According to Kant, pure intuition is an indispensable ingredient of mathematical proofs. Kant‘s thesis has been considered as obsolete since the advent of modern relational logic at the end of 19th century. Against this logicist orthodoxy Cassirer’s “critical idealism” insisted that formal logic alone could not make sense of the conceptual co-evolution of mathematical and scientific concepts. For Cassirer, idealizations, or, more precisely, idealizing completions, played a fundamental role in the formation of the mathematical and empirical concepts. The aim of (...) this paper is to outline the basics of Cassirer’s idealizational account, and to point at some interesting similarities it has with Kant’s and Peirce’s philosophies of mathematics based on the key notions of pure intuition and theorematic reasoning, respectively. (shrink)

This paper presents a formalization of the classical proof of completeness in Henkin-style developed by Troelstra and van Dalen for intuitionistic logic with respect to Kripke models. The completeness proof incorporates their insights in a fresh and elegant manner that is better suited for mechanization. We discuss details of our implementation in the Lean theorem prover with emphasis on the prime extension lemma and construction of the canonical model. Our implementation is restricted to a system of intuitionistic propositional (...) logic with implication, conjunction, disjunction, and falsity given in terms of a Hilbert-style axiomatization. As far as we know, our implementation is the first verified Henkin-style proof of completeness for intuitionistic logic following Troelstra and van Dalen's method in the literature. (shrink)

The first aim of this paper is to prove a topological completeness theorem for a weak version of Stalnaker’s logic KB of knowledge and belief. The weak version of KB is characterized by the assumption that the axioms and rules of KB have to be satisfied with the exception of the axiom (NI) of negative introspection. The proof of a topological completeness theorem for weak KB is based on the fact that nuclei (as defined in the framework of (...) point-free topology) give rise to a profusion of topological belief operators that are compatible with the familiar topological knowledge operator. Thereby a canonical topological model for weak KB can be constructed. For this canonical model a truth lemma for the K and B holds such that a completeness theorem for KB can be proved in the familiar way. The second aim of this paper is to show that the topological interpretation of knowledge K comes along with a complete Heyting algebra of belief operators B that all fit the knowledge operator K in the sense that the pairs (K, B) satisfy all axioms of weak KB. This amounts to a pluralistic relation between knowledge and belief: Knowledge does not fully determine belief, rather it designs a conceptual space for belief operators where different (competing) belief operators coexist. (shrink)

A construction principle for natural deduction systems for arbitrary, finitely-many-valued first order logics is exhibited. These systems are systematically obtained from sequent calculi, which in turn can be automatically extracted from the truth tables of the logics under consideration. Soundness and cut-free completeness of these sequent calculi translate into soundness, completeness, and normal-form theorems for natural deduction systems.

Modern fundamental science tends to avoid the principle of physical causality and realism, replacing it with heuristically postulated and separated mathematical constructions that impose their own rules before being adjusted to measurement results. While it is officially accepted as the single possible kind of rigorous knowledge, we argue that another, explicitly extended kind of science can provide the causally complete picture of reality avoiding the glaring gaps, growing problems and persisting stagnation of the artificially reduced knowledge paradigm. The logic of (...) science development and especially its current state call for the urgently needed transition to the extended, causally complete kind of knowledge and related sustainable progress at superior creativity levels of self-aware, reason-based society. (shrink)

Higher-order theories of properties, relations, and propositions are known to be essentially incomplete relative to their standard notions of validity. It turns out that the first-order theory of PRPs that results when first-order logic is supplemented with a generalized intensional abstraction operation is complete. The construction involves the development of an intensional algebraic semantic method that does not appeal to possible worlds, but rather takes PRPs as primitive entities. This allows for a satisfactory treatment of both the modalities and the (...) propositional attitudes, and it suggests a general strategy for developing a comprehensive treatment of intensional logic. (shrink)

Finding truth is an art that was learned and unlearned. Truth can only be found by looking for independent confirmation of our beliefs, by reality. This methodology is difficult to apply in personal- and social settings, because power and politics turn 'seeking independent confirmation' into 'avoiding dependent rejection'. A completely different social order is implied and the one keeps running the other into the ground like a tectonic plate. Philosophical Modernism showed us how dualism works, before Post-Modernism challenged it, regressing (...) to monism. (shrink)

