is NP-complete. We prove that there is a polynomial time translation of the equational theories of semilinear varieties into the existential theory of R. Then, if the variety is generated (up to isomorphism) by one semilinear algebra, the satisfiability problem is in NP, while the tautologicity problem is in co-NP. We apply this method in order to provide a comprehensive study of complexity of several varieties related to logics based on left-continuous conjunctive uninorms and left-continuous t-norms
Mixed Rational Assessments of Possibility and Probability Measures
No full textAbstractIn this paper we introduce the modal-fuzzy logic FPΠ(RŁΔ) for reasoning about probability and possibility at the same time. We will use such a logical formalism in order to treat mixed assessments of both those kinds of measures. The main result of this paper is a characterization of the coherence for rational mixed assessments by means of the logical consistency of a suitably defined FPΠ(RŁΔ)-theory. By means of this characterization, we will also prove that the problem of testing the coherence of a mixed assessment is NP-complete
T-norm-based logics with an independent involutive negation
No full textIn this paper we investigate the addition of arbitrary independent involutive negations to t-norm-based logics. We deal with several extensions of MTL and establish general completeness results. Indeed, we will show that, given any t-norm-based logic satisfying some basic properties, its extension by means of an involutive negation preserves algebraic and (finite) strong standard completeness. We will deal with both propositional and predicate logics
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