In this paper, we introduce the notion of intuitionistic fuzzy PMS-subalgebras under homomorphism and Cartesian product and investigate several properties. We study the homomorphic image and inverse image of the intuitionistic fuzzy PMS-subalgebras of a PMS-algebra, which are also intuitionistic fuzzy PMS-subalgebras of a PMS-algebra, and find some other interesting results. Furthermore, we also prove that the Cartesian product of intuitionistic fuzzy PMS-subalgebras is again an intuitionistic fuzzy PMS-subalgebra and characterize it in terms of its level (...) sets. Finally, we consider the strongest intuitionistic fuzzy PMS-relations on an intuitionistic fuzzy set in a PMS-algebra and demonstrate that an intuitionistic fuzzy PMS-relation on an intuitionistic fuzzy set in a PMS-algebra is an intuitionistic fuzzy PMS-subalgebra if and only if the corresponding intuitionistic fuzzy set in a PMS-algebra is an intuitionistic fuzzy PMS-subalgebra of a PMS-algebra. (shrink)

The purpose of this article is to introduce a Cartesian product structure into the social choice theoretical framework and to examine if new possibility results to Gibbard’s and Sen’s paradoxes can be developed thanks to it. We believe that a Cartesian product structure is a pertinent way to describe individual rights in the social choice theory since it discriminates the personal features comprised in each social state. First we define some conceptual and formal tools related to (...) the Cartesian product structure. We then apply these notions to Gibbard’s paradox and to Sen’s impossibility of a Paretian liberal. Finally we compare the advantages of our approach to other solutions proposed in the literature for both impossibility theorems. (shrink)

In this paper we show that for any set X ⊆ ω there exists a structure [MATHEMATICAL SCRIPT CAPITAL A] that has no presentation computable in X such that [MATHEMATICAL SCRIPT CAPITAL A]2 has a computable presentation. We also show that there exists a structure [MATHEMATICAL SCRIPT CAPITAL A] with infinitely many computable isomorphism types such that [MATHEMATICAL SCRIPT CAPITAL A]2 has exactly one computable isomorphism type.

Descartes argued that productivity, namely our ability to generate an unlimited number of new thoughts or ideas from previous ones, derives from a single undividable source in the human soul. Cognitive scientists, in contrast, have viewed productivity as a modular phenomenon. According to this latter view, syntactic, semantic, musical or visual productivity emerges each from their own generative engines in the human brain. Recent evidence has, however, led some authors to revitalize the Cartesian theory. According to this view, a (...) single source or a single mechanism in the human brain produces productivity in every cognitive domain, whether in the domain of music, semantics or syntax. In this article, we will address recent evidence concerning the single source hypothesis from brain-imaging studies, linguistics, cognitive theories of music perception, biology of cognition and cognitive development, along with several objections that have been presented against this hypothesis. We formulate two versions of the Cartesian theory which combine the more recent computational theory of cognition with Descartes' view on productivity. (shrink)

: Descartes provides an original and puzzling argument for the traditional theological doctrine that the world is continuously created by God. His key premise is that the parts of the duration of anything are "completely independent" of one another. I argue that Descartes derives this temporal independence thesis simply from the principle that causes are necessarily simultaneous with their effects. I argue further that it follows from Descartes's version of the continuous creation doctrine that God is the instantaneous and total (...) cause of everything that happens, and that this is just what his physics demands. But although God is the total cause of everything, he is not the only cause, since Descartes considers it obvious that finite minds have the power to move bodies. In the face of this apparent paradox, several recent commentators have suggested that Descartes accepted the late scholastic view that God and finite causes somehow collaborate or concur in the production of numerically identical effects. But close examination reveals that the case for Cartesian concurrentism is weak. Fortunately, there is a simpler and more fruitful solution to the problem of reconciling divine and human action. I argue that that in Descartes's world certain motions, such as voluntary movements of our bodies, are causally overdetermined This account allows Descartes to avoid occasionalism without embracing an elaborate metaphysics of secondary causality. Finally, I argue that the overdeterminist interpretation sheds new light on two longstanding problems of Cartesian metaphysics. First, it resolves a serious difficulty with a standard reading of Descartes's conception of human freedom. Second, it offers a novel approach to the old problem of reconciling genuine human action with the principle of the conservation of total quantity of motion. (shrink)

