Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological pluralism concerning the two foundational languages, and provide a theory that fruitfully (...) interrelates a `structural' perspective to a set-theoretic one. We present a set-theoretic system that is able to talk about structures more naturally, and argue that it provides an important perspective on plausibly structural properties such as cardinality. We conclude the language of set theory can provide useful information about the notion of mathematical structure. (shrink)

In this paper intuitionistic set theory INC# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.

According to Cantor (Mathematische Annalen 21:545–586, 1883 ; Cantor’s letter to Dedekind, 1899 ) a set is any multitude which can be thought of as one (“jedes Viele, welches sich als Eines denken läßt”) without contradiction—a consistent multitude. Other multitudes are inconsistent or paradoxical. Set theoretical paradoxes have common root—lack of understanding why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such multitudes do (...) exist. However we do not understand this logical truth so well as we understand, for example, the logical truth $${\forall x \, x = x}$$ . In this paper we formulate a logical truth which we call the productivity principle. Rusell (Proc Lond Math Soc 4(2):29–53, 1906 ) was the first one to formulate this principle, but in a restricted form and with a different purpose. The principle explicates a logical mechanism that lies behind paradoxical multitudes, and is understandable as well as any simple logical truth. However, it does not explain the concept of set. It only sets logical bounds of the concept within the framework of the classical two valued $${\in}$$ -language. The principle behaves as a logical regulator of any theory we formulate to explain and describe sets. It provides tools to identify paradoxical classes inside the theory. We show how the known paradoxical classes follow from the productivity principle and how the principle gives us a uniform way to generate new paradoxical classes. In the case of ZFC set theory the productivity principle shows that the limitation of size principles are of a restrictive nature and that they do not explain which classes are sets. The productivity principle, as a logical regulator, can have a definite heuristic role in the development of a consistent set theory. We sketch such a theory—the cumulative cardinal theory of sets. The theory is based on the idea of cardinality of collecting objects into sets. Its development is guided by means of the productivity principle in such a way that its consistency seems plausible. Moreover, the theory inherits good properties from cardinal conception and from cumulative conception of sets. Because of the cardinality principle it can easily justify the replacement axiom, and because of the cumulative property it can easily justify the power set axiom and the union axiom. It would be possible to prove that the cumulative cardinal theory of sets is equivalent to the Morse–Kelley set theory. In this way we provide a natural and plausibly consistent axiomatization for the Morse–Kelley set theory. (shrink)

Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the (...) axioms of ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by Löwe and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing Löwe and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF). (shrink)

A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section (...) I. Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them. Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines. The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV. Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations. (shrink)

I here investigate the sense in which diagonalization allows one to construct sentences that are self-referential. Truly self-referential sentences cannot be constructed in the standard language of arithmetic: There is a simple theory of truth that is intuitively inconsistent but is consistent with Peano arithmetic, as standardly formulated. True self-reference is possible only if we expand the language to include function-symbols for all primitive recursive functions. This language is therefore the natural setting for investigations of self-reference.

In this paprer a class of so called mathematically acceptable (shortly MA) languages is introduced First-order formal languages containing natural numbers and numerals belong to that class. MA languages which are contained in a given fully interpreted MA language augmented by a monadic predicate are constructed. A mathematical theory of truth (shortly MTT) is formulated for some of these languages. MTT makes them fully interpreted MA languages which posses their own truth predicates, yielding consequences to philosophy of mathematics. (...) MTT is shown to conform well with the eight norms presented for theories of truth in the paper 'What Theories of Truth Should be Like (but Cannot be)' by Hannes Leitgeb. MTT is also free from infinite regress, providing a proper framework to study the regress problem. (shrink)

Consider a variant of the usual story about the iterative conception of sets. As usual, at every stage, you find all the (bland) sets of objects which you found earlier. But you also find the result of tapping any earlier-found object with any magic wand (from a given stock of magic wands). -/- By varying the number and behaviour of the wands, we can flesh out this idea in many different ways. This paper's main Theorem is that any loosely constructive (...) way of fleshing out this idea is synonymous with a ZF-like theory. -/- This Theorem has rich applications; it realizes John Conway's (1976) Mathematicians' Liberation Movement; and it connects with a lovely idea due to Alonzo Church (1974). (shrink)

A theory of truth is introduced for a first--order language L of set theory. Fully interpreted metalanguages which contain their truth predicates are constructed for L. The presented theory is free from infinite regress, whence it provides a proper framework to study the regress problem. Only ZF set theory, concepts definable in L and classical two-valued logic are used.

Conceptual models as representations of real-world systems are based on diverse techniques in various disciplines but lack a framework that provides multidisciplinary ontological understanding of real-world phenomena. Concurrently, systems’ complexity has intensified, leading to a rise in developing models using different formalisms and diverse representations even within a single domain. Conceptual models have become larger; languages tend to acquire more features, and it is not unusual to use different modeling languages for different components. This diversity has caused problems with consistency (...) between models and incompatibly with designed systems. Two main solutions have been adopted over the last few years: (1) A currently dominant technology-based solution tries “to harmonize or unify” models, e.g., unifies EER and UML. This solution would solidify modeling achievements, reaping benefits from huge investments over the last thirty years. (2) A less prevalent solution is to pursuit deeper roots that reveal unifying modeling principles and apparatuses. An example of the second method is a “category theory”-based approach that utilizes the strengths of the graph and set theory, along with other topological tools. This manuscript is a sequel in a research venture that belongs to the second approach and uses a model called thinging machines (TMs) founded on Stoic ontology and Lupascian logic. TM modeling contests the thesis that there is no universal approach that covers all aspects of an application, and the paper demonstrates that pursuing such universality is anything but a dead-end method. This paper continues in this direction, with emphasis on TM foundation (e.g., existence and subsistence of things) and exemplifies this pursuit by proposing an alternative representation of set theory. (shrink)

Two prominent normative theories of business ethics are stakeholder and shareholder theory. Business ethicists generally favor the former, while business people prefer the latter. If the purpose of business ethics is “to produce a set of ethical principles that can be both expressed in language accessible to and conveniently applied by an ordinary business person” , then it is important to examine this dichotomy.While superficially attractive, the normative version of stakeholder theory contains numerous limitations. Since balancing multiple (...) stakeholder preferences is difficult, competing claims often become tests of political strength rather than justice. Furthermore, stakeholder theory has significant normative weaknesses.Although less attractive to academic ethicists, shareholder theory may provide superior results for society. The shareholder model focuses companies on meeting society’s material needs. Wise owners often balance other stakeholders’ views well since it is necessary for the business’s long-term success. Finally, shareholder theory has a strong normative basis in autonomy.In light of this analysis, it is incumbent upon academic business ethicists to emphasize the value of shareholder theory when teaching business ethics courses. (shrink)

Hannes Leitgeb formulated eight norms for theories of truth in his paper [5]: `What Theories of Truth Should be Like (but Cannot be)'. We shall present in this paper a theory of truth for suitably constructed languages which contain the first-order language of set theory, and prove that it satisfies all those norms.

