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Mathematical logic

New York: Springer Verlag (1976)

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  1. Probabilities defined on standard and non-standard cylindric set algebras.Miklós Ferenczi - 2015 - Synthese 192 (7):2025-2033.
    Cylindric set algebras are algebraizations of certain logical semantics. The topic surveyed here, i.e. probabilities defined on cylindric set algebras, is closely related, on the one hand, to probability logic (to probabilities defined on logical formulas), on the other hand, to measure theory. The set algebras occuring here are associated, in particular, with the semantics of first order logic and with non-standard analysis. The probabilities introduced are partially continous, they are continous with respect to so-called cylindric sums.
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  • A sequent calculus for relation algebras.Roger Maddux - 1983 - Annals of Pure and Applied Logic 25 (1):73-101.
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  • Some Boolean Algebras with Finitely Many Distinguished Ideals I.Regina Aragón - 1995 - Mathematical Logic Quarterly 41 (4):485-504.
    We consider the theory Thprin of Boolean algebras with a principal ideal, the theory Thmax of Boolean algebras with a maximal ideal, the theory Thac of atomic Boolean algebras with an ideal where the supremum of the ideal exists, and the theory Thsa of atomless Boolean algebras with an ideal where the supremum of the ideal exists. First, we find elementary invariants for Thprin and Thsa. If T is a theory in a first order language and α is a linear (...)
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  • Empiricism, Probability, and Knowledge of Arithmetic.Sean Walsh - 2014 - Journal of Applied Logic 12 (3):319–348.
    The topic of this paper is our knowledge of the natural numbers, and in particular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgements of probability. While considerations of probability have come to the forefront in recent epistemology, it seems safe to say that the thesis defended here is heterodox from the vantage point of traditional philosophy (...)
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  • Logicism, Interpretability, and Knowledge of Arithmetic.Sean Walsh - 2014 - Review of Symbolic Logic 7 (1):84-119.
    A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation (...)
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  • Untersuchungen zur Friedmanschen Theorie der Prädikate.Martin Kühnrich - 1986 - Mathematical Logic Quarterly 32 (1‐5):29-44.
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  • A common generalization for MV-algebras and Łukasiewicz–Moisil algebras.George Georgescu & Andrei Popescu - 2006 - Archive for Mathematical Logic 45 (8):947-981.
    We introduce the notion of n-nuanced MV-algebra by performing a Łukasiewicz–Moisil nuancing construction on top of MV-algebras. These structures extend both MV-algebras and Łukasiewicz–Moisil algebras, thus unifying two important types of structures in the algebra of logic. On a logical level, n-nuanced MV-algebras amalgamate two distinct approaches to many valuedness: that of the infinitely valued Łukasiewicz logic, more related in spirit to the fuzzy approach, and that of Moisil n-nuanced logic, which is more concerned with nuances of truth rather than (...)
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  • Decidability of General Extensional Mereology.Hsing-Chien Tsai - 2013 - Studia Logica 101 (3):619-636.
    The signature of the formal language of mereology contains only one binary predicate P which stands for the relation “being a part of”. Traditionally, P must be a partial ordering, that is, ${\forall{x}Pxx, \forall{x}\forall{y}((Pxy\land Pyx)\to x=y)}$ and ${\forall{x}\forall{y}\forall{z}((Pxy\land Pyz)\to Pxz))}$ are three basic mereological axioms. The best-known mereological theory is “general extensional mereology”, which is axiomatized by the three basic axioms plus the following axiom and axiom schema: (Strong Supplementation) ${\forall{x}\forall{y}(\neg Pyx\to \exists z(Pzy\land \neg Ozx))}$ , where Oxy means ${\exists (...)
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  • Models and theories I: The semantic view revisited.Chuang Liu - 1997 - International Studies in the Philosophy of Science 11 (2):147 – 164.
    The paper, as Part I of a two-part series, argues for a hybrid formulation of the semantic view of scientific theories. For stage-setting, it first reviews the elements of the model theory in mathematical logic (on whose foundation the semantic view rests), the syntactic and the semantic view, and the different notions of models used in the practice of science. The paper then argues for an integration of the notions into the semantic view, and thereby offers a hybrid semantic view, (...)
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  • How to Say Things with Formalisms.David Auerbach - 1992 - In Michael Detlefsen (ed.), Proof, Logic and Formalization. London, England: Routledge. pp. 77--93.
    Recent attention to "self-consistent" (Rosser-style) systems raises anew the question of the proper interpretation of the Gödel Second Incompleteness Theorem and its effect on Hilbert's Program. The traditional rendering and consequence is defended with new arguments justifying the intensional correctness of the derivability conditions.
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  • Scientific progress.Ilkka Niiniluoto - 2008 - Synthese.
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  • Theory evolution and reference kinematics.Bernhard Lauth - 1991 - Synthese 88 (3):279 - 307.
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  • Intensionality and the gödel theorems.David D. Auerbach - 1985 - Philosophical Studies 48 (3):337--51.
