Some paraconsistent sentential calculi

In [8] Jakowski defined by means of an appropriate interpretation a paraconsistent calculusD ₂ . In [9] J. Kotas showed thatD ₂ is equivalent to the calculusM(S5) whose theses are exactly all formulasa such thatMa is a thesis ofS5. The papers [11], [7], [3], and [4] showed that interesting paraconsistent calculi could be obtained using modal systems other thanS5 and modalities other thanM. This paper generalises the above work. LetA be an arbitrary modality (i.e. string ofM's,L's and negation signs). Then theA-extension of a set of formulasX is {¦A X}}. Various properties ofA-extensions of normal modal systems are examined, including a problem of their axiomatizability     

Some definitions of negation leading to paraconsistent logics

In positive logic the negation of a propositionA is defined byA X whereX is some fixed proposition. A number of standard properties of negation, includingreductio ad absurdum, can then be proved, but not the law of noncontradiction so that this forms a paraconsistent logic. Various stronger paraconsistent logics are then generated by putting in particular propositions forX. These propositions range from true through contingent to false.     

Models for a paraconsistent set theory

In this paper the existence of natural models for a paraconsistent version of naive set theory is discussed. These stand apart from the previous attempts due to the presence of some non-monotonic ingredients in the comprehension scheme they fulfill. Particularly, it is proved here that allowing the equality relation in formulae defining sets, within an extensional universe, compels the use of non-monotonic operators. By reviewing the preceding attempts, we show how our models can naturally be obtained as fixed points of some functor acting on a suitable category (stressing the use of fixed-point arguments in obtaining such alternative semantics).     

A reasoning method for a paraconsistent logic

A proof method for automation of reasoning in a paraconsistent logic, the calculus C1* of da Costa, is presented. The method is analytical, using a specially designed tableau system. Actually two tableau systems were created. A first one, with a small number of rules in order to be mathematically convenient, is used to prove the soundness and the completeness of the method. The other one, which is equivalent to the former, is a system of derived rules designed to enhance computational efficiency. A prototype based on this second system was effectively implemented.     

Depth relevance of some paraconsistent logics

The paper essentially shows that the paraconsistent logicDR satisfies the depth relevance condition. The systemDR is an extension of the systemDK of [7] and the non-triviality of a dialectical set theory based onDR has been shown in [3]. The depth relevance condition is a strengthened relevance condition, taking the form: If _DR- AB thenA andB share a variable at the same depth, where the depth of an occurrence of a subformulaB in a formulaA is roughly the number of nested 's required to reach the occurrence ofB inA. The method of proof is to show that a model structureM consisting of {M ₀ , M₁, ..., M}, where theM _i s are all characterized by Meyer's 6-valued matrices (c. f, [2]), satisfies the depth relevance condition. Then, it is shown thatM is a model structure for the systemDR.     

Elementary extensions of topological models in L t language

In this paper we define the relation _t of elementary extension of topological models in the language L _t and show a Back and Forth criterion for _t . We introduce some new operations on partial homeomorphisms preserving Back and Forth properties. Some properties of _t are proved by the Back and Forth technique.     

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.     

Hyperalgebraic primitive elements for relational algebraic and topological algebraic models

Using nonstandard methods, we generalize the notion of an algebraic primitive element to that of an hyperalgebraic primitive element, and show that under mild restrictions, such elements can be found infinitesimally close to any given element of a topological field.Presented by Robert Goldblatt     

Proof theory and mathematical meaning of paraconsistent C-systems

A proof-theoretic analysis and new arithmetical semantics are proposed for some paraconsistent C-systems, which are a relevant sub-class of Logics of Formal Inconsistency (LFIs) introduced by W.A. Carnielli et al. (2002, 2005) [8] and [9]. The sequent versions BC, CI, CIL of the systems bC, Ci, Cil presented in Carnielli et al. (2002, 2005) [8] and [9] are introduced and examined. BC, CI, CIL admit the cut-elimination property and, in general, a weakened sub-formula property. Moreover, a formal notion of constructive paraconsistent system is given, and the constructivity of CI is proven. Further possible developments of proof theory and provability logic of CI-based arithmetical systems are sketched, and a possible weakened Hilbert?s program is discussed. As to the semantical aspects, arithmetical semantics interprets C-system formulas into Provability Logic sentences of classical Arithmetic PA (Artemov and Beklemishev (2004) [2], Japaridze and de Jongh (1998) [19], Gentilini (1999) [15], Smorynski (1991) [22]): thus, it links the notion of truth to the notion of provability inside a classical environment. It makes true infinitely many contradictions B∧¬B and falsifies many arbitrarily complex instances of non-contradiction principle ¬(A∧¬A). Moreover, arithmetical models falsify both classical logic LK and intuitionistic logic LJ, so that a kind of metalogical completeness property of LFI-paraconsistent logic w.r.t. arithmetical semantics is proven. As a work in progress, the possibility to interpret CI-based paraconsistent Arithmetic PACI into Provability Logic of classical Arithmetic PA is discussed, showing the role that PACIarithmetical models could have in establishing new meta-mathematical properties, e.g. in breaking classical equivalences between consistency statements and reflection principles.     