The semantic view on theories has been much in vogue over four decades as the successor of the syntactic view. In the present paper, I take issue with this approach by arguing that theories and models must be separated and that a theory should be considered to be a linguistic systems consisting of a vocabulary and a set of rules for the use of that vocabulary.

The Turing machine is one of the simple abstract computational devices that can be used to investigate the limits of computability. In this paper, they are considered from several points of view that emphasize the importance and the relativity of mathematical languages used to describe the Turing machines. A deep investigation is performed on the interrelations between mechanical computations and their mathematical descriptions emerging when a human (the researcher) starts to describe a Turing machine (the object of the study) by (...) different mathematical languages (the instruments of investigation). Together with traditional mathematical languages using such concepts as ‘enumerable sets’ and ‘continuum’ a new computational methodology allowing one to measure the number of elements of different infinite sets is used in this paper. It is shown how mathematical languages used to describe the machines limit our possibilities to observe them. In particular, notions of observable deterministic and non-deterministic Turing machines are introduced and conditions ensuring that the latter can be simulated by the former are established. (shrink)

Here, we analyse some recent applications of set theory to topology and argue that set theory is not only the closed domain where mathematics is usually founded, but also a flexible framework where imperfect intuitions can be precisely formalized and technically elaborated before they possibly migrate toward other branches. This apparently new role is mostly reminiscent of the one played by other external fields like theoretical physics, and we think that it could contribute to revitalize the interest in (...) set theory in the future. (shrink)

Potentialists think that the concept of set is importantly modal. Using tensed language as an heuristic, the following bar-bones story introduces the idea of a potential hierarchy of sets: 'Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets.' Surprisingly, this story already guarantees well-foundedness and persistence. Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non-modal set theories; (...) specifically, with Level Theory, as developed in Part 1. (shrink)

The focus in this essay will be upon the paradoxes, and foremostly in set theory. A central result is that the librationist set theory £ extension \Pfund $\mathscr{HR}(\mathbf{D})$ of \pounds \ accounts for \textbf{Neumann-Bernays-Gödel} set theory with the \textbf{Axiom of Choice} and \textbf{Tarski's Axiom}. Moreover, \Pfund \ succeeds with defining an impredicative manifestation set $\mathbf{W}$, \emph{die Welt}, so that \Pfund$\mathscr{H}(\mathbf{W})$ %is a model accounts for Quine's \textbf{New Foundations}. Nevertheless, the points of view developed support the view that (...) the truth-paradoxes and the set-paradoxes have common origins, so that the librationist resolutions of the set theoretic paradoxes are at the same time resolutions of the truth theoretic paradoxes. Both the librationist resolutions of the set theoretic paradoxes and the truth theoretic paradoxes have non-trivial philosophical implications: librationist set theories have the consequence that there are no absolutely uncountable sets, and librationist truth theories allow the use of syntactical modalities in ways which circumvent limitations as those of \parencite{Montague1963}, and a truth predicate which is useful for more precise philosophical discourse. (shrink)

The book is a collection of papers and aims to unify the questions of syntax and semantics of language, which are included in logic, philosophy and ontology of language. The leading motif of the presented selection of works is the differentiation between linguistic tokens (material, concrete objects) and linguistic types (ideal, abstract objects) following two philosophical trends: nominalism (concretism) and Platonizing version of realism. The opening article under the title “The Dual Ontological Nature of Language Signs and (...) the Problem of Their Mutual Relations” provides a broad introduction into the problem area connected with this differentiation, while the logic-formal characteristics of the distinction are framed in the work entitled “On the Type-Token Relationships” (Chapter 1). The basic part of the book deals with issues relating to syntax (Chapters 2-4) and semantics of language (Chapters 5-6), as well as pertaining to syntactic-semantic pragmatic questions (Chapters 7-13). Throughout the book, language, categorial language, is characterized syntactically as generated by classical categorial grammar (Chapter 2) and formalized on two opposing levels: as language of expression-tokens (level of tokens) and language of expression-types (level of types). The author’s considerations contained in Chapters 2 and 4 lead to the important philosophical conclusion that in formal-logical syntactic studies on language the assumption that expression-types constitute the primary language layer while expression-tokens make the secondary one, can be neglected; thus, this speaks in favour of the opposing standpoint—the concretistic one—in the ontology of language syntax. In the works “Meaning and Interpretations”, Parts I and II (Chapters 5 and 6), it is underlined, however, that such semantic concepts as: meaning, denotation and interpretation are defined on the types level, yet their formal definitions require making use of notions of the tokens level. The semantic notions introduced in the above-mentioned articles are also used in the following works of the present selection, under the titles: “Three Principles of Compositionality” and “On Metaknowledge and Truth” (Chapters 7 and 8). They formalize two principles of compositionality that are well known in the literature on the subject, deriving from Frege, i.e. those of meaning and of denotation; they are related to the syntactic principle of compositionality which was introduced by the author. All the three principles are, at the same time, three conditions of homomorphism of categorial language algebra into three kinds of non-standard models of language (one syntactic and two semantic ones: intensional and extensional), which allows introducing three definitions of truthfulness into these models. The next two works in the collection, entitled: “On Language Adequacy” and “What is the Sense in Logic and Philosophy of Language” (Chapters 9 and 10) concern adequacy of categorial language syntax along with its dual semantics: intensional and extensional, and categorial compatibility of any of its syntactic categories with two corresponding semantic categories: intensional and extensional, based on the compatibility the syntactic category of each language expression with the ontological category assigned to its denotatum. The well-known problem of categorial compatibility for first-order quantifiers finds its solution in the paper “Categories of First-Order Quantifiers” (Chapter 11). In the work “Logic and Ontology of Language” (Chapter 12), being in a sense a summary of the considerations presented in the preceding chapters of the book, language is treated as an ontological being, characterized in compliance with the logical conception of language proposed by Ajdukiewicz. Application—like throughout the book—of tools of classical logic and set theory has resulted in emergence of a general formal logical theory of syntax, semantics and pragmatics of language, which takes into account duality in the understanding of linguistic expressions as tokens (concretes) and types (abstract objects). In terms that take into account a functional approach to language itself, there comes out an ontological neutrality of logic with respect to existential assumptions relating to the ontological nature of linguistic expressions and their extra-linguistic ontological counterparts. The issues connected with applying logic while explaining the manner of using linguistic tokens and linguistic types to determine notions of language communication are raised and illustrated in the last chapter of the work, bearing the title “A Logical Conceptualization of Knowledge on the Notion of Language Communication”. (shrink)