    Philosophers of language have drawn on metamathematical results in varied ways. Extensionalist philosophers have been particularly impressed with two, not unrelated, facts: the existence, due to Frege/Tarski, of a certain sort of semantics, and the seeming absence of intensional contexts from mathematical discourse. The philosophical import of these facts is at best murky. Extensionalists will emphasize the success and clarity of the model theoretic semantics; others will emphasize the relative poverty of the mathematical idiom; still others will question the aptness (...)
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  • Inductive inference in the limit for first-order sentences.Bernhard Lauth - 1993 - Studia Logica 52 (4):491 - 517.
    The paper investigates learning functions for first order languages. Several types of convergence and identification in the limit are defined. Positive and negative results on learning problems are presented throughout the paper.
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  • Predicative foundations of arithmetic.Solomon Feferman & Geoffrey Hellman - 1995 - Journal of Philosophical Logic 24 (1):1 - 17.
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  • A metaphysical foundation for mathematical philosophy.Wójtowicz Krzysztof & Skowron Bartłomiej - 2022 - Synthese 200 (4):1-28.
    Although mathematical philosophy is flourishing today, it remains subject to criticism, especially from non-analytical philosophers. The main concern is that even if formal tools serve to clarify reasoning, they themselves contribute nothing new or relevant to philosophy. We defend mathematical philosophy against such concerns here by appealing to its metaphysical foundations. Our thesis is that mathematical philosophy can be founded on the phenomenological theory of ideas as developed by Roman Ingarden. From this platonist perspective, the “unreasonable effectiveness of mathematics in (...)
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  • Against Fregean Quantification.Bryan Pickel & Brian Rabern - 2023 - Ergo: An Open Access Journal of Philosophy 9 (37):971-1007.
    There are two dominant approaches to quantification: the Fregean and the Tarskian. While the Tarskian approach is standard and familiar, deep conceptual objections have been pressed against its employment of variables as genuine syntactic and semantic units. Because they do not explicitly rely on variables, Fregean approaches are held to avoid these worries. The apparent result is that the Fregean can deliver something that the Tarskian is unable to, namely a compositional semantic treatment of quantification centered on truth and reference. (...)
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  • Advances in Natural Deduction: A Celebration of Dag Prawitz's Work.Luiz Carlos Pereira & Edward Hermann Haeusler (eds.) - 2012 - Dordrecht, Netherland: Springer.
    This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order (...)
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  • Supercover Semantics for Deontic Action Logic.Karl Nygren - 2019 - Journal of Logic, Language and Information 28 (3):427-458.
    The semantics for a deontic action logic based on Boolean algebra is extended with an interpretation of action expressions in terms of sets of alternative actions, intended as a way to model choice. This results in a non-classical interpretation of action expressions, while sentences not in the scope of deontic operators are kept classical. A deontic structure based on Simons’ supercover semantics is used to interpret permission and obligation. It is argued that these constructions provide ways to handle various problems (...)
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  • Boolean algebras in ast.Klaus Schumacher - 1992 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1):373-382.
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  • On the Expressive Power of Equality-Free First Order Languages.P. Ecsedi-Tóth - 1986 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (19-24):371-375.
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  • On the Algebraic Structure of Primitive Recursive Functions.István Szalkai - 1985 - Mathematical Logic Quarterly 31 (35-36):551-556.
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  • Saturated and Special Models in Modal Model Theory With Applications to the Modal and DE RE Hierarchies.Hirokazu Nishimura - 1980 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 26 (31):481-490.
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  • On Absoluteness.Karol Habart - 1989 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (5):469-480.
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  • The Decision Problem for Certain Nilpotent Closed Varieties.Stephen D. Comer - 1981 - Mathematical Logic Quarterly 27 (31-35):557-560.
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  • Applications of Many‐Sorted Robinson Consistency Theorem.Daniele Mundici - 1981 - Mathematical Logic Quarterly 27 (11-12):181-188.
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  • Model Theories of Set Theories and Type Theory.Robert Murray Jones - 2014 - Open Journal of Philosophy 4 (1):54-58.
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  • Preservation theorems in linear continuous logic.Seyed-Mohammad Bagheri & Roghieh Safari - 2014 - Mathematical Logic Quarterly 60 (3):168-176.
    Linear continuous logic is the fragment of continuous logic obtained by restricting connectives to addition and scalar multiplications. Most results in the full continuous logic have a counterpart in this fragment. In particular a linear form of the compactness theorem holds. We prove this variant and use it to deduce some basic preservation theorems.
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  • Modality and axiomatic theories of truth I: Friedman-Sheard.Johannes Stern - 2014 - Review of Symbolic Logic 7 (2):273-298.
    In this investigation we explore a general strategy for constructing modal theories where the modal notion is conceived as a predicate. The idea of this strategy is to develop modal theories over axiomatic theories of truth. In this first paper of our two part investigation we develop the general strategy and then apply it to the axiomatic theory of truth Friedman-Sheard. We thereby obtain the theory Modal Friedman-Sheard. The theory Modal Friedman-Sheard is then discussed from three different perspectives. First, we (...)
