Hybrid languages are introduced in order to evaluate the strength of “minimal” mereologies with relatively strong frame definability properties. Appealing to a robust form of nominalism, I claim that one investigated language \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {m}}$\end{document} is maximally acceptable for nominalistic mereology. In an extension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {gem}}$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {m}}$\end{document}, a modal (...) analog for the classical systems of Leonard and Goodman and Leśniewski is introduced and shown to be complete with respect to 0-deleted Boolean algebras. We characterize the formulas of first-order logic invariant for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {gem}}$\end{document}-bisimulations. (shrink)

Review of John Burgess' and Gideon Rosen's A Subject with no Object.

Mathematics, as Eugene Wigner noted, is unreasonably effective in physics. The argument of this paper is that the disproportionate attention that philosophers have paid to discrete structures such as the natural numbers, for which a nominalist construction may be possible, has deprived us of the best argument for Platonism, which lies in continuous structures—in fields and their derived algebras, such as Clifford algebras. The argument that Wigner was making is best made with respect to such structures—in a loose sense, with (...) respect to geometry rather than arithmetic. The purpose of the present paper is to make this connection between mathematical realism and geometrical entities. It thus constitutes an argument against formalism, for which mathematics is merely a game with humanly set rules; and nominalism, in which whatever mathematics is used is eliminable in the final analysis, by often insufficiently specified means. The hope is that light may be cast on the stubborn mysteries of the nature of quantum mechanics and its mathematical formulation, with particular reference to spinor representations—as they have been developed by Andrej Trautman. Thus, according to our argument, quantum mechanics (QM) may appear more natural, as we have better reasons to take spinor structures as irreducibly real, a view consonant with the work of Trautman and Penrose in particular. (shrink)

ABSTRACT‘expressionist’ accounts of applied mathematics seek to avoid the apparent Platonistic commitments of our scientific theories by holding that we ought only to believe their mathematics-free nominalistic content. The notion of ‘nominalistic content’ is, however, notoriously slippery. Yablo's account of non-catastrophic presupposition failure offers a way of pinning down this notion. However, I argue, its reliance on possible worlds machinery begs key questions against Platonism. I propose instead that abstract expressionists follow Geoffrey Hellman's lead in taking the assertoric content of (...) empirical science to be irreducibly modal, using the ‘non-interference’ of mathematical objects as justification for detaching nominalistic consequences. (shrink)

It is well known that over the eighteenth century the calculus moved away from its geometric origins; Euler, and later Lagrange, aspired to transform it into a “purely analytical” discipline. In the 1780 s, the Portuguese mathematician José Anastácio da Cunha developed an original version of the calculus whose interpretation in view of that process presents challenges. Cunha was a strong admirer of Newton (who famously favoured geometry over algebra) and criticized Euler’s faith in analysis. However, the fundamental propositions of (...) his calculus follow the analytical trend. This appears to have been possible due to a nominalistic conception of variable that allowed him to deal with expressions as names, rather than abstract quantities. Still, Cunha tried to keep the definition of fluxion directly applicable to geometrical magnitudes. According to a friend of Cunha’s, his calculus had an algebraic (analytical) branch and a geometrical branch, and it was because of this that his definition of fluxion appeared too complex to some contemporaries. (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)

Corresponding correlations is a method that allows us to infer formal causation from correlational data. In this paper, causal terms are traced to their philosophical and etymological roots. It is argued that causes are parts of their mutual whole . Nominalism, normal distributions and disjunctive causes are linked. Causal manifolds and sampling by potential are used to model conjunctive causes. Corresponding correlations are then demonstrated through simulations, in which causal relations are differentiated from spurious correlations. An algebraic method (...) for unraveling confounded variables is presented. Distinctions between laws and causes are made and related to corresponding correlations. The conclusion is that corresponding correlations should be a significant advance in causal inference. (shrink)

In this essay, I argue that Hume’s theory of Quantitative and Numerical Philosophical Relations can be interpreted in a way which allows mathematical knowledge to be about a body of objective and necessary truths, while preserving Hume’s nominalism and the basic principles of his theory of ideas. Attempts are made to clear up a number of obscure points about Hume’s claims concerning the abstract sciences of Arithmetic and Algebra by means of re-examining what he says and what he could (...) comfortably have said about relations of qualitative resemblance. (shrink)