How does our language relate to reality? This is a question that is especially pertinent in set theory, where we seem to talk of large infinite entities. Based on an analogy with the use of models in the natural sciences, we argue for a threefold correspondence between our language, models, and reality. We argue that so conceived, the existence of models can be underwritten by a weak notion of existence, where weak existence is to be understood as (...) existing in virtue of language. (shrink)

The creative aspect of language use provides a set of phenomena that a science of language must explain. It is the “central fact to which any signi- ficant linguistic theory must address itself” and thus “a theory of language that neglects this ‘creative’ aspect is of only marginal interest” (Chomsky 1964: 7–8). Therefore, the form and explanatory depth of linguistic science is restricted in accordance with this aspect of language. In this paper, the implications (...) of the creative aspect of language use for a scientific theory of language will be discussed, noting the possible further implications for a science of the mind. It will be argued that a corollary of the creative aspect of language use is that a science of language can study the mechanisms that make language use possible, but that such a science cannot explain how these mechanisms enter into human action in the form of language use. (shrink)

Human linguistic phenomenon is at one and the same time an individual, social, and political fact. As such, its study should bear in mind these complex interrelations, which are produced inside the framework of the sociocultural and historical ecosystem of each human community. Understanding this phenomenon is often no easy task, due to the range of elements involved and their interrelations. The absence of valid, clearly developed paradigms adds to the problem and means that the theoretical conclusions that emerge may (...) be unclear on certain points. It is true that in the last fifty years sociolinguistic studies have advanced considerably, and today we have access to an impressive set of data and a wide variety of theoretical reflections. But as a discipline sociolinguistics does not yet have unified, powerful theoretical models able to account rigorously and clearly for the phenomena it studies. Sociolinguistic studies are today a diverse set of contributions in which certain and theoretical schools and lines of research emerged; but as is to be expected in a relatively new field, there is not enough communication between the various schools and they cannot yet be said to be integrated in terms of their conceptual and theoretical postulates. Against this background, our work aims to contribute to the overall, integrated understanding of the processes of language contact. Via an interdisciplinary, eclectic approach, it also aims to aid the theoretical grounding and integration of a unified, common sociolinguistic paradigm. Our strategy will not be merely to combine the contributions from ongoing research lines, but to address the question from a more global viewpoint which, together with the more innovative contemporary scientific disciplines, permits a harmonious integration of the various sociolinguistic perspectives in a broad, deep and unitary approach to the reality. The materials used to construct this unified approach are taken from many sources: Theoretical physics, ecology, the philosophy of science and mind, anthropology, phenomenological and process sociology, cognitive sciences, political science, pragmatics, history, systems theory, approaches to complexity and obviously sociolinguistics, are all involved in a dialogue in this desire for integration. (shrink)

In this contribution, I explore the possibility of characterizing the emergence of time in causal set theory (CST) in terms of the growing block universe (GBU) metaphysics. I show that although GBU seems to be the most intuitive time metaphysics for CST, it leaves us with a number of interpretation problems, independently of which dynamics we choose to favor for the theory —here I shall consider the Classical Sequential Growth and the Covariant model. Discrete general covariance of the (...) CSG dynamics does not allow us to individuate a single history of the universe (defined by a causal history of different causal sets), thereby making the claim that ‘the past exists’ at best problematic. In addition, because the evolution of the universe in CSG dynamics leads to an outward branching causal tree, it becomes impossible to determine a proper ‘line of becoming’, thereby blurring the presentists’ claim that only the present exists. Similarly, the covariant approach runs into the same, if not even more severe problems, since each configuration of the universe would amount to a set of possible causal sets, thereby making the individuation of a single configuration of the universe —and thus the physical interpretation of the theory— implausible. (shrink)

The paper is a contribution to formal ontology. It seeks to use topological means in order to derive ontological laws pertaining to the boundaries and interiors of wholes, to relations of contact and connectedness, to the concepts of surface, point, neighbourhood, and so on. The basis of the theory is mereology, the formal theory of part and whole, a theory which is shown to have a number of advantages, for ontological purposes, over standard treatments of topology in (...) set-theoretic terms. One central goal of the paper is to provide a rigorous formulation of Brentano's thesis to the effect that a boundary can exist as a matter of necessity only as part of a whole of higher dimension which it is the boundary of. It concludes with a brief survey of current applications of mereotopology in areas such as natural-language analysis, geographic information systems, machine vision, naive physics, and database and knowledge engineering. (shrink)

In this paper a class of languages which are formal enough for mathematical reasoning is introduced. Its languages are called mathematically agreeable. Languages containing a given MA language L, and being sublanguages of L augmented by a monadic predicate, are constructed. A mathematical theory of truth (shortly MTT) is formulated for some of those languages. MTT makes them fully interpreted MA languages which posses their own truth predicates. MTT is shown to conform well with the eight norms formulated (...) for theories of truth in the paper 'What Theories of Truth Should be Like (but Cannot be)', by Hannes Leitgeb. MTT is also free from infinite regress, providing a proper framework to study the regress problem. Main tools used in proofs are Zermelo-Fraenkel (ZF) set theory and classical logic. (shrink)

