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  1. The Fundamental Theorem of Central Element Theory.Mariana Vanesa Badano & Diego Jose Vaggione - 2020 - Journal of Symbolic Logic 85 (4):1599-1606.
    We give a short proof of the fundamental theorem of central element theory. The original proof is constructive and very involved and relies strongly on the fact that the class be a variety. Here we give a more direct nonconstructive proof which applies for the more general case of a first-order class which is both closed under the formation of direct products and direct factors.
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  2. The (Greatest) Fragment of Classical Logic That Respects the Variable-Sharing Principle (in the Fmla-Fmla Framework).Damian Szmuc - forthcoming - Bulletin of the Section of Logic.
    We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined (...)
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  3. Double Negation Semantics for Generalisations of Heyting Algebras.Rob Arthan & Paulo Oliva - 2020 - Studia Logica 109 (2):341-365.
    This paper presents an algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations introduced by Kolmogorov, Gödel, Gentzen and Glivenko. We view these as variant semantics and present a semantic formulation of Troelstra’s syntactic criteria for a satisfactory negative translation. We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops. When a translation fails for a particular class of algebras, we (...)
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  4. Meaningless Divisions.Damian Szmuc & Thomas Macaulay Ferguson - forthcoming - Notre Dame Journal of Formal Logic.
    In this article we revisit a number of disputes regarding significance logics---i.e., inferential frameworks capable of handling meaningless, although grammatical, sentences---that took place in a series of articles most of which appeared in the Australasian Journal of Philosophy between 1966 and 1978. These debates concern (i) the way in which logical consequence ought to be approached in the context of a significance logic, and (ii) the way in which the logical vocabulary has to be modified (either by restricting some notions, (...)
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  5. Pooling Modalities and Pointwise Intersection: Axiomatization and Decidability.Frederik Van De Putte & Dominik Klein - 2021 - Studia Logica 109 (1):47-93.
    We establish completeness and the finite model property for logics featuring the pooling modalities that were introduced in Van De Putte and Klein. The definition of our canonical models combines standard techniques with a so-called “puzzle piece construction”, which we first illustrate informally. After that, we apply it to the weakest classical logics with pooling modalities and investigate the technique’s potential for the axiomatization of stronger logics, obtained by imposing well-known frame conditions on the models.
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  6. On Stalnaker’s Simple Theory of Propositions.Peter Fritz - 2021 - Journal of Philosophical Logic 50 (1):1-31.
    Robert Stalnaker recently proposed a simple theory of propositions using the notion of a set of propositions being consistent, and conjectured that this theory is equivalent to the claim that propositions form a complete atomic Boolean algebra. This paper clarifies and confirms this conjecture. Stalnaker also noted that some of the principles of his theory may be given up, depending on the intended notion of proposition. This paper therefore also investigates weakened constraints on consistency and the corresponding classes of Boolean (...)
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  7. The Open Future: Why Future Contingents Are All False.Patrick Todd - forthcoming - Oxford: Oxford University Press.
    In The Open Future: Why Future Contingents are all False, Patrick Todd launches a sustained defense of a radical interpretation of the doctrine of the open future, one according to which all claims about undetermined aspects of the future are simply false. Todd argues that this theory is metaphysically more parsimonious than its rivals, and that objections to its logical and practical coherence are much overblown. Todd shows how proponents of this view can maintain classical logic, and argues that the (...)
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  8. Transfinite Meta-inferences.Chris Scambler - 2020 - Journal of Philosophical Logic 49 (6):1079-1089.
    In Barrio et al. Barrio Pailos and Szmuc prove that there are systems of logic that agree with classical logic up to any finite meta-inferential level, and disagree with it thereafter. This article presents a generalized sense of meta-inference that extends into the transfinite, and proves analogous results to all transfinite orders.
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  9. Fractional Semantics for Classical Logic.Mario Piazza & Gabriele Pulcini - 2020 - Review of Symbolic Logic 13 (4):810-828.
    This article presents a new semantics for classical propositional logic. We begin by maximally extending the space of sequent proofs so as to admit proofs for any logical formula; then, we extract the new semantics by focusing on the axiomatic structure of proofs. In particular, the interpretation of a formula is given by the ratio between the number of identity axioms out of the total number of axioms occurring in any of its proofs. The outcome is an informational refinement of (...)
