Propositional System Research Articles

The SPAARS approach: implications for psychopathy

Schematic, propositional, analogical and associative representational Systems (SPAARS) is the integrated cognitive model of emotion proposed by Power and Dalgleish (Cognition and Emotion: from order to disorder. The Psychology Press, England, 1997). It is multi-level in nature and includes four different levels of representation. In SPAARS, emotions are described as appraisal-based according to an individual’s goals, thus making the theory functional in nature. Basic emotions possess an innate component and hence can be elicited automatically, since these emotions might already have been appraised in an individual’s evolutionary past. Thus, the theory proposes two routes for the generation of emotions. It provides an useful approach within which both basic and complex emotions can readily be understood. The theory can also be applied in order to explain emotional disorders as well as to generate novel therapeutic interventions for them. In the current review the SPAARS approach has been used to understand psychopathy, also called the disorder of empathy (a complex emotion). The theory provides a new perspective for looking at psychopaths. It suggests that psychopathy may arise either due to a problem generating the basic emotion of sadness, the complex emotion of empathy or a combination of the two. The interventions suggested would also vary depending upon the underlying nature of the problem.

Quantum deduction rules

We define propositional quantum Frege proof systems and compare them with classical Frege proof systems.

Quantitative logic

In this paper, we introduce the concept of the degree of the truth in the framework of so called many-valued propositional logic systems with the intention of measuring to what extent a given formula is true. Such a concept can indeed induce, in a very natural way, the degree of the similarity as well as to induce a pseudo-metric among formulas as the ‘graded’ version of the notion of the logical equivalence. The basic properties of such induced logic metric space hence are investigated in this paper. We further introduce different concepts of the degree of the divergence and the degree of the consistency in order to grade the extent of the consistency of a logic theory. To the end, all the basic logic notions are graded. Lastly, we propose in this paper three patterns of ‘approximate reasoning’ so far as the many-valued propositional logic systems are concerned. Hence, we can claim that we have laid the foundation for the “quantitative logic”. Since this theory has touched upon several key concepts of similarity, comparison, measurements, etc. in information theory, we are convinced that the theory may have profound impact in contemporary information technology. In addition, quantitative logic can be regarded as a highly representative example of mathematics of uncertainty capable of handling vagueness.

Canonical calculi with (n,k)-ary quantifiers

Propositional canonical Gentzen-type systems, introduced in 2001 by Avron and Lev, are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a connective is introduced and no other connective is mentioned. A constructive coherence criterion for the non-triviality of such systems was defined and it was shown that a system of this kind admits cut-elimination iff it is coherent. The semantics of such systems is provided using two-valued non-deterministic matrices (2Nmatrices). In 2005 Zamansky and Avron extended these results to systems with unary quantifiers of a very restricted form. In this paper we substantially extend the characterization of canonical systems to (n,k)-ary quantifiers, which bind k distinct variables and connect n formulas, and show that the coherence criterion remains constructive for such systems. Then we focus on the case of k∈{0,1} and for a canonical calculus G show that it is coherent precisely when it has a strongly characteristic 2Nmatrix, which in turn is equivalent to admitting strong cut-elimination.

Implications of neuroscientific evidence for the cognitive models of post-traumatic stress disorder

Brewin's dual representation theory, Ehlers and Clark's cognitive appraisal model, and Dalgleish's schematic, propositional, analogue and associative representational systems model are considered in the light of recent evidence on the neural substrates of post-traumatic stress disorder (PTSD). The models' proposals about the cognitive mechanism of memory dysfunction in PTSD are described and evaluated against current knowledge about the neural pathways and functions disrupted in PTSD. A dual pathway model of memory is consistent with neuroscience of memory. The appraisal model also provides an account of the top-down modulation of memory and arousal problems consistent with current neuroscientific evidence of PTSD. Dalgleish's model is less consistent with the evidence because it relies upon assumptions that cannot yet be tested neuroscientifically. All three models under-specify the causal and maintaining influence of hyperarousal relative to the role it plays in current neuroscientific models of PTSD. Implications of the evidence for improving treatment and prevention are discussed.

Axiomatization for 1-level universal AND operator

The aim of this article is the partial axiomatization for 1-level universal logic. A propositional calculus formal deductive system UL − hɛ(0,1) based on 1-level universal AND operator of universal logic is built up. The corresponding algebra ▪ is introduced. The soundness and the completeness of system UL − hɛ(0,1) are proved.