The target of Second- Language Acquisition (SLA), emerged in the second half of the 20th century, was to be helpful in foreign- language education/ teaching. It denotes mostly the study of individuals (or sometimes groups) who are learning a language consequent to learning their first language when they are young children. At the same time, it signifies the process of learning a second language. The added language is named a second language, but it (...) might indeed be the third, fourth or more which is going to be acquired. The range of SLA comprises informal Second- Language Learning occurring in natural milieus, formal second- language learning occurring in classroom or the one that contains a combination of them both, that is, settings and conditions. The three main aspects for the study of SLA process are the linguistic, psychological and social aspects. The Monitor Theory/ Model postulated by Krashen in the 1970s is a psychological approach in nature. With its five hypotheses (The Acquisition– Learning Hypothesis, The Monitor Hypothesis, The Natural Order Hypothesis, The Input Hypothesis, and The Affective Filter Hypothesis), it tries to find answers to the problems of SLA, such as what does a second- language learner come to know, how the acquisition process takes place, and why some learners are more successful than others? The Monitor Theory (MT) received extensively many criticisms after its appearance and was rejected. Its teaching implications were also at the centre of criticisms. What place does MT occupy in SLA today? This study aims to try to find an answer. The other questions are: How important is the MT for SLA? What kind of criticisms are expressed against it? How fair is the criticism by McLaughlin (1978, 1987)? The working hypotheses of the present work are: The hypotheses developed by Krashen are not/ will not be rejected. Because science is still lying in the so- called agony phase, and cannot find any answers to all questions in psychology (e.g. how exactly is the processing of language; in particular and of mind in general). Moreover, the problems related to memory etc., the thoughts emanated from the MT can probably not be refuted. They have evolved so far and will be evolved further, perhaps with small differences. This research is completely based on the literature written since the time the theory was developed. In other words, it was carried out using a descriptive method without using a special data collection tool. The sources written on the subject were reviewed and an answer to the research questions was tried to be found. Even though the theory is expressed with different names and different meanings today, it has survived all the criticisms made, and it has been concluded that it still occupies an important place in the discipline of second- language acquisition (SLA) and foreign- language teaching. Again, the inquiries carried out since the 1970s delineate that the implications in favour of language education are not very different from those stated by Krashen (1982), which were the products of his opinions in that period. There are still basic consequences grounded on MT for language teaching today. (shrink)

The origin of languages was a hotly debated topic in the eighteenth century. This paper reconstructs a distinctively Kantian account according to which the origination, progression, and diversification of languages is at bottom reason’s self-development under certain a priori constraints and external environments. The reconstruction builds on three sets of materials. The first is Herder’s famous prize essay on the origin of languages. The second includes Kant’s explicit remarks about language – especially his notion of “transcendental grammar,” his argument (...) that language cannot be innate, his contrast of “Oriental” symbolic (intuitive) and “Occidental” discursive languages, and his treatment of the latter as a sine qua non of humanity’s cultural and moral progress. The third includes the concepts that we need to make sense of those remarks, such as Kant’s epigenetic theory of biological formation and his account of categories as originally acquired. (shrink)

Standard approaches to proper names, based on Kripke's views, hold that the semantic values of expressions are (set-theoretic) functions from possible worlds to extensions and that names are rigid designators, i.e.\ that their values are \emph{constant} functions from worlds to entities. The difficulties with these approaches are well-known and in this paper we develop an alternative. Based on earlier work on a higher order logic that is \emph{truly intensional} in the sense that it does not validate the axiom scheme of (...) Extensionality, we develop a simple theory of names in which Kripke's intuitions concerning rigidity are accounted for, but the more unpalatable consequences of standard implementations of his theory are avoided. The logic uses Frege's distinction between sense and reference and while it accepts the rigidity of names it rejects the view that names have direct reference. Names have constant denotations across possible worlds, but the semantic value of a name is not determined by its denotation. (shrink)

This article develops a criticism of Alain Badiou’s assertion that “mathematics is ontology.” I argue that despite appearances to the contrary, Badiou’s case for bringing set theory and ontology together is problematic. To arrive at this judgment, I explore how a case for the identification of mathematics and ontology could work. In short, ontology would have to be characterised to make it evident that set theory can contribute to it fundamentally. This is indeed how Badiou proceeds in Being (...) and Event. I review his descriptions of the ontological problematic at some length here, only to argue that set theory is a poor fit. Although philosophers working on questions of being were certainly occupied with matters of oneness and the part-whole relationship, I argue that Badiou’s discussion of philosophical sources points towards a mereological treatment, not a set-theoretic one. Finally, I suggest that Badiou’s philosophical interpretations of key set-theoretic results are better understood as some sort of analogising between mathematics, ontology, and philosophical anthropology. (shrink)

The role of intuition in Kripke's arguments for the causal-historical theory of reference has been a topic of recent debate, particularly in light of empirical work on these intuitions. In this paper, I develop three interpretations of the role intuition might play in Kripke's arguments. The first aim of this exercise is to help clarify the options available to interpreters of Kripke, and the consequences for the experimental investigation of Kripkean intuitions. The second aim is to show that understanding (...) the role of intuition in Kripke's arguments, and in other arguments commonly interpreted as ?appeals to intuition? requires that we pay careful attention to the argumentative context and theoretical commitments of the work in question. The interpretations of Kripke developed here provide a set of options which might be adapted to other arguments to help clarify what role (if any) intuition plays. (shrink)

Since the 1960s, Kripke has been a central figure in several fields related to mathematical logic, language philosophy, mathematical philosophy, metaphysics, epistemology and set theory. He had influential and original contributions to logic, especially modal logic, and analytical philosophy, with a semantics of modal logic involving possible worlds, now called Kripke semantics. In Naming and Necessity, Kripke proposed a causal theory of reference, according to which a name refers to an object by virtue of a causal connection (...) with the object, mediated by the communities of speakers. DOI: 10.13140/RG.2.2.26557.20964. (shrink)

A possible world is a junky world if and only if each thing in it is a proper part. The possibility of junky worlds contradicts the principle of general fusion. Bohn (2009) argues for the possibility of junky worlds, Watson (2010) suggests that Bohn‘s arguments are flawed. This paper shows that the arguments of both authors leave much to be desired. First, relying on the classical results of Cantor, Zermelo, Fraenkel, and von Neumann, this paper proves the possibility of junky (...) worlds for certain weak set theories. Second, the paradox of Burali-Forti shows that according to the Zermelo-Fraenkel set theory ZF, junky worlds are possible. Finally, it is shown that set theories are not the only sources for designing plausible models of junky worlds: Topology (and possibly other "algebraic" mathematical theories) may be used to construct models of junky worlds. In sum, junkyness is a relatively widespread feature among possible worlds. (shrink)

