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Summary A logical connective is anything that joins smaller logical expressions into larger ones.  There are any number of logical connectives, depending on which logic one is using.  The subcategories here (with the obvious exception of the miscellaneous leaf node) are most appropriate for classical logic, and logics which depart from classical logic only modestly, where there is a widely held intuition (with the exception of the conditional) that the linguistic connectives, "and," "or," and "not," are, at least in most respects, the equivalents of the formal logical connectives, conjunction, disjunction, and negation.  However, even in classical propositional logic, there is the Sheffer stroke and the dagger, which allow the axiomatization of propositional logic with just one connective, but have no clear linguistic equivalent.   As one moves further afield from classical logic, along various dimensions, one will soon discover that the variety of logical connectives is limited only by the mathematical ingenuity of the human mind.  This might help explain why--with the exception of "conditionals"--there are (currently) far more entries in the miscellaneous category than there are in any of the more standard categories.
Key works Given the above variety, as discussed, there are separate key works for each logic, although there are a few multi-volume works which attempt to be all-inclusive and cover the enormous variety of logics, their operators, and their semantics.
Introductions See, key works, above.  Only the best-known logics have works that can fairly be called introductions.
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Related categories
Subcategories:
Negation* (70)
Conditionals* (1,058 | 125)
Disjunction* (41)
Conjunction* (19)
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  1. Joanna Golińska-Pilarek & Taneli Huuskonen (2012). Logic. Of Descriptions. A New Approach to the Foundations of Mathematics and Science. Studies in Logic, Grammar and Rhetoric 27:63-94.
    We study a new formal logic LD introduced by Prof. Grzegorczyk. The logic is based on so-called descriptive equivalence, corresponding to the idea of shared meaning rather than shared truth value. We construct a semantics for LD based on a new type of algebras and prove its soundness and complete- ness. We further show several examples of classical laws that hold for LD as well as laws that fail. Finally, we list a number of open problems.
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Logical Connectives, Misc
  1. Ken Akiba (2009). A New Theory of Quantifiers and Term Connectives. Journal of Logic, Language and Information 18 (3):403-431.
    This paper sets forth a new theory of quantifiers and term connectives, called shadow theory , which should help simplify various semantic theories of natural language by greatly reducing the need of Montagovian proper names, type-shifting, and λ-conversion. According to shadow theory, conjunctive, disjunctive, and negative noun phrases such as John and Mary , John or Mary , and not both John and Mary , as well as determiner phrases such as every man , some woman , and the boys (...)
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  2. Carlos E. Alchourrón & David Makinson (1986). Maps Between Some Different Kinds of Contraction Function: The Finite Case. Studia Logica 45 (2):187 - 198.
    In some recent papers, the authors and Peter Gärdenfors have defined and studied two different kinds of formal operation, conceived as possible representations of the intuitive process of contracting a theory to eliminate a proposition. These are partial meet contraction (including as limiting cases full meet contraction and maxichoice contraction) and safe contraction. It is known, via the representation theorem for the former, that every safe contraction operation over a theory is a partial meet contraction over that theory. The purpose (...)
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  3. Rafael-Andrés Alemañ-Berenguer (2011). Epistemologic Controversy on Quantum Operators. Principia 14 (2):241-253.
    Since the very begining of quantum theory there started a debate on the proper role of space and time in it. Some authors assumed that space and time have their own algebraic operators. On that basis they supposed that quantum particles had “coordinates of position”, even though those coordinates were not possible to determine with infinite precision. Furthermore, time in quantum physics was taken to be on an equal foot, by means of a so-called “Heisenberg’s fourth relation of indeterminacy” concerning (...)
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  4. Sergei Artëmov & Franco Montagna (1994). On First-Order Theories with Provability Operator. Journal of Symbolic Logic 59 (4):1139-1153.
    In this paper the modal operator "x is provable in Peano Arithmetic" is incorporated into first-order theories. A provability extension of a theory is defined. Presburger Arithmetic of addition, Skolem Arithmetic of multiplication, and some first order theories of partial consistency statements are shown to remain decidable after natural provability extensions. It is also shown that natural provability extensions of a decidable theory may be undecidable.
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  5. Axel Arturo Barceló Aspeitia (2008). Patrones inferenciales (Inferential Patterns). Crítica 40 (120):3 - 35.
