Quantifiers and quantification stanford encyclopedia of. 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. In particular, hol with henkin semantics has all the modeltheoretic properties of firstorder logic, and has a complete, sound, effective proof system inherited from firstorder logic. In logic, a manyvalued logic also multior multiplevalued logic is a propositional calculus in which there are more than two truth values. Model theory of modal logic 251 over the given frame in e. While the transition from the monosorted logic to manysorted one is a smooth process, we see our system as a step towards. Institutionalising manysorted coalgebraic modal logic. However, in 4, sorts are used to refer to the ingredients of an endofunctor on set, whereas here, manysortedness is. Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include. One may, for example, prove manysorted versions of compactness and the lowenheimskolem theorem. These systems will be described as coalgebras of socalled polynomial functors, built up from constants and identities, using products, coproducts and. Mar 30, 2001 logical options introduces the extensions and alternatives to classical logic which are most discussed in the philosophical literature. 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. Buehler based on firstorder modal logic by fitting and mendelsohn january 5, 2015.
Two families of manyvalued modal logics are investigated. 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. Institutionalising manysorted coalgebraic modal logic core. Pdf institutionalising manysorted coalgebraic modal logic. These notes are meant to present the basic facts about modal logic and so to provide a common ground for. However, the move to a manysorted language is largely a matter of convenience. On the degree of incompleteness in modal logic and the covering relations in the lattice of modal logics. Classical twovalued logic may be extended to nvalued logic for n greater than 2. This language is two sorted, containing formulas to be evaluated at points and at lines, respectively. A nice introduction to modal logic though somewhat outdated. 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. The other family considered in 3 allowed the accessibility relation itself to be many valued.
It attempts to distinguish good reasoning from bad reasoning aristotle defined logic as new and necessary reasoning, new because it. Jacobs 4 describes a modal logic for coalgebras of certain polynomial endofunctors on set. The primary goal is to provide students in philosophy 151 at stanford university with a study guide that will complement the lectures on modal logic. Modern modal logic was founded by gottlob frege, although he initially doubted its viability, and it was only later developed by rudolph carnap 1891 1970, kurt godel 1906 1978, c. The language has two diamonds whose intended accessibility relations are the two. They will be used to reprove completeness results, and hennesseymilner style characterisation results for the modal logic. Basic concepts in modal logic1 stanford university. Their approach is to turn twosorted structures into multidimensional onesorted structures.
A semantic perspective 3 chapters in this handbook. The present paper attempts to extend the results of l, in the domain of the propositional calculus, to a class of modal systems called normal. Our treatment of typed formulas resembles the concept of ingredients jacobs 2001, but the multisorted semantics avoids the use of the nextoperator of op. This language is twosorted, containing formulas to be evaluated at points and at lines, respectively. Our framework is based on a many sorted system of hybrid modal logic, for which we prove completeness results. Modal predicate logic an important topic in philosophical applications of modal logic that we have mostly ignored in this survey is modal predicate logic. However, the term modal logic may be used more broadly for a family of.
This paper gives a semantical underpinning for a manysorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in objectoriented languages. We present a new labeled tableau calculus to check the unsatisfiability and validity of sorted modal formulas. Semantically, one family is characterized using kripke models that allow formulas to take values in a. On the other hand, the passage from local to global semantics is achieved if one looks at truth in all states an abstraction through implicit universal. Second, many results can be understood more readily in the abstract setting. The algebraic semantics has led us to a manysorted generalization of boolean algebras with operators, for which we prove the analogue of the jonssontarski theorem. In logic, a many valued logic also multior multiplevalued logic is a propositional calculus in which there are more than two truth values. Chapter 1 modal logics of space institute for logic. A new textbook on modal logic providing a modern introduction to modal logic. Logic from the greek logos, which has a variety of meanings including word, thought, idea, argument, account, reason or principle is the study of reasoning, or the study of the principles and criteria of valid inference and demonstration. The following is a list of some texts this is not a complete list, but a pointer to books that i have found particularly useful.
We propose a general system that combines the powerful features of modal logic and manysorted reasoning. A view of its evolution 5 was a variable neither always true nor always false. Notes on modal logic notes for philosophy 151 eric pacuit january 28, 2009 these short notes are intended to supplement the lectures and text ntroduce some of the basic concepts of modal logic. This logic is axiomatized and some interesting fragments of it are investigated. Kuhn,manysorted modal logics, filosofiska studier 29, uppsala 1977. There are many sentences that you cant express in classic logic that can be expressed in modal logic. We present a new labeled tableau calculus to check the unsatis. Manyvalued modal logics ii 3 values in a manyvalued logic at possible worlds, but otherwise the general structure of a kripke frame was not altered. Building on and extending previous work on modal logic for coalgebras 28, 29, 35,64, a multisorted modal logic for use with process types is added to cocasl as syntactic sugar for formulae over. The second family generalizes this to allow the accessibility relation between worlds also to be manyvalued.