When Gödel developed his functional interpretation, also known as the Dialectica interpretation, his aim was to prove consistency of first order arithmetic by reducing it to a quantifier-free theory with finite types. Like other functional interpretations Gödel’s Dialectica interpretation gives rise to category theoretic constructions that serve both as new models for logic and semantics and as tools for analysing and understanding various aspects of the Dialectica interpretation itself. Gödel’s Dialectica interpretation gives rise to the Dialectica categories , in: Contemp. (...) Math., vol. 92, Amer. Math. Soc., Providence, RI, 1989, pp. 47–62] and J.M.E. Hyland in [J.M.E. Hyland, Proof theory in the abstract, Ann. Pure Appl. Logic 114 43–78, Commemorative Symposium Dedicated to Anne S. Troelstra ]). These categories are symmetric monoidal closed and have finite products and weak coproducts, but they are not Cartesian closed in general. We give an analysis of how to obtain weakly Cartesian closed and Cartesian closed Dialectica categories, and we also reflect on what the analysis might tell us about the Dialectica interpretation. (shrink)

It is proved that equations between arrows assumed for cartesian categories are maximal in the sense that extending them with any new equation in the language of free cartesian categories collapses a cartesian category into a preorder. An analogous result holds for categories with binary products, which may lack a terminal object. The proof is based on a coherence result for cartesian categories, which is related to model-theoretic methods of normalization. This maximality of cartesian categories, (...) which is analogous to Post completeness, shows that the usual equivalence between deductions in conjunctive logic induced by βη normalization in natural deduction is chosen optimally. (shrink)

The simplest combination of unimodal logics \ into a bimodal logic is their fusion, \, axiomatized by the theorems of \. Shehtman introduced combinations that are not only bimodal, but two-dimensional: he defined 2-d Cartesian products of 1-d Kripke frames, using these Cartesian products to define the frame product \. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman’s idea and introduced the topological product \, using Cartesian products of topological spaces rather than of Kripke (...) frames. Frame products have been extensively studied, but much less is known about topological products. The goal of the current paper is to give necessary and sufficient conditions for the topological product to match the frame product, for Kripke complete extensions of \. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Commentary on “Non-Cartesian Frameworks”Rom Harré (bio)There are three points in Dr. Berger’s paper that seem to me to call for immediate comment:1. There is the familiar (but in Berger’s case, only a partial) misunderstanding of the upshot of the third phase of Wittgenstein’s private-language argument. Having shown that expressive and descriptive discourse are radically different, and that expressive discourse can be learned only in contexts of action in (...) which there are natural expressions of personal feeling, experience, and so on, Wittgenstein goes on to show that whatever furnishes the mind, it is not thing-like. He does this by showing that “sameness” of thoughts and feelings cannot be spelled out in the same terms as sameness of things, that is, in terms of numerical and qualitative identity. Yet there is a proper use for the concept of sameness of thoughts and feelings. It is to be found in the role that expressions of feelings and declarations of thoughts play in the language games that constitute a form of life. There is no “unresolved tension,” as Berger seems to think. Although pains are not thing-like, they do have a location, do endure, and do differ from one another.The introduction of the powerful expressive/descriptive distinction is the key to the fallacies of representationalism. Expressive acts do not represent but descriptive acts might. But we can draw on the demonstration that mental “somethings” are not thing-like to eliminate that possibility. The root of representationalism is the seventeenth-century conception of the mind as a container of mental entities: ideas. Ideas are to psychology what atoms are to physics. According to Hume, ideas are copies of impressions. In that disastrous thesis, representationalism was born, as a generalization of Locke’s grounding principle that the ideas of primary qualities resembled primary qualities. The cure for this complex mesh of mistakes must be deeply philosophical. It must not be confined to easy criticisms of Locke’s resemblance doctrine, not yet to more subtle Berkleyan strictures on the supposed distinction between ideas and qualities. It must cut at the very deepest assumption of representationalism—that there are mental entities. And on that matter, Wittgenstein’s analysis will, one hopes, prove to be final.2. With representationalism cast onto the midden of history, what account are we to give of mentation? In a sense, Wittgenstein’s insights abolish the inner/outer distinction, an offshoot of the concept of the container-mind. But they leave standing the distinction between what is done privately and what is done publicly. Yet these domains must be of the same nature, since in the expressive use of signs, we find a holism encompassing what we feel and our disposition to express it. Linguistic substitutes for natural forms [End Page 185] of expression must share that holism. But there are no natural expressions of propositional thoughts. Why? Because we are capable of private propositional thinking just insofar as we have mastered public uses of the instruments of propositional thought. These instruments are cultural products.The practice of psychotherapy can be nothing other than discursive. The quotations concerning praxis that Berger offers are disconcertingly vague. To add, as Berger does, that we must take theory into account does not cure the vagueness. We do not know what Berger takes theory to be, only that it is not theory as physicists have it; i.e., systems of hypotheses about hidden processes which bring about that which can be observed. Following Wittgenstein one step farther, we can say that the only thing that could be theory-like in the enterprises of psychotherapists would be grammars. Grammars express the often unformulated norms of discourses. Psychoanalytic theories could then be addressed as expressions of the norms of certain kinds of discourses. Incorporating a “theory” would then be nothing but determining to adopt a certain grammar in one’s therapeutic discourses. Freed of the idea that one of some set of theories must be true and preferred to all others, one is free to adopt and adapt grammars to occasions. Some grammars are tied to natural expressions and so to the ethology of the human form of life... (shrink)