The incompleteness of set theory ZF C leads one to look for natural nonconservative extensions of ZF C in which one can prove statements independent of ZF C which appear to be “true”. One approach has been to add large cardinal axioms.Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski-Grothendieck set theory T G or It is a nonconservative extension of ZF C and is obtained from other axiomatic set theories by the inclusion (...) of Tarski’s axiom which implies the existence of inaccessible cardinals. See also related set theory with a filter quantifier ZF (aa). In this paper we look at a set theory NC# ∞# , based on bivalent gyper infinitary logic with restricted Modus Ponens Rule In this paper we deal with set theory NC# ∞# based on bivalent gyper infinitary logic with Restricted Modus Ponens Rule. Nonconservative extensions of the canonical internal set theories IST and HST are proposed. (shrink)

Semantics plays a role in grammar in at least three guises. (A) Linguists seek to account for speakers‘ knowledge of what linguistic expressions mean. This goal is typically achieved by assigning a model theoretic interpretation in a compositional fashion. For example, *No whale flies* is true if and only if the intersection of the sets of whales and fliers is empty in the model. (B) Linguists seek to account for the ability of speakers to make various inferences based on semantic (...) knowledge. For example, *No whale flies* entails *No blue whale flies* and *No whale flies high*. (C) The wellformedness of a variety of syntactic constructions depends on morpho-syntactic features with a semantic flavor. For example, *Under no circumstances would a whale fly* is grammatical, whereas *Under some circumstances would a whale fly* is not, corresponding to the downward vs. upward monotonic features of the preposed phrases. It is usually assumed that once a compositional model theoretic interpretation is assigned to all expressions, its fruits can be freely enjoyed by inferencing and syntax. What place might proof theory have in this picture? (shrink)

This paper explores the possibility of a standard legal language (e.g. English) for a principled evolution of law in line with technological development. In doing so, reference is made to blockchain networks and smart contracts to emphasise the discontinuity with the liberal legal tradition when it comes to decentralisation and binary code language. Methodologically, the argument is built on the underlying relation between law, semiotics and new forms of media adding to natural language; namely: code and symbols. (...) In what follows, I will concentrate on the study of factors that explain why such approach can be fruitful for the future of law and innovation. I have three reasons for selecting this topic. The first is a more pragmatic reason, based on my current research of law as a linguistic phenomenon. Secondly, the topic does also touch the matter on binary code language, rivalrous to legal alphabetic language. Lastly, the study aims at emphasizing the pivotal role of the jurist as an interpreter in a changing society accommodating diverging realms of reality. The study is structured as follows. Firstly, a quick exam of the traits of blockchain networks would provide the contextual link to establish the arguments in support of the need of a standard legal language. Secondly, a comparison between liberal legal institutions and theory of semiotics is set to perceive their functioning and ascertain their limits in the light of todays unprecedented changes. Thirdly, a summary on blockchain networks’ legal features would constitute the thrust behind the idea of a uniform legal language. Methodologically, the argument does also establish some relations with classical laws of physics and philosophy of media. Its aim is to demonstrate how the suggested legal interpretation and semiotic-based approach can contribute to overcome existing stumbling blocks including, but not limited to, the lack of cooperation at the international level as well as the gap in State norms when it comes to innovation. In this sense, the proposed strategies do not intend to replace current advances in the legal thought. In contrast, it seeks to harmonise their results providing a methodical approach that can concur to inform a new technique to address new controversial issues. In practice, the proposed method regarding the adoption of a uniform legal language would lower transactive costs in terms of normative coordination in the matter of international cooperation and in the definition of applicable law among different legal systems. Alternatively, it might contribute to the convergence of legal systems and/or their underlying concepts. Differently put, article’s contribution can be envisaged in fostering juridical consistency with regards to different forms of languages’ coexistence. (shrink)

The paper gives an overview of key themes of twentieth century philosophical treatment of the language of science, with special emphasis on the meaning variance of scientific terms and the comparison of alternative theories. These themes are dealt with via discussion of the topics of: (a) the logical positivist principle of verifiability and the problem of the meaning of theoretical terms, (b) the postpositivist thesis of semantic incommensurability, and (c) the scientific realist response to incommensurability based on the causal (...) theory of reference. (shrink)

Legal philosophers distinguish between a static and a dynamic interpretation of law. The former assumes that the meaning of the words used in a legal text is set at the moment of its enactment and does not change with time. The latter allows the interpreters to update the meaning and apply a contemporary understanding to the text. The dispute between these competing theories has significant ramifications for social and political life. To take an example, depending on the approach, the term (...) “cruel punishment” used in the US Constitution will be given an 18th century meaning or a contemporary one. -/- The philosophy of language seems to provide greater support to the static approach to legal interpretation. Within this approach the lawmaker is perceived as a speaker and legal texts are interpreted as utterances. As a consequence, interpretation is a quest for the speaker/lawmaker’s intention or the public meaning that prevailed at the time of enactment. Neither the intention nor the public meaning are considered to have changed in time. -/- In this paper I argue that the philosophy of language provides the dynamic approach with an equally robust support as the static one. This support comes from an externalist perspective in semantics, rooted in philosophical pragmatism and supported by Ruth Millikan’s concept of meaning as proper function. Grounding the dynamic approach in a well-founded linguistic philosophy rises to the challenge presented by the originalists’ declaration that “it takes a theory to beat a theory”. (shrink)

