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Journal articles on the topic 'Natural logic'

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Published: 4 June 2021

Last updated: 8 February 2022

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1

Avron, Arnon. "Natural 3-valued logics—characterization and proof theory." Journal of Symbolic Logic 56, no. 1 (March 1991): 276–94. http://dx.doi.org/10.2307/2274919.

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Many-valued logics in general and 3-valued logic in particular is an old subject which had its beginning in the work of Łukasiewicz [Łuk]. Recently there is a revived interest in this topic, both for its own sake (see, for example, [Ho]), and also because of its potential applications in several areas of computer science, such as proving correctness of programs [Jo], knowledge bases [CP] and artificial intelligence [Tu]. There are, however, a huge number of 3-valued systems which logicians have studied throughout the years. The motivation behind them and their properties are not always clear, and their proof theory is frequently not well developed. This state of affairs makes both the use of 3-valued logics and doing fruitful research on them rather difficult.Our first goal in this work is, accordingly, to identify and characterize a class of 3-valued logics which might be called natural. For this we use the general framework for characterizing and investigating logics which we have developed in [Av1]. Not many 3-valued logics appear as natural within this framework, but it turns out that those that do include some of the best known ones. These include the 3-valued logics of Łukasiewicz, Kleene and Sobociński, the logic LPF used in the VDM project, the logic RM3 from the relevance family and the paraconsistent 3-valued logic of [dCA]. Our presentation provides justifications for the introduction of certain connectives in these logics which are often regarded as ad hoc. It also shows that they are all closely related to each other. It is shown, for example, that Łukasiewicz 3-valued logic and RM3 (the strongest logic in the family of relevance logics) are in a strong sense dual to each other, and that both are derivable by the same general construction from, respectively, Kleene 3-valued logic and the 3-valued paraconsistent logic.

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Томова, Н. Е. "Natural three-valued logics and classical logic." Logical Investigations 19 (April 9, 2013): 344–52. http://dx.doi.org/10.21146/2074-1472-2013-19-0-344-352.

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In this paper implicative fragments of natural three- valued logic are investigated. It is proved that some fragments are equivalent by set of tautologies to implicative fragment of classical logic. It is also shown that some natural three-valued logics verify all tautologies of classical propositional logic.

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3

Wu, Kun, and Zhensong Wang. "Natural Philosophy and Natural Logic." Philosophies 3, no. 4 (September 21, 2018): 27. http://dx.doi.org/10.3390/philosophies3040027.