    El objetivo de este artículo es proponer un método de traducción de tablas de verdad a reglas de inferencia, para la lógica proposicional, que sea tan directo como el tradicional método inverso (de reglas a tablas). Este método, además, permitirá resolver de manera elegante el viejo problema, formulado originalmente por Prior en 1960, de determinar qué reglas de inferencia definen un conectivo. /// This article aims at setting forth a method to translate truth tables into inference rules, in propositional logic, (...)
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  6. A. Avron (1998). Multiplicative Conjunction and an Algebraic Meaning of Contraction and Weakening. Journal of Symbolic Logic 63 (3):831-859.
    We show that the elimination rule for the multiplicative (or intensional) conjunction $\wedge$ is admissible in many important multiplicative substructural logics. These include LL m (the multiplicative fragment of Linear Logic) and RMI m (the system obtained from LL m by adding the contraction axiom and its converse, the mingle axiom.) An exception is R m (the intensional fragment of the relevance logic R, which is LL m together with the contraction axiom). Let SLL m and SR m be, respectively, (...)
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  7. Arnon Avron (1986). On an Implication Connective of ${\Rm RM}$. Notre Dame Journal of Formal Logic 27 (2):201-209.
  8. Colin G. Bailey (2013). Some Jump-Like Operations in $\Mathbf \Beta $-Recursion Theory. Journal of Symbolic Logic 78 (1):57-71.
    In this paper we show that there are various pseudo-jump operators definable over inadmissible $J_{\beta}$ that relate to the failure of admissiblity and to non-regularity. We will use these ideas to construct some intermediate degrees.
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  9. Tomás Barrero & Walter Carnielli (2005). Tableaux sin refutación. Matemáticas: Enseñanza Universitaria 13 (2):81-99.
    Motivated by H. Curry’s well-known objection and by a proposal of L. Henkin, this article introduces the positive tableaux, a form of tableau calculus without refutation based upon the idea of implicational triviality. The completeness of the method is proven, which establishes a new decision procedure for the (classical) positive propositional logic. We also introduce the concept of paratriviality in order to contribute to the question of paradoxes and limitations imposed by the behavior of classical implication.
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  10. Howard Becker (1988). A Characterization of Jump Operators. Journal of Symbolic Logic 53 (3):708-728.
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  11. J. L. Bell (1993). Hilbert's Ɛ-Operator and Classical Logic. Journal of Philosophical Logic 22 (1):1 - 18.
  12. Ermanno Bencivenga & Peter W. Woodruff (1981). A New Modal Language with the Λ Operator. Studia Logica 40 (4):383 - 389.
    A system of modal logic with the operator is proposed, and proved complete. In contrast with a previous one by Stalnaker and Thomason, this system does not require two categories of singular terms.
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  13. Alexander Berenstein (2004). Dividing in the Algebra of Compact Operators. Journal of Symbolic Logic 69 (3):817-829.
    We interpret the algebra of finite rank operators as imaginaries inside a Hilbert space. We prove that the Hilbert space enlarged with these imaginaries has built-in canonical bases.
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  14. Katalin Bimbó (2010). Schönfinkel-Type Operators for Classical Logic. Studia Logica 95 (3):355-378.
    We briefly overview some of the historical landmarks on the path leading to the reduction of the number of logical connectives in classical logic. Relying on the duality inherent in Boolean algebras, we introduce a new operator ( Nallor ) that is the dual of Schönfinkel’s operator. We outline the proof that this operator by itself is sufficient to define all the connectives and operators of classical first-order logic ( Fol ). Having scrutinized the proof, we pinpoint the theorems of (...)
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  15. Patrick Blackburn & Maarten Marx (2002). Remarks on Gregory's “Actually” Operator. Journal of Philosophical Logic 31 (3):281-288.
    In this note we show that the classical modal technology of Sahlqvist formulas gives quick proofs of the completeness theorems in [8] (D. Gregory, Completeness and decidability results for some propositional modal logics containing "actually" operators, Journal of Philosophical Logic 30(1): 57-78, 2001) and vastly generalizes them. Moreover, as a corollary, interpolation theorems for the logics considered in [8] are obtained. We then compare Gregory's modal language enriched with an "actually" operator with the work of Arthur Prior now known under (...)
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  16. Robert B. Brandom (1979). A Binary Sheffer Operator Which Does the Work of Quantifiers and Sentential Connectives. Notre Dame Journal of Formal Logic 20 (2):262-264.
  17. Douglas S. Bridges (1995). Constructive Mathematics and Unbounded Operators — a Reply to Hellman. Journal of Philosophical Logic 24 (5):549 - 561.
    It is argued that Hellman's arguments purporting to demonstrate that constructive mathematics cannot cope with unbounded operators on a Hilbert space are seriously flawed, and that there is no evidence that his thesis is correct.