Leon Henkin (1921–2006) was not only an extraordinary logician, but also an excellent teacher, a dedicated professor and an exceptional person. The first two sections of this paper are biographical, discussing both his personal and academic life. In the last section we present three aspects of Henkin’s work. First we comment part of his work fruit of his emphasis on teaching. In a personal communication he affirms that On mathematical induction, published in 1969, was the favourite among his articles with (...) a somewhat panoramic nature and not meant exclusively to specialists. This subject is covered in the first subsection. Needless to say that we also analyse Henkin’s better known contribution: his completeness method. His renowned results on completeness for both type theory and first order logic were part of his thesis, The Completeness of Formal Systems, presented at Princeton in 1947 under the advise of Alonzo Church. It is interesting to note that he obtained the proof of completeness for first order logic readapting the argument for the theory of types. The last subsection is devoted to philosophy. The work most directly related to philosophy is an article entitled: Some Notes on Nominalism which appeared in the Journal of Symbolic Logic in 1953. Unfortunately, we are not covering his contribution to the field of cylindric algebras. As a matter of fact, Henkin spent many years investigating algebraic structures with Alfred Tarski and Donald Monk, among others. (shrink)

An analysis of the structural properties of collections or pluralities, homogeneous objects like water, and the semantics and philosophy of events.

Provability, Computability and Reflection.

This is the first in a series of papers on Predicative Algebraic Set Theory, where we lay the necessary groundwork for the subsequent parts, one on realizability [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory II: Realizability, Theoret. Comput. Sci. . Available from: arXiv:0801.2305, 2008], and the other on sheaves [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory III: Sheaf models, 2008 ]. We introduce the notion of a predicative (...) category with small maps and show that it provides a sound and complete semantics for constructive set theories like IZF and CZF. The main technical contribution of this paper is that it shows in detail that such categories can always be conservatively embedded in categories that are exact. These exactness properties play a crucial rôle in showing that predicative categories with small maps contain models of set theory and that they are closed under sheaves and realizability. We will prove the former statement in this paper as well, leaving a proof of the closure properties to the papers on realizability and sheaves as mentioned above. (shrink)

Shramko [. Truth, falsehood, information and beyond: The American plan generalized. In K. Bimbo, J. Michael Dunn on information based logics, outstanding contributions to logic...

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

Inquiry into the metaphysics of essence tends to be pursued in a realist and model-theoretic spirit, in the sense that metaphysical vocabulary is used in a metalanguage to model truth conditions for the object-language use of essentialist vocabulary. This essay adapts recent developments in proof-theoretic semantics to provide a nominalist analysis for a variety of essentialist vocabularies. A metalanguage employing explanatory inferences is used to individuate introduction and elimination rules for atomic sentences. The object-language assertions of sentences concerning essences are (...) then interpreted as devices for marking off structural features of the explanatory inferences that, under a given interpretation, constitute the contents of the atoms of the language. On this proposal, object-language essentialist vocabulary is mentioned in a proof-theoretic metalanguage that uses a vocabulary of explanation. The result is a nominalist interpretation of essence as a modality, understood in the grammatical sense as a modification of the copula, and a view of metaphysical inquiry that is closely connected to the explanatory commitments present in first-order inquiry into things like sets, chemicals, and organisms. This result illustrates that some of the presuppositions that have animated analytic metaphysics over the last few decades can be profitably substituted with more practice-oriented conceptions of the forms of reasoning at work in different domains of human knowledge. (shrink)

In the paper Busaniche and Cignoli we presented a quasivariety of commutative residuated lattices, called NPc-lattices, that serves as an algebraic semantics for paraconsistent Nelson’s logic. In the present paper we show that NPc-lattices form a subvariety of the variety of commutative residuated lattices, we study congruences of NPc-lattices and some subvarieties of NPc-lattices.

No-cloning theorem says that there is no unitary operation that makes perfect clones of non-orthogonal quantum states. The objective of the present paper is to examine whether an imperfect cloning operation exists or not in a C*-algebraic framework. We define a universal \ -imperfect cloning operation which tolerates a finite loss \ of fidelity in the cloned state, and show that an individual system’s algebra of observables is abelian if and only if there is a universal \ -imperfect cloning (...) operation in the case where the loss of fidelity is less than \ . Therefore in this case no universal \ -imperfect cloning operation is possible in algebraic quantum theory. (shrink)

Some of the basic substructural logics are shown by Ono to have the disjunction property (DP) by using cut elimination of sequent calculi for these logics. On the other hand, this syntactic method works only for a limited number of substructural logics. Here we show that Maksimova's criterion on the DP of superintuitionistic logics can be naturally extended to one on the DP of substructural logics over FL. By using this, we show the DP for some of the substructural logics (...) for which syntactic methods don't work well. (shrink)