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1. Nature has its own logic, which does not follow the human will. Nature is itself; it exists, moves, changes, and evolves according to its own intrinsic ways. Human and human society, as a product of a specific stage of natural development, can only be a concrete manifestation of the logic of nature. 2. In the broad sense, nature refers to all, both phenomena and processes, in the universe. It includes human society spiritual phenomena. In a narrow sense, nature refers to the world outside the society and opposed to society as well, or refers to the research objects of natural sciences 3. The narrow natural philosophy is in the intermediary position between the natural sciences and the overall philosophy (the supreme philosophy, an advocation of Kun Wu’s philosophy of information. For further detail, please refer to the subscript in the following.). Furthermore, it is an independent sub-level philosophical discipline; the broad natural philosophy is a meta-philosophy or supreme philosophy, stipulating the entire world from the dimensions of nature itself. 4. Natural philosophy reveals the laws of nature’s own existence, movement, change, and evolution. This determines that the way of expressing natural philosophy is necessarily natural ontology. The construction of the theoretical system of natural philosophy is inevitably a process of abandoning cognitive mediums of human beings through reflection. It is necessary for us to conclude that natural philosophy is the stipulation of nature itself, which comes out of the nature itself. So, we must explain the nature from the standpoint of the nature itself. 5. The true philosophy should move from the human world to the nature, finding back Husserl’s suspended things, and establish a brand-new philosophy in which man and nature, substance, information, and spirit are united. This kind of philosophy is able to provide contemporary ecological civilization with a reasonable philosophical foundation, rebuilding natural philosophy in a new era, which is a very urgent task for contemporary philosophers. 6. The unity of philosophy and science cannot be seen merely as an external convergence, but also as an intrinsic fusion; a true philosophy should have a scientific character, and science itself must have a philosophical basis. The unity of such an intrinsic fusion of science and philosophy can be fully demonstrated by the practical relationship of development between human philosophy and science. 7. In addition to the narrow path along epistemology, linguistics, and phenomenology, the development of human philosophy has another path. This is the development of philosophy itself that has been nurtured and demonstrated during the development of general science: On the one hand, the construction of scientific rationality requires philosophical thinking and exploration; On the other hand, the progress of science opens the way for the development of philosophy. 8. In the real process of the development of human knowledge, science and philosophy are regulated, contained, and merged with each other in the process of interaction. The two are inlaid together internally to form an interactive dynamic feedback loop. The unified relationship of mutual influence, regulation, promotion and transformation presented in the intrinsic interplay of interaction between science and philosophy profoundly breeds and demonstrates the general way of human knowledge development: the philosophicalization (a term used in Kun Wu’s philosophy of information. For more details please see in Kun Wu, 2016, The Interaction and Convergence of the Philosophy and Science of Information, https://doi.org/10.3390/philosophies1030228) of science and scientification (a term used in Kun Wu’s philosophy of information. For more detail, please see in Kun Wu, 2016, The Interaction and Convergence of the Philosophy and Science of Information, https://doi.org/10.3390/philosophies1030228) of philosophy. 9. We face two types of dogmatism: one is the dogmatism of naturalism, and the other is the dogmatism of the philosophy of consciousness. One of the best ways to overcome these tendencies of dogmatism is to unite natural ontology, and epistemic constructivism. The crisis of contemporary philosophy induced by the western consciousness philosophy seems like belonging to the field of epistemology, but the root of this crisis is deeply buried in the ontology. The key to solving the crisis of contemporary philosophy lies precisely in the reconstruction of the doctrine of natural philosophy centering to the nature itself and excluding God. The task to be accomplished by this new natural philosophy is how to regain the natural foundation of human consciousness after the God has left the field. 10. Since the 1980s, the philosophy of information established and developed in China has proposed a theory of objective information, as well as the dual existence and dual evolution of matter and information (a key advocation in the ontological theory of Kun Wu’s philosophy of information). It is this theory that has made up for the vacancy existing between matter and mind, which apparently exists in Cartesian dualism, after the withdrawal of the God’s from the field. Philosophy of information in China is first and foremost a natural philosophy that adheres to naturalistic attitudes. Second, this natural philosophy explains the human, human mind and human society in the interpretation of the process and mechanism of natural evolution. In this connection, philosophy of information (a key advocation of Kun Wu’s philosophy of information) in China is a system of meta-philosophy or supreme philosophy. This system undoubtedly has the nature of a new natural philosophy. At the same time, this philosophy can better reflect the philosophical spirit of the information age.

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4

BESSON, CORINE. "EXTERNALISM, INTERNALISM, AND LOGICAL TRUTH." Review of Symbolic Logic 2, no. 1 (March 2009): 1–29. http://dx.doi.org/10.1017/s1755020309090091.

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The aim of this paper is to show what sorts of logics are required by externalist and internalist accounts of the meanings of natural kind nouns. These logics give us a new perspective from which to evaluate the respective positions in the externalist--internalist debate about the meanings of such nouns. The two main claims of the paper are the following: first, that adequate logics for internalism and externalism about natural kind nouns are second-order logics; second, that an internalist second-order logic is a free logic—a second order logic free of existential commitments for natural kind nouns, while an externalist second-order logic is not free of existential commitments for natural kind nouns—it is existentially committed.