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  18. Berit Brogaard (2007). Span Operators. Analysis 67 (1):72–79.
    I. Tensed Plural Quantifiers Presentists typically assent to a range of tensed statements, for instance, that there were dinosaurs, that there was a president named Lincoln, and that my future grandchildren will be on their way to school.1 Past- and future-tensed claims are dealt with by introducing primitive, intensional tense operators, for instance, it has been 12 years ago that, it was the case when I was born that, and it will be the case that (Prior 1968). For example, ‘there (...)
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  19. Eric M. Brown, Logic II: The Theory of Propositions.
    This is part two of a complete exposition of Logic, in which there is a radically new synthesis of Aristotelian-Scholastic Logic with modern Logic. Part II is the presentation of the theory of propositions. Simple, composite, atomic, compound, modal, and tensed propositions are all examined. Valid consequences and propositional logical identities are rigorously proven. Modal logic is rigorously defined and proven. This is the first work of Logic known to unite Aristotelian logic and modern logic using scholastic logic as the (...)
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  20. M. W. Bunder (1979). Variable Binding Term Operators in $\Lambda $-Calculus. Notre Dame Journal of Formal Logic 20 (4):876-878.
  21. Xavier Caicedo & Roberto Cignoli (2001). An Algebraic Approach to Intuitionistic Connectives. Journal of Symbolic Logic 66 (4):1620-1636.
    It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting (...)
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  22. Carlos Caleiro, Luca Viganò & Marco Volpe (2013). On the Mosaic Method for Many-Dimensional Modal Logics: A Case Study Combining Tense and Modal Operators. [REVIEW] Logica Universalis 7 (1):33-69.
    We present an extension of the mosaic method aimed at capturing many-dimensional modal logics. As a proof-of-concept, we define the method for logics arising from the combination of linear tense operators with an “orthogonal” S5-like modality. We show that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of partial models, called mosaics, and apply the technique not only in obtaining a proof of decidability and a proof of completeness (...)
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  23. Enrique Casanovas (2007). Logical Operations and Invariance. Journal of Philosophical Logic 36 (1):33 - 60.
    I present a notion of invariance under arbitrary surjective mappings for operators on a relational finite type hierarchy generalizing the so-called Tarski-Sher criterion for logicality and I characterize the invariant operators as definable in a fragment of the first-order language. These results are compared with those obtained by Feferman and it is argued that further clarification of the notion of invariance is needed if one wants to use it to characterize logicality.
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  24. Sergio A. Celani & Hernán J. San Martín (2012). Frontal Operators in Weak Heyting Algebras. Studia Logica 100 (1-2):91-114.
    In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [ 10 ]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation $${\tau(a) \leq b \vee (b \rightarrow a)}$$, for all $${a, b \in A}$$. These operators were studied from an algebraic, logical and topological point of view by Leo Esakia (...)
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  25. William J. Collins & Paul Young (1983). Discontinuities of Provably Correct Operators on the Provably Recursive Real Numbers. Journal of Symbolic Logic 48 (4):913-920.
    In this paper we continue, from [2], the development of provably recursive analysis, that is, the study of real numbers defined by programs which can be proven to be correct in some fixed axiom system S. In particular we develop the provable analogue of an effective operator on the set C of recursive real numbers, namely, a provably correct operator on the set P of provably recursive real numbers. In Theorems 1 and 2 we exhibit a provably correct operator on (...)
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  26. Roger M. Cooke & Michiel Lambalgen (1983). The Representation of Takeuti's *20c ||_ -Operator. Studia Logica 42 (4):407 - 415.
    Gaisi Takeuti has recently proposed a new operation on orthomodular lattices L, ⫫: $\scr{P}(L)\rightarrow L$ . The properties of ⫫ suggest that the value of ⫫ $(A)(A\subseteq L)$ corresponds to the degree in which the elements of A behave classically. To make this idea precise, we investigate the connection between structural properties of orthomodular lattices L and the existence of two-valued homomorphisms on L.
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  27. S. B. Cooper (1973). Minimal Degrees and the Jump Operator. Journal of Symbolic Logic 38 (2):249-271.
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  28. Fabrice Correia (2007). Modality, Quantification, and Many Vlach-Operators. Journal of Philosophical Logic 36 (4):473 - 488.