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  10. The False Promises of Logic Textbooks - How Logic Has a Much More Limited Role Than People Usually Think.Matheus Silva - manuscript
    Logic textbooks advertise that they can teach how to spot a valid argument by its logical form alone. They also boast having collections with the most basic valid argumentative forms people of flesh and blood can use in deductive matters. Think about this for a moment. These are bold statements. If they were accurate, philosophers would be in higher demand than software engineers and no one would be able to make contributions to theoretical physics without ever taking a logic class. (...)
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  11. Untersuchungen über das logische Schließen. II.Gerhard Gentzen - 1935 - Mathematische Zeitschrift 39:405–431.
  12. Untersuchungen über das logische Schließen. I.Gerhard Gentzen - 1935 - Mathematische Zeitschrift 35:176–210.
  13. Restricted and Unrestricted Modus Ponens.Matheus Silva - manuscript
    In a typical modus ponens the reasoner will (a) assert that a premise materially implies a conclusion in a given world; (b) assert this premise and (c) infer the conclusion. But this restricted modus ponens has little in common with the unrestricted textbook modus ponens, since the latter claims that there are no possible worlds where: (a’) a premise materially implies a conclusion, (b’) that premise is true and (c’) the conclusion is false. It is clear that this textbook modus (...)
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  14. Is Classical Logic Monotonic?Matheus Silva - manuscript
    It is usually accepted that one of the properties of classical logic is monotonicity, which states that the validity of implication is not affected by the addition of new premises. In this piece, I will argue that this common notion is unjustified since it is motivated by a category mistake. The notion of monotonicity is primarily epistemic in character and can’t be meaningfully attributed to a system. This is acutely clear in the contrast of monotonicity with non-monotonicity, which we tend (...)
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  15. Modus Ponens, Conditional-Circularity and Material Implication.Matheus Silva - manuscript
    The modus ponens can be interpreted as an answer to a circularity charge, but this strategy is only feasible if the additional conditional premise is interpreted as a claim to a material implication. Here’s how it works.
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  16. Conditional Sentences as Implication Statements: A New Approach.Matheus Silva - manuscript
    It is usually accepted that conditional sentences are sui generis and enigmatic. In this paper I try to make them more accessible by interpreting them as claims to relations of implication restricted to a parameter world. This interpretation revives an old idea that fell into disuse, but in its improved version leads to refreshing solutions to known problems in conditional theory. The many benefits of this approach are evidenced by its insightful explanation of some apparent counter-examples to classical argumentative forms (...)
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  17. Two Degrees of Implication.Matheus Silva - manuscript
    In a material implication the relation of logical consequence is restricted to a world parameter, whereas in a formal implication the relation of logical consequence extends over many worlds. One could infer from this pattern that the material implication should be reduced to formal implication since it is just a restricted version of it, or, inversely, that formal implication should be reduced to material implication since it is an unrestricted version of it. But both reductionist claims would betray a superficial (...)
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  18. The Inextricable Link Between Conditionals and Logical Consequence.Matheus Silva - manuscript
    There is a profound, but frequently ignored, relationship between the classical conception of logical consequence and the material implication. The first repeats the patterns of the latter, but with a wider modal reach. This relationship suggests that there should be also a connection between the notion of logical consequence and the conditional connective of any given logical system. This implies, among other things, that it is incoherent to propose alternatives to the material implication while maintaining the classical conception of logical (...)
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  19. Silver Type Theorems for Collapses.Moti Gitik - 2020 - Annals of Pure and Applied Logic 171 (9):102825.
    Let κ be a cardinal of cofinality \omega_1 witnessed by a club of cardinals (κ_\alpha | \alpha < \omega_1) . We study Silver's type effects of collapsing of κ^+_\alphas 's on κ^+ . A model in which κ^+_\alphas 's (and also κ^+) are collapsed on a stationary co-stationary set is constructed.
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  20. Inner-Model Reflection Principles.Neil Barton, Andrés Eduardo Caicedo, Gunter Fuchs, Joel David Hamkins, Jonas Reitz & Ralf Schindler - 2020 - Studia Logica 108 (3):573-595.
    We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \varphi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subset A. A stronger principle, the ground-model reflection principle, asserts that any such \varphi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width (...)
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  21. The Explosion Calculus.Michael Arndt - 2020 - Studia Logica 108 (3):509-547.
    A calculus for classical propositional sequents is introduced that consists of a restricted version of the cut rule and local variants of the logical rules. Employed in the style of proof search, this calculus explodes a given sequent into its elementary structural sequents—the topmost sequents in a derivation thus constructed—which do not contain any logical constants. Some of the properties exhibited by the collection of elementary structural sequents in relation to the sequent they are derived from, uniqueness and unique representation (...)
