Journal articles: 'Propositional Quantifiers'

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FINE, KIT. "Propositional quantifiers in modal logic1." Theoria 36, no. 3 (February 11, 2008): 336–46. http://dx.doi.org/10.1111/j.1755-2567.1970.tb00432.x.

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Golińska-Pilarek, Joanna, and Taneli Huuskonen. "Non-Fregean Propositional Logic with Quantifiers." Notre Dame Journal of Formal Logic 57, no. 2 (2016): 249–79. http://dx.doi.org/10.1215/00294527-3470547.

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Artemov, Sergei N., and Lev D. Beklemishev. "On propositional quantifiers in provability logic." Notre Dame Journal of Formal Logic 34, no. 3 (June 1993): 401–19. http://dx.doi.org/10.1305/ndjfl/1093634729.

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Leivant, Daniel. "Propositional Dynamic Logic with Program Quantifiers." Electronic Notes in Theoretical Computer Science 218 (October 2008): 231–40. http://dx.doi.org/10.1016/j.entcs.2008.10.014.

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Crawford, Sean. "Quantifiers and propositional attitudes: Quine revisited." Synthese 160, no. 1 (February 15, 2007): 75–96. http://dx.doi.org/10.1007/s11229-006-9080-6.

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O'Hearn, Peter W., and David J. Pym. "The Logic of Bunched Implications." Bulletin of Symbolic Logic 5, no. 2 (June 1999): 215–44. http://dx.doi.org/10.2307/421090.

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AbstractWe introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers and which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predicate levels.

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Zhang, Cheng. "How to Deduce the Other 91 Valid Aristotelian Modal Syllogisms from the Syllogism IAI-3." Applied Science and Innovative Research 7, no. 1 (January 27, 2023): p46. http://dx.doi.org/10.22158/asir.v7n1p46.

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This paper firstly formalizes Aristotelian modal syllogisms by taking advantage of the trisection structure of (modal) categorical propositions. And then making full use of the truth value definition of (modal) categorical propositions, the transformable relations between an Aristotelian quantifier and its three negative quantifiers, the reasoning rules of classical propositional logic, and the symmetry of the two Aristotelian quantifiers (i.e. some and no), this paper shows that at least 91 valid Aristotelian modal syllogisms can be deduced from IAI-3 on the basis of possible world semantics and set theory. The reason why these valid Aristotelian modal syllogisms can be reduced is that any Aristotelian quantifier can be defined by the other three Aristotelian quantifiers, and the necessary modality ( ) and possible modality ( ) can also be defined mutually. This research method is universal. This innovative study not only provides a unified mathematical research paradigm for the study of generalized modal syllogistic and other kinds of syllogistic, but also makes contributions to knowledge representation and knowledge reasoning in computer science.

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Pascucci, Matteo. "Propositional quantifiers in labelled natural deduction for normal modal logic." Logic Journal of the IGPL 27, no. 6 (April 25, 2019): 865–94. http://dx.doi.org/10.1093/jigpal/jzz008.

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Abstract This article concerns the treatment of propositional quantification in a framework of labelled natural deduction for modal logic developed by Basin, Matthews and Viganò. We provide a detailed analysis of a basic calculus that can be used for a proof-theoretic rendering of minimal normal multimodal systems with quantification over stable domains of propositions. Furthermore, we consider variations of the basic calculus obtained via relational theories and domain theories allowing for quantification over possibly unstable domains of propositions. The main result of the article is that fragments of the labelled calculi not exploiting reductio ad absurdum enjoy the Church–Rosser property and the strong normalization property; such result is obtained by combining Girard’s method of reducibility candidates and labelled languages of lambda calculus codifying the structure of modal proofs.

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Montagna, Franco. "Δ-core Fuzzy Logics with Propositional Quantifiers, Quantifier Elimination and Uniform Craig Interpolation." Studia Logica 100, no. 1-2 (February 9, 2012): 289–317. http://dx.doi.org/10.1007/s11225-012-9379-x.

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Rönnedal, Daniel. "The Moral Law and The Good in Temporal Modal Logic with Propositional Quantifiers." Australasian Journal of Logic 17, no. 1 (April 7, 2020): 22. http://dx.doi.org/10.26686/ajl.v17i1.5674.

