About the open logic project the open logic text is an opensource, collaborative textbook of formal meta logic and formal methods, starting at an intermediate level i. The crucial point is that satisfaction of a formula in a modal. The fixed point property in modal logic sacchetti, lorenzo, notre dame journal of formal logic, 2001. In modern terms, his key idea was that we can take a formula a, pre. We define and investigate a new modal fixed point logic, called bisimulation safe fixedpoint logic bsfp, which is a calculus of binary relations that extends both pdl and the modal. Our analysis clarifies the latter systems, while also raising some new questions about fixed point logics. We present two modal typing systems with the approximation modality, which has been proposed by the author to capture selfreferences involved in computer programs and their speci. Basic concepts in modal logic1 stanford university. We show that propositional dynamic logic and the modal. Modal fixedpoint logic and changing models springerlink. The logic hfl includes negation as a firstclass construct and uses a simple type system to identify the monotonic functions on which the application of fixed point. Chapter 1 modal logics of space institute for logic. The logic of provability university of california, berkeley. Pdf the following notes are centered around multimodal logics extended by the possibility to introduce least and greatest fixed points.
Modal logic is the study of modal propositions and the logical relationships that they bear to one another. Our suggestions are backed up by some new results while we also refer to relevant work by earlier authors. Most of the ideas presented in this document are not my own, but rather boolos and should be treated accordingly. The following notes are centered around multi modal logics extended by the possibility to introduce least and greatest fixed points. We introduce a class of kripke models and prove that least fixed points of positive operators are definable in these. While it still retains a bit of this grandeur, today, modal logic sits at a crossroads of many academic disciplines, and thus, it provides a unique vantage point for students with broad interdisciplinary. Even though these features of modal logic have not evolved for specific spatial reasons, they are often congenial with thinking. The modal logic underlying our calculus is hennessymilner logic hm85.
The aim of this paper is to put this approach into the new context of modal fixed point logic. Definable fixed points in modal and temporal logics a survey. Preface what follows are my personal notes on george boolos the logic of provability. Various modal languages have been developed, in a uniform fashion, for coalgebras of arbitrary type, but none of these languages admit explicit xed point operators. The distinguishing feature of the calculus is the presence of fixed points that. Note that the only fact in this section that will be used in the proof of the sahlqvist theorem for modal fixed point logic section 5 is lemma 4. Two of kripkes earlier works, a completeness theorem in modal logic 1959 and semantical considerations on modal logic 1963, the former written when he was a teenager, were on modal logic. Fixedpoint logic with the approximation modality and its. Finite kripke models and predicate logics of provability artemov, sergei and dzhaparidze, giorgie, journal of symbolic.
Comments on modal fixed point logic and changing models. Bisimilar states must share the same local properties and any transition from vto win tmust have a transition of the same kind from v0to w 0in t and vice versa such that wand w0are again bisimilar. Modal frame correspondences and fixedpoints springerlink. Prooftheoretic contributions to modal fixed point logics. We begin with discussing a range of traditional results and turn to more recent approaches dealing with. In each case, the set defined is the least fixed point of a monotone operator on. The modal system gl is well known as the logic of provability, since it has the connection with arithmetical theories, for instance, peano arithmetic pa per solovay. This paper deals with the modal logics associated with possibly nonstandard provability predicates of peano arithmetic. The shifting process above is the key to understand fixed point number representation. Modal fixed point logics mic the modal inationary calculus. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Montagues paradox, informal provability, and explicit modal logic dean, walter, notre dame journal of formal logic, 2014. Preliminary report ronald fagin ibm research laboratory.
First order logic with fixed point operators is a classical matter, as evidenced by the wellknown textbook by ebbinghaus and flum 1. Researchers in areas ranging from economics to computational linguistics have since realised its worth. Fixed point logics are extensions of first order predicate logic with fixed point operators. Least and inflationary fixed point operators have been studied and compared in other contexts, particularly in. Refutation of the fixedpoint property of selfproving for. Modal logic wasborn in the earlypart ofthe 20th century as a branchof logic applied to the analysis of philosophical notions and issues. We begin with discussing a range of traditional results and turn towards a more explicit and operational approach in the fourth lecture. We prove that, for every has the explicit fixed point property. We will then simply adhere to this implicit convention when we represent numbers.
Fixedpoint logics and computation university of cambridge. Automata and fixed point logic institute for logic, language and. We present a higher order modal fixed point logic hfl that extends the modal. Modal logic can be given semantics over topological spaces. Spatial logic of modal mucalculus and tangled closure operators. We show a natural connection between scanlons test for justifiability and the computation of the smallest fixed point. Comments on modal fixed point logic and changing models jan van eijck august 2007 this is indeed a very nice draft that i have read with great pleasure, and that has helped me to better understand the completeness proof for lcc. Moreover, s is a least fixed point or greatest fixed point if for. We present a higher order modal fixed point logic hfl that extends the modal calculus to allow predicates on states sets of states to be specified using recursively defined higher order. In this paper we use fixed point modal logic to study the logical properties of justified norms in scanlonian contractualism. We begin with discussing a range of traditional results and. A higher order modal fixed point logic springerlink. Mardaev, least fixed points in grzegorczyk logic and in intuitionistic propositional logic, algebra i logika 32 no 5 1993, pp.
T0are bisimilar if the set of possible traces from these states are equivalent in a strong sense. The following notes are centered around multi modal logics extended by the possibility to introduce least and greatest. The class is widest of the known ones in which least fixed points of positive operators are definable. Sahlqvist theorem for modal fixed point logic sciencedirect. These four lectures are centered around multi modal logics extended by the possibility to introduce least and largest fixed points. Modal logic with an inflationary fixed point operator is more expressive than l. An infinitary system for the least fixedpoint logic. We determine the modal logic of fixed point models of truth and their axiomatizations by solomon feferman via solovaystyle completeness results. A number of such logics arose in finite model theory but they are of interest to much larger audience, e. This is an advanced 2001 textbook on modal logic, a field which caught the attention of computer scientists in the late 1970s. Though aimed at a nonmathematical audience in particular, students of philosophy and computer science, it is rigorous. To represent a real number in computers or any hardware in general, we can define a fixed point number type simply by implicitly fixing the binary point to be at some position of a numeral. One of our goals is to present some modal systems having the fixed point property and not extending the godellob system gl.
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