The strong proof from hypotheses and conditionals: Some theorems of deduction for relevant systems

The aim of this paper is to present a modified version of the notion of strong proof from hypotheses (definition D2), and to give three deduction theorems for the relevant logicsR (theoremsT1, andT2) andE (theoremT3).     

Embedding the elementary ontology of stanisław Leśniewski into the monadic second-order calculus of predicates

LetEO be the elementary ontology of Le?niewski formalized as in Iwanu? [1], and letLS be the monadic second-order calculus of predicates. In this paper we give an example of a recursive function ?, defined on the formulas of the language ofEO with values in the set of formulas of the language of LS, such that ? _EO A iff ? _LS ?(A) for each formulaA.     

An Inconsistency-Adaptive Deontic Logic for Normative Conflicts

We present the inconsistency-adaptive deontic logic DP ^r , a nonmonotonic logic for dealing with conflicts between normative statements. On the one hand, this logic does not lead to explosion in view of normative conflicts such as O A?∧?O ～A, O A?∧?P ～A or even O A?∧?～O A. On the other hand, DP ^r still verifies all intuitively reliable inferences valid in Standard Deontic Logic (SDL). DP ^r interprets a given premise set ‘as normally as possible’ with respect to SDL. Whereas some SDL-rules are verified unconditionally by DP ^r , others are verified conditionally. The latter are applicable unless they rely on formulas that turn out to behave inconsistently in view of the premises. This dynamic process is mirrored by the proof theory of DP ^r .     

On the extension of intuitionistic propositional logic with Kreisel-Putnam's and Scott's schemes

LetSKP be the intermediate prepositional logic obtained by adding toI (intuitionistic p.l.) the axiom schemes:S = ((? ?α→α)→α∨ ?α)→ ?α∨ ??α (Scott), andKP = (?α→β∨γ)→(?α→β)∨(?α→γ) (Kreisel-Putnam). Using Kripke's semantics, we prove:

1. SKP has the finite model property;
2. SKP has the disjunction property.

In the last section of the paper we give some results about Scott's logic S = I+S.     

Checking quasi-identities in a finite semigroup may be computationally hard

We exhibit a 10-element semigroup Q such that the question “Does a given quasi-identity hold in Q?” is co-NP-complete while the question “Does a given identity hold in Q?” can be answered in linear time.     

Axiomatic extensions of the constructive logic with strong negation and the disjunction property

We study axiomatic extensions of the propositional constructive logic with strong negation having the disjunction property in terms of corresponding to them varieties of Nelson algebras. Any such varietyV is characterized by the property: (PQWC) ifA,B εV, thenA×B is a homomorphic image of some well-connected algebra ofV. We prove:

• each varietyV of Nelson algebras with PQWC lies in the fibre σ^?1(W) for some varietyW of Heyting algebras having PQWC,
• for any varietyW of Heyting algebras with PQWC the least and the greatest varieties in σ^?1(W) have PQWC,
• there exist varietiesW of Heyting algebras having PQWC such that σ^?1(W) contains infinitely many varieties (of Nelson algebras) with PQWC.

     

Where gamma fails

A major question for the relevant logics has been, “Under what conditions is Ackermann's ruleγ from -A ∨B andA to inferB, admissible for one of these logics?” For a large number of logics and theories, the question has led to an affirmative answer to theγ problem itself, so that such an answer has almost come to be expected for relevant logics worth taking seriously. We exhibit here, however, another large and interesting class of logics-roughly, the Boolean extensions of theW — free relevant logics (and, precisely, the well-behaved subsystems of the 4-valued logicBN4) — for which γ fails.     

Boolean Skeletons of MV-algebras and ?-groups

Let ?? be Mundici??s functor from the category ${\mathcal{LG}}$ whose objects are the lattice-ordered abelian groups (?-groups for short) with a distinguished strong order unit and the morphisms are the unital homomorphisms, onto the category ${\mathcal{MV}}$ of MV-algebras and homomorphisms. It is shown that for each strong order unit u of an ?-group G, the Boolean skeleton of the MV-algebra ??(G, u) is isomorphic to the Boolean algebra of factor congruences of G.     