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The Moral Law is fulfilled (in a possible world w at a time t) iff (if and only if) everything that ought to be the case is the case (in w at t), and The Good (or The Highest Possible Good) is realised in a possible world w' at a time t' iff w' is deontically accessible from w at t. In this paper, I will introduce a set of temporal alethic deontic systems with propositional quantifiers that can be used to prove some interesting theorems about The Moral Law and The Good. First, I will describe a set of systems without any propositional quantifiers. Then, I will show how these systems can be extended by a couple of propositional quantifiers. I will use a kind of TxW semantics to describe the systems semantically and semantic tableaux to describe them syntactically. Every system will include a constant · that stands for The Good. ‘·’ is read as ‘The Good is realised’. All systems that contain the propositional quantifiers will also include a constant '*' that stands for The Moral Law. '*' is read as ‘The Moral Law is fulfilled’. I will prove that all systems (without the propositional quantifiers) are sound and complete with respect to their semantics and that all systems (including the extended systems) are sound with respect to their semantics. It is left as an open question whether or not the extended systems are complete.

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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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Bílková, Marta. "Uniform Interpolation and Propositional Quantifiers in Modal Logics." Studia Logica 85, no. 1 (February 1, 2007): 1–31. http://dx.doi.org/10.1007/s11225-007-9021-5.

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Fitting, Melvin. "Strict/Tolerant Family Continued: Quantifiers and Modalities." Australasian Journal of Logic 18, no. 6 (August 24, 2021): 616–44. http://dx.doi.org/10.26686/ajl.v18i6.6832.

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This paper continues my earlier work, which showed there is a broad family of propositional many valued logics that have a strict/tolerant counterpart. Here we generalize those results from propositional to a range of both modal and quantified many valued logics, providing strict/tolerant counterparts for all. This paper is not self-contained; some results from earlier papers are called on, and are not reproved here. The key new machinery added to earlier work, allowing modalities and quantifiers to be handled in similar ways, is the central use of bilattices that are function spaces, and more generally lattices that are function spaces. Two versions of the central proofs are considered, one at length and the other in outline.

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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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Antonelli, G. Aldo, and Richmond H. Thomason. "Representability in second-order propositional poly-modal logic." Journal of Symbolic Logic 67, no. 3 (September 2002): 1039–54. http://dx.doi.org/10.2178/jsl/1190150147.

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AbstractA propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.In this paper we generalize this framework by allowing multiple modalities. While this does not affect the undecidability of K, B, T, K4 and S4, poly-modal second-order S5 is dramatically more expressive than its mono-modal counterpart. As an example, we establish the definability of the transitive closure of finitely many modal operators. We also take up the decidability issue, and, using a novel encoding of sets of unordered pairs by partitions of the leaves of certain graphs, we show that the second-order propositional logic of two S5 modalitities is also equivalent to full second-order logic.

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NAUMOV, PAVEL, and JIA TAO. "EVERYONE KNOWS THAT SOMEONE KNOWS: QUANTIFIERS OVER EPISTEMIC AGENTS." Review of Symbolic Logic 12, no. 2 (January 9, 2019): 255–70. http://dx.doi.org/10.1017/s1755020318000497.

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AbstractModal logic S5 is commonly viewed as an epistemic logic that captures the most basic properties of knowledge. Kripke proved a completeness theorem for the first-order modal logic S5 with respect to a possible worlds semantics. A multiagent version of the propositional S5 as well as a version of the propositional S5 that describes properties of distributed knowledge in multiagent systems has also been previously studied. This article proposes a version of S5-like epistemic logic of distributed knowledge with quantifiers ranging over the set of agents, and proves its soundness and completeness with respect to a Kripke semantics.

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Kremer, Philip. "Quantifying over propositions in relevance logic: nonaxiomatisability of primary interpretations of ∀p and ∃p." Journal of Symbolic Logic 58, no. 1 (March 1993): 334–49. http://dx.doi.org/10.2307/2275341.

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A typical approach to semantics for relevance (and other) logics: specify a class of algebraic structures and take a model to be one of these structures, α, together with some function or relation which associates with every formula A a subset of α. (This is the approach of, among others, Urquhart, Routley and Meyer and Fine.) In some cases there are restrictions on the class of subsets of α with which a formula can be associated: for example, in the semantics of Routley and Meyer [1973], a formula can only be associated with subsets which are closed upwards. It is natural to take a proposition of α to be such a subset of α, and, further, to take the propositional quantifiers to range over these propositions. (Routley and Meyer [1973] explicitly consider this interpretation.) Given such an algebraic semantics, we call (following Routley and Meyer [1973], who follow Henkin [1950]) the above-described interpretation of the quantifiers the primary interpretation associated with the semantics.

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18

Ghilardi, Silvio, and Marek Zawadowski. "Undefinability of propositional quantifiers in the modal system S4." Studia Logica 55, no. 2 (1995): 259–71. http://dx.doi.org/10.1007/bf01061237.

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Bale, Alan Clinton. "Quantifiers and verb phrases: An exploration of propositional complexity." Natural Language & Linguistic Theory 25, no. 3 (September 1, 2007): 447–83. http://dx.doi.org/10.1007/s11049-007-9019-8.

