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Melvin Fitting [135]Melvin C. Fitting [3]Melvin Chris Fitting [1]
  1.  43
    The Logic of Proofs, Semantically.Melvin Fitting - 2005 - Annals of Pure and Applied Logic 132 (1):1-25.
    A new semantics is presented for the logic of proofs (LP), [1, 2], based on the intuition that it is a logic of explicit knowledge. This semantics is used to give new proofs of several basic results concerning LP. In particular, the realization of S4 into LP is established in a way that carefully examines and explicates the role of the + operator. Finally connections are made with the conventional approach, via soundness and completeness results.
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  2.  22
    First-Order Modal Logic.Melvin Fitting, R. Mendelsohn & Roderic A. Girle - 2002 - Bulletin of Symbolic Logic 8 (3):429-430.
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  3.  5
    Proof Methods for Modal and Intuitionistic Logics.Melvin Fitting - 1985 - Journal of Symbolic Logic 50 (3):855-856.
  4.  27
    Bilattices and the Semantics of Logic Programming.Melvin Fitting - unknown
    Bilattices, due to M. Ginsberg, are a family of truth value spaces that allow elegantly for missing or conflicting information. The simplest example is Belnap’s four-valued logic, based on classical two-valued logic. Among other examples are those based on finite many-valued logics, and on probabilistic valued logic. A fixed point semantics is developed for logic programming, allowing any bilattice as the space of truth values. The mathematics is little more complex than in the classical two-valued setting, but the result provides (...)
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  5.  32
    Bilattices In Logic Programming.Melvin Fitting - unknown
    Bilattices, introduced by M. Ginsberg, constitute an elegant family of multiple-valued logics. Those meeting certain natural conditions have provided the basis for the semantics of a family of logic programming languages. Now we consider further restrictions on bilattices, to narrow things down to logic programming languages that can, at least in principle, be implemented. Appropriate bilattice background information is presented, so the paper is relatively self-contained.
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  6.  63
    Many-Valued Modal Logics.Melvin C. Fitting - unknown
    Two families of many-valued modal logics are investigated. Semantically, one family is characterized using Kripke models that allow formulas to take values in a finite many-valued logic, at each possible world. The second family generalizes this to allow the accessibility relation between worlds also to be many-valued. Gentzen sequent calculi are given for both versions, and soundness and completeness are established.
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  7.  4
    First-Order Modal Logic.Roderic A. Girle, Melvin Fitting & Richard L. Mendelsohn - 2002 - Bulletin of Symbolic Logic 8 (3):429.
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  8.  50
    Bilattices Are Nice Things.Melvin Fitting - 2006 - In T. Bolander, V. Hendricks & S. A. Pedersen (eds.), Self-Reference. CSLI Publications.
    One approach to the paradoxes of self-referential languages is to allow some sentences to lack a truth value (or to have more than one). Then assigning truth values where possible becomes a fixpoint construction and, following Kripke, this is usually carried out over a partially ordered family of three-valued truth-value assignments. Some years ago Matt Ginsberg introduced the notion of bilattice, with applications to artificial intelligence in mind. Bilattices generalize the structure Kripke used in a very natural way, while making (...)
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  9.  80
    Many-Valued Modal Logics II.Melvin Fitting - unknown
    Suppose there are several experts, with some dominating others (expert A dominates expert B if B says something is true whenever A says it is). Suppose, further, that each of the experts has his or her own view of what is possible — in other words each of the experts has their own Kripke model in mind (subject, of course, to the dominance relation that may hold between experts). How will they assign truth values to sentences in a common modal (...)
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  10.  68
    Prefixed Tableaus and Nested Sequents.Melvin Fitting - 2012 - Annals of Pure and Applied Logic 163 (3):291 - 313.
    Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tableaus go back to 1972, and are modal tableau systems with extra machinery to represent accessibility in a purely syntactic way. We show that modal nested sequents and prefixed modal tableaus are notational variants (...)
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  11.  11
    Intuitionistic Logic, Model Theory and Forcing.Melvin Fitting - 1969 - Amsterdam: North-Holland Pub. Co..