    Consider two standard quantified modal languages A and P whose vocabularies comprise the identity predicate and the existence predicate, each endowed with a standard S5 Kripke semantics where the models have a distinguished actual world, which differ only in that the quantifiers of A are actualist while those of P are possibilist. Is it possible to enrich these languages in the same manner, in a non-trivial way, so that the two resulting languages are equally expressive-i.e., so that for each sentence (...)
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  29. Janusz Czelakowski (2003). The Suszko Operator. Part I. Studia Logica 74 (1-2):181 - 231.
    The paper is conceived as a first study on the Suszko operator. The purpose of this paper is to indicate the existence of close relations holding between the properties of the Suszko operator and the structural properties of the model class for various sentential logics. The emphasis is put on generality both of the results and methods of tackling the problems that arise in the theory of this operator. The attempt is made here to develop the theory for non-protoalgebraic logics.
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  30. René David & Karim Nour (1995). Storage Operators and Directed Lambda-Calculus. Journal of Symbolic Logic 60 (4):1054-1086.
    Storage operators have been introduced by J. L. Krivine in [5] they are closed λ-terms which, for a data type, allow one to simulate a "call by value" while using the "call by name" strategy. In this paper, we introduce the directed λ-calculus and show that it has the usual properties of the ordinary λ-calculus. With this calculus we get an equivalent--and simple--definition of the storage operators that allows to show some of their properties: $\bullet$ the stability of the set (...)
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  31. Maarten de Rijke & Yde Venema (1995). Sahlqvist's Theorem for Boolean Algebras with Operators with an Application to Cylindric Algebras. Studia Logica 54 (1):61-78.
    For an arbitrary similarity type of Boolean Algebras with Operators we define a class ofSahlqvist identities. Sahlqvist identities have two important properties. First, a Sahlqvist identity is valid in a complex algebra if and only if the underlying relational atom structure satisfies a first-order condition which can be effectively read off from the syntactic form of the identity. Second, and as a consequence of the first property, Sahlqvist identities arecanonical, that is, their validity is preserved under taking canonical embedding algebras. (...)
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  32. Stéphane Demri (1999). A Logic with Relative Knowledge Operators. Journal of Logic, Language and Information 8 (2):167-185.
    We study a knowledge logic that assumes that to each set of agents, an indiscernibility relation is associated and the agents decide the membership of objects or states up to this indiscernibility relation. Its language contains a family of relative knowledge operators. We prove the decidability of the satisfiability problem, we show its EXPTIME-completeness and as a side-effect, we define a complete Hilbert-style axiomatization.
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  33. Stéphane Demri (1997). A Completeness Proof for a Logic with an Alternative Necessity Operator. Studia Logica 58 (1):99-112.
    We show the completeness of a Hilbert-style system LK defined by M. Valiev involving the knowledge operator K dedicated to the reasoning with incomplete information. The completeness proof uses a variant of Makinson's canonical model construction. Furthermore we prove that the theoremhood problem for LK is co-NP-complete, using techniques similar to those used to prove that the satisfiability problem for propositional S5 is NP-complete.
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  34. Harry Deutsch (2010). Diagonalization and Truth Functional Operators. Analysis 70 (2):215-217.
    (No abstract is available for this citation).
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  35. Jürgen Dix & David Makinson (1992). The Relationship Between KLM and MAK Models for Nonmonotonic Inference Operations. Journal of Logic, Language and Information 1 (2):131-140.
    The purpose of this note is to make quite clear the relationship between two variants of the general notion of a preferential model for nonmonotonic inference: the models of Kraus, Lehmann and Magidor (KLM models) and those of Makinson (MAK models).On the one hand, we introduce the notion of the core of a KLM model, which suffices to fully determine the associated nonmonotonic inference relation. On the other hand, we slightly amplify MAK models with a monotonic consequence operation as additional (...)
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  36. Fred I. Dretske (1970). ``Epistemic Operators&Quot;. Journal of Philosophy 67:1007-1023.
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  37. M. Fattorosi-Barnaba & G. Amati (1987). Modal Operators with Probabilistic Interpretations, I. Studia Logica 46 (4):383 - 393.
    <span class='Hi'></span> We present a class of normal modal calculi PFD,<span class='Hi'></span> whose syntax is endowed with operators M r <span class='Hi'></span>(and their dual ones,<span class='Hi'></span> L r)<span class='Hi'></span>, one for each r <span class='Hi'></span>[0,1]<span class='Hi'></span>: if a is sentence,<span class='Hi'></span> M r is to he read the probability that a is true is strictly greater than r and to he evaluated as true or false in every world of a F-restricted probabilistic kripkean model.<span class='Hi'></span> Every such a model is (...)