In this research article, we presented the idea of intuitionistic fuzzy incidence graphs along with their certain properties. The number of operations including Cartesian product, composition, tensor product, and normal product in an IFIGs are also investigated. The method to compute the degree of IFIGs obtained by CP, composition, tensor product, and the normal product is discussed. Some important theorems to calculate the degree of the vertices of IFIGs acquired by CP, composition, tensor (...) class='Hi'>product, and normal product are elaborated. An application of CP and composition of two IFIGs in the textile industry to find the best combinations of departments expressing the highest percentage of progress and the lowest percentage of nonprogress is provided. A comparative analysis of our study with the existing study is discussed. Our study will be beneficial to comprehend and understand the further characteristics of IFIGs in detail. Another advantage of our study is that it will be helpful to find the maximum percentage of progress and minimum percentage of nonprogress in different departments of universities, garment factories, and hospitals. (shrink)

Working in the fragment of Martin-Löfs extensional type theory [12] which has products (but not sums) of dependent types, we consider two additional assumptions: firstly, that there are (strong) equality types; and secondly, that there is a type which is universal in the sense that terms of that type name all types, up to isomorphism. For such a type theory, we give a version of Russell's paradox showing that each type possesses a closed term and (hence) that all terms of (...) each type are provably equal. We consider the kind of category theoretic structure which corresponds to this kind of type theory and obtain a categorical version of the paradox. A special case of this result is the degeneracy of a locally cartesian closed category with a morphism which is generic in the sense that every other morphism in the category can be obtained from it via pullback. (shrink)

In an earlier article [Found. Phys. 30, 1191 (2000)], a quasiclassical phase space approximation for quantum projection operators was presented, whose accuracy increases in the limit of large basis size (projection subspace dimensionality). In a second paper [J. Chem. Phys. 111, 4869 (1999)], this approximation was used to generate a nearly optimal direct-product basis for representing an arbitrary (Cartesian) quantum Hamiltonian, within a given energy range of interest. From a few reduced-dimensional integrals, the method determines the optimal 1D (...) marginal Hamiltonians, whose eigenstates comprise the direct-product basis. In the present paper, this phase space optimized direct-product basis method is generalized to incorporate non-Cartesian coordinate spaces, composed of radii and angles, that arise in molecular applications. Analytical results are presented for certain standard systems, including rigid rotors, and three-body vibrators. (shrink)

This study provides an innovative approach to neutrosophic algebraic structures by introducing a new structure called Neutrosophic Soft Cubic T-ideal (NSCTID), which combines T-ideal (TID) and neutrosophic Soft Cubic Sets (NSCSs) within the framework of PS-Algebra. Within the already-existing neutrosophic cubic structures, the addition of soft sets with the characteristics of TID makes this structure more desirable. The theoretical development of the proposed structure includes the application of fundamental ideas as union, intersection, the Cartesian product, and homomorphism. We (...) also introduce the notions of NSCTID-translation and NSCTID-multiplication to further enhance the structure of NSCTID. By applying the idea of translation and multiplication, we offer improved algorithm for neutrosophic cubic sets to deal with different parameters that are satisying the TID’s distinctive characteristics. Through this thorough research, we offer an elementary understanding of NSCTID and its capabilities, providing the way to new algebraic structures. (shrink)

Recently a new attempt to go beyond QM was performed in the form of so-called prequantum classical statistical field theory (PCSFT). In this approach quantum systems are described by classical random fields, e.g., the electron field or the neutron field. Averages of quantum observables arise as approximations of averages of classical variables (functionals of “prequantum fields”) with respect to fluctuations of fields. For classical variables given by quadratic functionals of fields, quantum and prequantum averages simply coincide. In this paper we (...) generalize PCSFT to the composite quantum system. The main discovery is that, opposite to a rather common opinion, a composite system can be described by the Cartesian product of state spaces (like in classical physics) and not by the tensor product of them (like in conventional QM). A natural interpretation of the quantum pure state for a composite system is proposed: it is the nondiagonal block in the covariance matrix of the random field describing a composite system. The interpretation of a pure state due to Dirac and von Neumann seems to be an artifact of the conventional mathematical description of micro-systems. PCSFT provides a new possibility for interpretation of entanglement. (shrink)

A new combined temporal logic called synchronized linear-time temporal logic (SLTL) is introduced as a Gentzen-type sequent calculus. SLTL can represent the n -Cartesian product of the set of natural numbers. The cut-elimination and completeness theorems for SLTL are proved. Moreover, a display sequent calculus δ SLTL is defined.