  12.  26
    Fixpoint Semantics for Logic Programming A Survey.Melvin Fitting - unknown
    The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and many-valued logic, lattice theory, game theory, and topology. One source of this richness is the inherent non-monotonicity of its negation, something that does not have close parallels with the machinery of other programming paradigms. Nonetheless, much of the work on logic programming semantics seems to exist side by side with similar work done for imperative and functional programming, with relatively minimal contact (...)
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  13.  60
    First-Order Intensional Logic.Melvin Fitting - 2004 - Annals of Pure and Applied Logic 127 (1-3):171-193.
    First - order modal logic is very much under current development, with many different semantics proposed. The use of rigid objects goes back to Saul Kripke. More recently, several semantics based on counterparts have been examined, in a development that goes back to David Lewis. There is yet another line of research, using intensional objects, that traces back to Richard Montague. I have been involved with this line of development for some time. In the present paper, I briefly sketch several (...)
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  14.  54
    Kleene's Three Valued Logics and Their Children.Melvin Fitting - unknown
    Kleene’s strong three-valued logic extends naturally to a four-valued logic proposed by Belnap. We introduce a guard connective into Belnap’s logic and consider a few of its properties. Then we show that by using it four-valued analogs of Kleene’s weak three-valued logic, and the asymmetric logic of Lisp are also available. We propose an extension of these ideas to the family of distributive bilattices. Finally we show that for bilinear bilattices the extensions do not produce any new equivalences.
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  15.  22
    Realizations and LP.Melvin Fitting - 2009 - Annals of Pure and Applied Logic 161 (3):368-387.
    LP can be seen as a logic of knowledge with justifications. See [S. Artemov, The logic of justification, The Review of Symbolic Logic 1 477–513] for a recent comprehensive survey of justification logics generally. Artemov’s Realization Theorem says justifications can be extracted from validities in the more conventional Hintikka-style logic of knowledge S4, in which they are not explicitly present. Justifications, however, are far from unique. There are many ways of realizing each theorem of S4 in the logic LP. If (...)
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  16.  11
    The Stable Model Semantics for Logic Programming. [REVIEW]Melvin Fitting - 1992 - Journal of Symbolic Logic 57 (1):274-277.
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  17.  6
    Paraconsistent Logic, Evidence, and Justification.Melvin Fitting - 2017 - Studia Logica 105 (6):1149-1166.
    In a forthcoming paper, Walter Carnielli and Abilio Rodrigues propose a Basic Logic of Evidence whose natural deduction rules are thought of as preserving evidence instead of truth. BLE turns out to be equivalent to Nelson’s paraconsistent logic N4, resulting from adding strong negation to Intuitionistic logic without Intuitionistic negation. The Carnielli/Rodrigues understanding of evidence is informal. Here we provide a formal alternative, using justification logic. First we introduce a modal logic, KX4, in which \ can be read as asserting (...)
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  18.  55
    Kleene's Logic, Generalized.Melvin Fitting - unknown
    Kleene’s well-known strong three-valued logic is shown to be one of a family of logics with similar mathematical properties. These logics are produced by an intuitively natural construction. The resulting logics have direct relationships with bilattices. In addition they possess mathematical features that lend themselves well to semantical constructions based on fixpoint procedures, as in logic programming.
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  19.  44
    A Quantified Logic of Evidence.Melvin Fitting - 2008 - Annals of Pure and Applied Logic 152 (1):67-83.
    A propositional logic of explicit proofs, LP, was introduced in [S. Artemov, Explicit provability and constructive semantics, The Bulletin for Symbolic Logic 7 1–36], completing a project begun long ago by Gödel, [K. Gödel, Vortrag bei Zilsel, translated as Lecture at Zilsel’s in: S. Feferman , Kurt Gödel Collected Works III, 1938, pp. 62–113]. In fact, LP can be looked at in a more general way, as a logic of explicit evidence, and there have been several papers along these lines. (...)
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  20.  28
    Modal Logics, Justification Logics, and Realization.Melvin Fitting - 2016 - Annals of Pure and Applied Logic 167 (8):615-648.