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  22. Interpolation in Extensions of First-Order Logic.Guido Gherardi, Paolo Maffezioli & Eugenio Orlandelli - 2020 - Studia Logica 108 (3):619-648.
    We prove a generalization of Maehara’s lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig’s interpolation property. As a corollary, we obtain a direct proof of interpolation for first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, partial and linear orders, and various intuitionistic order theories such as apartness and positive partial and linear orders.
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  23. Formal Notes on the Substitutional Analysis of Logical Consequence.Volker Halbach - 2020 - Notre Dame Journal of Formal Logic 61 (2):317-339.
    Logical consequence in first-order predicate logic is defined substitutionally in set theory augmented with a primitive satisfaction predicate: an argument is defined to be logically valid if and only if there is no substitution instance with true premises and a false conclusion. Substitution instances are permitted to contain parameters. Variants of this definition of logical consequence are given: logical validity can be defined with or without identity as a logical constant, and quantifiers can be relativized in substitution instances or not. (...)
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  24. Factorials of Infinite Cardinals in Zf Part II: Consistency Results.Guozhen Shen & Jiachen Yuan - 2020 - Journal of Symbolic Logic 85 (1):244-270.
    For a set x, let S(x) be the set of all permutations of x. We prove by the method of permutation models that the following statements are consistent with ZF: (1) There is an infinite set x such that |p(x)|<|S(x)|<|seq^1-1(x)|<|seq(x)|, where p(x) is the powerset of x, seq(x) is the set of all finite sequences of elements of x, and seq^1-1(x) is the set of all finite sequences of elements of x without repetition. (2) There is a Dedekind infinite set (...)
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  25. Herbrand's Theorem as Higher Order Recursion.Bahareh Afshari, Stefan Hetzl & Graham E. Leigh - 2020 - Annals of Pure and Applied Logic 171 (6):102792.
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  26. Três Vezes Não: Um Estudo Sobre as Negações Clássica, Paraconsistente e Paracompleta.Kherian Gracher - 2020 - Dissertation, Federal University of Santa Catarina
    Could there be a single logical system that would allow us to work simultaneously with classical, paraconsistent, and paracomplete negations? These three negations were separately studied in logics whose negations bear their names. Initially we will restrict our analysis to propositional logics by analyzing classical negation, ¬c, as treated by Classical Propositional Logic (LPC); the paraconsistent negation, ¬p, as treated through the hierarchy of Paraconsistent Propositional Calculi Cn (0 ≤ n ≤ ω); and the paracomplete negation, ¬q, as treated by (...)
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  27. A Recovery Operator for Nontransitive Approaches.Eduardo Alejandro Barrio, Federico Pailos & Damian Szmuc - 2020 - Review of Symbolic Logic 13 (1):80-104.
    In some recent articles, Cobreros, Egré, Ripley, & van Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble caused by semantic paradoxes. For that purpose, they develop the Strict-Tolerant approach, which leads them to entertain a nontransitive theory of truth, where the structural rule of Cut is not generally valid. However, that Cut fails in general in the target theory of truth does not mean that there are not certain safe instances of Cut (...)
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  28. Keep All Your Textbooks.Matheus Silva - manuscript
    Akman (2017) argued that our logic textbooks should be burned, since they present a propositional analysis of necessary and sufficient conditions that leads to a contradiction. According to Akman, we should instead adopt a first-order analysis where conditions are interpreted as one-place predicates. I will argue that (1) Akman’s argument fails to show that the propositional analysis of conditions leads to a contradiction, since the negation of a conjunction is not a conjunction with negated conjuncts, but rather a disjunction with (...)
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  29. In Defense of Hypothetical Syllogism.Matheus Silva - manuscript
    Moti Mizrahi (2013) presented a putative counterexample to Hypothetical Syllogism (HS) for indicative conditionals aiming to succeed where previous attempts to refute HS have failed. Lee Walters (2014a) objected that Mizrahi’s putative counterexample results from an inadequate analysis of conditionals with embedded modals, but advanced new putative counterexamples to HS for subjunctive conditionals that are supposed to bypass this issue (Walters, 2014a; 2014b). It is argued that Walter’s analysis of embedded modals is unnecessary to prevent Mizrahi’s putative counterexample, since the (...)