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5

Hodges, Wilfrid. "Traditional Logic, Modern Logic and Natural Language." Journal of Philosophical Logic 38, no. 6 (October 14, 2009): 589–606. http://dx.doi.org/10.1007/s10992-009-9113-y.

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6

Peregrin, Jaroslav. "Logic and Natural Selection." Logica Universalis 4, no. 2 (July 20, 2010): 207–23. http://dx.doi.org/10.1007/s11787-010-0018-x.

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7

Uckelman, Sara L. "A Quantified Temporal Logic for Ampliation and Restriction." Vivarium 51, no. 1-4 (2013): 485–510. http://dx.doi.org/10.1163/15685349-12341259.

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Abstract Temporal logic as a modern discipline is separate from classical logic; it is seen as an addition or expansion of the more basic propositional and predicate logics. This approach is in contrast with logic in the Middle Ages, which was primarily intended as a tool for the analysis of natural language. Because all natural language sentences have tensed verbs, medieval logic is inherently a temporal logic. This fact is most clearly exemplified in medieval theories of supposition. As a case study, we look at the supposition theory of Lambert of Lagny (Auxerre), extracting from it a temporal logic and providing a formalization of that logic.

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KREMER, PHILIP. "COMPLETENESS OF SECOND-ORDER PROPOSITIONAL S4 AND H IN TOPOLOGICAL SEMANTICS." Review of Symbolic Logic 11, no. 3 (September 2018): 507–18. http://dx.doi.org/10.1017/s1755020318000229.

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AbstractWe add propositional quantifiers to the propositional modal logic S4 and to the propositional intuitionistic logic H, introducing axiom schemes that are the natural analogs to axiom schemes typically used for first-order quantifiers in classical and intuitionistic logic. We show that the resulting logics are sound and complete for a topological semantics extending, in a natural way, the topological semantics for S4 and for H.

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9

Avron, Arnon. "Gentzenizing Schroeder-Heister's natural extension of natural deduction." Notre Dame Journal of Formal Logic 31, no. 1 (December 1989): 127–35. http://dx.doi.org/10.1305/ndjfl/1093635337.

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10

Brady, Ross T. "Normalized natural deduction systems for some relevant logics I: The logic DW." Journal of Symbolic Logic 71, no. 1 (March 2006): 35–66. http://dx.doi.org/10.2178/jsl/1140641162.

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Fitch-style natural deduction was first introduced into relevant logic by Anderson in [1960], for the sentential logic E of entailment and its quantincational extension EQ. This was extended by Anderson and Belnap to the sentential relevant logics R and T and some of their fragments in [ENT1], and further extended to a wide range of sentential and quantified relevant logics by Brady in [1984]. This was done by putting conditions on the elimination rules, →E, ~E, ⋁E and ∃E, pertaining to the set of dependent hypotheses for formulae used in the application of the rule. Each of these rules were subjected to the same condition, this condition varying from logic to logic. These conditions, together with the set of natural deduction rules, precisely determine the particular relevant logic. Generally, this is a simpler representation of a relevant logic than the standard Routley-Meyer semantics, with its existential modelling conditions stated in terms of an otherwise arbitrary 3-place relation R, which is defined over a possibly infinite set of worlds. Readers are urged to refer to Brady [1984], if unfamiliar with the above natural deduction systems, but we will introduce in §2 a modified version in full detail.Natural deduction for classical logic was invented by Jaskowski and Gentzen, but it was Prawitz in [1965] who normalized natural deduction, streamlining its rules so as to allow a subformula property to be proved. (This key property ensures that each formula in the proof of a theorem is indeed a subformula of that theorem.)

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11

Byrd, Michael. "The Logic of Natural Language." International Studies in Philosophy 18, no. 3 (1986): 98–100. http://dx.doi.org/10.5840/intstudphil198618336.

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12

Murray,, Bertram G. "Natural History and Deductive Logic." BioScience 36, no. 8 (September 1986): 513–14. http://dx.doi.org/10.2307/1310147.