Reviews of studies on sexual orientation reveal that researchers have not developed a clear conceptual definition of sexual orientation and that most researchers have failed to provide a theoretical framework for their studies. While some scholars recognize the conceptual confusion that has plagued the development of a theory of sexual orientation, most researchers study sexual orientation as if it were an immutable, essential nature of an individual without questioning how social context influences categorization. The problem with this approach is that (...) components have been hypothesized to be part of the construct, but there has been no systematic study of how to judge the accuracy of the construct. The present study suggests that, because categories of sexual orientation are socially constructed, the study of sexual orientation would benefit from an understanding of how the ordinary language community defines and categorizes sexual orientation. The purpose of the present study was to test the semantics of various category labels of sexual orientation used in ordinary language. Three hypothesized components of sexual orientation , involving either same-sex or opposite-sex oriented behavior, along with a fourth stimulus condition were manipulated in a 2 x 2 x 2 x 2 factorial design. Subjects were 371 college students who completed a questionnaire in which they were asked to give a one-word description in response to written statements about hypothetical individuals described as male or female who engage in various same- or opposite-sex behavior. Contrary to what was hypothesized, results demonstrate that there were some necessary and sufficient conditions for categorizing sexual orientation. Also contrary to a proto-behavioral hypothesis, the results of the relative strength of the three components in predicting category label indicate that sexual attraction was the strongest predictor, followed by overt sexual behavior, and sexual fantasy, respectively. The most impressive finding was the overwhelming support for the impurity hypothesis in which descriptions with any same-sex oriented behavior appeared to be judged to contaminate the individual's sexual orientation resulting in greater frequencies of responses of "homosexual" compared to other category labels. Finally, as predicted, overt sexual behavior was least predictive of the sexual orientation of females with sexual fantasy being the most predictive. For descriptions of males, sexual attraction was the most important predictor. Provided in the discussion are possible explanations for the current findings, a description of the limitations and implications of the study, and recommendations for future research. (shrink)

From the time of Confucius and Aristotle up until the present day, theorists have argued that the individual self exists only as an aspect of social structures. The claim is not merely that the self is causally affected by social structures; but that the self is just social structure. The most recent iteration of this claim comes in book-length from Brian Lowery, though the argument was made more completely by Charles Taylor and Kenneth Gergen in the preceding decades. The most (...) rigorous version of the argument draws on Wittgenstein, claiming that thought itself is impossible without socially constructed meanings. Wittgenstein argued that a “private language” would be incoherent because the person using the language could not have any criteria by which to judge the correctness of the symbols used. However, this reasoning fails as an argument against individualized thought because it applies equally well to the use of public languages. So this paper argues that eliminative social constructivism has never made any sense, least of all in its latest form. (shrink)

Principles are central to physical reasoning, particularly in the search for a theory of quantum gravity (QG), where novel empirical data is lacking. One principle widely adopted in the search for QG is UV completion: the idea that a theory should (formally) hold up to all possible high energies. We argue---/contra/ standard scientific practice---that UV-completion is poorly-motivated as a guiding principle in theory-construction, and cannot be used as a criterion of theory-justification in the search for QG. For this, we explore (...) the reasons for expecting, or desiring, a UV-complete theory, as well as analyse how UV completion is used, and how it should be used, in various specific approaches to QG. (shrink)

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition (...) for granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined. (shrink)

The book "Theory of Forms: The Construction of Plato and Aristotle’s Criticism" focuses on two main aspects, construction and criticism. The constriction of Forms theory is the basis on which Plato built all of his philosophy and which influenced all forms of ideas philosophy that emerged after Plato. The research topic was completed by adding Aristotle's critique of the theory of Forms in order to put a clear picture in front of the reader, which was presented by Plato himself and (...) Aristotle's criticism of this theory. -/- Therefore, the book consists four chapters, the first chapter covers the following topics; The philosophical roots of Plato's theory of Forms, The numbers in the Pythagorean school, The Heraclitus' idea of change, Anaxagoras’ idea of Mind, Sophists. Chapter two, Plato's theory of Forms. Definitions of the Forms. The importance of the Forms, The existential and logical significance, Evidence of Forms, Evidence that knowledge is remembering. Chapter Three, Forms and Levels of Knowledge, Dialectic ,Inferential knowledge, Mental knowledge The fourth chapter, The nature of the Forms existence and its being Causes. Forms as numbers and their relationship to movement. The problem of Participation. The final Evaluation of Forms. (shrink)

This article develops an axiom system to justify an additive representation for a preference relation ${\succsim}$ on the product ${\prod_{i=1}^{n}A_{i}}$ of extensive structures. The axiom system is basically similar to the n-component (n ≥ 3) additive conjoint structure, but the independence axiom is weakened in the system. That is, the axiom exclusively requires the independence of the order for each of single factors from fixed levels of the other factors. The introduction of a concatenation operation on each factor A i (...) makes it possible to yield a special type of restricted solvability, i.e., additive solvability and the usual cancellation on ${\prod_{i=1}^{n}A_{i}}$ . In addition, the assumption of continuity and completeness for A i implies a stronger type of solvability on A i . The additive solvability, cancellation, and stronger solvability axioms allow the weakened independence to be effective enough in constructing the additive representation. (shrink)