Some deterministic models of concept identification

It is shown that deterministic models can compete effectively with stochastic models in summarizing concept identification behavior. Three groups of deterministic models are examined. Examination of individual learners' trial by trial behavior in a concept experiment shows: (1) One person exhibited behavior consistent with a Hypothesis Permutation (HP) model despite being a nonlearner who showed no evidence of improvement over a period of 24 trials. However, when all 50 persons studied in each of two treatment groups were examined, only 22 members of one group and 10 of the other showed no inconsistencies with deterministic local consistency assumptions. (2) Certain deterministic computer programs could find at least one satisfactory order for predicting all responses by 18 of the 22 consistent solvers and 6 of the 10 consistent solvers, respectively, in the two groups just mentioned. For these 24 persons, then, a less restrictive deterministic model is adequate than for the others. (3) Those 38 original members of the first treatment group who met a stringent learning criterion were compared with respect to predictions generated by stochastic and mathematized deterministic models. One deterministic model (RSS-U 9-state) is in some respects the best of the models examined, but this success is a partial reflection of estimating eight parameters from the data.     

Some properties of binormamin

Three properties of the binormamin criterion for analytic transformation in factor analysis are discussed. Particular reference is made to Carroll's oblimin class of criteria.The research reported in this note was supported in part by the Program in Applied Mathematics and Statistics. National Science Foundation, and by the National Institute of Education, Department of Health, Education, and Welfare.     

Some evidence on additive learning models

Deterministic and probabilistic additive learning models forsignal detection/recognition replace the fixed criterion of classical detection models with one which shifts from trial to trial in the light of the preceding trial events. Data from a sinusoid in noise detection task without feedback and an auditory amplitude recognition task with feedback are used to test these models with respect to their predictions about asymptotic response frequency, and, where possible, by likelihood ratio tests. These and some previous experiments show that whether or not feedback is given subjects do not universally probability match, overmatch, undermatch, or keep response probability constant over discriminability, so that none of the testable special models can fit more than a proportion of subjects. The likelihood ratio tests confirm this conclusion for the special deterministic models. The six-parameter general deterministic model does nonsignificantly better than an ad hoc six-parameter response runs model in fitting the recognition data and significantly better than the five-parameter memory recognition model of Tanner, Rauk, andAtkinson (1970). Monte Carlo methods are used to confirm the applicability of asymptotic response frequency results to practically feasible sample sizes.     

Some multi-attribute models in examination assessment

Examination assessment can be likened to multi-attribute models in value theory by treating candidates as alternatives that have to be compared and examination components as the different attributes on which they are assessed. Then the mark combination process is equivalent to a value function, the main difference being that in this context the examiners' judgement is modelled rather than their preferences. This paper consists of five parts. In the first, a formal representation of the problem is given. In the next three sections, three additive representations are studied, each involving a different set of properties for the marks and the combination process. Finally, there is a short discussion and comparison of the three representations.     

Some local models for correlation experiments

This paper constructs two classes of models for the quantum correlation experiments used to test the Bell-type inequalities, synchronization models and prism models. Both classes employ deterministic hidden variables, satisfy the causal requirements of physical locality, and yield precisely the quantum mechanical statistics. In the synchronization models, the joint probabilities, for each emission, do not factor in the manner of stochastic independence, showing that such factorizability is not required for locality. In the prism models the observables are not random variables over a common space; hence these models throw into question the entire random variables idiom of the literature. Both classes of models appear to be testable.Work on this paper was supported, in part, by National Science Foundation Grant SES 79-25917.