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By
Fitting, Melvin; Mendelsohn, Richard L.
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Most presentations of firstorder classical logic allow constant and function symbols to appear in formulas. This is the appropriate point for us to introduce them into our treatment of firstorder modal logic. But doing so brings some unexpected complications with it. On the other hand, the rewards are great. The material that follows is, perhaps, the central part of this book.
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By
Fitting, Melvin; Mendelsohn, Richard L.
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In this chapter we combine the machinery of predicate abstraction with other fundamental machinery, such as equality and an existence predicate. And we develop tableau systems that take predicate abstracts into account.
By
Fitting, Melvin; Mendelsohn, Richard L.
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Informally, a proof is an argument that convinces. Formally, a proof is of a formula, and is a finite object constructed according to fixed syntactic rules that refer only to the structure of formulas and not to their intended meaning. The syntactic rules that define proofs are said to specify a proof procedure. A proof procedure is sound for a particular logic if any formula that has a proof must be a valid formula of the logic. A proof procedure is complete for a logic if every valid formula has a proof. Then a sound and complete proof procedure allows us to produce “witnesses,” namely proofs, that formulas are valid.
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By
Fitting, Melvin
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A generalization of conventional Horn clause logic programming is proposed in which the space of truth values is a pseudoBoolean or Heyting algebra, whose members may be thought of as evidences for propositions. A minimal model and an operational semantics is presented, and their equivalence is proved, thus generalizing the classic work of Van Emden and Kowalski.
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By
Fitting, Melvin; Mendelsohn, Richard L.
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In Section 2.2 we gave prefixed tableau rules for several propositional modal logics. Now we extend these to deal with quantifiers. But we have considered two versions of quantifier semantics: varying domain and constant domain. Not surprisingly, these correspond to different versions of tableau rules for quantifiers. Also not surprisingly, the rules corresponding to constant domain semantics are somewhat simpler, so we will start with them.
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By
Fitting, Melvin; Mendelsohn, Richard L.
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There is a natural confusion between existence and designation, but these are really orthogonal issues. Terms designate; objects exist. For instance the phrase “the first President of the United States” designates George Washington, though thinking temporally, the person being designated is no longer with us—the person designated does not exist, though he once did. The nonexistent George Washington is designated by the phrase now. On the other hand the phrase, “the present King of France,” does not designate anybody now, living or dead, though at certain past instances it did designate.
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By
Fitting, Melvin; Mendelsohn, Richard L.
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We saw, in Chapter 4, that two basic kinds of quantification were natural in firstorder modal logics: possibilist and actualist. Possibilist quantifiers range over what might exist. This corresponds semantically to constant domain models where the common domain is, intuitively, the set of things that could exist. We also saw that in the possibilist approach we could introduce an existence primitive (Section 4.8) and relativize quantifiers to it, permitting the possibilist approach to paraphrase the actualist version.
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By
Fitting, Melvin; Mendelsohn, Richard L.
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We have used phrases like “the King of France” several times, though we always treated them like nonrigid constant symbols. But such phrases have more structure than constant symbols—they do not arbitrarily designate. The King of France, for instance, has the property of being King of France, provided there is one, and the phrase “the King of France” designates him because he alone has that property. Phrases of the form “the soandso” are called definite descriptions. In this chapter we examine the behavior of definite descriptions in modal contexts.
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By
Fitting, Melvin; Mendelsohn, Richard L.
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For analytic philosophy, formalization is a fundamental tool for clarifying language, leading to better understanding of thoughts expressed through language. Formalization involves abstraction and idealization. This is true in the sciences as well as in philosophy. Consider physics as a representative example. Newton’s laws of motion formalize certain basic aspects of the physical universe. Mathematical abstractions are introduced that strip away irrelevant details of the real universe, but which lead to a better understanding of its “deep” structure. Later Einstein and others proposed better models than that of Newton, reflecting deeper understanding made possible by the experience gained through years of working with Newton’s model.
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By
Fitting, Melvin; Mendelsohn, Richard L.
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1 Citations
This is a book on firstorder modal logic. Adding quantifier machinery to classical propositional logic yields firstorder classical logic, fully formed and ready to go. For modal logic, however, adding quantifiers is far from the end of the story, as we will soon see. But certainly adding quantifiers is the place to start. As it happens, even this step presents complications that do not arise classically. We say what some of these are after we get language syntax matters out of the way.
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By
Fitting, Melvin; Mendelsohn, Richard L.
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When we say that two people own the same car, we might mean either they own the same type of car, e.g., a Honda, or they are joint owners of a single car. In the former case, same means qualitative identity or equivalence; in the latter case, it means quantitative or numerical identity, or, as we shall call it, equality. Equality is actually a special case of equivalence, as we shall see after we have introduced the notion of an equivalence relation.
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By
Fitting, Melvin; Mendelsohn, Richard L.
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The axiomatic approach to a logic is quite different from that of tableaus. Certain formulas are simply announced to be theorems (they are called axioms), and various rules are adopted for adding to the stock of theorems by deducing additional ones from those already known. An axiom system is, perhaps, the most traditional way of specifying a logic, though proof discovery can often be something of a fine art. Historically, almost all of the bestknown modal logics had axiomatic characterizations long before either tableau systems or semantical approaches were available. While early modal axiom systems were somewhat circuitous by today’s standards, Gödel (1933) introduced the modern axiomatic approach and this is now used by almost everybody, including us.