The Reality of Knowledge completes a trilogy begun with The Architecture of Knowledge (1980) and Processes of Knowledge (2001). It presents a holistic analysis of knowledge and the reality that is known. The book shows how living things, including humans, construct reality in specific ways that maximize their ability to know it. Different species construct different areas of reality, but they all use the same methods: objectification, categorization, and generalization. Support for this analysis comes from examining certain logical protocols in (...) computer technology, because computer architectures have evolved to emulate human knowledge. (shrink)

This paper argues that in addition to the familiar approach using formal contexts, there is now a need in artificial intelligence to study contexts as social constructs. As a successful example of the latter approach, I draw attention to 'interpretation' (in the sense of literary theory), viz. the reconstruction of the intended meaning of a literary text that takes into account the context in which the author assumed the reader would place the text. An important contribution here comes from Wendell (...) Harris, enumerating the seven crucial dimensions of context: knowledge of reality, knowledge of language, and the authorial, generic, collective, specific, and textual dimensions. Finally, two recent approaches to interpretation, due to Jon Barwise and Jerry Hobbs, are analyzed as useful attempts which also come to grips with the notion of context. It must be noted that there has been a considerable body of contributions connecting linguistic structure with social context. For example, anthropological linguistics, from Bronislaw Malinowski onwards, has underlined the cultural context of discourse as essential to meaning. This viewpoint became prominent with the emergence of the ethnography of speaking in anthropology. Thus, conversation analysis represents a consistent formal effort to contribute to an analysis of the nature of context. While this paper emphasizes and reviews the literary theory approach, it makes various contacts with works of the latter kind (e.g., the landmark contributions of Erving Goffman, John Gumperz, William Hanks, John Heritage, Dell Hymes, et al.) in order to deliver a more balanced and complete study of the dimensions of context. (shrink)

Research has shown that the construction of visual representations may have a positive effect on cognitive skills, including argumentation. In this paper we present a study on learning argumentation through computer-supported argument diagramming. We specifically focus on whether students, when provided with an argument-diagramming tool, create better diagrams, are more motivated, and learn more when working with other students or on their own. We use learning analytics to evaluate a variety of student activities: pre and post questionnaires to explore motivational (...) changes; the argument diagrams created by students to evaluate richness, complexity and completion; and pre and post knowledge tests to evaluate learning gains. (shrink)

In this paper it is shown how a formal theory of interpretation in Montague’s style can be reconciled with a view on meaning as a social construct. We sketch a formal theory in which agents can have their own theory of interpretation and in which groups can have common theories of interpretation. Frege solved the problem how different persons can have access to the same proposition by placing the proposition in a Platonic realm, independent from all language users but accessible (...) to all of them. Here we explore the alternative of letting meaning be socially constructed. The meaning of a sentence is accessible to each member of a linguistic community because the way the sentence is to be interpreted is common knowledge among the members of that community. Misunderstandings can arise when the semantic knowledge of two or more individuals is not completely in sync. (shrink)

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then (...) adopt what may be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. -/- We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. -/- We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. -/- We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences. -/- . (shrink)

Once I had explored the research issue of North and South unification with a focus on the legal integration for uniform constitution and various statutes. It pushed me to deal with a big question, and looked like a semi-textbook with an inchoate idea and baby theory upon the completion of research project. The literature review thankfully had allowed the space of creativity and originality of my work product, and can also be a typical way of foreign graduate legal researchers in (...) the process of his dissertation work. They would be beneficial with new statutes and codes, for example, from the Korean Congress, that are enacted recently, which would be their research topic certainly original and contributive. This could be viewed as one of qualitative enclaves enmeshed into a larger scope of qualitative inquiry. This interplay is fairly consequential in excavating a new knowledge and research findings. It could be aboriginal -- one trait word to characterize the qualitative studies-- from the viewpoint of host country, but became original as a crop to expand the horizon of new knowledge. I have once argued that the quantitative studies are on generality of subjects and commonness for involved strands. This profile would vary that the qualitative studies tend to be strong with the story and enclaves. The e-age and rapid compression of space with the development of telecommunication technology also factor to change a paradigm of knowledge and research activities. At least, the contemporary society certainly would be more specific as if we can trace a person with the satellite spotting and the electronic identifier on the leg of sex criminals in Korea guarantees to be safeguarded from his potential danger. That may be favorable concerning the elements and environments of qualitative studies. For example, we may follow up by comparing the research work of Korean statute with the e-government of Korea displaying the statute. We can entertain many on-line sources of information and stories with the dissertations of aboriginal culture. With my time on the qualitative studies, I came across the nocebo effect illustrated with the fear of MERs-CoV in Korea. The effect looks into the negative dimension of human mind and his personality, which could aggravate the current challenge of national crisis, June, 2015. Paradoxically, the nocebo effect can be said to characterize the qualitative context of inquiry. As Freud suggested, the vast of human element is of unexplainable dimension that is less easily quantifiable or generizable over the population (Broome, 2010). This does not say that the continental researchers had been affected with a late nationalization or late development of region leaving them to search for an ego. This is to say that the qualitative studies could explain the part of truths for the universe and human. The researchers interested in deep knowledge could employ their method, but in ethics, who would not vulture the objects of his theme and conduct on due scientific ground. (shrink)

