Barcan formulas in second-order modal logic pdf

Firstorder justification logic with constant domain semantics. These theoretical innovations may be developed in a syntactically secondorder, quanti ed s5 modal logic with both rst and secondorder barcan formulas that has two kinds of atomic formulas fnx 1x n n 0 x 1x. First order extensions of classical modal logic eric pacuit university of maryland, college park tilburg institute for logic and philosophy of science ai. Secondorder logic and modal logic are both, separately, major topics of philosophical discussion. A hypersequent approach to modal logic introduces a new framework for the proof theory of various modal logics. This dissertation develops an inferentialist theory of meaning. With the plenist constraints 1, 2 and 3, quantificational modal logics with the barcan formula and its converse are straightforwardly accounted for.

Saul krike 1959 a completeness theorem in modal logic. On modality and reference ruth barcan marcus 19212012. In first approximation, modal logic im using the term loosely can be understood as an interesting fragment of first order logic for simplicity i ignore e. Modal logic in this form aims to discover which generalizations in such terms are true. For secondorder modal logic there are both firstorder and secondorder barcan formulas. A normal logic is a set of formulas in l such that it is rst of all a normal modal logic propositional modal logic and that it contains 8p. Some modal formulas impose conditions on frames that cannot be expressed in a first order language, thus even propositional modal logic is fundamentally second order in nature. A modala word that expresses a modalityqualifies a statement.

He rejects the search for a metaphysically neutral logic as futile. Objects, properties and contingent existence to appear. As a result, secondorder logic has much more expressive power than fol does. Ruth barcan marcus 1947 the identity of individuals in a strict functional calculus of second order. The axiom b raises an important point about the interpretation of modal formulas. The logic underlying the theory of objects can now be summarized. An introduction to its syntax and semantics 9780195366570. Higherorder free logic and the priorkaplan paradox andrew bacon, john hawthorne and gabriel uzquiano april 11, 2016 1 introduction a central theme from modal logic as metaphysics is the idea that higherorder logic is a fruitful framework for formulating and assessing some of the somewhat elusive debates in the metaphysics of properties and. Contingentism about individuals and higherorder necessitism theoria 78 20. This is particularly serious, since their standard applications depend on there being sufficiently many of them. A modal is an expression like necessarily or possibly that is used to qualify the truth of a judgement.

Priors basic system of temporal logic, and discuss some of the fundamental logical questions pertaining to it. First order extensions of classical modal logic 162. In particular, there are two features of the use of a sentence that. Based on firstorder modal logic by fitting and mendelsohn.

The first order models we present permit the study of. First, we present the language of second order propositional modal logic sopml, some of its fragments, and their interpretation on kripke frames and models. Objects, properties and contingent existence to appear as. This success is due to several reasons, including an expressive and flexible formal language, which enjoys nice computational properties. Williamson forthcoming barcan formulas in second order modal logic, i. Contingentism about individuals and higherorder necessitism.

It takes as a starting point that the sense of a sentence is determined by the rules governing its use. If the barcan formula is assumed as an axiom, it implies that. Labelled proofs for quantified modal logic uq espace. Converse barcan formula cbf seem to be valid for higherorder modal quantificational logic. Higher order free logic and the priorkaplan paradox andrew bacon, john hawthorne and gabriel uzquiano april 11, 2016 1 introduction a central theme from modal logic as metaphysics is the idea that higher order logic is a fruitful framework for formulating and assessing some of the somewhat elusive debates in the metaphysics of properties and. Publications in reverse chronological order in preparation a with paul boghossian debating the apriori, volume of our published exchanges and new ones, including reply to boghossian on the distinction between the a priori and the a posteriori and reply to boghossian on intuition, understanding and the a priori. Modal logic is, strictly speaking, the study of the deductive behavior of the.

Insofar as the notion of validity on a frame abstracts from the interpretation function, it implicitly involves a higher order quantification over propositions. Firstorder extensions of classical modal logic eric pacuit university of maryland, college park. Sellars, secondorder logic, and ontological commitment clarifies sellars arguments that secondorder quantification is ontologically inocuous and offers an account of quantification that can meet the demands of sellarss arguments. We start today with a recap of the syntax and semantics of first.

Hypersequent system d a study of a hypersequent system for the modal logic d. Modal logic is, strictly speaking, the study of the deductive behavior of the expressions it is necessary that and it is possible that. We assume the usual primitive logical notions standardly represented using. The book contains detailed historical discussion of how the metaphysical issues emerged in the twentieth century development of quantified modal logic, through the work of such figures as rudolf. On modality and reference ruth barcan marcus 1921 2012 genoveva marti it is difficult to think of ruth barcan marcus without almost automatically thinking about her pioneering work in modal logic and, in particular, about the long lasting impact of the barcan formula, a formula that she in. Modal logic as metaphysics timothy williamson download.

An important part of williamsons case for necessitism is a powerful new development of this style of argument, for secondorder rather than rstorder quanti ed modal logic. Second order modal logic andrew parisi, phd university of connecticut, 2017 abstract. Unifiable formulas in some extensions of qk4 are characterized and an explicit basis for the passive rules those with nonunifiable premises is provided. However, the term modal logic may be used more broadly for a family of. Formulas are built up from atomic formulas in the usual way, using propositional connectives, modal operators, and two kinds of quantifiers. 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. A familiar observation is that virtually every putatively fundamental principle of logic has been challenged over the last century on broadly metaphysical grounds. In modal logic as metaphysics, timothy williamson argues for positive answers to those questions on the basis of an integrated approach to the issues, applying the technical resources of modal logic to provide structural cores for metaphysical theories. Barcan up the wrong tree an argument that the validity of the barcan formulas is neither necessary nor sufficient to capture the dispute between necessitists and contingentists. Both unifiability and passive rules depend on the number of. Pdf a general semantics for quantified modal logic. I then discuss modal logic and counterfactual conditionals.

