Based on firstorder modal logic by fitting and mendelsohn. Topologicalsheaf semantics for first order modal logic 25 i. Everyday low prices and free delivery on eligible orders. Buehler based on firstorder modal logic by fitting and mendelsohn january 5, 2015. We present a new way of formulating rst order modal logic which circumvents the usual di culties associated with variables changing their reference on moving between states. Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality. Some widely discussed examples include there could have been things other than there actually are and everyone who is actually rich could have been poor. Mathematics and computer science lehman college cuny, bronx, ny 10468 email. In what follows, syntactic objects languages, theories, sentences are generally written in roman or greek letters for example l, t. Alexander kocurek, on the expressive power of firstorder.
Lecture notes on firstorder reductions of firstorder modal. Firstorder modal logic1 kohei kishida 1draft of november 14, 2010. In fact, there is no way of formalizing, using standard. I use prefix terms for worlds and sequent calculus inference, following the comprehensive treatment of the first order modal logic using prefix terms and analytic tableaux or, seen upsidedown. This combination results in higher order modal logic, the subject. Mathematics and computer science lehman college cuny, bronx, ny 10468 depts. A semantic perspective 3 chapters in this handbook.
In this paper we give an overview of results for modal logic which can be shown using techniques and methods from. Firstorder modal logic, topological semantics, completeness. Lecture notes on firstorder reductions of firstorder. The focus here is on rstorder modal logic as opposed to propositional modal logic which is the focus of most of the. In addition, for the case of bimodal logic, we show that there is a naturally occurring. The book covers such issues as quantification, equality including a treatment of freges morning starevening star puzzle, the notion of existence, nonrigid constants and function symbols, predicate abstraction, the distinction between nonexistence and nondesignation, and definite descriptions, borrowing from both fregean and russellian paradigms. An advanced, but very accessible, textbook focusing on the main technical results in the area. A first order modal logic and its sheaf models barnaby p. Buy first order modal logic synthese library softcover reprint of the original 1st ed. It includes deontic logic the logic of duty and the logic of the law, plus epistemic logic. This is an advanced 2001 textbook on modal logic, a field which caught the attention of computer scientists in the late 1970s. Combining the monodic restriction with a decidable fragment of fo we often. First order modal logic, topological semantics, completeness. I use prefix terms for worlds and sequent calculus inference, following the comprehensive treatment of the firstorder modal logic using prefix terms and analytic tableaux or, seen upsidedown.
Buy firstorder modal logic synthese library softcover reprint of the original 1st ed. Fitting and mendelsohn present a thorough treatment of first order modal logic, together with some propositional background. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, timothy williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. We present syntactic proofs for all the metatheoretical results that were proved modeltheoretically in loc. This is the socalled first order or secondary interpretation of propositional. For example, the statement john is happy might be qualified by saying that john is usually happy, in which. Firstorder modal logics are modal logics in which the underlying propositional logic is replaced by a firstorder predicate logic. Thinker is an automated natural deduction firstorder theorem proving program.
The succinctness of firstorder logic over modal logic via. Now we must combine the semantics of predicate logic, which has. It can be viewed as a di rect translation from the satisfiability definition 2. Melvin mel fitting born january 24, 1942 is a logician with special interests in philosophical logic and tableau proof systems. Firstorder logicalso known as predicate logic, quantificational logic, and firstorder predicate calculusis a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. See fitting and mendelson 5 for details on tableau procedures for first order modal. This sequent calculus of cutfree proofs is chosen as a proxy to develop the prooftheory of the logics introduced in 14, 15, 4. A modal logic for ceteris paribus preferences, journal of philosophical logic, 38.
This chapter surveys basic firstorder modal logics and examines recent attempts to find a general mathematical setting in which to. Repairing the interpolation theorem in firstorder modal logic. All professors consider the dean a friend or dont know him. It elegantly straddles the line between philosophy and mathematics, without getting bogged down in the details of either as much of the rest of the modal logic literature seems to. Firstorder modal logic is very much under current development, with many different. Mathematics and computer science lehman college cuny, 250 bedford park boulevard west bronx, ny 104681589 email. Higherorder logics with their standard semantics are more expressive, but their modeltheoretic properties are less wellbehaved than those of firstorder logic the term higherorder logic, abbreviated as hol.
This allows us to model cases where, for example, alex is a. The term modal logic refers to an enrichment of standard formal logic where the standard operations and, or, not, implication and perhaps forall, etc. February 18, 1991 abstract firstorder modal logics, as traditionally formulated, are not expressive enough. Pelletier identity in modal logic theorem proving abstract. First order modal logic introduction ps pdf authors. Using a firstorder reformulation of the property of complete additivity, we prove that the modal logic that starred in van benthems article resolves the open question.
Fitting, first order intensional logic, apal, 2004. Modal logic is a simplified form of the first order predicate logic. This is a thorough treatment of firstorder modal logic. Firstorder logic uses quantified variables over nonlogical objects and allows the use of sentences that contain variables, so that rather than propositions such as socrates is a man. People only criticize people that are not their friends.