This work studies some problems connected to the role of negation in logic, treating the positive fragments of propositional calculus in order to deal with two main questions: the proof of the completeness theorems in systems lacking negation, and the puzzle raised by positive paradoxes like the well-known argument of Haskel Curry. We study the constructive com- pleteness method proposed by Leon Henkin for classical fragments endowed with implication, and advance some reasons explaining what makes difficult to extend (...) this constructive method to non-classical fragments equipped with weaker implications (that avoid Curry's objection). This is the case, for example, of Jan Lukasiewicz's n-valued logics and Wilhelm Ackermann's logic of restricted implication. Besides such problems, both Henkin's method and the triviality phenomenon enable us to propose a new positive tableau proof system which uses only positive meta-linguistic resources, and to mo- tivate a new discussion concerning the role of negation in logic proposing the concept of paratriviality. In this way, some relations between positive reasoning and infinity, the possibilities to obtain a ¯first-order positive logic as well as the philosophical connection between truth and meaning are dis- cussed from a conceptual point of view. (shrink)

Complete deductive systems are constructed for the non-valid (refutable) formulae and sequents of some propositional modal logics. Thus, complete syntactic characterizations in the sense of Lukasiewicz are established for these logics and, in particular, purely syntactic decision procedures for them are obtained. The paper also contains some historical remarks and a general discussion on refutation systems.

A complete axiomatic system CTL$_{rp}$ is introduced for a temporal logic for finitely branching $\omega^+$-trees in a temporal language extended with so called reference pointers. Syntactic and semantic interpretations are constructed for the branching time computation tree logic CTL$^{*}$ into CTL$_{rp}$. In particular, that yields a complete axiomatization for the translations of all valid CTL$^{*}$-formulae. Thus, the temporal logic with reference pointers is brought forward as a simpler (with no path quantifiers), but in a way more expressive medium for reasoning (...) about branching time. (shrink)

Classical interpretations of Goedels formal reasoning, and of his conclusions, implicitly imply that mathematical languages are essentially incomplete, in the sense that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is, both, non-algorithmic, and essentially unverifiable. However, a language of general, scientific, discourse, which intends to mathematically express, and unambiguously communicate, intuitive concepts that correspond to scientific investigations, cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth (...) verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Goedel's reasoning, under which any formal system of Peano Arithmetic---classically accepted as the foundation of all our mathematical Languages---is verifiably complete in the above sense. We show how some paradoxical concepts of Quantum mechanics can, then, be expressed, and interpreted, naturally under a constructive definition of mathematical truth. (shrink)

Textbook for students in mathematical logic. Part 1. Total formalization is possible! Formal theories. First order languages. Axioms of constructive and classical logic. Proving formulas in propositional and predicate logic. Glivenko's theorem and constructive embedding. Axiom independence. Interpretations, models and completeness theorems. Normal forms. Tableaux method. Resolution method. Herbrand's theorem.

ABSTRACT A complete axiomatic system CTLrp is introduced for a temporal logic for finitely branching ω+ -trees in a language extended with so called reference pointers. Syntactic and semantic interpretations are constructed for the branching time computation tree logic CTL* into CTLrp. In particular, that yields a complete axiomatization for the translations of all valid CTL*-formulae. Thus, the temporal logic with reference pointers is brought forward as a simpler (with no path quantifiers), but in a way more expressive medium for (...) reasoning about branching time. (shrink)

In the present article we attempt to show that Aristotle's syllogistic is an underlying logiC which includes a natural deductive system and that it isn't an axiomatic theory as had previously been thought. We construct a mathematical model which reflects certain structural aspects of Aristotle's logic. We examine the relation of the model to the system of logic envisaged in scattered parts of Prior and Posterior Analytics. Our interpretation restores Aristotle's reputation as a logician of consummate imagination and skill. Several (...) attributions of shortcomings and logical errors to Aristotle are shown to be without merit. Aristotle's logic is found to be self-sufficient in several senses: his theory of deduction is logically sound in every detail. (His indirect deductions have been criticized, but incorrectly on our account.) Aristotle's logic presupposes no other logical concepts, not even those of propositional logic. The Aristotelian system is seen to be complete in the sense that every valid argument expressible in his system admits of a deduction within his deductive system: every semantically valid argument is deducible. (shrink)

