Precisely what axioms and rules must be added to the propositional calculus to create a usable system of modal logic is a matter of philosophical opinion, often driven by the theorems one wishes to prove. Consequently formulae are given by the grammar a p j. 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. Modal operators 323 nontruthfunctionality 323 modal and nonmodal propositions.
In contrast we are interested in primarily natural deduction and sequent calculus formu lations, and their metatheoretic properties. Perhaps, the most influential is kripkes based on his theory of rigid designation, which is a revival of aristotelian essentialism some properties of objects are essential, and pick out the same objects in all possible worlds, others are not. A view of its evolution 5 was a variable neither always true nor always false. Thm joubkal oj symbolic loglc volume 12, number 2, june 1947 the problem of interpreting modal logic w. Given any formula, it is straigtforward to make a truth table and determine whether the formula is a. In these principles we use a and b as metavariables ranging over formulas of the language. The distribution axiom says that if it is necessary that if a then b, then if necessarily a. So the acceptability of axioms for modal logic depends on which of these. The first two are straightforward and are left as an exercise tutorial sheet.
As i mentioned in class, there have been a great many modal logics. Now in this, our last chapter, we concentrate our attention on the kind of propositional logic modal propositional logic within which modal concepts feature overtly. What are the objections to the axioms of modal logic. These systems have both intuitionistic and modal aspects. Modal logic axioms valid in quotient spaces of finite cw. A modal is an expression like necessarily or possibly that is used to qualify the truth of a judgement. Systems of modal logic department of computing imperial. Every effort has been made to simplify the presentation by using diagrams in place of more complex mathematical apparatus. Theexplanation of modal logic thus afforded is adequate so long as modalities are not used inside thescopes of quan tifiers. This logic is then compared with the system in kripkes semantical considerations on modal logic. We will define several varieties of modal logic, providing both their semantics and their axiomatic proof systems, and prove their standard soundness and.
Since k4, kt and s4 are decidable logics, an immediate question is. Most work on intuitionistic modal logics simply gives the logic using an axiomatic formulation, the primary interest being provability. A modal epistemic logic for agents is obtained by joining together modal logics, one for each agent. The language of the classical logic is simple, straightforward, and easy to work with.
Modal logic is, strictly speaking, the study of the deductive behavior of the. Canonical kripke models play a role similar to the lindenbaumtarski algebra construction in algebraic semantics. It was first conceived for modal logics, and later adapted to intuitionistic logic and other nonclassical systems. Feb 04, 2016 a summary of all of the axioms that we have investigated in regards to all of the different kinds of modal logics that we have looked at. Independence of the dual axiom in modal k with primitive. How is kripkestyle modal logic distinct from classical. Research article modal logic axioms valid in quotient. Examples for convenience, we reproduce the item logic modal logic of principia metaphysica in which the modal logic is defined. A modal a word that expresses a modalityqualifies a statement.
In their standard survey article, bull and segerberg ascribe the comparative lack of influence of his work to two factors, the unassuming mode of publication and the fact that his work is difficult to decipher. Godel 1933, where he introduces translations from intuitionistic propositional logic into modal logic more precisely, into the system nowadays called s4, and briefly mentions that provability can be viewed as a modal operator. In classical modal logic, it is wellknown that some properties of kripke frames are characterized by modal axioms. Together, these axioms partially axiomatize the modal system k, including all the usual axioms, but not the dual axiom. Stronger systems of modal logic can be obtained by adding additional axioms. For proving theorem t1, only the left to right direction of axiom a1 is needed. An equation a is necessarily valid, if it is valid for every possible assignment of values, which we would denote a in modal logic. Modal logic axioms valid in quotient spaces of nite cwcomplexes and other families of topological spaces maria nogin bing xu california state university, fresno amsmaa joint mathematics meetings atlanta, ga january 5, 2017. The new rule of inference in this system is the rule of necessitation. Modal logic for philosophers designed for use by philosophy students, this book provides an accessible yet technically sound treatment of modal logic and its philosophical applications.
Judgemental modal logic, pfenning et al, from 2001 separation logic, reynolds and ohearn modalities as monads, moggi et al, lax logic, mendler et al, from. Like other logical systems, issues of soundness and completeness of a model with respect to an axiom system, as well as various decidability questions are also studied in modal logic. Axioms are formulas that are considered to be selfevidently true, for which no proof is required. Thus, whenever we say modal logic, we always mean propositional unimodal logic, modal logics without quanti. This book is an introduction to logic for students of contemporary philosophy.