This thesis is concerned with the ways in which meaning is generically mediated in the novel. In particular it addresses the productive diversity of meanings generated by critical interpretation and asks how, given this diversity, comprehension and consensus might be possible. I argue that the construction of subject, object, space and time is achieved in the novel through different manifestations of four key metaphors: voice, view, setting and event. These metaphors supply meanings that rely on a common experience of embodiment. (...) Embodiment supplies the basis for an intersubjectively established consensus which identifies subjects and objects in terms that correspond to the semiotic, syntagmatic and semantic uses of metaphor. I begin by surveying and reviewing certain constructionist accounts of language and meaning in order to establish the importance of metaphor in conveying meaning in rhetorical, rather than linguistic, terms. I then discuss two examples of continental philosophy on speech: Derrida on voice and Foucault on discourse. I demonstrate how meaning is rhetorically constructed according to the use to which metaphors are put—whether they denote formal, semantic, or functional aspects of meaning—through an explication of Foucault’s theory, reserving Derrida’s grammatology for my analysis of the stabilisation of meaning in the theory of certain narratologists (Gérard Genette and Marie-Laure Ryan). Voice and view are key terms in literary theory, and I examine refinements to these metaphors in Genette and Ryan. The insistence of these narratologists on restricting the rhetorical dimension of meaning undermines the value of the literary work, prompting me to argue for a relaxing of such constraints. I read Peter Carey’s True History of the Kelly Gang for its uses of voice and point of view, many of which expose the limitations of the narratological approach. This is a theoretical thesis, which engages a wide range of perspectives—constructionist, deconstructionist, sociological and structuralist—together with their various accounts of language, metaphor, speech and meaning. In terms of sociological approaches to genre and discourse, I employ the chronotope as a critical tool, examining it in the context of the theme of the country house to demonstrate the role of two of my key metaphors, setting and event, which are important for the construction of space and time in abstract terms. Focusing on novels by Jane Austen, Virginia Woolf, and Kazuo Ishiguro, in which the country house defines the space and the temporality through which thematic and “material” elements are mediated, I reveal how the chronotope supplies an overarching coherence that unites thought and event in narrative. My readings of these novels describe a theory that is concerned with the rhetorical capacities of genre rather than with taxonomic constructions of form. To this end I argue that the metaphorical dimensions of narrative—voice, view, setting and event—be loosely drawn, according to the acknowledged mutability of genre. I also emphasise the constructive effect of metaphor whilst according it a rhetorical versatility consistent with the mutability and instability of genre as a semiotic tool, concluding that consensus is achieved in our readings of novels through the form’s reliance on these four key metaphors of embodiment. These metaphors have ordinary, everyday, widespread usage beyond narrative theory; in appreciating their rhetorical value, it must be noted nonetheless that any confusion that ensues is productive, for it results in the diversities of meaning and form celebrated in the literary work. (shrink)

The notion of computability is developed through the study of the behavior of a set of languages interpreted over the natural numbers which contain their own fully defined satisfaction predicate and whose only other vocabulary is limited to "0", individual variables, the successor function, the identity relation and operators for disjunction, conjunction, and existential quantification.

Easy to understand philosophy papers in all areas. Table of contents: Three Short Philosophy Papers on Human Freedom The Paradox of Religions Institutions Different Perspectives on Religious Belief: O’Reilly v. Dawkins. v. James v. Clifford Schopenhauer on Suicide Schopenhauer’s Fractal Conception of Reality Theodore Roszak’s Views on Bicameral Consciousness Philosophy Exam Questions and Answers Locke, Aristotle and Kant on Virtue Logic Lecture for Erika Kant’s Ethics Van Cleve on Epistemic Circularity Plato’s Theory of Forms Can we trust our senses? (...) Yes we can Descartes on What He Believes Himself to Be The Role of Values in Science Modern Science Kant’s Moral Philosophy Plato’s Republic as Pol Potist Bureaucracy Schopenhauer on Human Suffering Bertrand Russell on the Value of Philosophy The Philosophical Value of Uncertainty Logic Homework: Theorems and Models Searle vs. Turing on the Imitation Game Hume, Frankfurt, and Holbach on Personal Freedom Manifesto of the University of Wisconsin, Madison Secular Society Michael’s Analysis of the Limits of Civil Protections Bentham and Mill on Different Types of Pleasure Set Theory Homework Aristotle on Virtue Nagel On the Hard Problem Wittgenstein on Language and Thought Camus and Schopenhauer on the Meaning of Life Camus’ Hero as Rebel without a Cause My Little Finger: Camus’ Absurdism Illustrated Are Late-term Abortions Ethical? Does Mathematics Assume the Truth of Platonism? The Self-defeating Nature of Utilitarianism and Consequentialism Generally What is The Good Life? Bentham and Mill regarding types of pleasures Kant’s Moral Philosophy Five Short Papers on Mind-body Dualism Tracy Latimer’s Father had the Right to Kill Her: Towards a doctrine of generalized self-defense Arguments Concerning God and Morality Goldman, Rousseau and von Hayek on the Ideal State J.S Mill on Liberty and Personal Freedom A Kantian Analysis of a Borderline Date-rape Situation Living Well as Flourishing: Aristotle’s Conception of the Good Life Three Essays on Medical Ethics: Answers to Exam Questions on Elective Amputation, Vaccination, and Informed Consent Hobbes, Marx, Rousseau, Nietzsche: Their Central Themes De Tocqueville on Egoism Mill vs. Hobbes on Liberty Exam-Essays on the Moral Systems of Mill, Bentham, and Kant Kant’s Moral System Aristotle on Virtue Plato’s Cave Allegory An Ethical Quandary Superorganisms The Tuskegee Experiment A Rawlsian Analysis Why Moore’s Proof of an External World Fails A Defense of Nagel’s Argument Against Materialism A Utilitarian Analysis of a Case of Theft The Paradox of the Self-aware Wretch: An Analysis of Pascal’s Moral Philosophy Jean-Paul Sartre: Decline and Fall of a Marxist Sell-out A One Page Proof of Plato’s Theory of Forms Plato’s Republic as Pol Potist Bureaucracy The problem of the one and the many Four Short Essays on Truth and Knowledge What is ‘the Good Life?’ The Ontological Argument Different Political Philosophies: Plato, Locke, Madison, Rousseau, Hayek, and Mill on the State What do I know with certainty? Skepticism about skepticism Neuroscience and Freewill Operant Conditioning What makes us special? Are Late-term Abortions Ethical? No Two Papers on Epistemology: Gettier and Bostrom Examination Nietzsche on Punishment God’s Foreknowledge and Moral Responsibility . (shrink)