This paper and its sequel “look under the hood” of the usual sorts of proof-theoretic systems for certain well-known intuitionistic and classical propositional modal logics. Section 1 is preliminary. Of most importance: a marked formula will be the result of prefixing a formula in a propositional modal language with a step-marker, for this paper either 0 or 1. Think of 1 as indicating the taking of “one step away from 0.” Deductions will be constructed using marked formulas. Section 2 presents (...) the model-theoretic concepts, based on those in [7], that guide the rest of this paper. Section 3 presents Natural Deduction systems IK and CK, formalizations of intuitionistic and classical one-step versions of K. In these systems, occurrences of step-markers allow deductions to display deductive structure that is covered over in familiar “no step” proof-theoretic systems for such logics. Box and Diamond are governed by Introduction and Elimination rules; the familiar K rule and Necessitation are derived (i.e. admissible) rules. CK will be the result of adding the 0-version of the Rule of Excluded Middle to the rules which generate IK. Note: IK is the result of merely dropping that rule from those generating CK, without addition of further rules or axioms (as was needed in [7]). These proof-theoretic systems yield intuitionistic and classical consequence relations by the obvious definition. Section 4 provides some examples of what can be deduced in IK. Section 5 defines some proof-theoretic concepts that are used in Section 6 to prove the soundness of the consequence relation for IK (relative to the class of models defined in Section 2.) Section 7 proves its completeness (relative to that class). Section 8 extends these results to the consequence relation for CK. (Looking ahead: Part 2 will investigate one-step proof-theoretic systems formalizing intuitionistic and classical one-step versions of some familiar logics stronger than K.). (shrink)

Hyperboolean algebras are Boolean algebras with operators, constructed as algebras of complexes (or, power structures) of Boolean algebras. They provide an algebraic semantics for a modal logic (called here a {\em hyperboolean modal logic}) with a Kripke semantics accordingly based on frames in which the worlds are elements of Boolean algebras and the relations correspond to the Boolean operations. We introduce the hyperboolean modal logic, give a complete axiomatization of it, and show that it lacks the finite model property. The (...) method of axiomatization hinges upon the fact that a "difference" operator is definable in hyperboolean algebras, and makes use of additional non-Hilbert-style rules. Finally, we discuss a number of open questions and directions for further research. (shrink)

This paper investigates and develops generalizations of two-dimensional modal logics to any finite dimension. These logics are natural extensions of multidimensional systems known from the literature on logics for a priori knowledge. We prove a completeness theorem for propositional n-dimensional modal logics and show them to be decidable by means of a systematic tableau construction.

The aim of this paper is to show that every topological space gives rise to a wealth of topological models of the modal logic S4.1. The construction of these models is based on the fact that every space defines a Boolean closure algebra (to be called a McKinsey algebra) that neatly reflects the structure of the modal system S4.1. It is shown that the class of topological models based on McKinsey algebras contains a canonical model that can be used to (...) prove a completeness theorem for S4.1. Further, it is shown that the McKinsey algebra MKX of a space X endoewed with an alpha-topologiy satisfies Esakia's GRZ axiom. (shrink)

Violence is one of the most pervasive problems in the world today. Despite all efforts to apply the powers of reason in order to contain, if not completely eliminate violence, violence proves to be capable of escaping capture and re-emerging in new and unexpected forms. Reason and rationality appear to be powerless against violence. The paper explores some philosophical issues that shed new light on the persistence of violence in the modern world. It argues that the failure of modernity to (...) recognize and come to grips with the process of construction that constitutes the basis of our relationship with reality plays a critical role in the continued survival of violence. (shrink)

We generalize intuitionistic tense logics to the multi-modal case by placing grammar logics on an intuitionistic footing. We provide axiomatizations for a class of base intuitionistic grammar logics as well as provide axiomatizations for extensions with combinations of seriality axioms and what we call "intuitionistic path axioms". We show that each axiomatization is sound and complete with completeness being shown via a typical canonical model construction.