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20

Kremer, Philip. "Defining relevant implication in a propositionally quantified S4." Journal of Symbolic Logic 62, no. 4 (December 1997): 1057–69. http://dx.doi.org/10.2307/2275626.

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AbstractR. K. Meyer once gave precise form to the question of whether relevant implication can be defined in any modal system, and his answer was ‘no’. In the present paper, we extend S4, first with propositional quantifiers, to the system S4π+; and then with definite propositional descriptions, to the system S4π+ip. We show that relevant implication can in some sense be defined in the modal system S4π+ip, although it cannot be defined in S4π+.

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Båve, Arvid. "CONCEPT DESIGNATION." American Philosophical Quarterly 56, no. 4 (October 1, 2019): 331–44. http://dx.doi.org/10.2307/48563047.

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Abstract The paper proposes a way for adherents of Fregean, structured propositions to designate propositions and other complex senses/concepts using a special kind of functor. I consider some formulations from Peacocke’s works and highlight certain problems that arise as we try to quantify over propositional constituents while referring to propositions using “that”-clauses. With the functor notation, by contrast, we can quantify over senses/concepts with objectual, first-order quantifiers and speak without further ado about their involvement in propositions. The functor notation also turns out to come with an important kind of expressive strengthening, and is shown to be neutral on several controversial issues.

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Holliday, Wesley H. "A Note on Algebraic Semantics for $\mathsf{S5}$ with Propositional Quantifiers." Notre Dame Journal of Formal Logic 60, no. 2 (May 2019): 311–32. http://dx.doi.org/10.1215/00294527-2019-0001.

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23

Połacik, Tomasz. "Propositional quantification in the monadic fragment of intuitionistic logic." Journal of Symbolic Logic 63, no. 1 (March 1998): 269–300. http://dx.doi.org/10.2307/2586601.

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AbstractWe study the monadic fragment of second order intuitionistic propositional logic in the language containing the standard propositional connectives and propositional quantifiers. It is proved that under the topological interpretation over any dense-in-itself metric space, the considered fragment collapses to Heyting calculus. Moreover, we prove that the topological interpretation over any dense-in-itself metric space of fragment in question coincides with the so-called Pitts' interpretation. We also prove that all the nonstandard propositional operators of the form q ↦ ∃p (q ↔ F(p)), where F is an arbitrary monadic formula of the variable p, are definable in the language of Heyting calculus under the topological interpretation of intuitionistic logic over sufficiently regular spaces.

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Friedman, Harvey M., and Andrej Ščedrov. "On the quantificational logic of intuitionistic set theory." Mathematical Proceedings of the Cambridge Philosophical Society 99, no. 1 (January 1986): 5–10. http://dx.doi.org/10.1017/s0305004100063854.

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Formal propositional logic describing the laws of constructive (intuitionistic) reasoning was first proposed in 1930 by Heyting. It is obtained from classical pro-positional calculus by deleting the Law of Excluded Middle, and it is usually referred to as Heyting's (intuitionistic) propositional calculus ([9], §§23, 19) (we write HPP in short). Formal logic involving predicates and quantifiers based on HPP is called Heyting's (intuitionistic) predicate calculus ([9], §§31, 19) (we write HPR in short).

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Hausmann, Marco. "The Consequence of the Consequence Argument." KRITERION – Journal of Philosophy 34, no. 4 (December 1, 2020): 45–70. http://dx.doi.org/10.1515/krt-2020-340406.

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Abstract The aim of my paper is to compare three alternative formal reconstructions of van Inwagen's famous argument for incompatibilism. In the first part of my paper, I examine van Inwagen's own reconstruction within a propositional modal logic. I point out that, due to the expressive limitations of his propositional modal logic, van Inwagen is unable to argue directly (that is, within his formal framework) for incompatibilism. In the second part of my paper, I suggest to reconstruct van Inwagen's argument within a first-order predicate logic. I show, however, that even though this reconstruction is not susceptible to the same objection, this reconstruction can be shown to be inconsistent (given van Inwagen's own assumptions). At the end of my paper, I suggest to reconstruct van Inwagen's argument within a quantified counterfactual logic with propositional quantifiers. I show that within this formal framework van Inwagen would not only be able to argue directly for incompatibilism, he would also be able to argue for crucial assumptions of his argument.

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Niiniluoto, Ilkka. "Perception, memory, and imagination as propositional attitudes." Logical Investigations 26, no. 1 (August 6, 2020): 36–47. http://dx.doi.org/10.21146/2074-1472-2020-26-1-36-47.