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13

Over, David E. "The logic of natural sampling." Behavioral and Brain Sciences 30, no. 3 (June 2007): 277. http://dx.doi.org/10.1017/s0140525x07001859.

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AbstractBarbey & Sloman (B&S) relegate the logical rule of the excluded middle to a footnote. But this logical rule is necessary for natural sampling. Making the rule explicit in a logical tree can make a problem easier to solve. Examples are given of uses of the rule that are non-constructive and not reducible to a domain-specific module.

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14

Brauner, T. "Natural Deduction for Hybrid Logic." Journal of Logic and Computation 14, no. 3 (June 1, 2004): 329–53. http://dx.doi.org/10.1093/logcom/14.3.329.

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15

Maldonato, Mauro, and Silvia Dell’Orco. "The Natural Logic of Action." World Futures 69, no. 3 (January 2013): 174–83. http://dx.doi.org/10.1080/02604027.2013.788865.

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16

Purdy, William C. "A logic for natural language." Notre Dame Journal of Formal Logic 32, no. 3 (June 1991): 409–25. http://dx.doi.org/10.1305/ndjfl/1093635837.

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17

Pistoia-Reda, Salvatore, and Luca San Mauro. "On logicality and natural logic." Natural Language Semantics 29, no. 3 (July 14, 2021): 501–6. http://dx.doi.org/10.1007/s11050-021-09184-0.

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AbstractIn this paper we focus on the logicality of language, i.e. the idea that the language system contains a deductive device to exclude analytic constructions. Puzzling evidence for the logicality of language comes from acceptable contradictions and tautologies. The standard response in the literature involves assuming that the language system only accesses analyticities that are due to skeletons as opposed to standard logical forms. In this paper we submit evidence in support of alternative accounts of logicality, which reject the stipulation of a natural logic and assume instead the meaning modulation of nonlogical terms.

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18

Martin, Christopher J. "The Theory of Natural Consequence." Vivarium 56, no. 3-4 (October 15, 2018): 340–66. http://dx.doi.org/10.1163/15685349-12341357.

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Abstract The history of thinking about consequences in the Middle Ages divides into three periods. During the first of these, from the eleventh to the middle of the twelfth century, and the second, from then until the beginning of the fourteenth century, the notion of natural consequence played a crucial role in logic, metaphysics, and theology. The first part of this paper traces the development of the theory of natural consequence in Abaelard’s work as the conditional of a connexive logic with an equivalent connexive disjunction and the crisis precipitated by the discovery of inconsistency in this system. The second part considers the accounts of natural consequence given in the thirteenth century as a special case of the standard modal definition of consequence, one for which the principle ex impossibili quidlibet does not hold, in logics in which disjunction is understood extensionally.

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Gerla, Giangiacomo, and Roberto Tortora. "Fuzzy natural deduction." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 36, no. 1 (1990): 67–77. http://dx.doi.org/10.1002/malq.19900360108.

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20

Bilanová, Zuzana, Ján Perháč, Eva Chovancová, and Martin Chovanec. "Logic Analysis of Natural Language Based on Predicate Linear Logic." Acta Polytechnica Hungarica 17, no. 6 (2020): 239–52. http://dx.doi.org/10.12700/aph.17.6.2020.6.14.

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21

Cheng, Xian Yi, Chen Cheng, and Qian Zhu. "The Applications of Description Logics in Natural Language Processing." Advanced Materials Research 204-210 (February 2011): 381–86. http://dx.doi.org/10.4028/www.scientific.net/amr.204-210.381.

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As a sort of formalizing tool of knowledge representation, Description Logics have been successfully applied in Information System, Software Engineering and Natural Language processing and so on. Description Logics also play a key role in text representation, Natural Language semantic interpretation and language ontology description. Description Logics have been logical basis of OWL which is an ontology language that is recommended by W3C. This paper discusses the description logic basic ideas under vocabulary semantic, context meaning, domain knowledge and background knowledge.