A Topological Study of Contextuality and Modality in Quantum Mechanics

Kochen-Specker theorem rules out the non-contextual assignment of values to physical magnitudes. Here we enrich the usual orthomodular structure of quantum mechanical propositions with modal operators. This enlargement allows to refer consistently to actual and possible properties of the system. By means of a topological argument, more precisely in terms of the existence of sections of sheaves, we give an extended version of Kochen-Specker theorem over this new structure. This allows us to prove that contextuality remains a central feature even in the enriched propositional system.

Some properties of the roots of theories in propositional logic systems

Some properties of the roots of theories, and the relationship between the sets D ( Γ ) of all conclusions of two different theories in the classical propositional logic system, Łukasiewicz propositional logic system, and the R 0 -propositional logic system are studied. It is proved that D ( Γ ) is completely determined by its root whenever Γ has a root, and the construction of the roots is given.

The Logic of Classes

An extension of the Quantified Propositional Calculus 1 obtained by the addition of two binary propositional functions is put forward as an inheritor of E. Schroder's “Algebra der Logik”. The formal system is itself not new, in fact it forms part of A. P. Morse's “A Theory of Sets”; although the latter is considered as a first-order system (of a rather non-standard type).Since the additional propositional functions are not invariant under the logical biconditional, this system-and many others naturally obtained from it-give us a collection of examples of non-standard, but mathematically meaningful, propositional systems.

SPAARS Approach: Integrated cognitive model of emotion of Attention Deficit/Hyperactivity Disorder

SPAARS (Schematic, Propositional, Analogical and Associative Representational Systems) is the integrated cognitive model of emotion proposed by Power & Dalgleish (1997). In SPAARS, emotions are described as appraisal based against an individual’s goals and this makes the theory functional in nature. The theory proposes two routes for the generation of emotions, namely a direct one in which the appraisals have become automatized and the other where these are still not automatized. It provides a useful approach within which both basic and complex emotions can readily be understood. The theory can also be applied for explaining the disorders of emotion and it can be used to generate novel therapeutic interventions for them. In the current review, the SPAARS approach in conjunction with the theory proposed by Nigg and Casey (2005) has been applied for understanding ADHD. According to Nigg & Casey, affective deficits in ADHD could be due to problems with either approach or avoidance emotions. These problems have been explained with the help of appraisals and reinforcement learning within the SPAARS framework. Interventions that follow logically from the SPAARS framework have also been suggested.

The Birth of quantum logic

By quoting extensively from unpublished letters written by John von Neumann to Garret Birkhoff during the preparatory phase (in 1935) of their ground-breaking 1936 paper that established quantum logic, the main steps in the thought process leading to the 1936 Birkhoff–von Neumann paper are reconstructed. The reconstruction makes it clear why Birkhoff and von Neumann rejected the notion of quantum logic as the projection lattice of an infinite dimensional complex Hilbert space and why they postulated in their 1936 paper that the quantum propositional system should be isomorphic to an abstract projective geometry. Looking at the paper now I see, that I forgot to say this, which should be said somewhere in the first §: That while common logics did apply to quantum mechanics, if the notion of simultaneous measurability is introduced as an auxiliary notion, we wished to construct a logical system, which applies directly to quantum mechanics – without any extraneous secondary notions like simultaneous measurability. And in order to have such a consequent, one-piece system of logics, we must change the classical class calculus of logics. (J. von Neumann to G. Birkhoff, November 21, 1935)

A Tale of Two Natures: Worried Reflections on the Study of Literature and Culture in an Age of Neuroscience and Neo-Darwinism

Bliss was it in that dawn to be alive, But to be young was very heaven! O times, In which the meagre, stale, forbidding ways Of custom, law, and statute, took at once The attraction of a country in romance! When Reason seemed the most to assert her rights, When most intent on making of herself A prime enchantress – to assist the work, Which then was going forward in her name! (Wordsworth 1850, 401)

Incorporating Planning into BDI Systems

Many architectures of autonomous agent have been proposed throughout AI research. The most common architectures, BDI, are procedural in that they do no planning, seriously curtailing an agent's ability to cope with unforeseen events. In this paper, we explore the relationship between propositional planning systems and the process of means-ends reasoning used by BDI agents and define a mapping from BDI mental states to propositional planning problems and from propositional plans back to mental states. In order to test the viability of such a mapping, we have implemented it in an extension of a BDI agent model through the use of Graphplan as the propositional planning algorithm. The implemented prototype was applied to model a case study of an agent controlled production cell.