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Jaakko Hintikka started in 1969 the study of the logic of perception as a spe- cial case of his more general approach to propositional attitudes by means of the possible worlds semantics. His students and co-workers extended this study to the logic of memory and imagination. The key elements of this approach are the distinction between physical and perspectival cross-identification and the related two kinds of quantifiers, which allow a formulation of the syntax and semantics of various types of statements about perceiving, re- membering and imagining. This paper surveys the main results of these logical investigations.

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D’Agostino, Giovanna. "Uniform interpolation for propositional and modal team logics." Journal of Logic and Computation 29, no. 5 (May 7, 2019): 785–802. http://dx.doi.org/10.1093/logcom/exz006.

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Abstract In this paper we consider modal team logic, a generalization of classical modal logic in which it is possible to describe dependence phenomena between data. We prove that most known fragments of full modal team logic allow the elimination of the so called ‘existential bisimulation quantifiers’, where the existence of a certain set is required only modulo bisimulation (i.e. not in the model itself but possibly in a bisimilar model). As a consequence, we prove that these fragments enjoy the uniform interpolation property.

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WEISS, YALE. "CUT AND GAMMA I: PROPOSITIONAL AND CONSTANT DOMAIN R." Review of Symbolic Logic 13, no. 4 (August 29, 2019): 887–909. http://dx.doi.org/10.1017/s1755020319000388.

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AbstractThe main object of this article is to give two novel proofs of the admissibility of Ackermann’s rule (γ) for the propositional relevant logic R. The results are established as corollaries of cut elimination for systems of tableaux for R. Cut elimination, in turn, is established both nonconstructively (as a corollary of completeness) and constructively (using Gentzen-like methods). The extensibility of the techniques is demonstrated by showing that (γ) is admissible for RQ* (R with constant domain quantifiers). The status of the admissibility of (γ) for RQ* was, to the best of the author’s knowledge, an open problem. Further extensions of these results will be explored in the sequel(s).

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Inoué, Takao, and Riku Hanaoka. "Intuitionistic Propositional Calculus in the Extended Framework with Modal Operator. Part II." Formalized Mathematics 30, no. 1 (April 1, 2022): 1–12. http://dx.doi.org/10.2478/forma-2022-0001.

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Summary This paper is a continuation of Inoué [5]. As already mentioned in the paper, a number of intuitionistic provable formulas are given with a Hilbert-style proof. For that, we make use of a family of intuitionistic deduction theorems, which are also presented in this paper by means of Mizar system [2], [1]. Our axiom system of intuitionistic propositional logic IPC is based on the propositional subsystem of H1-IQC in Troelstra and van Dalen [6, p. 68]. We also owe Heyting [4] and van Dalen [7]. Our treatment of a set-theoretic intuitionistic deduction theorem is due to Agata Darmochwał’s Mizar article “Calculus of Quantifiers. Deduction Theorem” [3].

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Dyckhoff, Roy, and Sara Negri. "Admissibility of structural rules for contraction-free systems of intuitionistic logic." Journal of Symbolic Logic 65, no. 4 (December 2000): 1499–518. http://dx.doi.org/10.2307/2695061.

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AbstractWe give a direct proof of admissibility of cut and contraction for the contraction-free sequent calculus G4ip for intuitionistic propositional logic and for a corresponding multi-succedent calculus: this proof extends easily in the presence of quantifiers, in contrast to other, indirect, proofs, i.e., those which use induction on sequent weight or appeal to admissibility of rules in other calculi.

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Czajka, Łukasz. "Higher-Order Illative Combinatory Logic." Journal of Symbolic Logic 78, no. 3 (September 2013): 837–72. http://dx.doi.org/10.2178/jsl.7803080.

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AbstractWe show a model construction for a system of higher-order illative combinatory logic thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.

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SCHOENBAUM, LUCIUS T. "ON THE SYNTAX OF LOGIC AND SET THEORY." Review of Symbolic Logic 3, no. 4 (September 15, 2010): 568–99. http://dx.doi.org/10.1017/s1755020310000122.

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We introduce an extension of the propositional calculus to include abstracts of predicates and quantifiers, employing a single rule along with a novel comprehension schema and a principle of extensionality, which are substituted for the Bernays postulates for quantifiers and the comprehension schemata of ZF and other set theories. We prove that it is consistent in any finite Boolean subset lattice. We investigate the antinomies of Russell, Cantor, Burali-Forti, and others, and discuss the relationship of the system to other set-theoretic systems ZF, NBG, and NF. We discuss two methods of axiomatizing higher order quantification and abstraction, and then very briefly discuss the application of one of these methods to areas of mathematics outside of logic.

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Fitting, Melvin. "Interpolation for first order S5." Journal of Symbolic Logic 67, no. 2 (June 2002): 621–34. http://dx.doi.org/10.2178/jsl/1190150101.

