Higher order modal logic pdf

We take internal adjoints between certain internal frames within a topos, which provides semantics for intuitionistic higher oder modal logic. Tilburg university higher order modal logic muskens, r. Naturally the tableau rules are not complete, but they are with respect to a henkinization of the \true semantics. Intensional and higherorder modal logic with applications to. After defining the syntax and possible worlds semantics of some higherorder modal logics, we show that they can be embedded into classical higherorder logic by systematically lifting the types of propositions, making them depend on a new atomic type for possible.

Though aimed at a nonmathematical audience in particular, students of philosophy and computer science, it is rigorous. First order modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like freges morning starevening star puzzle arise. Northholland mathematics studies intensional and higher. In mathematics and logic, a higher order logic is a form of predicate logic that is distinguished from first order logic by additional quantifiers and, sometimes, stronger semantics. The formalization is thus essentially done in classical higherorder logic where quanti. From firstorder to higherorder modal logic oxford scholarship. Bressans logic is not only modal but also higher order, as it essentially replaces set theory and concepts such as natural number and real number should therefore be definable within the logic for example, the natural number n is defined as the property of having n elements. Borrow ebooks, audiobooks, and videos from thousands of public libraries worldwide. Higherorder modal logic introduction ps pdf author. Higher order logic in relation to computing and programming. The text then examines higher order modal logic and algebraic semantics. Modal linear logic in higher order logic an experiment. Lloyd computer sciences laboratory research school of information sciences and engineering australian national university abstract this paper introduces a modal higher order logic for representing belief states of agents. If we are to think of an intension as designating different things under different circumstances, we need things.

This is the logic that the current paper investigates. Pdf a relational modal logic for higherorder stateful. A logic is called higher order if it allows for quantification and possibly ab straction over higher order objects, such as functions of individuals, relations. Lloyd computer sciences laboratory research school of information sciences and engineering australian national university abstract this paper introduces a modal higherorder logic for representing belief states of agents. With applications to montague semantics focuses on an approach to the problem of providing a precise account of natural language syntax and semantics, including the settheoretic semantical methods, boolean models, and twosorted type theory. Firstorder modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like freges morning starevening star puzzle arise. A relational modal logic for higher order stateful adts. A logic is called higher order if it allows for quantification and possibly ab straction over. In this paper we present a sketch of just such a higher order modal logic. Secondorder logic permits quantification into predicate or sentence position too. Lloyd computer sciences laboratory college of engineering and computer science the australian national university august 23, 2007 abstract this paper studies knowledge representation and reasoning in a polymorphicallytyped, multimodal, probabilistic, higherorder logic. These are the lecture notes of a tutorial on higher order modal logics held at the 11th reasoning web summer school. However, modal higherorder logic has been studied as well.

Then, in section 3, richard montagues system of intensional logic, by far the most in. Now let us further observe that, since f is a topos, it in fact has enough structure to also interpret higherorder logic, and so a geometric morphism f. A relational modal logic for higherorder stateful adts. Firstorder logic permits quantification into name position. This combination results in higher order modal logic, the subject of this chapter. Homl extends higherorder classical logic with a multimodal logic using a countable set of parameterized dual pairs of abstract modal boxdiamond operators, where the. The modal calculus is found in hfl as formulas only using the base type, and consequently they denote predicates over the states of a transition system.

Higher order modal logic is not a large field, but it is a significant one, and over the years an impressive body of work has explored it in interestingly different directions for a useful survey. Embedding of quantified higherorder nominal modal logic. They have incorporated all the new developments that have taken place since 1968 in both modal propositional logic and modal predicate logic, without sacrificing tha clarity of. Pdf these are the lecture notes of a tutorial on higherorder modal logics held at the 11th reasoning web summer school. General topos semantics for higherorder modal logic. It can be ordered now for delivery when back in stock. Intensional and higherorder modal logic mathematics nonfiction. Intensional logic stanford encyclopedia of philosophy. Thf stands for typed higherorder form and refers to a family of syntax formats for higherorder logic hol.

In spite of the philosophical significance of higherorder modal logic, the modal logicians main concern has been with sentential logic. E, but rather by a suitable complete heyting algebra h. Abstract in this paper, we present an embedding of higherorder nominal modal logic into classical higherorder logic, and study its automation. But this machinery can also be thought of as part of a move to a full higher order modal logic. At the propositional level truth values play the role of things, but at the first order level something more is needed. In this thesis a tool which converts a higherorder modal problem into a higherorder problem by applying this method is. We use cookies for statistical and other functions to give you a superfast browsing experience. Capturing bisimulationinvariant complexity classes with.

Pdf modal logic as higherorder logic mircea dumitru. Hauptsatz for higherorder modal logic the journal of. The logic differs from non modal higher order logic in that the principles of functional and propositional extensionality are no longer valid but may be replaced by modalized versions. Jul 28, 2014 topostheoretic semantics for modal logic usually uses structures induced by a surjective geometric morphism between toposes. The introduction of predicate abstraction machinery provides a natural extension in which such difficulties can be addressed.

The syntax and semantics of the logic and a tableau system for proving theo. Download book a new introduction to modal logic in pdf format. Dec 14, 2015 abstract in this paper, we present an embedding of higher order nominal modal logic into classical higher order logic, and study its automation. The growth of higherorder modal logic is traced, starting with lewis and langfords quantification into sentence position in propositional modal logic, and on to the higherorder modal logics of barcan marcus, carnap, montague, gallin, and others. Higherorder modal logic is not a large field, but it is a significant one, and over the years an impressive body of work has explored it in interestingly different directions for a useful survey.