We explore a possibility of generalization of classical truth values by distinguishing between their ontological and epistemic aspects and combining these aspects within a joint semantical framework. The outcome is four generalized classical truth values implemented by Cartesian product of two sets of classical truth values, where each generalized value comprises both ontological and epistemic components. This allows one to define two unary twin connectives that can be called “semi-classical negations”. Each of these negations deals only with one (...) of the above mentioned components, and they may be of use for a logical reconstruction of argumentative reasoning. (shrink)

We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product of one-dimensional definable group-intervals. We discuss the general open question of elimination of imaginaries in an o-minimal structure.

We introduce a ranking of multidimensional alternatives, including uncertain prospects as a particular case, when these objects can be given a matrix form. This ranking is separable in terms of rows and columns, and continuous and monotonic in the basic quantities. Owing to the theory of additive separability developed here, we derive very precise numerical representations over a large class of domains (i.e., typically notof the Cartesian product form). We apply these representationsto (1)streams of commodity baskets through time, (...) (2)uncertain social prospects, (3)uncertain individual prospects. Concerning(1), we propose a finite horizon variant of Koopmans’s (1960) axiomatization of infinite discounted utility sums. The main results concern(2). We push the classic comparison between the exanteand expostsocial welfare criteria one step further by avoiding any expected utility assumptions, and as a consequence obtain what appears to be the strongest existing form of Harsanyi’s (1955) Aggregation Theorem. Concerning(3), we derive a subjective probability for Anscombe and Aumann’s (1963) finite case by merely assuming that there are two epistemically independent sources of uncertainty. (shrink)

This paper presents an information-based logic that is applied to the analysis of entailment, implicature and presupposition in natural language. The logic is very fine-grained and is able to make distinctions that are outside the scope of classical logic. It is independently motivated by certain properties of natural human reasoning, namely partiality, paraconsistency, relevance, and defeasibility: once these are accounted for, the data on implicature and presupposition comes quite naturally.The logic is based on the family of semantic spaces known as (...) bilattices, originally proposed by Ginsberg (1988), and used extensively by Fitting (1989, 1992). Specifically, the logic is based on a subset of bilattices that I call evidential bilattices, constructed as the Cartesian product of certain algebras with themselves. The specific details of the epistemic agent approach of the logical system is derived from the work of Belnap (1975, 1977), augmented by the use of evidential links for inferencing. An important property of the system is that it has been implemented using an extension of Fitting's work on bilattice logic programming (1989, 1991) to build a model-based inference engine for the augmented Belnap logic. This theorem prover is very efficient for a reasonably wide range of inferences. (shrink)

Given a positive universal formula in the language of residuated lattices, we construct a recursive basis of equations for a variety, such that a subdirectly irreducible residuated lattice is in the variety exactly when it satisfies the positive universal formula. We use this correspondence to prove, among other things, that the join of two finitely based varieties of commutative residuated lattices is also finitely based. This implies that the intersection of two finitely axiomatized substructural logics over FL + is also (...) finitely axiomatized. Finally, we give examples of cases where the join of two varieties is their Cartesian product. (shrink)

The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [2] for sets and functions definable in models of weakly o-minimal theories. We pay special attention to large subsets (...) of Cartesian products of definable sets, showing that if X, Y and S are non-empty definable sets and S is a large subset of X × Y, then for a large set of tuples $\langle \overline{a}_{1},\ldots,\overline{a}_{2^{k}}\rangle \in X^{2^{k}}$ , where k = dim(Y), the union of fibers $S_{\overline{a}_{1}}\cup \cdots \cup S_{\overline{a}_{2^{k}}}$ is large in Y. Finally, given a weakly o-minimal structure ������, we find various conditions equivalent to the fact that the topological dimension in ������ enjoys the addition property. (shrink)

The paper introduces the notion of size of countable sets, which preserves the Part-Whole Principle. The sizes of the natural and the rational numbers, their subsets, unions, and Cartesian products are algorithmically enumerable as sequences of natural numbers. The method is similar to that of Numerosity Theory, but in comparison it is motivated by Bolzano’s concept of infinite series, it is constructive because it does not use ultrafilters, and set sizes are uniquely determined. The results mostly agree, but some (...) differ, such as the size of rational numbers. However, set sizes are only partially, not linearly, ordered. Quid pro quo. (shrink)