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AbstractAn interpolation theorem holds for many standard modal logics, but first order S5 is a prominent example of a logic for which it fails. In this paper it is shown that a first order S5 interpolation theorem can be proved provided the logic is extended to contain propositional quantifiers. A proper statement of the result involves some subtleties, but this is the essence of it.

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Lewitzka, Steffen. "Denotational Semantics for Modal Systems S3–S5 Extended by Axioms for Propositional Quantifiers and Identity." Studia Logica 103, no. 3 (September 21, 2014): 507–44. http://dx.doi.org/10.1007/s11225-014-9577-9.

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Baltag, Alexandru, and Johan van Benthem. "A Simple Logic of Functional Dependence." Journal of Philosophical Logic 50, no. 5 (March 24, 2021): 939–1005. http://dx.doi.org/10.1007/s10992-020-09588-z.

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AbstractThis paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized assignment semantics for first order logic. The expressive strength, complete proof calculus and meta-properties of LFD are explored. Various language extensions are presented as well, up to undecidable modal-style logics for independence and dynamic logics of changing dependence models. Finally, more concrete settings for dependence are discussed: continuous dependence in topological models, linear dependence in vector spaces, and temporal dependence in dynamical systems and games.

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Jin, Dawei. "Why-questions, topicality and intervention effects in Chinese." Yearbook of the Poznan Linguistic Meeting 2, no. 1 (September 1, 2016): 91–113. http://dx.doi.org/10.1515/yplm-2016-0005.

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Abstract This paper revisits intervention effects in Mandarin Chinese why-questions. I present new data showing that the ability for quantifiers to induce intervention hinges upon their monotonicity and their ability to be interpreted as topics. I then develop a semantic account that correlates topicality with monotone properties. Furthermore, I propose that why-questions in Chinese are idiosyncratic in that why directly merges at a high scope position that stays above a propositional argument. Combining the semantic idiosyncrasies of why-questions with the wide scope behaviors of topicality, I conclude that my account explains a wide range of intervention phenomena in terms of interpretation failure.

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Konikowska, Beata, Andrzej Tarlecki, and Andrzej Blikle. "A Three-Valued Logic for Software Specification and Validation. Tertium tamen datur." Fundamenta Informaticae 14, no. 4 (April 1, 1991): 411–53. http://dx.doi.org/10.3233/fi-1991-14403.

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Different calculi of partial or three-valued predicates have been used and studied by several authors in the context of software specification, development and validation. This paper offers a critical survey on the development of three-valued logics based on such calculi. In the first part of the paper we review two three-valued predicate calculi, based on, respectively, McCarthy’s and Kleene’s propositional connectives and quantifiers, and point out that in a three-valued logic one should distinguish between two notions of validity: strong validity (always true) and weak validity (never false). We define in model-theoretic terms a number of consequence relations for three-valued logics. Each of them is determined by the choice of the underlying predicate calculus and of the weak or strong validity of axioms and of theorems. We discuss mutual relationships between consequence relations defined in such a way and study some of their basic properties. The second part of the paper is devoted to the development of a formal deductive system of inference rules for a three-valued logic. We use the method of semantic tableaux (slightly modified to deal with three-valued formulas) to develop a Gentzen-style system of inference rules for deriving valid sequents, from which we then derive a sound and complete system of natural deduction rules. We have chosen to study the consequence relation determined by the predicate calculus with McCarthy’s propositional connectives and Kleene’s quantifiers and by the strong interpretation of both axioms and theorems. Although we find this choice appropriate for applications in the area of software specification, verification and development, we regard this logic merely as an example and use it to present some general techniques of developing a sequent calculus and a natural deduction system for a three-valued logic. We also discuss the extension of this logic by a non-monotone is-true predicate.

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Pratt-Hartmann, Ian. "On the Computational Complexity of the Numerically Definite Syllogistic and Related Logics." Bulletin of Symbolic Logic 14, no. 1 (March 2008): 1–28. http://dx.doi.org/10.2178/bsl/1208358842.

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AbstractThe numerically definite syllogistic is the fragment of English obtained by extending the language of the classical syllogism with numerical quantifiers. The numerically definite relational syllogistic is the fragment of English obtained by extending the numerically definite syllogistic with predicates involving transitive verbs. This paper investigates the computational complexity of the satisfiability problem for these fragments. We show that the satisfiability problem (= finite satisfiability problem) for the numerically definite syllogistic is strongly NP-complete, and that the satisfiability problem (= finite satisfiability problem) for the numerically definite relational syllogistic is NEXPTIME-complete, but perhaps not strongly so. We discuss the related problem of probabilistic (propositional) satisfiability, and thereby demonstrate the incompleteness of some proof-systems that have been proposed for the numerically definite syllogistic.