(1) This is Part 2 of the semantic theory I call TM. In Part 1, I developed TM as a theory in the analytic philosophy of language, in lexical semantics, and in the sociology of relating occasions of statement production and comprehension to formal and informal lexicographic conclusions about statements and lexical items – roughly, as showing how synchronic semantics is a sociological derivative of diachronic, person-relative acts of linguistic behavior. I included descriptions of new cognitive psychology (...) experimental paradigms which would allow us to precisely measure the two constituents of semantics – meaning and reference – both at the level of individual speech acts and at the level of societal convergences, i.e. at both the token and type levels. -/- (2) In the Introduction, I recapitulate the arguments of Part 1. The Introduction also develops some analytic philosophical and lexical semantics themes not discussed in Part 1. -/- (3) After the Introduction, I present neural TM (nTM) as a theory of the neural mechanisms and processes which give rise to these person/occasion-relative acts of linguistic behavior. I develop nTM at three levels, the first two of which describe linguistic/semantic functions independently of their cortical locations. At the first level, I describe individual word-to-word and word-to-object connections. At the second level, I describe the corresponding structuralist networks of which they are the individual components. At this level, I introduce some key linguistic concepts of TM – its graded meaning, reference, and generalization sets, and the types of statements which express various levels of word-to-word and word-to-object relationships among lexical items which, because of the constraints they impose on the use of those lexical items in statements we produce and comprehend, are concepts. This constitutes the second structural level of nTM. -/- (4) At the third level, I associate the non-localized structures of the previous levels with cortically located neural structures and with the fasciculi that connect them. I distinguish neural areas in which primary (phonetic) and secondary (orthographic) lexicons are stored in long-term memory. I also describe the embodied concepts which co-exist in the anterior temporal lobes with the images they lexicalize. These concepts are often said to name physical objects and their features, although what they in fact name are kinds of physical objects and features. I describe how conceptual constraints and referential constraints interact to channel our intentions to say how things are into statements which are semantically well-formed, and which consequently successfully communicate information. -/- (5) Following this presentation of nTM, I examine five prominent neural semantic theories. I point out what is wrong with each of them as far as their explanations of semantics are concerned, and I also indicate how nTM can replace the “semantic cores” of those theories. -/- (6) The two basic mistakes made by neuroscience semantic theories, as I will explain, are (i) that all but one of them regard semantics as a matter of the association of words with perceptual images, and of generalizations from those associations; and (ii) that they all rely on an unspecified set of neural structures which purportedly encode the meaning of concepts in abstraction from their phonological and orthographic forms. nTM maintains, in contrast, that there are no abstract neural representations of semantic content. Neural constraints on our linguistic behavior, especially on our ascriptive and co-ascriptive use of words, express the semantic constraints on those words which make them concepts. That is the semantic content of words. -/- (7) I next consider several results from neuroscience experimental data which have been given one interpretation by one or another of the standard neurosemantic theories, but to which nTM gives a different interpretation. I include several predictions which I have found neither confirmed nor disconfirmed in the experimental neuroscience literature. -/- (8) After a concluding section in which I summarize the major changes to neurosemantic theory introduced by TM, and the analytic philosophy of language and lexical semantics contexts within which TM is situated, there follows an appendix in which I discuss neural net AI, and make some recommendations for implementing nTM in silicon. (shrink)

The thesis deals with the concept of truth and the paradoxes of truth. Philosophical theories usually consider the concept of truth from a wider perspective. They are concerned with questions such as - Is there any connection between the truth and the world? And, if there is - What is the nature of the connection? Contrary to these theories, this analysis is of a logical nature. It deals with the internal semantic structure of language, the mutual semantic connection of (...) sentences, above all the connection of sentences that speak about the truth of other sentences and sentences whose truth they speak about. Truth paradoxes show that there is a problem in our basic understanding of the language meaning and they are a test for any proposed solution. It is important to make a distinction between the normative and analytical aspect of the solution. The former tries to ensure that paradoxes will not emerge. The latter tries to explain paradoxes. Of course, the practical aspect of the solution is also important. It tries to ensure a good framework for logical foundations of knowledge, for related problems in Artificial Intelligence and for the analysis of the natural language. Tarski’s analysis emphasized the T-scheme as the basic intuitive principle for the concept of truth, but it also showed its inconsistency with the classical logic. Tarski’s solution is to preserve the classical logic and to restrict the scheme: we can talk about the truth of sentences of a language only inside another essentially richer metalanguage. This solution is in harmony with the idea of reflexivity of thinking and it has become very fertile for mathematics and science in general. But it has normative nature | truth paradoxes are avoided in a way that in such frame we cannot even express paradoxical sentences. It is also too restrictive because, for the same reason we cannot express a situation in which there is a circular reference of some sentences to other sentences, no matter how common and harmless such a situation may be. Kripke showed that there is no natural restriction to the T-scheme and we have to accept it. But then we must also accept the riskiness of sentences | the possibility that under some circumstances a sentence does not have the classical truth value but it is undetermined. This leads to languages with three-valued semantics. Kripke did not give any definite model, but he gave a theoretical frame for investigations of various models | each fixed point in each three-valued semantics can be a model for the concept of truth. The solutions also have normative nature | we can express the paradoxical sentences, but we escape a contradiction by declaring them undetermined. Such a solution could become an analytical solution only if we provide the analysis that would show in a substantial way that it is the solution that models the concept of truth. Kripke took some steps in the direction of finding an analytical solution. He preferred the strong Kleene three-valued semantics for which he wrote it was "appropriate" but did not explain why it was appropriate. One reason for such a choice is probably that Kripke finds paradoxical sentences meaningful. This eliminates the weak Kleene three valued semantics which corresponds to the idea that paradoxical sentences are meaningless, and thus indeterminate. Another reason could be that the strong Kleene three valued semantics has the so-called investigative interpretation. According to this interpretation, this semantics corresponds to the classical determination of truth, whereby all sentences that do not have an already determined value are temporarily considered indeterminate. When we determine the truth value of these sentences, then we can also determine the truth value of the sentences that are composed of them. Kripke supplemented this investigative interpretation with an intuition about learning the concept of truth. That intuition deals with how we can teach someone who is a competent user of an initial language (without the predicate of truth T) to use sentences that contain the predicate T. That person knows which sentences of the initial language are true and which are not. We give her the rule to assign the T attribute to the former and deny that attribute to the latter. In that way, some new sentences that contain the predicate of truth, and which were indeterminate until then, become determinate. So the person gets a new set of true and false sentences with which he continues the procedure. This intuition leads directly to the smallest fixed point of strong Kleene semantics as an analytically acceptable model for the logical notion of truth. However, since this process is usually saturated only on some transfinite ordinal, this intuition, by climbing on ordinals, increasingly becomes a metaphor. This thesis is an attempt to give an analytical solution to truth paradoxes. It gives an analysis of why and how some sentences lack the classical truth value. The starting point is basic intuition according to which paradoxical sentences are meaningful (because we understand what they are talking about well, moreover we use it for determining their truth values), but they witness the failure of the classical procedure of determining their truth value in some "extreme" circumstances. Paradoxes emerge because the classical procedure of the truth value determination does not always give a classically supposed (and expected) answer. The analysis shows that such an assumption is an unjustified generalization from common situations to all situations. We can accept the classical procedure of the truth value determination and consequently the internal semantic structure of the language, but we must reject the universality of the exterior assumption of a successful ending of the procedure. The consciousness of this transforms paradoxes to normal situations inherent to the classical procedure. Some sentences, although meaningful, when we evaluate them according to the classical truth conditions, the classical conditions do not assign them a unique value. We can assign to them the third value, \undetermined", as a sign of definitive failure of the classical procedure. An analysis of the propagation of the failure in the structure of sentences gives exactly the strong Kleene three-valued semantics, not as an investigative procedure, as it occurs in Kripke, but as the classical truth determination procedure accompanied by the propagation of its own failure. An analysis of the circularities in the determination of the classical truth value gives the criterion of when the classical procedure succeeds and when it fails, when the sentences will have the classical truth value and when they will not. It turns out that the truth values of sentences thus obtained give exactly the largest intrinsic fixed point of the strong Kleene three-valued semantics. In that way, the argumentation is given for that choice among all fixed points of all monotone three-valued semantics for the model of the logical concept of truth. An immediate mathematical description of the fixed point is given, too. It has also been shown how this language can be semantically completed to the classical language which in many respects appears a natural completion of the process of thinking about the truth values of the sentences of a given language. Thus the final model is a language that has one interpretation and two systems of sentence truth evaluation, primary and final evaluation. The language through the symbol T speaks of its primary truth valuation, which is precisely the largest intrinsic fixed point of the strong Kleene three valued semantics. Its final truth valuation is the semantic completion of the first in such a way that all sentences that are not true in the primary valuation are false in the final valuation. (shrink)