Semantical analysis of modal logic i kornal modal propositional calcull 73. If the tableau t is hi an ordered set 9, it is clear that. Pdf intuitionistic modal logic based on neighborhood. For logical aspects, adding axiom schemata is a simple and popular way to. These notes are meant to present the basic facts about modal logic and so to provide a common. All the more important systems have in fact but a single additional rule of necessitation, permitting inference from a to a. One of these was stig kanger, who presented a model theory for modal logic in his dissertation. Two strands of research have led to the birth of provability logic. Lewissare not intuitively clear until explained in nonmodalterms. Including alethic, deontic, temporal, doxastic, epistemic. W is called our universe and elements of w are called worlds r is a relation on w.
Gl is a normal modal logic like the systems k, t, s4, s5, and others, meaning that it is at least as strong as the logic k. This relationship between properties of frames and modal axioms is called correspondence. In this paper we investigate certain systems of propositional logic defined semantically in terms of neighborhood structures. A brief introduction to modal logic introduction consider. For simplicitys sake it is usually assumed that the agents are homogeneous, i. A summary of all of the axioms that we have investigated in regards to all of the different kinds of modal logics that we have looked at. There are multiple systems of axioms for modal logic.
Quine there are logicians, myself among them, to \,hom the ideas of modal logic e. I suppose that some such conception underlies the intuition where by axioms are evaluated andadoptedfor modal logic. Reductio ad absurdum tests 315 summary 320 6 modal propositional logic 323 1. Let a set of inference rules and axioms s be given. Basic concepts in modal logic1 stanford university. On the one hand we are guided by the reading of the. Modal logic is, strictly speaking, the study of the deductive behavior of the expressions it is necessary that and it is possible that. 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. Of particular interest are socalled normal systems of modal logics. In this section we give an axiomatic, or hilbertstyle, formulation of is4. The choice of logical connectives depends on the development of propositional logic one wants to follow.
An axiomatic formulation of is4 in this paper we shall only consider propositional is4. A modal logic is just a set of axioms and rules that govern the modal operators. Notice that adding axioms to system k may even result in the inconsistent logic. Basic concepts in this chapter we recollect some basic facts concerning modal logic, concentrating on completeness theory. Modal logic axioms valid in quotient spaces of finite cwcomplexes and other families of topological spaces marianoginandbingxu department of mathematics, california state university, fresno, ca, usa correspondence should be addressed t o maria nogin. This has to be distinguished from the validity of the equation for a specific assignment of values. If we also only make use of axioms as the premises of our proof, then the conclusion of the proof is just as selfevident as the axioms. Our previous work see the nonrefereed, invited paper 3 has already demonstrated the feasibility of the. In the example above, the operator satisfies the axioms of the modal logic system s5. 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. Chapter 1 modal logics of space institute for logic. This relationship between properties of frames and modal axioms.
In this tutorial, we give examples of the axioms, consider some rules of inference and in particular, the derived rule of necessitation, and then draw out some consequences. Keywords modal logic s5, proof theory, deep inference, calculus of. The logic of provability university of california, berkeley. Pdf the axiomatic translation principle for modal logic. A modal logic is normal if its axiom system consists of the axiom schema k and the rule rn. In the stream of studies on intuitionistic modal logic, we can find mainly three kinds of natural deduction systems. According to the necessitation rule, any theorem of logic is necessary. K our variablefree modal notation has no laws like the above. To some extent, modal logics also take a more finegrained. However, the term modal logic may be used more broadly for a family of. Axioms are complete for boolean propositional logic. First, i will discuss the technical work that accessibility does in unifying and providing an extensional formal semantics for the plethora of modal logics that have come into existence since aristotle. Provability logic stanford encyclopedia of philosophy.
Proceedings of the 6th workshop on intuitionistic modal logic and applications imla 20, electronic notes in theoretical computer science, volume 300, 2014 n. We discuss various restrictions imposed on our models i. Given any formula, it is straigtforward to make a truth table and determine whether the formula is a tautology, a contradiction, or neither. Though aimed at a nonmathematical audience in particular, students of philosophy and computer science, it is rigorous. 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. So an epistemic logic for agents consists of copies of a. 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. For any normal modal logic, l, a kripke model called the canonical model can be constructed that refutes precisely the nontheorems of l, by an adaptation of the standard technique of using maximal consistent sets as models. In the example above, the operator satisfies the axioms of the modal logic.
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