  21.  16
    Possible World Semantics for First-Order Logic of Proofs.Melvin Fitting - 2014 - Annals of Pure and Applied Logic 165 (1):225-240.
    In the tech report Artemov and Yavorskaya [4] an elegant formulation of the first-order logic of proofs was given, FOLP. This logic plays a fundamental role in providing an arithmetic semantics for first-order intuitionistic logic, as was shown. In particular, the tech report proved an arithmetic completeness theorem, and a realization theorem for FOLP. In this paper we provide a possible-world semantics for FOLP, based on the propositional semantics of Fitting [5]. We also give an Mkrtychev semantics. Motivation and intuition (...)
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  22.  3
    Nested Sequents for Intuitionistic Logics.Melvin Fitting - 2014 - Notre Dame Journal of Formal Logic 55 (1):41-61.
  23.  34
    Reasoning About Games.Melvin Fitting - 2011 - Studia Logica 99 (1-3):143-169.
    is used to give a formalization of Artemov’s knowledge based reasoning approach to game theory, (KBR), [ 4 , 5 ]. Epistemic states of players are represented explicitly and reasoned about formally. We give a detailed analysis of the Centipede game using both proof theoretic and semantic machinery. This helps make the case that PDL + E can be a useful basis for the logical investigation of game theory.
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  24. Types Tableaus and Gödel's God.Melvin Fitting - 2005 - Studia Logica 81 (3):425-427.
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  25.  26
    Modal Logics Between Propositional and First Order.Melvin Fitting - unknown
    One can add the machinery of relation symbols and terms to a propositional modal logic without adding quantifiers. Ordinarily this is no extension beyond the propositional. But if terms are allowed to be non-rigid, a scoping mechanism (usually written using lambda abstraction) must also be introduced to avoid ambiguity. Since quantifiers are not present, this is not really a first-order logic, but it is not exactly propositional either. For propositional logics such as K, T and D, adding such machinery produces (...)
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  26.  17
    Notes on the Mathematical Aspects of Kripke's Theory of Truth.Melvin Fitting - 1986 - Notre Dame Journal of Formal Logic 27 (1):75-88.
  27.  48
    Tableaus for Many-Valued Modal Logic.Melvin Fitting - 1995 - Studia Logica 55 (1):63 - 87.
    We continue a series of papers on a family of many-valued modal logics, a family whose Kripke semantics involves many-valued accessibility relations. Earlier papers in the series presented a motivation in terms of a multiple-expert semantics. They also proved completeness of sequent calculus formulations for the logics, formulations using a cut rule in an essential way. In this paper a novel cut-free tableau formulation is presented, and its completeness is proved.
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  28.  22
    Bilattices and the Theory of Truth.Melvin Fitting - 1989 - Journal of Philosophical Logic 18 (3):225 - 256.
    While Kripke's original paper on the theory of truth used a three-valued logic, we believe a four-valued version is more natural. Its use allows for possible inconsistencies in information about the world, yet contains Kripke's development within it. Moreover, using a four-valued logic makes it possible to work with complete lattices rather than complete semi-lattices, and thus the mathematics is somewhat simplified. But more strikingly, the four-valued version has a wide, natural generalization to the family of interlaced bilattices. Thus, with (...)
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  29.  45
    A Logic of Explicit Knowledge.Melvin Fitting - unknown
    A well-known problem with Hintikka-style logics of knowledge is that of logical omniscience. One knows too much. This breaks down into two subproblems: one knows all tautologies, and one’s knowledge is closed under consequence. A way of addressing the second of these is to move from knowledge simpliciter, to knowledge for a reason. Then, as consequences become ‘further away’ from one’s basic knowledge, reasons for them become more complex, thus providing a kind of resource measurement. One kind of reason is (...)
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  30.  33
    Reasoning with Justifications.Melvin Fitting - unknown
    This is an expository paper in which the basic ideas of a family of Justification Logics are presented. Justification Logics evolved from a logic called LP, introduced by Sergei Artemov [1, 3], which formed the central part of a project to provide an arithmetic semantics for propositional intuitionistic logic. The project was successful, but there was a considerable bonus: LP came to be understood as a logic of knowledge with explicit justifications and, as such, was capable of addressing in a (...)