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  • Representable cylindric algebras.Leon Henkin, J. Donald Monk & Alfred Tarski - 1986 - Annals of Pure and Applied Logic 31:23-60.
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  • On varieties of cylindric algebras with applications to logic.I. Németi - 1987 - Annals of Pure and Applied Logic 36:235-277.
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  • Algorithms for sentences over integral domains.Shih Ping Tung - 1990 - Annals of Pure and Applied Logic 47 (2):189-197.
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  • Elementary descent recursion and proof theory.Harvey Friedman & Michael Sheard - 1995 - Annals of Pure and Applied Logic 71 (1):1-45.
    We define a class of functions, the descent recursive functions, relative to an arbitrary elementary recursive system of ordinal notations. By means of these functions, we provide a general technique for measuring the proof-theoretic strength of a variety of systems of first-order arithmetic. We characterize the provable well-orderings and provably recursive functions of these systems, and derive various conservation and equiconsistency results.
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  • A Note on the Interpolation Theorem in First Order Logic.George Weaver - 1982 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 28 (14-18):215-218.
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  • On Absoluteness.Karol Habart - 1989 - Mathematical Logic Quarterly 35 (5):469-480.
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  • On the Expressive Power of Equality‐Free First Order Languages.P. Ecsedi‐Tóth - 1986 - Mathematical Logic Quarterly 32 (19‐24):371-375.
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  • The Decision Problem for Certain Nilpotent Closed Varieties.Stephen D. Comer - 1981 - Mathematical Logic Quarterly 27 (31‐35):557-560.
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  • Applications of Many‐Sorted Robinson Consistency Theorem.Daniele Mundici - 1981 - Mathematical Logic Quarterly 27 (11‐12):181-188.
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  • Saturated and Special Models in Modal Model Theory With Applications to the Modal and DE RE Hierarchies.Hirokazu Nishimura - 1980 - Mathematical Logic Quarterly 26 (31):481-490.
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  • Artificial Intelligence as a Possible Tool for Discovering Laws of Logic.David Isles - 1978 - Cognitive Science 2 (4):329-360.
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  • Boolean algebras in ast.Klaus Schumacher - 1992 - Mathematical Logic Quarterly 38 (1):373-382.
    In this paper we investigate Boolean algebras and their subalgebras in Alternative Set Theory . We show that any two countable atomless Boolean algebras are isomorphic and we give an example of such a Boolean algebra. One other main result is, that there is an infinite Boolean algebra freely generated by a set. At the end of the paper we show that the sentence “There is no non-trivial free group which is a set” is consistent with AST.
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  • On the Algebraic Structure of Primitive Recursive Functions.István Szalkai - 1985 - Mathematical Logic Quarterly 31 (35‐36):551-556.
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  • On Meaningfulness and Truth.Brian Edison McDonald - 2000 - Journal of Philosophical Logic 29 (5):433 - 482.
    We show how to construct certain " $[Unrepresented Character]_{M,T}$ -type" interpreted languages, with each such language containing meaningfulness and truth predicates which apply to itself. These languages are comparable in expressive power to the $[Unrepresented Character]_{T}$ -type, truth-theoretic languages first considered by. Kripke, yet each of our $[Unrepresented Character]_{M,T}$ -type languages possesses the additional advantage that, within it, the meaninglessness of any given meaningless expression can itself be meaningfully expressed. One therefore has, for example, the object level truth (and meaningfulness) (...)
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  • On the logical structure of some value systems of classical economics: Marx and Sraffa.David Pearce & Michele Tucci - 1982 - Theory and Decision 14 (2):155-175.
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  • A Comprehensive Picture of the Decidability of Mereological Theories.Hsing-Chien Tsai - 2013 - Studia Logica 101 (5):987-1012.
    The signature of the formal language of mereology contains only one binary predicate which stands for the relation “being a part of” and it has been strongly suggested that such a predicate must at least define a partial ordering. Mereological theories owe their origin to Leśniewski. However, some more recent authors, such as Simons as well as Casati and Varzi, have reformulated mereology in a way most logicians today are familiar with. It turns out that any theory which can be (...)
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  • The compactness of first-order logic:from gödel to lindström.John W. Dawson - 1993 - History and Philosophy of Logic 14 (1):15-37.
    Though regarded today as one of the most important results in logic, the compactness theorem was largely ignored until nearly two decades after its discovery. This paper describes the vicissitudes of its evolution and transformation during the period 1930-1970, with special attention to the roles of Kurt Gödel, A. I. Maltsev, Leon Henkin, Abraham Robinson, and Alfred Tarski.
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  • Ein neuer “strukturtyp” Von logikbuch? [REVIEW]Ulrich Nortmann - 1987 - Erkenntnis 27 (1):113 - 145.
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  • Scientific progress.Ilkka Niiniluoto - 1980 - Synthese 45 (3):427 - 462.
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  • Der deskriptive glaubensbegriff.Y. Nakayama - 1988 - Erkenntnis 28 (1):29 - 53.
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