Recent experiments allowed concluding that Bell-type inequalities are indeed violated thus it is important to understand what it means and how can we explain the existence of strong correlations between outcomes of distant measurements. Do we have to announce that: Einstein was wrong, Nature is nonlocal and nonlocal correlations are produced due to the quantum magic and emerge, somehow, from outside space-time? Fortunately such conclusions are unfounded because if supplementary parameters describing measuring instruments are correctly incorporated in a theoretical model (...) then Bell-type inequalities may not be proven .We construct a simple probabilistic model explaining these correlations in a locally causal way. In our model measurement outcomes are neither predetermined nor produced in irreducibly random way. We explain in detail why, contrary to the general belief; an introduction of setting dependent parameters does not restrict experimenters’ freedom of choice. Since the violation of Bell-type inequalities does not allow concluding that Nature is nonlocal and that quantum theory is complete thus the Bohr-Einstein quantum debate may not be closed. The continuation of this debate is not only important for a better understanding of Nature but also for various practical applications of quantum phenomena. (shrink)

In this paper we prove the completeness of three logical systems I LI, IL2 and IL3. IL1 deals solely with identities {a = b), and its deductions are the direct deductions constructed with the three traditional rules: (T) from a = b and b = c infer a = c, (S) from a = b infer b = a and (A) infer a = a(from anything). IL2 deals solely with identities and inidentities {a ± b) and its deductions include (...) both the direct and the indirect deductions constructed with the three traditional rules. IL3 is a hybrid of IL1 and IL2: its deductions are all direct as in IL1 but it deals with identities and inidentities as in IL2. IL1 and IL2 have a high degree of naturalness. Although the hybrid system IL3 was constructed as an artifact useful in the mathematical study of IL1 and IL2, it nevertheless has some intrinsically interesting aspects. The main motivation for describing and studying such simple systems is pedagogical. In teaching beginning logic one would like to present a system of logic which has the following properties. First, it exemplifies the main ideas of logic: implication, deduction, non-implication, counterargument(or countermodel), logical truth, self-contradiction, consistency,satisfiability, etc. Second, it exemplifies the usual general metaprinciples of logic: contraposition and transitivity of implication, cut laws, completeness,soundness, etc. Third, it is simple enough to be thoroughly grasped by beginners. Fourth, it is obvious enough so that its rules do not appear to be arbitrary or purely conventional. Fifth, it does not invite confusions which must be unlearned later. Sixth, it involves a minimum of presuppositions which are no longer accepted in mainstream contemporary logic. (shrink)

A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information.

Scholars agree that Schelling’s critique of Hegel consists in charging reason with an inability to account for its own possibility. This is not an attack on reason’s project of constructing a logical system, but rather on the pretense of doing so with complete justification and so without presuppositions, as if it were obvious why there is a logical system or why there is anything meaningful at all. Scholars accordingly cite the question ‘why is there something rather than nothing’ as emblematic (...) of Schelling’s critique and the source of his claim against Hegel that reason is bounded by something other than itself. But what sort of claim is this? For what sort of argument could it be the conclusion? On these questions, the literature is lacking. I propose that the question ‘why something’ motivates Schelling to give a transcendental argument to the effect that past and future are a priori conditions of the possibility of reason. As I reconstruct it from the Weltalter, this argument construes the past as the pre-rational decision to construct a logical system, while the future represents the purpose that guides the construction of the same. Schelling’s claim against Hegel that reason is bounded by something other is thus the discovery of inescapable presuppositions that condition the possibility of reason as such. (shrink)

This is a forthcoming section for the book "Theism and Atheism: Opposing Arguments in Philosophy", edited by Graham Oppy, Gregory Dawes, Evan Fales, Joseph Koterski, Mashhad Al-Allaf, Robert Fastiggi, and David Shatz. I was asked to write a brief essay on divine hiddenness and divine inscrutability. I argue that theism is trapped between two opposite poles. On one end, we encounter an argument developed by John Schellenberg. God is understood as a being who, in virtue of God's perfect love, would (...) have relationship with any of Her creatures who are open to loving relationship with God. If so, we would not expect there to exist non-resistant non-believers. Yet there are non-resistant non-believers. So, this sort of God does not exist. On the other pole, God is completely incomprehensible. But if God is incomprehensible, then this leads to a series of theological problems. For example, if God is incomprehensible, then it is difficult to understand how any human could rationally endorse God's existence. I argue that both poles are problematic for the theist and that it is difficult to see how to construct a road through the middle that would not inherit the problems of either pole. (shrink)

A history of the construction of number has been in line with the process of recognition about the properties of geometry. Natural number representing countability is exhibited on a straight line and the completeness of real number is also originated from the continuous property of the number line. Complex number on a plane off the number line is established and thereafter, the whole number system is completed. When the process of constructing a number with geometric features is investigated from (...) different perspectives, it provides a new insight into the fundamental principle of number, which leads to a novel methodology in mathematics. (shrink)