Homl extends higherorder classical logic with a multimodal logic. Hence the method enables automatic reasoning in the desired logic. This talk develops an algebraic generalization of this framework. Discussions focus on cohens independence results, topological models of mlp, modal independence results, boolean models of mlp, relative strength of intensional logic and mlp, propositional operators, modal predicate logic, and propositions in mlp. In this paper we present a sketch of just such a higherorder modal logic. Introduction this tutorial is about narrow picture i higherorder modal logic homl i classical higherorder logic hol i embedding of homl in hol i mechanisation and automation with hol atps i various applications, including metaphysics. Pdl and modal calculus introduction ps pdf authors. Intensional and higherorder modal logic with applications.

Higherorder fixpoint logic hfl 1 is a modal logic obtained by combining the modal calculus 2 and the simply typed calculus. Except for standard material concerning propositional modal logics, the paper is essentially selfcontained. There exists no automated theorem prover for firstorder or higherorder nominal logic at the moment, hence, this is the first automation for this kind of logic. Knowledge representation and reasoning in modal higherorder logic j.

Intensional and higherorder modal logic by daniel gallin. In this paper we do not intend to go into philosophical details, but we only remark that higherorder modal logic has a close relationship with montagues wellknown idea of universal grammar, which is an ambitious attempt to build a logical theory of. You can read online a new introduction to modal logic here in pdf, epub, mobi or docx formats. We define the notion of a model of higherorder modal logic in an arbitrary elementary topos e. The encoding has been done using twolevel meta reasoning in coq. Introduction a little way into the twentieth century, both bertrand russell and ernst. Higher order logics with their standard semantics are more expressive, but their modeltheoretic properties are less wellbehaved than those of first order logic the term higher order logic, abbreviated as hol. In this thesis a tool which converts a higher order modal problem into a higher order problem by applying this method is. Homl extends higherorder classical logic with a multimodal logic using a countable set of. Knowledge representation and reasoning in modal higher. Intensional and higherorder modal logic 1st edition.

Naturally the tableau rules are not complete, but they are with respect to a henkinization of the true semantics. Pdf a relational modal logic for higherorder stateful adts. In the next section we will look at possible motivations behind the idea of combining modality and higher order logic. In contrast to the wellknown interpretation of nonmodal higherorder logic, the type of propositions is not interpreted by the subobject classifier. Higherorder logic takes the generalization even further. In our ongoing computerassisted study of godels proof we have obtained the following results. There exists no automated theorem prover for first order or higher order nominal logic at the moment, hence, this is the first automation for this kind of logic. Converting higherorder modal logic problems into classical.

An appropriate notion of semantics for homl is obtained by adapting henkin semantics for hol cf. Knowledge representation and reasoning in modal higherorder. Firstorder modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like freges morning. Embedding of quantified higherorder nominal modal logic into. Download pdf a new introduction to modal logic free. These are the lecture notes of a tutorial on higherorder modal logics held at the 11th reasoning web summer school. But this machinery can also be thought of as part of a move to a full higherorder modal logic. Lloyd computer sciences laboratory college of engineering and computer science the australian national university august 23, 2007 abstract this paper studies knowledge representation and reasoning in a polymorphicallytyped, multi modal, probabilistic, higher order logic. The sequent calculus of classical modal linear logic kdt4 lin is coded in the higher order logic using the proof assistant coq. Higherorder logic 243 for their own sake, and countable models of set theory are at the base of the independence proofs. The chapter discusses richard montagues system of intensional logic, which is by far the most influential of higher order modal logics to date.

Higherorder modal logic is proposed as a new setting in which to resolve such metaphysical questions scientifically, by the construction of systematic logical theories embodying rival answers and their comparison by normal scientific standards, more specifically by abduction inference to. During checkout, you can choose an additional, free ebook. Translating higherorder modal logic from ruleml to tptp ceur. After defining the syntax and possible worlds semantics of some higher order modal logics, we show that they can be embedded into classical higher order logic by systematically lifting the types of propositions, making them depend on a new atomic type for possible. Higherorder and modal logic as a framework for explanation. 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.

Knowledge representation and reasoning in modal higher order logic j. The logic differs from nonmodal higherorder logic in that the principles of functional and propositional extensionality are no longer valid but may be replaced by modalized versions. Modal logic is, strictly speaking, the study of the deductive behavior of the. In classical logic each model has a domain, the things of that model, and quantifiers are understood as ranging. The text then examines higherorder modal logic and algebraic semantics. 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. Topos semantics for higherorder modal logic request pdf.

The growth of higherorder modal logic is traced, starting with lewis and langfords quantification into sentence position in propositional modal logic, and on to the higherorder modal logics. Steve awodey, kohei kishida, hanschristoph kotzsch download pdf. Certain tasks, such as formal program development and theorem proving, are in. Higher order and modal logic as a framework for explanationbased generalization scott dietzen frank pfenning cmucs89160 october 16, 1989 school of computer science carnegie mellon university pittsburgh, pa 1523890 abstract. We take internal adjoints between certain internal frames within a topos, which provides semantics for intuitionistic higheroder modal logic. Translating higherorder modal logic from ruleml to tptp. The term higherorder logic is assumed in some context to refer to classical higherorder logic. A natural way to extend the modal calculus to include higher order functions is to add the operations of. Topostheoretic semantics for modal logic usually uses structures induced by a surjective geometric morphism between toposes. Automation and applications christoph benzmuller1 and bruno woltzenlogelpaleo 1supported by dfg heisenberg fellowship be 2501912 c. Furthermore, thedevelopmentofanimproved syntactical hiding for the utilized logic embedding technique allows the refutation to be presented in a humanfriendly way, suitable for nonexperts in the technicalities of higherorder theorem proving.