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  30. 'If-Then' as a Version of 'Implies'.Matheus Silva - manuscript
    Russell’s role in the controversy about the paradoxes of material implication is usually presented as a tale of how even the greatest minds can fall prey of basic conceptual confusions. Quine accused him of making a silly mistake in Principia Mathematica. He interpreted ‘if-then’ as a version of ‘implies’ and called it material implication. Quine’s accusation is that this decision involved a use-mention fallacy because the antecedent and consequent of ‘if-then’ are used instead of being mentioned as the premise and (...)
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  31. A Contextualist Defence of the Material Account of Indicative Conditionals.Matheus Silva - manuscript
    The material account of indicative conditionals faces a legion of counterexamples that are the bread and butter in any entry about the subject. For this reason, the material account is widely unpopular among conditional experts. I will argue that this consensus was not built on solid foundations, since these counterexamples are contextual fallacies. They ignore a basic tenet of semantics according to which when evaluating arguments for validity we need to maintain the context constant, otherwise any argumentative form can be (...)
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  32. The Material Account of Conditionals and the Clash Between Intensional and Extensional Evidence.Matheus Silva - manuscript
    Intensional evidence is any reason to accept a proposition that is not the truth values of the proposition accepted or, if it is a complex proposition, its propositional contents. Extensional evidence is non-intensional evidence. Someone can accept a complex proposition, but deny its logical consequences in two circumstances: (1) when her acceptance is based on intensional evidence, while the logical consequences of the proposition presuppose the acceptance of extensional evidence, e.g., she can refuse the logical consequence of a proposition she (...)
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  33. Subjunctive Conditionals Are Material.Matheus Silva - manuscript
    The material account claims that indicative conditionals are material. However, the conventional wisdom even among material account enthusiasts is that the material account cannot be extended to subjunctive conditionals. There are mainly three reasons that motivate this consensus: (1) the belief that if subjunctives were material, most subjunctive conditionals would be vacuously true, which is implausible; (2) its inconsistency with Adam pairs, which suggest that indicative and subjunctive conditionals have different truth conditions; and (3) the belief that it is an (...)
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  34. Indicative Conditionals Are Material - Expanding the Survey.Matheus Martins Silva - manuscript
    The material account of indicative conditionals states that indicative conditional sentences and the material implication have the same truth conditions. Recently, Adam Rieger has carried out a survey of arguments in favour of the material account. In this paper, I extend this survey by presenting yet more arguments for the material account. On top of presenting more arguments, I also want to argue that it is plausible to extend the material account to subjunctive conditionals. For that reason, the arguments here (...)
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  35. Computational Logic. Vol. 1: Classical Deductive Computing with Classical Logic. 2nd Ed.Luis M. Augusto - 2020 - London: College Publications.
    This is the 2nd edition of Computational logic. Vol. 1: Classical deductive computing with classical logic. This edition has a wholly new chapter on Datalog, a hard nut to crack from the viewpoint of semantics when negation is included.
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  36. Fregean Quantification Theory.Saul A. Kripke - 2014 - Journal of Philosophical Logic 43 (5):879-881.
    Frege’s system of first-order logic is presented in a contemporary framework. The system described is distinguished by economy of expression and an unusual syntax.
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  37. Formal Logic: Classical Problems and Proofs.Luis M. Augusto - 2019 - London, UK: College Publications.
    Not focusing on the history of classical logic, this book provides discussions and quotes central passages on its origins and development, namely from a philosophical perspective. Not being a book in mathematical logic, it takes formal logic from an essentially mathematical perspective. Biased towards a computational approach, with SAT and VAL as its backbone, this is an introduction to logic that covers essential aspects of the three branches of logic, to wit, philosophical, mathematical, and computational.
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  38. Classical Logic and the Strict Tolerant Hierarchy.Chris Scambler - 2020 - Journal of Philosophical Logic 49 (2):351-370.
    In their recent article “A Hierarchy of Classical and Paraconsistent Logics”, Eduardo Barrio, Federico Pailos and Damien Szmuc present novel and striking results about meta-inferential validity in various three valued logics. In the process, they have thrown open the door to a hitherto unrecognized domain of non-classical logics with surprising intrinsic properties, as well as subtle and interesting relations to various familiar logics, including classical logic. One such result is that, for each natural number n, there is a logic which (...)
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  39. Spaces of Types in Positive Model Theory.Levon Haykazyan - 2019 - Journal of Symbolic Logic 84 (2):833-848.
    We introduce a notion of the space of types in positive model theory based on Stone duality for distributive lattices. We show that this space closely mirrors the Stone space of types in the full first-order model theory with negation (Tarskian model theory). We use this to generalise some classical results on countable models from the Tarskian setting to positive model theory.