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Grädel, Erich. "On the Restraining Power of Guards." Journal of Symbolic Logic 64, no. 4 (December 1999): 1719–42. http://dx.doi.org/10.2307/2586808.

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AbstractGuarded fragments of first-order logic were recently introduced by Andréka, van Benthem and Németi; they consist of relational first-order formulae whose quantifiers are appropriately relativized by atoms. These fragments are interesting because they extend in a natural way many propositional modal logics, because they have useful model-theoretic properties and especially because they are decidable classes that avoid the usual syntactic restrictions (on the arity of relation symbols, the quantifier pattern or the number of variables) of almost all other known decidable fragments of first-order logic.Here, we investigate the computational complexity of these fragments. We prove that the satisfiability problems for the guarded fragment (GF) and the loosely guarded fragment (LGF) of first-order logic are complete for deterministic double exponential time. For the subfragments that have only a bounded number of variables or only relation symbols of bounded arity, satisfiability is Exptime-complete. We further establish a tree model property for both the guarded fragment and the loosely guarded fragment, and give a proof of the finite model property of the guarded fragment.It is also shown that some natural, modest extensions of the guarded fragments are undecidable.

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Simons, Peter. "Term Logic." Axioms 9, no. 1 (February 10, 2020): 18. http://dx.doi.org/10.3390/axioms9010018.

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The predominant form of logic before Frege, the logic of terms has been largely neglected since. Terms may be singular, empty or plural in their denotation. This article, presupposing propositional logic, provides an axiomatization based on an identity predicate, a predicate of non-existence, a constant empty term, and term conjunction and negation. The idea of basing term logic on existence or non-existence, outlined by Brentano, is here carried through in modern guise. It is shown how categorical syllogistic reduces to just two forms of inference. Tree and diagram methods of testing validity are described. An obvious translation into monadic predicate logic shows the system is decidable, and additional expressive power brought by adding quantifiers enables numerical predicates to be defined. The system’s advantages for pedagogy are indicated.

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Rett, Jessica. "Manner implicatures and how to spot them." International Review of Pragmatics 12, no. 1 (February 13, 2020): 44–79. http://dx.doi.org/10.1163/18773109-01201105.

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Abstract The goal of this paper is to help develop a general picture of conversational implicature (Grice, 1975) by looking beyond scalar implicature to see how the phenomenon behaves in a general sense. I focus on non-scalar Quantity implicatures and Manner implicatures. I review canonical examples of Manner implicature, as well as a more recent, productive one involving gradable adjective antonym pairs (Rett, 2015). Based on these data, I argue that Manner implicatures—and conversational implicatures generally—are distinguishable primarily by their calculability; their reinforceability; their discourse sensitivity (to the Question Under Discussion; Roberts, 1990; van Kuppevelt, 1995; Simons et al., 2011); and their embeddability (under negation, propositional attitude verbs, quantifiers, etc.). I use these data to draw conclusions about the usefulness of implicature-specific operators and about ways to compositionally represent conversational implicatures.

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Rönnedal, Daniel. "Boulesic-Doxastic Logic." Australasian Journal of Logic 16, no. 3 (June 26, 2019): 83. http://dx.doi.org/10.26686/ajl.v16i3.4158.

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In this paper, I will develop a set of boulesic-doxastic tableau systems and prove that they are sound and complete. Boulesic-doxastic logic consists of two main parts: a boulesic part and a doxastic part. By ‘boulesic logic’ I mean ‘the logic of the will’, and by ‘doxastic logic’ I mean ‘the logic of belief’. The first part deals with ‘boulesic’ concepts, expressions, sentences, arguments and theorems. I will concentrate on two types of boulesic expression: ‘individual x wants it to be the case that’ and ‘individual x accepts that it is the case that’. The second part deals with ‘doxastic’ concepts, expressions, sentences, arguments and theorems. I will concentrate on two types of doxastic expression: ‘individual x believes that’ and ‘it is imaginable to individual x that’. Boulesic-doxastic logic investigates how these concepts are related to each other. Boulesic logic is a new kind of logic. Doxastic logic has been around for a while, but the approach to this branch of logic in this paper is new. Each system is combined with modal logic with two kinds of modal operators for historical and absolute necessity and predicate logic with necessary identity and ‘possibilist’ quantifiers. I use a kind of possible world semantics to describe the systems semantically. I also sketch out how our basic language can be extended with propositional quantifiers. All the systems developed in this paper are new.

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Du, Guoping. "Parenthesis Notation." Journal of Research in Philosophy and History 5, no. 1 (February 22, 2022): p44. http://dx.doi.org/10.22158/jrph.v5n1p44.