Effective Cut-elimination for a Fragment of Modal mu-calculus

A non-effective cut-elimination proof for modal mu-calculus has been given by G. J?ger, M. Kretz and T. Studer. Later an effective proof has been given for a subsystem M ₁ with non-iterated fixpoints and positive endsequents. Using a new device we give an effective cut-elimination proof for M ₁ without restriction to positive sequents.     

Algebraic Functions

Let A be an algebra. We say that the functions f ₁, . . . , f _m : A ⁿ ?? A are algebraic on A provided there is a finite system of term-equalities ${{\bigwedge t_{k}(\overline{x}, \overline{z}) = s_{k}(\overline{x}, \overline{z})}}$ satisfying that for each ${{\overline{a} \in A^{n}}}$ , the m-tuple ${{(f_{1}(\overline{a}), \ldots , f_{m}(\overline{a}))}}$ is the unique solution in A ^m to the system ${{\bigwedge t_{k}(\overline{a}, \overline{z}) = s_{k}(\overline{a}, \overline{z})}}$ . In this work we present a collection of general tools for the study of algebraic functions, and apply them to obtain characterizations for algebraic functions on distributive lattices, Stone algebras, finite abelian groups and vector spaces, among other well known algebraic structures.     

Extraction of relief from visual motion

We quantified the ability of human subjects to discriminate the relative distance of two points from a slanted plane when viewing the projected velocities of this scene (orthographic projection). The relative distance from a plane (called relief) is a 3-D property that is invariant under linear (affine) transformations. As such, relief canin principle be extracted from the instantaneous projected velocity field; a metric representation, which requires the extraction of visual acceleration, is not required. The stimulus consisted of a slanted planeP (specified by three points) and two pointsQ ₁ andQ ₂ that are non-coplanar withP. This configuration of points oscillated rigidly around the vertical axis. We have measured thesystematic error andaccuracy with which human subjects estimate the relative distance of pointsQ ₁ andQ ₂ from planeP as a function of the slant of_P. The systematic error varies with slant: it is low for small slant values, reaches a maximum for medium slant values, and drops again for high slant values. The accuracy covaries with the systematic error and is thus high for small and large slant values and low for medium slant values. These results are successfully modeled by a simple relief-from-motion computation based on local estimates of projected velocities. The data are well predicted by assuming (1) a measurement error in velocity estimation that varies proportionally to velocity (Weber’s law) and (2) an eccentricity-dependent underestimation of velocity.     

The d-Logic of the Rational Numbers: A Fruitful Construction

We present a geometric construction that yields completeness results for modal logics including K4, KD4, GL and GL _n with respect to certain subspaces of the rational numbers. These completeness results are extended to the bimodal case with the universal modality.     

Algebraic study of Sette's maximal paraconsistent logic

The aim of this paper is to study the paraconsistent deductive systemP ¹ within the context of Algebraic Logic. It is well known due to Lewin, Mikenberg and Schwarse thatP ¹ is algebraizable in the sense of Blok and Pigozzi, the quasivariety generated by Sette's three-element algebraS being the unique quasivariety semantics forP ¹. In the present paper we prove that the mentioned quasivariety is not a variety by showing that the variety generated byS is not equivalent to any algebraizable deductive system. We also show thatP ¹ has no algebraic semantics in the sense of Czelakowski. Among other results, we study the variety generated by the algebraS. This enables us to prove in a purely algebraic way that the only proper non-trivial axiomatic extension ofP ¹ is the classical deductive systemPC. Throughout the paper we also study those abstract logics which are in a way similar toP ¹, and are called hereabstract Sette logics. We obtain for them results similar to those obtained for distributive abstract logics by Font, Verdú and the author.     

The adequacy of the theories of deductive systems with respect to sentential calculi

Summary The sentential calculiR, under discussion, are axiomatizable and implication is among their primitive terms. The modus ponens and the rule of substitution are their primitive rules. ByS _r is denoted the set of sentences obtained from the formulae of the calculusR by substituting sentences of a given language for all variables. The variablesx, y, z ... represent the elements of the setS _r , the variablesX, Y, Z ... represent the subsets ofS _R . The formulacxy designates an implication withx as its antecedent andy as its consequent,cxy is always an element ofS _R δ(X) means, thatX is closed with respect to the modus ponens rule.A _R designates the class of allS _R -substitutions of the axioms of the sentential calculusR.