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  40. Quine’s Fluted Fragment Revisited.Ian Pratt-Hartmann, Wiesław Szwast & Lidia Tendera - forthcoming - Journal of Symbolic Logic:1-30.
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  41. A Simple Treatment of Truth Functions.Alan Ross Anderson & Nuel D. Belnap - 1959 - Journal of Symbolic Logic 24 (4):301-302.
    In this note we present an axiomatization of the classical two-valued propositional calculus, for which proofs of decidability, consistency, completeness, and independence, are almost trivial (given an understanding of truth tables).
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  42. Michael P. Slattery and Tadeusz Gierymski. A Propositions. The Modern Schoolman, Vol. 36 No. 2 , Pp. 91–107.Charles A. Baylis - 1964 - Journal of Symbolic Logic 29 (2):112.
  43. Tsutomu Hosoi. Algebraic Proof of the Separation Theorem on Classical Propositional Calculus. Proceedings of the Japan Academy, Vol. 42 , Pp. 67–69. - Tsutomu Hosoi. Algebraic Proof of the Separation Theorem on Dummett's LC. Proceedings of the Japan Academy, Vol. 42 , Pp. 693–695. [REVIEW]Alfred Horn - 1968 - Journal of Symbolic Logic 33 (1):128-129.
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  44. J. A. Faris. Truth-Functional Logic. Routledge & Kegan Paul, London, and The Free Press of Glencoe, New York, 1962, Vi + 122 Pp. [REVIEW]Gene F. Rose - 1966 - Journal of Symbolic Logic 31 (1):108-108.
  45. Yasuyuki Imai and Kiyoshi Iséki. On Axiom Systems of Propositional Calculi. I. Proceedings of the Japan Academy, Vol. 41 , Pp. 436–439. - Yoshinari Arai. On Axiom Systems of Propositional Calculi. II.Proceedings of the Japan Academy, Vol. 41 , Pp. 440–442. - Yoshinari Arai. On Axiom Systems of Propositional Calculi. III.Proceedings of the Japan Academy, Vol. 41 , Pp. 570–574. - Kiyoshi Iséki. On Axiom Systems Ofpropositional Calculi. IV.Proceedings of the Japan Academy, Vol. 41 , Pp. 575–577. - Kiyoshi Iséki and Shôtarô Tanaka. On Axiom Systems of Propositional Calculi. V.Proceedings of the Japan Academy, Vol. 41 , Pp. 661–662. - Shôtarô Tanaka. On Axiom Systems Ofpropositional Calculi. VI.Proceedings of the Japan Academy, Vol. 41 , Pp. 663–666. - Yoshinari Arai and Kiyoshi Iséki. On Axiom Systems of Propositional Calculi. VII.Proceedings of the Japan Academy, Vol. 41 , Pp. 667–669. - Shôtarô Tanaka. On Axiom Systems of Propositional Calculi. VIII.Proceedings of the Japan Academy, Vol. [REVIEW]Alan Rose - 1969 - Journal of Symbolic Logic 34 (1):122-124.
  46. T. Thacher Robinson. Independence of Two Nice Sets of Axioms for the Propositional Calculus.The Journal of Symbolic Logic, Vol. 33 , Pp. 265–270. [REVIEW]John R. Chidgey - 1970 - Journal of Symbolic Logic 35 (1):139-140.
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  47. Don Davis Roberts. The Existential Graphs and Natural Deduction. Studies in the Philosophy of Charles Sanders Peirce, Second Series, Edited by Edward C. Moore and Richard S. Robin, The University of Massachusetts Press, Amherst1964, Pp. 109–121. [REVIEW]J. Jay Zeman - 1970 - Journal of Symbolic Logic 35 (2):320-321.
  48. Per Lindström. First Order Predicate Logic with Generalized Quantifiers. Theoria, Vol. 32 , Pp. 186–195.G. Fuhrken - 1969 - Journal of Symbolic Logic 34 (4):650.
  49. C. A. Meredith and A. N. Prior. Notes on the Axiomatics of the Prepositional Calculus. Notre Dame Journal of Formal Logic, Vol. 4 , Pp. 171–187. [REVIEW]John Bacon - 1968 - Journal of Symbolic Logic 33 (2):306-307.
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  50. R. A. Bull. Some Results for Implicational Calculi. The Journal of Symbolic Logic, Vol. 29 No. 1 , Pp. 33–39.John Bacon - 1968 - Journal of Symbolic Logic 33 (2):306-306.
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