[This is the complete issue of the second issue of Mɛtascience] -/- This second issue of the journal Mεtascience continues the char acterization of this new branch of knowledge that is metasci ence. If it is new, it is not in a radical sense since Mario Bunge practiced it in an exemplary way, since logical positivists were accused of practicing only a mere metascience, since scientists have always practiced it implicitly, and since some philosophers no longer practice philosophy but rather (...) metascience, but without characterizing it or theorizing it, that is, without realizing that they have abandoned one general discourse for another. The novelty therefore lies in this aware ness that a general discourse without philosophy is possible: a scien tific general discourse. The twelve contributions gathered in this volume illustrate the metascientific approach to knowledge of the world as well as to knowledge of knowledge of the world, that is, science. And like Bunge’s project, they are neither part of the analytical movement nor the continental movement in philosophy. We will read here studies about the Bungean system, some applications of Bungean thought, some metascientific contributions, and some reflections around meta science. Among metascientific disciplines, ontology occupies a prominent place in this issue of Mεtascience. Metascience differs from philoso phy in its rejection of the fundamental philosophical distinction be tween appearance and reality. Metascientific ontology therefore does not postulate the existence of any metaphysical reality. But metasci entific ontology, no more than philosophical ontology, is a factual sci ence. The first, because it studies scientific constructs and not concrete objects, the second, because it is interested in transcendent or meta physical objects. (shrink)

Oliver Marchart constructs an elaborate ontologization of the political that builds on theories developed by the Essex School while relying on Heideggerianism and Hegelianism. This original thought is a powerful and convincing attempt to think the ontology of the political without lapsing into a celebration of essentialist grounding or complete groundlessness, which are equally metaphysical and mutually supporting positions. Tensions arise within Marchart’s own thought when the notion of instrumentality appears to be inscribed solely on the side of politics or (...) the ontic. I suggest that a theory of practical judgment that is inchoate in Marchart’s own position can resolve the tensions toward constructing a genuinely materialist ontology. (shrink)

The aim of the research is to identify the relationship between the performance criteria and the achievement of the objectives of supervision which is represented in the performance of the job at the Islamic University in Gaza Strip. To achieve the objectives of the research, the researchers used the descriptive analytical approach to collect information. The questionnaire consisted of (22) paragraphs distributed to three categories of employees of the Islamic University (senior management, faculty members, their assistants and members of the (...) administrative board). A random sample of 314 employees was selected, 276 responses were retrieved with a return rate of 88.1%. The SPSS program was used to enter, process, and analyze the data. The results of the study showed a positive relationship between the performance criteria and the achievement of the control objectives represented by the job performance in the Islamic University from the point of view of the members (senior management, faculty and their assistants and the administrative board). The researchers also recommended a number of recommendations, including the provision of an appropriate level of control system components today through the continuous updating and development of performance standards and the need to provide the necessary physical and financial resources to continue the development and achievement within the university. Expand the development of technology in the various activities of the university through the construction of a complete and integrated system to support the control systems in the university to suit its size. The researchers also recommended the follow-up, review of the performance standards and work to modify them in line with the mission of the university and the goals that the university seeks to reach. (shrink)

In the late summer of 1998, the authors, a cognitive scientist and a logician, started talking about the relevance of modern mathematical logic to the study of human reasoning, and we have been talking ever since. This book is an interim report of that conversation. It argues that results such as those on the Wason selection task, purportedly showing the irrelevance of formal logic to actual human reasoning, have been widely misinterpreted, mainly because the picture of logic current in psychology (...) and cognitive science is completely mistaken. We aim to give the reader a more accurate picture of mathematical logic and, in doing so, hope to show that logic, properly conceived, is still a very helpful tool in cognitive science. The main thrust of the book is therefore constructive. We give a number of examples in which logical theorizing helps in understanding and modeling observed behavior in reasoning tasks, deviations of that behavior in a psychiatric disorder (autism), and even the roots of that behavior in the evolution of the brain. (shrink)

This study aimed to construct structural equation model of students’ competence in mathematics through selected students profile variables. The structural model revealed interesting influence of the profile variables to the competency in mathematics. It can be conveyed that better mother’s work status, higher educational level expected to complete, more confident and did not repeat kinder, have better competency in mathematics. The four variables that directly influenced the competence variables were also influenced with other profile variables such as family background. The (...) family background and confidence level consistently had the highest total effect and indirect effect to the competence in mathematics. Hence, this model can serve as a guide in making programs in the classroom or curriculum in mathematics. (shrink)

It is common to assume that the problem of induction arises only because of small sample sizes or unreliable data. In this paper, I argue that the piecemeal collection of data can also lead to underdetermination of theories by evidence, even if arbitrarily large amounts of completely reliable experimental and observational data are collected. Specifically, I focus on the construction of causal theories from the results of many studies (perhaps hundreds), including randomized controlled trials and observational studies, where the studies (...) focus on overlapping, but not identical, sets of variables. Two theorems reveal that, for any collection of variables V, there exist fundamentally different causal theories over V that cannot be distinguished unless all variables are simultaneously measured. Underdetermination can result from piecemeal measurement, regardless of the quantity and quality of the data. Moreover, I generalize these results to show that, a priori, it is impossible to choose a series of small (in terms of number of variables) observational studies that will be most informative with respect to the causal theory describing the variables under investigation. This final result suggests that scientific institutions may need to play a larger role in coordinating differing research programs during inquiry. (shrink)