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  31.  58
    Intensional Logic — Beyond First Order.Melvin Fitting - unknown
    Classical first-order logic can be extended in two different ways to serve as a foundation for mathematics: introduce higher orders, type theory, or introduce sets. As it happens, both approaches have natural analogs for quantified modal logics, both approaches date from the 1960’s, one is not very well-known, and the other is well-known as something else. I will present the basic semantic ideas of both higher order intensional logic, and intensional set theory. Before doing so, I’ll quickly sketch some necessary (...)
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  32. First-Order Logic and Automated Theorem Proving.Melvin Fitting - 1998 - Studia Logica 61 (2):300-302.
     
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  33.  28
    Intensional Logic.Melvin Fitting - 2008 - Stanford Encyclopedia of Philosophy.
    There is an obvious difference between what a term designates and what it means. At least it is obvious that there is a difference. In some way, meaning determines designation, but is not synonymous with it. After all, “the morning star” and “the evening star” both designate the planet Venus, but don't have the same meaning. Intensional logic attempts to study both designation and meaning and investigate the relationships between them.
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  34.  22
    Term-Modal Logics.Melvin Fitting, Lars Thalmann & Andrei Voronkov - 2001 - Studia Logica 69 (1):133-169.
    Many powerful logics exist today for reasoning about multi-agent systems, but in most of these it is hard to reason about an infinite or indeterminate number of agents. Also the naming schemes used in the logics often lack expressiveness to name agents in an intuitive way.To obtain a more expressive language for multi-agent reasoning and a better naming scheme for agents, we introduce a family of logics called term-modal logics. A main feature of our logics is the use of modal (...)
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  35.  78
    Justification Logics, Logics of Knowledge, and Conservativity.Melvin Fitting - unknown
    Several justification logics have been created, starting with the logic LP, [1]. These can be thought of as explicit versions of modal logics, or of logics of knowledge or belief, in which the unanalyzed necessity (knowledge, belief) operator has been replaced with a family of explicit justification terms. We begin by sketching the basics of justification logics and their relations with modal logics. Then we move to new material. Modal logics come in various strengths. For their corresponding justification logics, differing (...)
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  36.  46
    Justification Logics and Hybrid Logics.Melvin Fitting - 2010 - Journal of Applied Logic 8 (4):356-370.
    Hybrid logics internalize their own semantics. Members of the newer family of justification logics internalize their own proof methodology. It is an appealing goal to combine these two ideas into a single system, and in this paper we make a start. We present a hybrid/justification version of the modal logic T. We give a semantics, a proof theory, and prove a completeness theorem. In addition, we prove a Realization Theorem, something that plays a central role for justification logics generally. Since (...)
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  37.  57
    A Simple Propositional S5 Tableau System.Melvin Fitting - 1999 - Annals of Pure and Applied Logic 96 (1-3):107-115.
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  38.  20
    Metric Methods Three Examples and a Theorem.Melvin Fitting - unknown
    £ The existence of a model for a logic program is generally established by lattice-theoretic arguments. We present three examples to show that metric methods can often be used instead, generally in a direct, straightforward way. One example is a game program, which is not stratified or locally stratified, but which has a unique supported model whose existence is easily established using metric methods. The second example is a program without a unique supported model, but having a part that is (...)
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  39.  18
    Tableau Methods of Proof for Modal Logics.Melvin Fitting - 1972 - Notre Dame Journal of Formal Logic 13 (2):237-247.
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  40.  48
    Interpolation for First Order S5.Melvin Fitting - 2002 - Journal of Symbolic Logic 67 (2):621-634.
    An 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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  41.  49
    Many-Valued Non-Monotonic Modal Logics.Melvin Fitting - unknown
    Among non-monotonic systems of reasoning, non-monotonic modal logics, and autoepistemic logic in particular, have had considerable success. The presence of explicit modal operators allows flexibility in the embedding of other approaches. Also several theoretical results of interest have been established concerning these logics. In this paper we introduce non-monotonic modal logics based on many-valued logics, rather than on classical logic. This extends earlier work of ours on many-valued modal logics. Intended applications are to situations involving several reasoners, not just one (...)