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Cheng, Xian Yi, Chen Cheng, and Qian Zhu. "The Applications of Description Logics in Natural Language Processing." Advanced Materials Research 181-182 (January 2011): 236–41. http://dx.doi.org/10.4028/www.scientific.net/amr.181-182.236.

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As a sort of formalizing tool of knowledge representation, Description Logics have been successfully applied in Information System, Software Engineering and Natural Language processing and so on. Description Logics also play a key role in text representation, Natural Language semantic interpretation and language ontology description. Description Logics have been logical basis of OWL which is an ontology language that is recommended by W3C. This paper discusses the description logic basic ideas under vocabulary semantic, context meaning, domain knowledge and background knowledge.

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23

Lipparini, Paolo. "Some transfinite natural sums." Mathematical Logic Quarterly 64, no. 6 (December 2018): 514–28. http://dx.doi.org/10.1002/malq.201600092.

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24

Тамминга, А. "Correspondence analysis for strong three-valued logic." Logical Investigations 20 (May 8, 2014): 253–66. http://dx.doi.org/10.21146/2074-1472-2014-20-0-253-266.

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I apply Kooi and Tamminga’s (2012) idea of correspondence analysis for many-valued logics to strong three-valued logic (K3). First, I characterize each possible single entry in the truth-table of a unary or a binary truth-functional operator that could be added to K3 by a basic inference scheme. Second, I define a class of natural deduction systems on the basis of these charac- terizing basic inference schemes and a natural deduction system for K3. Third, I show that each of the resulting natural deduction systems is sound and complete with respect to its particular semantics. Among other things, I thus obtain a new proof system for _ukasiewicz’s three-valued logic.

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25

Murinová, Petra, and Vilém Novák. "Intermediate Syllogisms in Fuzzy Natural Logic." Journal of Fuzzy Set Valued Analysis 2016, no. 2 (2016): 99–111. http://dx.doi.org/10.5899/2016/jfsva-00271.

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26

Zhang, Lifeng. "The Logic of Natural Kind Terms." Philosophical Forum 45, no. 3 (July 25, 2014): 199–216. http://dx.doi.org/10.1111/phil.12036.

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27

Hawthorn, John. "Natural deduction in normal modal logic." Notre Dame Journal of Formal Logic 31, no. 2 (March 1990): 263–73. http://dx.doi.org/10.1305/ndjfl/1093635420.

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28

Fyodorov, Y. "Order-Based Inference in Natural Logic." Logic Journal of IGPL 11, no. 4 (July 1, 2003): 385–416. http://dx.doi.org/10.1093/jigpal/11.4.385.

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29

HSIEH, CHIH HSUN. "THE NATURAL OPERATIONS OF LINGUISTIC LOGIC." New Mathematics and Natural Computation 04, no. 01 (March 2008): 77–86. http://dx.doi.org/10.1142/s1793005708000969.

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A linguistic truth set in which each element is a linguistic truth value is discussed. Ranking linguistic truth values based on Graded Mean Integration Representation method is discussed also. We then give a decreasing linguistic truth set and an increasing linguistic truth set by using the above ranking method, and present a Not function of linguistic truth value combined by the above decreasing linguistic truth set and the increasing linguistic truth set. A minimum function and a maximum function based on representations of linguistic truth values are introduced. In addition, some natural operations of linguistic logic combined by minimum function and maximum function, and Not function are presented. Some properties of our presented natural operations are presented, and are proved. Furthermore, some application examples of linguistic logical statements are discussed finally.

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30

Troelstra, A. S. "Natural deduction for intuitionistic linear logic." Annals of Pure and Applied Logic 73, no. 1 (May 1995): 79–108. http://dx.doi.org/10.1016/0168-0072(93)e0078-3.

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31

Tranchini, Luca. "Natural deduction for bi-intuitionistic logic." Journal of Applied Logic 25 (December 2017): S72—S96. http://dx.doi.org/10.1016/j.jal.2017.12.001.