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  38. Hércules A. Feitosa, Mauri C. Do Nascimento & Maria Claudia C. Grácio (2011). Logic TK: Algebraic Notions From Tarski's Consequence Operator. Principia 14 (1):47-70.
    Tarski apresentou sua definição de operador de consequência com a intenção de expor as concepções fundamentais da consequência lógica. Um espaço de Tarski é um par ordenado determinado por um conjunto não vazio e um operador de consequência sobre este conjunto. Esta estrutura matemática caracteriza um espaço quase topológico. Este artigo mostra uma visão algébrica dos espaços de Tarski e introduz uma lógica proposicional modal que interpreta o seu operador modal nos conjuntos fechados de algum espaço de Tarski. DOI:10.5007/1808-1711.2010v14n1p47.
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  39. Eduardo L. Fermé & Sven Ove Hansson (1999). Selective Revision. Studia Logica 63 (3):331-342.
    We introduce a constructive model of selective belief revision in which it is possible to accept only a part of the input information. A selective revision operator ο is defined by the equality K ο α = K * f(α), where * is an AGM revision operator and f a function, typically with the property ⊢ α → f(α). Axiomatic characterizations are provided for three variants of selective revision.
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  40. Branden Fitelson, Notes on Gibbard's Theorem.
    Let L be a sentential (object) language containing atoms ‘A’, ‘B’, . . . , and two logical connectives ‘&’ and ‘→’. In addition to these two logical connectives, L will also contain another binary connective ‘ ’, which is intended to be interpreted as the English indicative. In the meta-language for L , we will have two meta-linguistic operations: ‘ ’ and ‘ ’. ‘ ’ is a binary relation between individual sentences in L . It will be interpreted (...)
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  41. Melvin Fitting (1969). Logics With Several Modal Operators. Theoria 35 (3):259-266.
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  42. Josep M. Font & Ventura Verdú (1993). The Lattice of Distributive Closure Operators Over an Algebra. Studia Logica 52 (1):1 - 13.
    In our previous paper Algebraic Logic for Classical Conjunction and Disjunction we studied some relations between the fragmentL of classical logic having just conjunction and disjunction and the varietyD of distributive lattices, within the context of Algebraic Logic. The central tool in that study was a class of closure operators which we calleddistributive, and one of its main results was that for any algebraA of type (2,2) there is an isomorphism between the lattices of allD-congruences ofA and of all distributive (...)
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  43. Michael Freund & Daniel Lehmann (1994). Nonmonotonic Reasoning: From Finitary Relations to Infinitary Inference Operations. Studia Logica 53 (2):161 - 201.
    A. Tarski [22] proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the monotonicity requirement and consider more general operations, inference operations. These operations describe the nonmonotonic logics both humans and machines seem to be using when infering defeasible information from incomplete knowledge. We single out a number of interesting families of inference operations. This study of infinitary inference operations (...)
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  44. Joseph S. Fulda (1993). Exclusive Disjunction and the Biconditional: An Even-Odd Relationship. Mathematics Magazine 66 (2):124.
    Proves two simple identities relating the biconditional and exclusive disjunction. -/- The PDF has been made available gratis by the publisher.
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  45. James Andrew Fulton (1979). An Intensional Logic of Predicates and Predicate Modifiers Without Modal Operators. Notre Dame Journal of Formal Logic 20 (4):807-834.
  46. Dov M. Gabbay (1973). Applications of Scott's Notion of Consequence to the Study of General Binary Intensional Connectives and Entailment. Journal of Philosophical Logic 2 (3):340 - 351.
  47. Dov M. Gabbay (1972). A General Theory of the Conditional in Terms of a Ternary Operator. Theoria 38 (3):97-104.
  48. Pietro Galliani (2013). Epistemic Operators in Dependence Logic. Studia Logica 101 (2):367-397.
    The properties of the ${\forall^{1}}$ quantifier defined by Kontinen and Väänänen in [13] are studied, and its definition is generalized to that of a family of quantifiers ${\forall^{n}}$ . Furthermore, some epistemic operators δ n for Dependence Logic are also introduced, and the relationship between these ${\forall^{n}}$ quantifiers and the δ n operators are investigated.The Game Theoretic Semantics for Dependence Logic and the corresponding Ehrenfeucht- Fraissé game are then adapted to these new connectives.Finally, it is proved that the ${\forall^{1}}$ quantifier (...)
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  49. N. Georgiewa (1971). A Logical System Which has ≡ and V as Primitive Connectives. Studia Logica 28 (1):76.
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