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  42.  34
    Bisimulations and Boolean Vectors.Melvin Fitting - 2003 - In Philippe Balbiani, Nobu-Yuki Suzuki, Frank Wolter & Michael Zakharyaschev (eds.), Advances in Modal Logic, Volume 4. CSLI Publications. pp. 97-125.
    A modal accessibility relation is just a transition relation, and so can be represented by a {0, 1} valued transition matrix. Starting from this observation, I first show that the machinery of matrices, over Boolean algebras more general than the two-valued one, is appropriate for investigating multi-modal semantics. Then I show that bisimulations have a rather elegant theory, when expressed in terms of transformations on Boolean vector spaces. The resulting theory is a curious hybrid, fitting between conventional modal semantics and (...)
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  43.  30
    The Realization Theorem for S5 a Simple, Constructive Proof.Melvin Fitting - unknown
    Justification logics are logics of knowledge in which explicit reasons are formally represented. Standard logics of knowledge have justification logic analogs. Connecting justification logics and logics of knowledge are Realization Theorems. In this paper we give a new, constructive proof of the Realization Theorem connecting S5 and its justification analog, JS5. This proof is, I believe, the simplest in the literature.
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  44. Justification Logics and Hybrid Logics.Melvin Fitting - 2010 - Journal of Applied Logic 8 (4):356-370.
  45.  43
    A Theory of Truth That Prefers Falsehood.Melvin Fitting - 1997 - Journal of Philosophical Logic 26 (5):477-500.
    We introduce a subclass of Kripke's fixed points in which falsehood is the preferred truth value. In all of these the truthteller evaluates to false, while the liar evaluates to undefined (or overdefined). The mathematical structure of this family of fixed points is investigated and is shown to have many nice features. It is noted that a similar class of fixed points, preferring truth, can also be studied. The notion of intrinsic is shown to relativize to these two subclasses. The (...)
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  46. Fundamentals of Generalized Recursion Theory.Melvin Fitting - 1986 - Journal of Symbolic Logic 51 (4):1078-1079.
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  47.  35
    Modal Logic Should Say More Than It Does.Melvin Fitting - unknown
    First-order modal logics, as traditionally formulated, are not expressive enough. It is this that is behind the difficulties in formulating a good analog of Herbrand’s Theorem, as well as the well-known problems with equality, non-rigid designators, definite descriptions, and nondesignating terms. We show how all these problems disappear when modal language is made more expressive in a simple, natural way. We present a semantic tableaux system for the enhanced logic, and (very) briefly discuss implementation issues.
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  48.  31
    Herbrand's Theorem for a Modal Logic.Melvin Fitting - unknown
    Herbrand’s theorem is a central fact about classical logic, [9, 10]. It provides a constructive method for associating, with each first-order formula X, a sequence of formulas X1, X2, X3, . . . , so that X has a first-order proof if and only if some Xi is a tautology. Herbrand’s theorem serves as a constructive alternative to..
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  49.  25
    The Family of Stable Models.Melvin Fitting - unknown
    The family of all stable models for a logic program has a surprisingly simple overall structure, once two naturally occurring orderings are made explicit. In a so-called knowledge ordering based on degree of definedness, every logic program P has a smallest stable model, sk P — it is the well-founded model. There is also a dual largest stable model, S k P, which has not been considered before. There is another ordering based on degree of truth. Taking the meet and (...)
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  50.  14
    A Program to Compute G¨Odel-L¨Ob Fixpoints.Melvin Fitting - unknown
    odel-L¨ ob computability logic. In order to make things relatively self-contained, I sketch the essential ideas of GL, and discuss the significance of its fixpoint theorem. Then I give the algorithm embodied in the program in a little more detail. It should be emphasized that nothing new is presented here — all the theory and methodology are due to others. The main interest is, in a sense, psychological. The approach taken here has been declared in the literature, more than once, (...)
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