Barcan formulas in second-order modal logic pdf

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. Second order logic and modal logic are both, separately, major topics of philosophical discussion. An introduction to its syntax and semantics amazon site. In short, it teaches the logic necessary for being a contemporary philosopher.

In addition to the usual formulacreating machinery, we have the following. Logical vocabulary name instances category individual variables x, y z, etc. Mar 27, 2012 the role of possible worlds in philosophy is hard to overestimate. However, the term modal logic may be used more broadly for a family of. In particular, there are two features of the use of a sentence that. Objects, properties and contingent existence to appear. Timothy williamson has been the wykeham professor of logic at oxford since 2000. Firstorder modal logic and the barcan formula stanford university. Converse barcan formula cbf seem to be valid for higherorder modal quantificational logic. On modality and reference ruth barcan marcus 19212012.

Pdf a general semantics for quantified modal logic. In modal logic, sahlqvist formulas are a certain kind of modal formula with remarkable properties. For example, there is no way in fol to say that a and b have some property in common. 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.

As we shall see, under the graphbased perspective discussed here, modal logic is closely linked to both. Introduction in an earlier chapter, we saw that certain sentences of english can be formalized using the actuality operator. Simplest secondorder quanti ed s5 modal logic linsky and zalta, 1994, including 1st and 2nd order barcan formulas i. We start today with a recap of the syntax and semantics of first. Labelled proofs for quantified modal logic uq espace. First, we present the language of second order propositional modal logic sopml, some of its fragments, and their interpretation on kripke frames and models.

I begin with a sketch of standard propositional and predicate logic. A modala word that expresses a modalityqualifies a statement. On the logics with propositional quantifiers extending s5. Second order barcan formulas and transcendent universals. Objects, properties and contingent existence to appear as.

Both unifiability and passive rules depend on the number of. 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. Simplest second order quanti ed s5 modal logic linsky and zalta, 1994, including 1st and 2nd order barcan formulas i. Ruth barcan marcus 1947 the identity of individuals in a strict functional calculus of second order. As a result, secondorder logic has much more expressive power than fol does. Firstorder extensions of classical modal logic eric pacuit university of maryland, college park.

The system provides a uniform prooftheoretical treatment of firstorder. It is known that modal logic can be interpreted in firstorder logic via standard translation. Transitivity is needed in order for the formula in 4 to come out valid. Modal logic in this form aims to discover which generalizations in such terms are true. First order extensions of classical modal logic eric pacuit university of maryland, college park tilburg institute for logic and philosophy of science ai. Modal logic is, strictly speaking, the study of the deductive behavior of the expressions it is necessary that and it is possible that. 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. Indeed, s4 may also be shown to be the modal logic of the partial orders.

A functional calculus of first order based on strict implication volume 11 issue 1 ruth c. Introduction one of the most fertile ideas in modal logic and metaphysics traces to leibniz, who proposed the following famous account of necessity. Thomason, gabbay, esakia, van benthem, blok and myself. Based on firstorder modal logic by fitting and mendelsohn. We assume the usual primitive logical notions standardly represented using. Second order modal logic andrew parisi, phd university of connecticut, 2017 abstract. 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. Modal logic is, strictly speaking, the study of the deductive behavior of the. A familiar observation is that virtually every putatively fundamental principle of logic has been challenged over the last century on broadly metaphysical. If the barcan formula is assumed as an axiom, it implies that. Secondorder modal logic andrew parisi, phd university of connecticut, 2017 abstract. Barcan later ruth barcan marcus developed the first axiomatic systems of quantified modal logic first and second order extensions of lewis s2, s4, and s5. Formulas are built up from atomic formulas in the usual way, using propositional connectives, modal operators, and two kinds of quantifiers. Intensional logic stanford encyclopedia of philosophy.

This is particularly serious, since their standard applications depend on there being sufficiently many of them. The smallest normal logic containing a normal modal logic l is called l. 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. Secondorder logic and modal logic are both, separately, major topics of philosophical discussion. In fact, there is no way of formalizing, using standard. In an intended interpretation of a formula of slogic, the propositional variables are assigned to subsystems of secondorder arithmetic,jis interpreted as.

Specifically, modal logic is intended to help account for the valid. 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. For second order modal logic there are both first order and second order barcan formulas. Secondorder barcan formulas and transcendent universals. Unifiable formulas in some extensions of qk4 are characterized and an explicit basis for the passive rules those with nonunifiable premises is provided.

Pdf secondorder barcan formulas and transcendent universals. An introduction to its syntax and semantics 9780195366570. However, this translation needs a unary predicate for every propositional variable. Justification logic, firstorder logic of proofs, completeness, epistemic logic. 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.

Timothy williamson, modal logic as metaphysics philpapers. 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. 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. 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. 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.

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. A b s t r a c t in a series of writings timothy williamson has argued for necessitism cf. It prepares students to read the logically sophisticated articles in todays philosophy journals, and helps them resist bullying by symbolmongerers. The true generalizations constitute a quanti ed modal logic, but we do not know ahead of enquiry. 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. A hypersequent approach to modal logic introduces a new framework for the proof theory of various modal logics.

Barcan formulas in secondorder modal logic university of oxford. 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. 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. The role of possible worlds in philosophy is hard to overestimate. First order extensions of classical modal logic 162.

With the plenist constraints 1, 2 and 3, quantificational modal logics with the barcan formula and its converse are straightforwardly accounted for. Contingentism about individuals and higherorder necessitism theoria 78 20. The paper develops an account of possible worlds on which it is particularly easy to believe in their existence. We also saw that, in cases in which this operator is a termmodifying adverb, the formalization. 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. For secondorder modal logic there are both firstorder and secondorder barcan formulas.

A familiar observation is that virtually every putatively fundamental principle of logic has been challenged over the last century on broadly metaphysical grounds. 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. 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. 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. Converse barcan formula cbf seem to be valid for higher order modal quantificational logic. The axiom b raises an important point about the interpretation of modal formulas. I then discuss modal logic and counterfactual conditionals. Similarly, secondorder logic recognizes as formally valid certain inferences that are not fovalid. Insofar as the notion of validity on a frame abstracts from the interpretation function, it implicitly involves a higher order quantification over propositions. For formulas of monadic quantified modal logic we have valuationatomicity. The first order models we present permit the study of. Hypersequent system d a study of a hypersequent system for the modal logic d.

Williamson forthcoming barcan formulas in secondorder modal logic, i. Nevertheless, their nature and existence is very controversial. The logic underlying the theory of objects can now be summarized. Contrary to the widespread assumption that logic and metaphysics are disjoint, he argues that modal logic provides a structural core for metaphysics.

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. Propositional justification logics are similar to modal logics, except. 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. Unification in firstorder transitive modal logic logic.

Probably the most common version of the barcan formula is bf1. 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. 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. Priors basic system of temporal logic, and discuss some of the fundamental logical questions pertaining to it.

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. We introduce unification in firstorder transitive modal logics, i. He rejects the search for a metaphysically neutral logic as futile.

A modal is an expression like necessarily or possibly that is used to qualify the truth of a judgement. Firstorder extensions of classical modal logic 1462. This dissertation develops an inferentialist theory of meaning. Saul krike 1959 a completeness theorem in modal logic. 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. Thus, the barcan formula expresses an interaction between r and d. With the plenist constraints 1, 2 and 3, quantificational modal logics with the barcan formula and its. This success is due to several reasons, including an expressive and flexible formal language, which enjoys nice computational properties. Firstorder justification logic with constant domain semantics.