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The formal language of a logical system usually contains several types of symbols. In infix notation, two different kinds of symbols are used to construct compound formula and to indicate the order of combination. The logical constants such as Ø, Ú are used to construct compound formula, and auxiliary symbols such as ( ) are used to indicate the order of a combination. In Polish notation, there is no need for auxiliary symbols such as ( ), and only one class of symbols, N, C, K, etc., is used as a conjunction to make it function as a parenthesis. Contrary to Polish notation, a new parenthesis notation is put forward in this paper. Parenthesis notation uses only parentheses, and empowers them the function of connectives. More importantly, it is proved in this paper that we can define logical constants such as propositional connectives, quantifiers, modal operators and temporal operators in the same formula by using only parenthesis, which can greatly simplify the initial connectives needed to construct the formal system.

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Blikle, Andrzej. "Three-Valued Predicates for Software Specification and Validation." Fundamenta Informaticae 14, no. 4 (April 1, 1991): 387–410. http://dx.doi.org/10.3233/fi-1991-14402.

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Partial functions, hence also partial predicates, cannot be avoided in algorithms. However, in spite of the fact that partial functions have been formally introduced into the theory of software very early, partial predicates are still not quite commonly recognized. In many programming- and software-specification languages partial Boolean expressions are treated in a rather simplistic way: the evaluation of a Boolean sub-expression to an error leads to the evaluation of the hosting Boolean expression to an error and, in the consequence, to the abortion of the whole program. This technique is known as an eager evaluation of expressions. A more practical approach to the evaluation of expressions – gaining more interest today among both theoreticians and programming-language designers – is lazy evaluation. Lazily evaluated Boolean expressions correspond to (non-strict) three-valued predicates where the third value represents both an error and an undefinedness. On the semantic ground this leads to a three-valued propositional calculus, three-valued quantifiers and an appropriate logic. This paper is a survey-essay devoted to the discussion and the comparison of a few three-valued propositional and predicate calculi and to the discussion of the author’s claim that a two-valued logic, rather than a three-valued logic, is suitable for the treatment of programs with three-valued Boolean expressions. The paper is written in a formal but not in a formalized style. All discussion is carried on a semantic ground. We talk about predicates (functions) and a semantic consequence relation rather than about expressions and inference rules. However, the paper is followed by more formalized works which carry our discussion further on a formalized ground, and where corresponding formal logics are constructed and discussed.

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Reis, Róbson Ramos dos. "Ways of being and expressivity." Estudios de Filosofía, no. 61 (February 4, 2020): 11–33. http://dx.doi.org/10.17533/udea.ef.n61a03.

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In this paper, I present a hermeneutic version of ontological pluralism, addressing the question of the discursive articulation of ways of being. The first section presents the notion of a pluralism of ways of being as a restriction of an ontological monism. The second section puts forward a criticism of Kris McDaniel’s proposal of understanding ways of being as kinds of quantifiers. The third section analyses the notion of way of being as a modal concept, explaining ways of being as internal possibilities endowed with a normative force regarding the identity-conditions of entities. The fourth one is a statement about the need of developing a pluralist account of the propositional reference to entities based on ontological pluralism. The fifth section deals with the issue of the discursive articulation of ways of being. The two last sections present a hypothesis concerning a semantic condition for an adequate articulation of ways of being. I argue for a kind of finitude-sensitivity in the semantics of the discursive articulation of internal possibilities, which implies the requirement of developing a hermeneutic notion of silence that may properly work in the discursive articulation of ways of being.

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Baaz, M., and N. Preining. "Quantifier Elimination for Quantified Propositional Logics on Kripke Frames of Type." Journal of Logic and Computation 18, no. 4 (November 22, 2007): 649–68. http://dx.doi.org/10.1093/logcom/exn004.

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FRITZ, PETER. "LOGICS FOR PROPOSITIONAL CONTINGENTISM." Review of Symbolic Logic 10, no. 2 (March 20, 2017): 203–36. http://dx.doi.org/10.1017/s1755020317000028.

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AbstractRobert Stalnaker has recently advocated propositional contingentism, the claim that it is contingent what propositions there are. He has proposed a philosophical theory of contingency in what propositions there are and sketched a possible worlds model theory for it. In this paper, such models are used to interpret two propositional modal languages: one containing an existential propositional quantifier, and one containing an existential propositional operator. It is shown that the resulting logic containing an existential quantifier is not recursively axiomatizable, as it is recursively isomorphic to second-order logic, and a natural candidate axiomatization for the resulting logic containing an existential operator is shown to be incomplete.