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By
Fitting, Melvin; Mendelsohn, Richard L.
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In order to keep this book to a reasonable size we have decided not to give an extensive treatment of subjects that can be found elsewhere and that are somewhat peripheral to our primary interests. That policy begins to have an effect now. Proving completeness of firstorder axiomatizations can be quite complex—see (Garson, 1984) for a full discussion of the issues involved. Indeed, a common completeness proof that can cover constant domains, varying domains, and models meeting other conditions, does not seem available, so a thorough treatment would have to cover things separately for each version. Instead we simply present some appropriate axiomatizations, make a few pertinent remarks, and provide references. For machinery presented in later chapters we omit an axiomatic treatment altogether.
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By
Fitting, Melvin
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5 Citations
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
$\mathsf{LP}$
, introduced by Sergei Artemov (Technical Report MSI 9529, 1995; The Bulletin for Symbolic Logic 7(1): 1–36, 2001), 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:
$\mathsf{LP}$
came to be understood as a logic of knowledge with explicit justifications and, as such, was capable of addressing in a natural way longstanding problems of logical omniscience. Since then,
$\mathsf{LP}$
has become one member of a family of related logics, all logics of knowledge with explicit knowledge terms. In this paper the original problem of intuitionistic foundations is discussed only briefly. We concentrate entirely on issues of reasoning about knowledge.
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By
Fitting, Melvin
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18 Citations
We continue a series of papers on a family of manyvalued modal logics, a family whose Kripke semantics involves manyvalued accessibility relations. Earlier papers in the series presented a motivation in terms of a multipleexpert 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 cutfree tableau formulation is presented, and its completeness is proved.
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By
Fitting, Melvin
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3 Citations
A mixture of propositional dynamic logic and epistemic logic that we call PDL + E 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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By
Fitting, Melvin
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2 Citations
In a forthcoming paper, Walter Carnielli and Abilio Rodrigues propose a Basic Logic of Evidence (BLE) 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
$$\square X$$
can be read as asserting there is implicit evidence for X, where we understand evidence to permit contradictions. We show BLE embeds into KX4 in the same way that Intuitionistic logic embeds into S4. Then we formulate a new justification logic, JX4, in which the implicit evidence motivating KX4 is made explicit. KX4 embeds into JX4 via a realization theorem. Thus BLE has both implicit and explicit possibly contradictory evidence interpretations in a formal sense.
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By
Fitting, Melvin
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2 Citations
In an earlier paper, [5], I gave semantics and tableau rules for a simple firstorder intensional logic called FOIL, in which both objects and intensions are explicitly present and can be quantified over. Intensions, being nonrigid, are represented in FOIL as (partial) functions from states to objects. Scoping machinery, predicate abstraction, is present to disambiguate sentences like that asserting the necessary identity of the morning and the evening star, which is true in one sense and not true in another.
In this paper I address the problem of axiomatizing FOIL. I begin with an interesting sublogic with predicate abstraction and equality but no quantifiers. In [2] this sublogic was shown to be undecidable if the underlying modal logic was at least K4, though it is decidable in other cases. The axiomatization given is shown to be complete for standard logics without a symmetry condition. The general situation is not known. After this an axiomatization for the full FOIL is given, which is straightforward after one makes a change in the point of view.
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By
Fitting, Melvin; Thalmann, Lars; Voronkov, Andrei
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12 Citations
Many powerful logics exist today for reasoning about multiagent 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 multiagent reasoning and a better naming scheme for agents, we introduce a family of logics called termmodal logics. A main feature of our logics is the use of modal operators indexed by the terms of the logics. Thus, one can quantify over variables occurring in modal operators. In termmodal logics agents can be represented by terms, and knowledge of agents is expressed with formulas within the scope of modal operators.
This gives us a flexible and uniform language for reasoning about the agents themselves and their knowledge. This article gives examples of the expressiveness of the languages and provides sequentstyle and tableaubased proof systems for the logics. Furthermore we give proofs of soundness and completeness with respect to the possible world semantics.
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By
Fitting, Melvin
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2 Citations
This is a largely expository paper in which the following simple idea is pursued. Take the truth value of a formula to be the set of agents that accept the formula as true. This means we work with an arbitrary (finite) Boolean algebra as the truth value space. When this is properly formalized, complete modal tableau systems exist, and there are natural versions of bisimulations that behave well from an algebraic point of view. There remain significant problems concerning the proper formalization, in this context, of natural language statements, particularly those involving negative knowledge and common knowledge. A case study is presented which brings these problems to the fore. None of the basic material presented here is new to this paper—all has appeared in several papers over many years, by the present author and by others. Much of the development in the literature is more general than here—we have confined things to the Boolean case for simplicity and clarity. Most proofs are omitted, but several of the examples are new. The main virtue of the present paper is its coherent presentation of a systematic point of view—identify the truth value of a formula with the set of those who say the formula is true.
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