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32

Tranchini, Luca. "Natural Deduction for Dual-intuitionistic Logic." Studia Logica 100, no. 3 (June 2012): 631–48. http://dx.doi.org/10.1007/s11225-012-9417-8.

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33

von Plato, Jan. "From Axiomatic Logic to Natural Deduction." Studia Logica 102, no. 6 (June 25, 2014): 1167–84. http://dx.doi.org/10.1007/s11225-014-9565-0.

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34

van Benthem, Johan. "Natural Language and Logic of Agency." Journal of Logic, Language and Information 23, no. 3 (March 2, 2014): 367–82. http://dx.doi.org/10.1007/s10849-014-9188-x.

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35

Dahl, Veronica. "Natural language processing and logic programming." Journal of Logic Programming 19-20 (May 1994): 681–714. http://dx.doi.org/10.1016/0743-1066(94)90036-1.

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36

Maldonato. "Between Formal Logic and Natural Logic: Prolegomena for a Middle Way." American Journal of Psychology 125, no. 3 (2012): 387. http://dx.doi.org/10.5406/amerjpsyc.125.3.0387.

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37

Janssen, Theo M. V. "Compositional Natural Language Semantics using Independence Friendly Logic or Dependence Logic." Studia Logica 101, no. 2 (March 13, 2013): 453–66. http://dx.doi.org/10.1007/s11225-013-9480-9.

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VON PLATO, JAN. "KURT GÖDEL’S FIRST STEPS IN LOGIC: FORMAL PROOFS IN ARITHMETIC AND SET THEORY THROUGH A SYSTEM OF NATURAL DEDUCTION." Bulletin of Symbolic Logic 24, no. 3 (September 2018): 319–35. http://dx.doi.org/10.1017/bsl.2017.42.

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AbstractWhat seem to be Kurt Gödel’s first notes on logic, an exercise notebook of 84 pages, contains formal proofs in higher-order arithmetic and set theory. The choice of these topics is clearly suggested by their inclusion in Hilbert and Ackermann’s logic book of 1928, the Grundzüge der theoretischen Logik. Such proofs are notoriously hard to construct within axiomatic logic. Gödel takes without further ado into use a linear system of natural deduction for the full language of higher-order logic, with formal derivations closer to one hundred steps in length and up to four nested temporary assumptions with their scope indicated by vertical intermittent lines.

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Topal, Selçuk. "On Logics of Transitive Verbs With and Without Intersective Adjectives." Studia Humana 7, no. 1 (March 1, 2018): 31–43. http://dx.doi.org/10.2478/sh-2018-0003.

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Abstract The purpose of this paper is to contribute to the natural logic program which invents logics in natural language. This study presents two logics: a logical system called d R(∀,∃) containing transitive verbs and a more expressive logical system R(∀,∃, IA) containing both transitive verbs and intersective adjectives. The paper offers three different set-theoretic semantics which are equivalent for the logics.

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Gardner, Philippa. "Equivalences between logics and their representing type theories." Mathematical Structures in Computer Science 5, no. 3 (September 1995): 323–49. http://dx.doi.org/10.1017/s0960129500000785.

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We propose a new framework for representing logics, called LF+, which is based on the Edinburgh Logical Framework. The new framework allows us to give, apparently for the first time, general definitions that capture how well a logic has been represented. These definitions are possible because we are able to distinguish in a generic way that part of the LF+ entailment corresponding to the underlying logic. This distinction does not seem to be possible with other frameworks. Using our definitions, we show that, for example, natural deduction first-order logic can be well-represented in LF+, whereas linear and relevant logics cannot. We also show that our syntactic definitions of representation have a simple formulation as indexed isomorphisms, which both confirms that our approach is a natural one and provides a link between type-theoretic and categorical approaches to frameworks.

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Banaschewski, B. "Fixpoints Without the Natural Numbers." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 37, no. 8 (1991): 125–28. http://dx.doi.org/10.1002/malq.19910370804.

