Higherorder modal logic introduction ps pdf author. Intensional and higherorder modal logic by daniel gallin. The formalization is thus essentially done in classical higherorder logic where quanti. Homl extends higherorder classical logic with a multimodal logic using a countable set of parameterized dual pairs of abstract modal boxdiamond operators, where the. 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. Certain tasks, such as formal program development and theorem proving, are in. Homl extends higherorder classical logic with a multimodal logic using a countable set of. 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. Thf stands for typed higherorder form and refers to a family of syntax formats for higherorder logic hol. But this machinery can also be thought of as part of a move to a full higher order modal logic.
Secondorder logic permits quantification into predicate or sentence position too. Higherorder fixpoint logic hfl 1 is a modal logic obtained by combining the modal calculus 2 and the simply typed calculus. This combination results in higher order modal logic, the subject of this chapter. From firstorder to higherorder modal logic oxford scholarship. Embedding of quantified higherorder nominal modal logic into. Intensional and higherorder modal logic with applications. Higherorder logic 243 for their own sake, and countable models of set theory are at the base of the independence proofs. 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. But this machinery can also be thought of as part of a move to a full higherorder modal logic. Pdf these are the lecture notes of a tutorial on higherorder modal logics held at the 11th reasoning web summer school.
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. Jul 28, 2014 topostheoretic semantics for modal logic usually uses structures induced by a surjective geometric morphism between toposes. General topos semantics for higherorder modal logic. 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. 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.
If we are to think of an intension as designating different things under different circumstances, we need things. Homl extends higherorder classical logic with a multimodal logic. At the propositional level truth values play the role of things, but at the first order level something more is needed. 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. In contrast to the wellknown interpretation of nonmodal higherorder logic, the type of propositions is not interpreted by the subobject classifier. Capturing bisimulationinvariant complexity classes with. Knowledge representation and reasoning in modal higher. Introduction a little way into the twentieth century, both bertrand russell and ernst. In this paper we present a sketch of just such a higher order modal logic. 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.
Pdl and modal calculus introduction ps pdf authors. These are the lecture notes of a tutorial on higherorder modal logics held at the 11th reasoning web summer school. In this paper we present a sketch of just such a higherorder modal logic. Northholland mathematics studies intensional and higher. An appropriate notion of semantics for homl is obtained by adapting henkin semantics for hol cf. In this thesis a tool which converts a higher order modal problem into a higher order problem by applying this method is. The sequent calculus of classical modal linear logic kdt4 lin is coded in the higher order logic using the proof assistant coq. Though aimed at a nonmathematical audience in particular, students of philosophy and computer science, it is rigorous. 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. Then, in section 3, richard montagues system of intensional logic, by far the most in. We take internal adjoints between certain internal frames within a topos, which provides semantics for intuitionistic higher oder modal logic.
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. Pdf modal logic as higherorder logic mircea dumitru. Higherorder logic takes the generalization even further. Abstract in this paper, we present an embedding of higherorder nominal modal logic into classical higherorder logic, and study its automation. The chapter discusses richard montagues system of intensional logic, which is by far the most influential of higher order modal logics to date. 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. Topos semantics for higherorder modal logic request pdf. The text then examines higher order modal logic and algebraic semantics. Translating higherorder modal logic from ruleml to tptp. In our ongoing computerassisted study of godels proof we have obtained the following results. You can read online a new introduction to modal logic here in pdf, epub, mobi or docx formats. It can be ordered now for delivery when back in stock.
Intensional and higherorder modal logic mathematics nonfiction. In spite of the philosophical significance of higherorder modal logic, the modal logicians main concern has been with sentential logic. Tilburg university higher order modal logic muskens, r. The introduction of predicate abstraction machinery provides a natural extension in which such difficulties can be addressed. Naturally the tableau rules are not complete, but they are with respect to a henkinization of the true semantics. Converting higherorder modal logic problems into classical. Automation and applications christoph benzmuller1 and bruno woltzenlogelpaleo 1supported by dfg heisenberg fellowship be 2501912 c. In this thesis a tool which converts a higherorder modal problem into a higherorder problem by applying this method is. 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.
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. This is the logic that the current paper investigates. Translating higherorder modal logic from ruleml to tptp ceur. The encoding has been done using twolevel meta reasoning in coq. 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. 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 term higherorder logic is assumed in some context to refer to classical higherorder logic.
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. 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. 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. Modal logic is, strictly speaking, the study of the deductive behavior of the. Intensional and higherorder modal logic 1st edition. Knowledge representation and reasoning in modal higherorder. Higher order logic in relation to computing and programming. 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. In classical logic each model has a domain, the things of that model, and quantifiers are understood as ranging. Firstorder modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like freges morning starevening star puzzle arise.
Knowledge representation and reasoning in modal higher order logic j. 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. Embedding of quantified higherorder nominal modal logic. These are the lecture notes of a tutorial on higher order modal logics held at the 11th reasoning web summer school. Firstorder logic permits quantification into name position. We take internal adjoints between certain internal frames within a topos, which provides semantics for intuitionistic higheroder modal logic. We use cookies for statistical and other functions to give you a superfast browsing experience. 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. 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. A relational modal logic for higherorder stateful adts. 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. 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. Steve awodey, kohei kishida, hanschristoph kotzsch download pdf.
We define the notion of a model of higherorder modal logic in an arbitrary elementary topos e. This talk develops an algebraic generalization of this framework. Download book a new introduction to modal logic in pdf format. Higherorder and modal logic as a framework for explanation. 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. Intensional logic stanford encyclopedia of philosophy. Topostheoretic semantics for modal logic usually uses structures induced by a surjective geometric morphism between toposes. In the next section we will look at possible motivations behind the idea of combining modality and higher order logic. According to several logicians, godels ontological proof is best studied from a technical perspective in such a context.
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. 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. However, modal higherorder logic has been studied as well. Hauptsatz for higherorder modal logic the journal of. 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.
Knowledge representation and reasoning in modal higherorder logic j. Modal linear logic in higher order logic an experiment. Except for standard material concerning propositional modal logics, the paper is essentially selfcontained. 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.
The syntax and semantics of the logic and a tableau system for proving theo. Download pdf a new introduction to modal logic free. 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. Hence the method enables automatic reasoning in the desired logic. Firstorder modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like freges morning.
Intensional and higherorder modal logic with applications to. Pdf a relational modal logic for higherorder stateful. The text then examines higherorder modal logic and algebraic semantics. Borrow ebooks, audiobooks, and videos from thousands of public libraries worldwide. Naturally the tableau rules are not complete, but they are with respect to a henkinization of the \true semantics. A natural way to extend the modal calculus to include higher order functions is to add the operations of. A relational modal logic for higher order stateful adts. E, but rather by a suitable complete heyting algebra h. First order modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like freges morning starevening star puzzle arise. A logic is called higher order if it allows for quantification and possibly ab straction over. Pdf a relational modal logic for higherorder stateful adts.