The pessimistic induction holds that successful past scientific theories are completely false, so successful current ones are completely false too. I object that past science did not perform as poorly as the pessimistic induction depicts. A close study of the history of science entitles us to construct an optimistic induction that would neutralize the pessimistic induction. Also, even if past theories were completely false, it does not even inductively follow that the current theories will also turn out to be completely (...) false because the current theories are more successful and have better birth qualities than the past theories. Finally, the extra success and better birth qualities justify an anti-induction in favor of the present theories. (shrink)

In this paper I begin by extending two results of Solovay; the first characterizes the possible Turing degrees of models of True Arithmetic (TA), the complete first-order theory of the standard model of PA, while the second characterizes the possible Turing degrees of arbitrary completions of P. I extend these two results to characterize the possible Turing degrees of m-diagrams of models of TA and of arbitrary complete extensions of PA. I next give a construction showing that the conditions Solovay (...) identified for his characterization of degrees of models of arbitrary completions of PA cannot be dropped (I showed that these conditions cannot be simplified in the paper. (shrink)

In recent years, the e ffort to formalize erotetic inferences (i.e., inferences to and from questions) has become a central concern for those working in erotetic logic. However, few have sought to formulate a proof theory for these inferences. To fill this lacuna, we construct a calculus for (classes of) sequents that are sound and complete for two species of erotetic inferences studied by Inferential Erotetic Logic (IEL): erotetic evocation and regular erotetic implication. While an attempt has been made to (...) axiomatize the former in a sequent system, there is currently no proof theory for the latter. Moreover, the extant axiomatization of erotetic evocation fails to capture its defeasible character and provides no rules for introducing or eliminating question-forming operators. In contrast, our calculus encodes defeasibility conditions on sequents and provides rules governing the introduction and elimination of erotetic formulas. We demonstrate that an elimination theorem holds for a version of the cut rule that applies to both declarative and erotetic formulas and that the rules for the axiomatic account of question evocation in IEL are admissible in our system. (shrink)

In 1980 F. Wattenberg constructed the Dedekind completion

d of the Robinson non-archimedean field

and established basic algebraic properties of

d [6]. In 1985 H. Gonshor established further fundamental properties of

d [7].In [4] important construction of summation of countable sequence of Wattenberg numbers was proposed and corresponding basic properties of such summation were considered. In this paper the important applications of the Dedekind completion

d in transcendental number theory were considered. We dealing using set theory ZFC
(
-model of ZFC).Given (...) an class of analytic functions of one complex variable f  

z, we investigate the arithmetic nature of the values of fz at transcendental points en, n  .Main results are: (i) the both numbers e
 and e   are irrational, (ii) number ee is transcendental. Nontrivial generalization of the Lindemann- Weierstrass theorem is obtained. (shrink)

Molyneux's Question (MQ) concerns whether a newly sighted man would recognize/distinguish a sphere and a cube by vision, assuming he could previously do this by touch. We argue that (MQ) splits into questions about (a) shared representations of space in different perceptual systems, and about (b) shared ways of constructing higher dimensional spatiotemporal features from information about lower dimensional ones, most of the technical difficulty centring on (b). So understood, MQ resists any monolithic answer: everything depends on the constraints faced (...) by particular perceptual systems in extracting features of higher dimensionality from those of lower. Each individual question of this type is empirical and must be investigated separately. We present several variations on MQ based on different levels of dimensional integration—some of these are familiar, some novel adaptations of problems known elsewhere, and some completely novel. Organizing these cases in this way is useful because it unifies a set of disparate questions about intermodal transfer that have held philosophical and psychological interest, suggests a new range of questions of the same type, sheds light on similarities and differences between members of the family, and allows us to formulate a much-augmented set of principles and questions concerning the intermodal transfer of spatiotemporal organization. (shrink)

The paper addresses the formation of striking patterns within originally near-homogenous tissue, the process prototypical for embryology, and represented in particularly purist form by cut sections of hydra regenerating, by internal reorganisation of the pre-existing tissue, a complete animal with head and foot. The essential requirements are autocatalytic, self-enhancing activation, combined with inhibitory or depletion effects of wider range – “lateral inhibition”. Not only de-novo-pattern formation, but also well known, striking features of developmental regulation such as induction, inhibition, and proportion (...) regulation can be explained on this basis. The theory provides a mathematical recipe for the construction of molecular models with criteria for the necessary non-linear interactions. It has since been widely applied to different developmental processes. (shrink)