Certain commentators on Russell's “no class” theory, in which apparent reference to classes or sets is eliminated using higher-order quantification, including W. V. Quine and (recently) Scott Soames, have doubted its success, noting the obscurity of Russell’s understanding of so-called “propositional functions”. These critics allege that realist readings of propositional functions fail to avoid commitment to classes or sets (or something equally problematic), and that nominalist readings fail to meet the demands placed on classes by mathematics. I show that (...) Russell did thoroughly explore these issues, and had good reasons for rejecting accounts of propositional functions as extra-linguistic entities. I argue in favor of a reading taking propositional functions to be nothing over and above open formulas which addresses many such worries, and in particular, does not interpret Russell as reducing classes to language. (shrink)

The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the standard probabilistic interpretation of quantum (...) class='Hi'>theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness. (shrink)

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In (...) defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

A set theory model of reality, representation and language based on the relation of completeness and incompleteness is explored. The problem of completeness of mathematics is linked to its counterpart in quantum mechanics. That model includes two Peano arithmetics or Turing machines independent of each other. The complex Hilbert space underlying quantum mechanics as the base of its mathematical formalism is interpreted as a generalization of Peano arithmetic: It is a doubled infinite set of doubled Peano arithmetics having (...) a remarkable symmetry to the axiom of choice. The quantity of information is interpreted as the number of elementary choices (bits). Quantum information is seen as the generalization of information to infinite sets or series. The equivalence of that model to a quantum computer is demonstrated. The condition for the Turing machines to be independent of each other is reduced to the state of Nash equilibrium between them. Two relative models of language as game in the sense of game theory and as ontology of metaphors (all mappings, which are not one-to-one, i.e. not representations of reality in a formal sense) are deduced. (shrink)

Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u_{F}={x|F(x)} for a property F() could never be self-predicative in the sense of u_{F}∈u_{F}. But the mathematical (...) theory of categories, dating from the mid-twentieth century, includes a theory of always-self-predicative universals--which can be seen as forming the "other bookend" to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. (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)

1971. Discourse Grammars and the Structure of Mathematical Reasoning II: The Nature of a Correct Theory of Proof and Its Value, Journal of Structural Learning 3, #2, 1–16. REPRINTED 1976. Structural Learning II Issues and Approaches, ed. J. Scandura, Gordon & Breach Science Publishers, New York, MR56#15263. -/- This is the second of a series of three articles dealing with application of linguistics and logic to the study of mathematical reasoning, especially in the setting of a concern for improvement (...) of mathematical education. The present article presupposes the previous one. Herein we develop our ideas of the purposes of a theory of proof and the criterion of success to be applied to such theories. In addition we speculate at length concerning the specific kinds of uses to which a successful theory of proof may be put vis-a-vis improvement of various aspects of mathematical education. The final article will deal with the construction of such a theory. The 1st is the 1971. Discourse Grammars and the Structure of Mathematical Reasoning I: Mathematical Reasoning and Stratification of Language, Journal of Structural Learning 3, #1, 55–74. https://www.academia.edu/s/fb081b1886?source=link . (shrink)

Ways of Scope Taking is concerned with syntactic, semantic and computational aspects of scope. Its starting point is the well-known but often neglected fact that different types of quantifiers interact differently with each other and other operators. The theoretical examination of significant bodies of data, both old and novel, leads to two central claims. (1) Scope is a by-product of a set of distinct Logical Form processes; each quantifier participates in those that suit its particular features. (2) Scope interaction is (...) further constrained by the semantics of the interacting operators. The arguments are developed using Minimalist syntax, Generalized Quantify theory, Discourse Representation Theory, and algebraic semantics. The contributors (Beghelli, Ben-Shalom, Doetjes, Farkas, Gutiérrez Rexach, Honcoop, Stabler, Stowell, Szabolcsi and Zwarts) make tightly related theoretical assumptions and focus on related empirical phenomena, which include the direct and inverse scope of quantifiers, distributivity, negation, modal and intensional contexts, weak islands, event-related readings, interrogatives, wh/quantifier interactions, and Hungarian syntax. An introduction to the formal semantics background is provided. Audience: Linguists, philosophers, computational and psycholinguists; advanced undergraduates, graduate students and researchers in these fields. (shrink)