Two classes of π are studied whose properties are similar to those of the protoalgebraic deductive systems of Blok and Pigozzi. The first is the class of N-protoalgebraic π-institutions and the second is the wider class of N-prealgebraic π-institutions. Several characterizations are provided. For instance, N-prealgebraic π-institutions are exactly those π-institutions that satisfy monotonicity of the N-Leibniz operator on theory systems and N-protoalgebraic π-institutions those that satisfy monotonicity of the N-Leibniz operator on theory families. Analogs of the correspondence property of (...) Blok and Pigozzi for π-institutions are also introduced and their connections with preand protoalgebraicity are explored. Finally, relations of these two classes with the (, N)-algebraic systems, introduced previously by the author as an analog of the -algebras of Font and Jansana, and with an analog of the Suszko operator of Czelakowski for π-institutions are also investigated. (shrink)

A category theoretic generalization of the theory of algebraizable deductive systems of Blok and Pigozzi is developed. The theory of institutions of Goguen and Burstall is used to provide the underlying framework which replaces and generalizes the universal algebraic framework based on the notion of a deductive system. The notion of a term -institution is introduced first. Then the notions of quasi-equivalence, strong quasi-equivalence and deductive equivalence are defined for -institutions. Necessary and sufficient conditions are given for the quasi-equivalence (...) and the deductive equivalence of two term -institutions, based on the relationship between their categories of theories. The results carry over without any complications to institutions, via their associated -institutions. The -institution associated with a deductive system and the institution of equational logic are examined in some detail and serve to illustrate the general theory. (shrink)

This paper surveys the issue of relativistic causality within the framework of algebraic quantum field theory . In doing so, we distinguish various notions of causality formulated in the literature and study their relationships, and thereby we offer what we hope to be a useful taxonomy. We propose that the most direct expression of relativistic causality in AQFT is captured not by the spectrum condition but rather by the axiom of local primitive causality, in that it entails a form (...) of local determinism for quantum fields which generalizes the constraint of no superluminal propagation of classical field theories to relativistic quantum field theory. We discuss the status of the axiom of micro-causality by locating its place within a large family of separability/independence/locality conditions developed for AQFT and also by relating it to so-called no-signalling theorems. And we also provide a critical survey of attempts to understand the implications for relativistic causality of the distant correlations endemic to the states in models of AQFT satisfying the standard axioms, and we provide an assessment of attempts to employ Reichenbach's common cause principle in AQFT to defuse worries that these distant correlations implicate direct causal connections between relatively spacelike events. (shrink)

In the paper it will be shown that Reichenbach’s Weak Common Cause Principle is not valid in algebraic quantum field theory with locally finite degrees of freedom in general. Namely, for any pair of projections A, B supported in spacelike separated double cones ${\mathcal{O}}_{a}$ and ${\mathcal{O}}_{b}$ , respectively, a correlating state can be given for which there is no nontrivial common cause (system) located in the union of the backward light cones of ${\mathcal{O}}_{a}$ and ${\mathcal{O}}_{b}$ and commuting with the (...) both A and B. Since noncommuting common cause solutions are presented in these states the abandonment of commutativity can modulate this result: noncommutative Common Cause Principles might survive in these models. (shrink)

divisibility in Physics VI. I had been assuming at that time that Aristotle's elimination of reference to the infinitely large in his account of the potential inf inite--like the elimination of the infinitely small from nineteenth century accounts of limits and continuity--gave us everything that was important in a theory of the infinite. Hilbert's paper showed me that this was not obviously so. Suddenly other certainties about Aristotle's (apparently) judicious toning down of (supposed) Platonic extremisms began to crumble. The upshot (...) of work I had been doing earlier on Plato's 'Third Man Argument' began to look different from the way it had before. I was confronted with a possibility I had not till then so much as entertained. What if the more extreme posi tions of Plato on these issues were the more likely to be correct? The present work is the first instalment of the result ing reassessment of Plato's metaphysics, and especially of his theory of Forms. It has occupied much of my teaching and scholarly time over the past fifteen years and more. The central question wi th which I concern myself is, "How does Plato argue for the existence of his Forms (if he does )7" The idea of making this the central question is that if we know how he argues for the existence of Forms, we may get a better sense of what they are. (shrink)

When we understand that something is a pot, is it because of one property that all pots share? This seems unlikely, but without this common essence, it is difficult to see how we could teach someone to use the word "pot" or to see something as _a_ pot. The Buddhist apoha theory tries to resolve this dilemma, first, by rejecting properties such as "potness" and, then, by claiming that the element uniting all pots is their very difference from all non-pots. (...) In other words, when we seek out a pot, we select an object that is not a non-pot, and we repeat this practice with all other items and expressions. Writing from the vantage points of history, philosophy, and cognitive science, the contributors to this volume clarify the nominalist apoha theory and explore the relationship between apoha and the scientific study of human cognition. They engage throughout in a lively debate over the theory's legitimacy. Classical Indian philosophers challenged the apoha theory's legitimacy, believing instead in the existence of enduring essences. Seeking to settle this controversy, essays explore whether apoha offers new and workable solutions to problems in the scientific study of human cognition. They show that the work of generations of Indian philosophers can add much toward the resolution of persistent conundrums in analytic philosophy and cognitive science. (shrink)

The objective of this paper is to develop neutrosophic quadruple algebraic hyperstructures. Specifically, we develop neutrosophic quadruple semihypergroups, neutrosophic quadruple canonical hypergroups and neutrosophic quadruple hyperrings and we present elementary properties which characterize them.