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42

Fenstad, Jens Erik. "Tarski, truth and natural languages." Annals of Pure and Applied Logic 126, no. 1-3 (April 2004): 15–26. http://dx.doi.org/10.1016/j.apal.2003.10.017.

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43

Velleman, Daniel J. "Variable declarations in natural deduction." Annals of Pure and Applied Logic 144, no. 1-3 (December 2006): 133–46. http://dx.doi.org/10.1016/j.apal.2006.05.009.

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Seuren, Pieter A. M. "The natural logic of language and cognition." Pragmatics. Quarterly Publication of the International Pragmatics Association (IPrA) 16, no. 1 (March 1, 2006): 103–38. http://dx.doi.org/10.1075/prag.16.1.02seu.

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This paper aims at an explanation of the discrepancies between natural intuitions and standard logic in terms of a distinction between NATURAL and CONSTRUCTED levels of cognition, applied to the way human cognition deals with sets. NATURAL SET THEORY (NST) restricts standard set theory cutting it down to naturalness. The restrictions are then translated into a theory of natural logic. The predicate logic resulting from these restrictions turns out to be that proposed in Hamilton (1860) and Jespersen (1917). Since, in this logic, NO is a quantifier in its own right, different from NOT-SOME, and given the assumption that natural lexicalization processes occur at the level of basic naturalness, single-morpheme lexicalizations for NOT-ALL should not occur, just as there is no single-morpheme lexicalization for NOT-SOME at that level. An analogous argument is developed for the systematic absence of lexicalizations for NOT-AND in propositional logic.

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45

Crossley, John N., and Jane Bridge Kister. "Natural well-orderings." Archiv für Mathematische Logik und Grundlagenforschung 26, no. 1 (December 1987): 57–76. http://dx.doi.org/10.1007/bf02017491.

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MILNE, PETER. "SUBFORMULA AND SEPARATION PROPERTIES IN NATURAL DEDUCTION VIA SMALL KRIPKE MODELS." Review of Symbolic Logic 3, no. 2 (June 2010): 175–227. http://dx.doi.org/10.1017/s175502030999030x.

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Various natural deduction formulations of classical, minimal, intuitionist, and intermediate propositional and first-order logics are presented and investigated with respect to satisfaction of the separation and subformula properties. The technique employed is, for the most part, semantic, based on general versions of the Lindenbaum and Lindenbaum–Henkin constructions. Careful attention is paid (i) to which properties of theories result in the presence of which rules of inference, and (ii) to restrictions on the sets of formulas to which the rules may be employed, restrictions determined by the formulas occurring as premises and conclusion of the invalid inference for which a counterexample is to be constructed. We obtain an elegant formulation of classical propositional logic with the subformula property and a singularly inelegant formulation of classical first-order logic with the subformula property, the latter, unfortunately, not a product of the strategy otherwise used throughout the article. Along the way, we arrive at an optimal strengthening of the subformula results for classical first-order logic obtained as consequences of normalization theorems by Dag Prawitz and Gunnar Stålmarck.

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Ding, Yifeng. "On the Logic of Belief and Propositional Quantification." Journal of Philosophical Logic 50, no. 5 (April 5, 2021): 1143–98. http://dx.doi.org/10.1007/s10992-021-09595-8.

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AbstractWe consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is something that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss.

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Levitz, Hilbert, and Warren Nichols. "A Natural Variant of Ackermann's Function." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 34, no. 5 (1988): 399–401. http://dx.doi.org/10.1002/malq.19880340504.

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Bedürftig, Thomjas. "Another Characterization of the Natural Numbers." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 35, no. 2 (1989): 185–86. http://dx.doi.org/10.1002/malq.19890350208.

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Zimmermann, Ernst. "Full Lambek Calculus in natural deduction." MLQ 56, no. 1 (January 2010): 85–88. http://dx.doi.org/10.1002/malq.200810042.

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