College park with a study guide that will complement the lectures on modal logic. Lewis 1883 1964 and then saul kripke 1940 who established system k, the form of modal logic that most scholars use today. In this setting we will see lindenbaum mbao models as initial objects, and canonical coalgebraic models of maximally consistent sets of formulas as nal objects. An introduction to modal logic geared towards philosophy students.
We believe that our approach to program verification improves over the existing approaches within modal logic. There are many textbooks that you can consult for more information. Many valued modal logics ii 3 values in a many valued logic at possible worlds, but otherwise the general structure of a kripke frame was not altered. Each logic is introduced with a brief description of some aspect of its philosophical significance, and wherever possible semantic and. Ioana leustean, natalia moanga, traian florin serbanuta submitted on 26 mar 2018 v1, last revised 30 nov 2018 this version, v3.
The structure of the endofunctors considered is then exploited in order to define ways of moving from coalgebras of one endofunctor to coalgebras of another, and to equip them with translations between the associated modal languages. The choice of logical connectives depends on the development of propositional logic one wants to follow. Existential rigidity and many modalities in ordersorted logic. The present paper attempts to extend the results of l, in the domain of the. Prominent modal logics are constructed from a weak logic called k after saul kripke. Modal logic is, strictly speaking, the study of the deductive behavior of the expressions it is necessary that and it is possible that. Each logic is introduced with a brief description of some aspect of its philosophical significance, and wherever possible. The metatheory for manysorted logic is closely related to the metatheory for classical quantificational logic.
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. General structures and frames built on pmstructures. In this study, we extend order sorted logic by introducing existential rigidity and many modalities. Such logics, in fact, have a long history, 14, 12, 5, 7, 6, 8. In this chapter we recollect some basic facts concerning modal logic, concentrating on completeness theory. This item appears in the following collections academic publications 176654 academic output radboud university. The other family considered in 3 allowed the accessibility relation itself to be manyvalued. Logical options introduces the extensions and alternatives to classical logic which are most discussed in the philosophical literature.
Operational semantics and program verification using many. 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. Technical report 7807, department of mathematics, university of amsterdam, 1978. Makinson, a warning about the choice of primitive operators in modal logic, journal of philosophical logic 2 1973, pp.
Purchase handbook of modal logic, volume 3 1st edition. Apr 15, 2002 this paper gives a semantical underpinning for a many sorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in objectoriented languages. In this study, we extend ordersorted logic by introducing existential rigidity and many modalities. Chapter 1 topology and epistemic logic rohit parikh department of computer science, brooklyn college, and departments of computer sci. I get burned if i lie in the sun for too long, but you cant express the possibility of getting burned. Traditionally, in aristotles logical calculus, there were only two possible values i. About the open logic project the open logic text is an opensource, collaborative textbook of formal metalogic and formal methods, starting at an intermediate level i. Its algebraic semantics leads to a manysorted generalization of boolean algebras with operators, for which we prove the analogue of the jonssontarski theorem. Modular algorithms for heterogeneous modal logics 3 sults make the multisorted nature of heterogeneous logics explicit by considering multisorted models. This paper presents a manysorted polyadic modal logic that generalizes some of the existing approaches.
Buehler based on first order modal logic by fitting and mendelsohn january 5, 2015. In this paper we show that classical quantifiers can, in fact, be regarded as s5like operators in a purely propositional modal logic. 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. In the extended logic, sorted modal formulas are interpreted over the cartesian product of sets of possible worlds.
Open access publications 51688 freely accessible full text publications. Fundamenta informaticae volume 173, issue 23 journals. The structure of the endofunctors considered is then exploited in order to define ways of moving from coalgebras of one endofunctor to coalgebras of another, and to equip. Modal logic wasborn in the earlypart ofthe 20th century as a branchof logic applied to the analysis of philosophical notions and issues. A modala word that expresses a modalityqualifies a statement. Notes on modal logic notes for phil370 eric pacuit. A modal is an expression like necessarily or possibly that is used to qualify the truth of a judgement. Recall that modal logics tend to be much easier than. The chellas text in uenced me the most, though the order of presentation is inspired more by goldblatt. While the transition from the monosorted logic to manysorted one is a smooth process, we see our. This logic is here generalised to endofunctors on categories of sorted sets. Hol with these semantics is equivalent to many sorted firstorder logic, rather than being stronger than firstorder logic.
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