In response to this lack of expressive power, many authors have discussed extensions of firstorder modal logic with twodimensional operators. On the prooftheory of two formalisations of modal first. The set of first order formulas and free variable occurrences are as follows. For example, the statement john is happy might be qualified by saying that john is usually happy, in which case the term usually is functioning as a modal. This very extensive volume represents the current statofa airs in modal logic. Thestructurem,w is called a pointed kripkemodel for. Firstorder modal logic theorem proving and functional. Firstorder modal logic is a big area with a great number of di erent logics. Kwell so much associating is a good start, but the interesting property. Kx j x m it is true of kay that jay believes that she is the murderer. Researchers in areas ranging from economics to computational linguistics have since realised its worth. First order modal logics are modal logics in which the underlying propositional logic is replaced by a first order predicate logic.
Neighborhood semantics for first order modal logic 31 i. In mathematics and logic, a higherorder logic is a form of predicate logic that is distinguished from firstorder logic by additional quantifiers and, sometimes, stronger semantics. This is a great place to get a clear introduction to first order modal logic. Firstorder modal logic in the necessary framework of objects. It is philosophically motivated by the epistemic reading of modal operators and, in particular, three desiderata in.
The remaining cases are treated by the usual homomorphic extension. Firstorder model theory stanford encyclopedia of philosophy. While this is faithful to the field as a whole technically, modal predicate logic is just one of many system combinations, it is a serious omission for many purposes, and we will only. Lecture 12 february 25, 2010 1 introduction to this lecture in this lecture, we will introduce. First order modal logic by melvin fitting and elliot mehdelsohn. Buehler based on first order modal logic by fitting and mendelsohn january 5, 2015. Hustadt2 1 the university of manchester, uk, renate. Kwell so much associating is a good start, but the interesting property is that the reduction preserves truth. We can formulate the first reading within our logical system as follows.
This is a thorough treatment of first order modal logic. Manual of intensional logic van benthem, 1988a extends the canvas. In part i of this chapter we give an introduction to. Computational modal logic introduction ps pdf authors. A modala word that expresses a modalityqualifies a statement.
Modal logic is an extension of classic propositional and predicate logic that allows the use of modal operators. This chapter surveys basic first order modal logics and examines recent attempts to find a general mathematical setting in which to analyze them. Mathematical model theory carries a heavy load of notation, and html is not the best container for it. This paper reports on how it was adapted so as to prove theorems in modal logic. With first order modal logic we have a domain function that assigns each possible world its own domain, so that each predicate gets an extension only relative to these possible worlds.
We introduce a gentzenstyle modal predicate logic and prove the cutelimination theorem for it. Firstorder logic godels completeness theorem showed that a proof procedure exists but none was demonstrated until robinsons 1965 resolution algorithm. The method can be explained by means of a simple example taken from carnielli et al. It is shown that the modal logic s4, simple calculus and modal calculus admit a realization in a very simple propositional logical system lp, which has an exact provability semantics. Complexity of modal logic introduction ps pdf author. With standard first order logic we have a single domain and each predicate is assigned one extension. The resulting logic is defined in a language whose formulas are obtained by replacing. The book covers such issues as quantification, equality including a treatment of freges morning starevening star puzzle, the notion of existence, nonrigid constants and function symbols, predicate abstraction, the distinction between nonexistence and nondesignation, and definite. This is a great place to get a clear introduction to firstorder modal logic. Firstorder modal logic introduction ps pdf authors.
They are general enough to also apply to other modal systems. Higherorder logic is the union of first, second, third, n thorder logic. This formulation allows a very general notion of model sheaf models. The second aim of this paper is to study lindstrom type theorems for. Firstorder modal logic viii3 jay believes of kay that she is the murderer jay believes the proposition. He was a professor at city university of new york, lehman college and the graduate center from 1968 to 20. Modal logic should say more than it does lehman college. Ian horrocks, ullrich hustadt, ulrike sattler, renate schmidt. Modal predicate logic an important topic in philosophical applications of modal logic that we have mostly ignored in this survey is modal predicate logic. The book is for novices and for more experienced readers, with two distinct tracks clearly signposted at the start of each chapter.
There are two possible semantics for higher order logic. Modal logics between propositional and first order melvin fitting dept. They pose some of the most difficult mathematical challenges. Fitting and mendelsohn present a thorough treatment of firstorder modal logic, together with some propositional background. A firstorder predicate logic 323 b modal algebra 333. In the standard or full semantics, quantifiers over highertype objects range over all possible objects of that. Nov 14, 2010 we introduce a gentzenstyle modal predicate logic and prove the cutelimination theorem for it. In this post, ill demonstrate this, show what model it corresponds to, and then discuss the role of n rule and various modal axioms. Computer science, philosophy, mathematics graduate center cuny, 33 west 42nd street, nyc, ny 10036. Contents vii february 2, 2010 answers and hints to selected exercises 341 guide to further literature 371 references 373.
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