In the first section of this paper I present a well known objection to meaning holism, according to which holism is inconsistent with natural language being learnable. Then I show that the objection fails if language acquisition includes stages of partial grasp of the meaning of at least some expressions, and I argue that standard model theoretic semantics cannot fully capture such stages. In the second section the above claims are supported through a review of current research into language acquisition. (...) Finally, in the third section it is argued that contemporary algebraic logical systems consist in a superior formal vehicle through which to capture stages of partial grasp of meaning; this claim is supported by concrete examples. (shrink)

There is a constructive method to define a structure of simple k-cyclic Post algebra of order p, L p,κ, on a given finite field F, and conversely. There exists an interpretation Ф₁ of the variety V generated by L p,κ into the variety V) generated by F and an interpretation Ф₂ of V) into V such that Ф₂Ф₁ = B for every B ϵ V and Ф₁₂ = R for every R ϵ V). In this paper we show how we (...) can solve an algebraic system of equations over an arbitrary cyclic Post algebra of order p, p prime, using the above interpretation, Gröbner bases and algorithms programmed in Maple. (shrink)

We characterize when the elementary diagram of a mutually algebraic structure has a model complete theory, and give an explicit description of a set of existential formulas to which every formula is equivalent. This characterization yields a new, more constructive proof that the elementary diagram of any model of a strongly minimal, trivial theory is model complete.

Recently, Caleiro, Gon¸calves and Martins introduced the notion of behaviorally algebraizable logic. The main idea behind their work is to replace, in the traditional theory of algebraizability of Blok and Pigozzi, unsorted equational logic with multi-sorted behavioral logic. The new notion accommodates logics over many-sorted languages and with non-truth-functional connectives. Moreover, it treats logics that are not algebraizable in the traditional sense while, at the same time, shedding new light to the equivalent algebraic semantics of logics that are algebraizable (...) according to the original theory. In this paper, the notion of an abstract multi-sorted π-institution is introduced so as to transfer elements of the theory of behavioral algebraizability to the categorical setting. Institutions formalize a wider variety of logics than deductive systems, including logics involving multiple signatures and quantifiers. The framework developed has the same relation to behavioral algebraizability as the classical categorical abstract algebraic logic framework has to the original theory of algebraizability of Blok and Pigozzi. (shrink)

We review, and in the process unify two techniques , for proving results concerning amalgamation in several classes studied in algebraic logic. The logical counterpart of these results adress interpolation and definability properties in modal and algebraic logic. Presenting them in a functorial context as adjoint situations, we show that both techniques can indeed be seen as instances of the use of the Keisler-Shelah ultrapower Theorem in proving Robinson's Joint Consistency Theorem. Some new results are surveyed. The results (...) of this paper are presented in such a way that further establishes the hitherto existing well developed duality theory between boolean algebras with operators and modal logic. (shrink)

We develop a general algebraic and proof-theoretic study of substructural logics that may lack associativity, along with other structural rules. Our study extends existing work on substructural logics over the full Lambek Calculus [34], Galatos and Ono [18], Galatos et al. [17]). We present a Gentzen-style sequent system that lacks the structural rules of contraction, weakening, exchange and associativity, and can be considered a non-associative formulation of . Moreover, we introduce an equivalent Hilbert-style system and show that the logic (...) associated with and is algebraizable, with the variety of residuated lattice-ordered groupoids with unit serving as its equivalent algebraic semantics. Overcoming technical complications arising from the lack of associativity, we introduce a generalized version of a logical matrix and apply the method of quasicompletions to obtain an algebra and a quasiembedding from the matrix to the algebra. By applying the general result to specific cases, we obtain important logical and algebraic properties, including the cut elimination of and various extensions, the strong separation of , and the finite generation of the variety of residuated lattice-ordered groupoids with unit. (shrink)

The fields K) of generalized power series with coefficients in a field K and exponents in an additive abelian ordered group G play an important role in the study of real closed fields. The subrings K) consisting of series with non-positive exponents find applications in the study of models of weak axioms for arithmetic. Berarducci showed that the ideal JK) generated by the monomials with negative exponents is prime when is the additive group of the reals, and asked whether the (...) same holds for any G. We prove that this is the case and that in the quotient ring K)/J, each element admits at least one factorization into irreducibles. (shrink)