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MIMRAM, SAMUEL. "The structure of first-order causality." Mathematical Structures in Computer Science 21, no. 1 (January 24, 2011): 65–110. http://dx.doi.org/10.1017/s0960129510000459.

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Game semantics describe the interactive behaviour of proofs by interpreting formulas as games on which proofs induce strategies. Such a semantics is introduced here for capturing dependencies induced by quantifications in first-order propositional logic. One of the main difficulties that has to be faced during the elaboration of this kind of semantics is to characterise definable strategies, that is, strategies that actually behave like a proof. This is usually done by restricting the model to strategies satisfying subtle combinatorial conditions, whose preservation under composition is often difficult to show. In this paper we present an original methodology to achieve this task, which requires a combination of advanced tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of the monoidal category of definable strategies of our model using generators and relations: these strategies can be generated from a finite set of atomic strategies, and the equality between strategies admits a finite axiomatisation, and this equational structure corresponds to a polarised variation of the bialgebra notion. The work described in this paper thus forms a bridge between algebra and denotational semantics in order to reveal the structure of dependencies induced by first-order quantifiers, and lays the foundations for a mechanised analysis of causality in programming languages.

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CUI, LICONG, YONGMING LI, and XIAOHONG ZHANG. "INTUITIONISTIC FUZZY LINGUISTIC QUANTIFIERS BASED ON INTUITIONISTIC FUZZY-VALUED FUZZY MEASURES AND INTEGRALS." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 17, no. 03 (June 2009): 427–48. http://dx.doi.org/10.1142/s0218488509005966.

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In this paper, we generalize Ying's model of linguistic quantifiers [M.S. Ying, Linguistic quantifiers modeled by Sugeno integrals, Artificial Intelligence, 170 (2006) 581-606] to intuitionistic linguistic quantifiers. An intuitionistic linguistic quantifier is represented by a family of intuitionistic fuzzy-valued fuzzy measures and the intuitionistic truth value (the degrees of satisfaction and non-satisfaction) of a quantified proposition is calculated by using intuitionistic fuzzy-valued fuzzy integral. Description of a quantifier by intuitionistic fuzzy-valued fuzzy measures allows us to take into account differences in understanding the meaning of the quantifier by different persons. If the intuitionistic fuzzy linguistic quantifiers are taken to be linguistic fuzzy quantifiers, then our model reduces to Ying's model. Some excellent logical properties of intuitionistic linguistic quantifiers are obtained including a prenex norm form theorem. A simple example is presented to illustrate the use of intuitionistic linguistic quantifiers.

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Toffano, Zeno, and François Dubois. "Quantum eigenlogic observables applied to the study of fuzzy behaviour of Braitenberg vehicle quantum robots." Kybernetes 48, no. 10 (November 4, 2019): 2307–24. http://dx.doi.org/10.1108/k-11-2018-0603.

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Purpose The purpose of this paper is to apply the quantum “eigenlogic” formulation to behavioural analysis. Agents, represented by Braitenberg vehicles, are investigated in the context of the quantum robot paradigm. The agents are processed through quantum logical gates with fuzzy and multivalued inputs; this permits to enlarge the behavioural possibilities and the associated decisions for these simple vehicles. Design/methodology/approach In eigenlogic, the eigenvalues of the observables are the truth values and the associated eigenvectors are the logical interpretations of the propositional system. Logical observables belong to families of commuting observables for binary logic and many-valued logic. By extension, a fuzzy logic interpretation is proposed by using vectors outside the eigensystem of the logical connective observables. The fuzzy membership function is calculated by the quantum mean value (Born rule) of the logical projection operators and is associated to a quantum probability. The methodology of this paper is based on quantum measurement theory. Findings Fuzziness arises naturally when considering systems described by state vectors not in the considered logical eigensystem. These states correspond to incompatible and complementary systems outside the realm of classical logic. Considering these states allows the detection of new Braitenberg vehicle behaviours related to identified emotions; these are linked to quantum-like effects. Research limitations/implications The method does not deal at this stage with first-order logic and is limited to different families of commuting logical observables. An extension to families of logical non-commuting operators associated to predicate quantifiers could profit of the “quantum advantage” due to effects such as superposition, parallelism, non-commutativity and entanglement. This direction of research has a variety of applications, including robotics. Practical implications The goal of this research is to show the multiplicity of behaviours obtained by using fuzzy logic along with quantum logical gates in the control of simple Braitenberg vehicle agents. By changing and combining different quantum control gates, one can tune small changes in the vehicle’s behaviour and hence get specific features around the main basic robot’s emotions. Originality/value New mathematical formulation for propositional logic based on linear algebra. This methodology demonstrates the potentiality of this formalism for behavioural agent models (quantum robots).

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Journal articles on the topic 'Propositional Quantifiers'

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