Theory of truth degrees of propositions in the logic system L n *

By means of infinite product of uniformly distributed probability spaces of cardinal n the concept of truth degrees of propositions in the n-valued generalized Lukasiewicz propositional logic system L * n is introduced in the present paper. It is proved that the set consisting of truth degrees of all formulas is dense in [0,1], and a general expression of truth degrees of formulas as well as a deduction rule of truth degrees is then obtained. Moreover, similarity degrees among formulas are proposed and a pseudo-metric is defined therefrom on the set of formulas, and hence a possible framework suitable for developing approximate reasoning theory in n-valued generalized Lukasiewicz proposi-tional logic is established.

Generalized consistency degrees of theories w.r.t. formulas in several standard complete logic systems

The present paper carries out a deep analysis on the inconsistency of theories by means of deduction theorems, completeness theorems and satisfiability degrees of formulas, and introduces the concept of the degree of entailment of a contradiction from a theory in classical two-valued logic system, Łukasiewicz fuzzy logic system, Gödel fuzzy logic system, product fuzzy logic system and the R 0 -fuzzy logic system. It is proved that in classical two-valued logic system, Łukasiewicz fuzzy logic system and the R 0 -fuzzy logic system, respectively, the concept of consistency degrees of theories defined by the divergence degrees of theories in [X.N. Zhou, G.J. Wang, Consistency degrees of theories in some systems of propositional fuzzy logic, Fuzzy Sets and Systems 152 (2005) 321–331; H.J. Zhou, G.J. Wang, A new theory index based on deduction theorems in several logic systems, Fuzzy Sets and Systems 157(2006) 427–443] is reasonable and can accurately measure the consistency degrees of theories. The concept of the degree of entailment of a contradiction from a theory is generalized by replacing the contradiction with a general formula and then the generalized consistency degrees of theories w.r.t. general formulas in the above-mentioned logic systems is established.

Cut-Elimination and Quantification in Canonical Systems

Canonical Propositional Gentzen-type systems are systems which in addition to the standard axioms and structural rules have only pure logical rules with the sub-formula property, in which exactly one occurrence of a connective is introduced in the conclusion, and no other occurrence of any connective is mentioned anywhere else. In this paper we considerably generalize the notion of a “canonical system” to first-order languages and beyond. We extend the Propositional coherence criterion for the non-triviality of such systems to rules with unary quantifiers and show that it remains constructive. Then we provide semantics for such canonical systems using 2-valued non-deterministic matrices extended to languages with quantifiers, and prove that the following properties are equivalent for a canonical system G: (1) G admits Cut-Elimination, (2) G is coherent, and (3) G has a characteristic 2-valued non-deterministic matrix.

Comments on Predicative Logic

We show how to interpret intuitionistic propositional logic into a predicative second-order intuitionistic propositional system having only the conditional and the universal second-order quantifier. We comment on this fact. We argue that it supports the legitimacy of using classical logic in a predicative setting, even though the philosophical cast of predicativism is non-realistic. We also note that the absence of disjunction and existential quantifications allows one to have a process of normalization of proofs that avoids the use of "commuting conversions".

Non-deterministic Multiple-valued Structures

The ordinary concept of a multiple-valued matrix is generalized by introducing non-deterministic matrices (Nmatrices), in which non-deterministic computations of truth-values are allowed. It is shown that some important logics for reasoning under uncertainty can be characterized by finite Nmatrices (and so they are decidable), although they have only infinite characteristic ordinary (deterministic) matrices. A generalized compactness theorem that applies to all finite Nmatrices is then proved. Finally, a strong connection is established between the admissibility of the cut rule in canonical Gentzen-type propositional systems, non-triviality of such systems, and the existence of sound and complete non-deterministic two-valued semantics for them. This connection is used for providing a complete solution for the old 'Tonk' problem of Prior.

A cost-reducing question-selection algorithm for propositional knowledge-based systems

In many applications of knowledge-based systems, initially given data are often not sufficient to reach a conclusion and more data are needed. A question-selection algorithm is to identify missing information and select proper questions to ask. We present a question-selection algorithm for propositional knowledge-based systems, which aims at asking more relevant and less expensive questions. Comparing to those algorithms currently used in many expert systems, the new algorithm is capable of reaching a conclusion more economically in our computational experiments.

A unified approximate reasoning theory suitable for both propositional calculus system L* and predicate calculus system K*

The concepts of metric R 0-algebra and Hilbert cube of type R 0 are introduced. A unified approximate reasoning theory in propositional caculus system \(\mathcal{L}^* \) and predicate calculus system \(\mathcal{K}^* \) is established semantically as well as syntactically, and a unified complete theorem is obtained.