Modern origins of modal logic stanford encyclopedia of. In an intended interpretation of a formula of slogic, the propositional variables are assigned to subsystems of secondorder arithmetic,jis interpreted as. In fact, there is no way of formalizing, using standard. Second order barcan formulas and transcendent universals. Williamson forthcoming barcan formulas in secondorder modal logic, i. It prepares students to read the logically sophisticated articles in todays philosophy journals, and helps them resist bullying by symbolmongerers. The smallest normal logic containing a normal modal logic l is called l. Introduction in an earlier chapter, we saw that certain sentences of english can be formalized using the actuality operator. Nevertheless, their nature and existence is very controversial. I begin with a sketch of standard propositional and predicate logic. As we shall see, under the graphbased perspective discussed here, modal logic is closely linked to both. After an undergraduate degree in mathematics and philosophy and a doctorate in philosophy, both at oxford, he was a lecturer in philosophy at trinity college dublin, a fellow and tutor at university college oxford, and professor of logic and metaphysics at the university of. 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. Introduction one of the most fertile ideas in modal logic and metaphysics traces to leibniz, who proposed the following famous account of necessity.

We also saw that, in cases in which this operator is a termmodifying adverb, the formalization. Unification in firstorder transitive modal logic logic. We introduce unification in firstorder transitive modal logics, i. For systems with the barcan formula it is possible to preserve the usual tarskian. Thus, the barcan formula expresses an interaction between r and d. However, this translation needs a unary predicate for every propositional variable. Indeed, s4 may also be shown to be the modal logic of the partial orders.

The role of possible worlds in philosophy is hard to overestimate. But modal logic is not the only tool for talking about graphs, and this brings us to one of the major themes of the chapter. The paper develops an account of possible worlds on which it is particularly easy to believe in their existence. In short, it teaches the logic necessary for being a contemporary philosopher. Pdf secondorder barcan formulas and transcendent universals. For second order modal logic there are both first order and second order barcan formulas. Intensional logic stanford encyclopedia of philosophy. The sahlqvist correspondence theorem states that every sahlqvist formula is canonical, and corresponds to a firstorder definable class of kripke frames sahlqvists definition characterizes a decidable set of modal formulas with firstorder correspondents. Probably the most common version of the barcan formula is bf1. Timothy williamson has been the wykeham professor of logic at oxford since 2000. To introduce the language of second order propositional modal logic, we fix a set ap of atomic propositions and a finite set i of indices. Similarly, secondorder logic recognizes as formally valid certain inferences that are not fovalid. Firstorder extensions of classical modal logic 1462. Barcan formulas in secondorder modal logic university of oxford.

Contrary to the widespread assumption that logic and metaphysics are disjoint, he argues that modal logic provides a structural core for metaphysics. Transitivity is needed in order for the formula in 4 to come out valid. Mar 27, 2012 the role of possible worlds in philosophy is hard to overestimate. Secondorder barcan formulas and transcendent universals. For formulas of monadic quantified modal logic we have valuationatomicity.

The true generalizations constitute a quanti ed modal logic, but we do not know ahead of enquiry. Converse barcan formula cbf seem to be valid for higher order modal quantificational logic. An introduction to its syntax and semantics amazon site. With the plenist constraints 1, 2 and 3, quantificational modal logics with the barcan formula and its. In addition to the usual formulacreating machinery, we have the following. Modal logic is nowadays a wellestablished area in mathematical logic, which has also become one of the most popular formal frameworks in artificial intelligence for knowledge representation and reasoning. In modal logic, sahlqvist formulas are a certain kind of modal formula with remarkable properties.

Secondorder modal logic andrew parisi, phd university of connecticut, 2017 abstract. Timothy williamson gives an original and provocative treatment of deep metaphysical questions about existence, contingency, and change, using the latest resources of quantified modal logic. Simplest secondorder quanti ed s5 modal logic linsky and zalta, 1994, including 1st and 2nd order barcan formulas i. A familiar observation is that virtually every putatively fundamental principle of logic has been challenged over the last century on broadly metaphysical. A functional calculus of first order based on strict implication volume 11 issue 1 ruth c. Although both have been criticized by quine and others, increasingly many philosophers find their strictures uncompelling, and regard both branches of logic as valuable resources for the articulation and investigation of significant issues in logical metaphysics and elsewhere.

With the exception of the logic of descriptions, the modal closures of all of the following are axioms. The barcan formula, introduced in barcan 1946, raises fundamental issues about the contingency or otherwise of existence, issues that arise neither in firstorder nonmodal logic nor in unquantified modal logic. Firstorder classical modal logic barcan formulas and neighborhood frames. Barcan skip to main content we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Specifically, modal logic is intended to help account for the valid. Thomason, gabbay, esakia, van benthem, blok and myself. It is known that modal logic can be interpreted in firstorder logic via standard translation. Firstorder modal logic and the barcan formula stanford university. A b s t r a c t in a series of writings timothy williamson has argued for necessitism cf. Timothy williamson, modal logic as metaphysics philpapers. Books notes on modal logic stanford university preface these notes were composed while teaching a class at stanford and studying the work of brian chellas modal. Propositional justification logics are similar to modal logics, except. The barcan formula, introduced in barcan 1946, raises fundamental issues about the contingency or otherwise of existence, issues that arise neither in first order non modal logic nor in unquantified modal logic. Logical vocabulary name instances category individual variables x, y z, etc.