Dekker and others published possible worlds, belief, and modal logic. For simplicitys sake it is usually assumed that the agents are homogeneous, i. 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. An overview of applications of modal logic in linguistics can be found in. Researchers in areas ranging from economics to computational linguistics have since realised its worth. Modal logic, philosophy society and department of philosophy, university of uppsala, vol. Before moving on, i give a few definitions that will be important as we move on, and list a few well known modal logics together with their axioms. Knowing all this, i have implemented a solver for propositional modal logic s4 and it also terminates with a finite model. The meaning of set forth in this section is that of epistemic modal. There were, and still are, some recurring complaints. A modal is an expression like necessarily or possibly that is used to qualify the truth of a judgement.
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. This is an advanced 2001 textbook on modal logic, a field which caught the attention of computer scientists in the late 1970s. Axiom t will hold in all pws models just in case the relation r is re. It is easy to prove modal modus ponens, given axiom 1 of modal logic. Chapter 1 modal logics of space institute for logic. A modal epistemic logic for agents is obtained by joining together modal logics, one for each agent. So an epistemic logic for agents consists of copies of a. Axiomatic theory modal logic kt axioms necessity axiom, called t p p axiom k p q p q 16 40 sg models and formalisms 33. Thm joubkal oj symbolic loglc volume 12, number 2, june 1947 the problem of interpreting modal logic w. 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 exposition and approachability that were essential features of their. A explanation of the basics of modal logic, including the difference between the k, t, b, s4 and s5 systems of modal logic 100 days of. Modal logic is, strictly speaking, the study of the deductive behavior of the expressions it is necessary that and it is possible that.
A new introduction to modal logic is an entirely new work, completely rewritten by the authors. We assume that we possess a denumerably infinite list. 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. On intuitionistic modal and tense logics and their. Is there any further reduction possible by proving the distribution axiom. This algebraic perspective dominated the semantical approach to modal logic until the publication of the groundbreaking works of kripke 30, 31. It is a normal modal logic, and one of the oldest systems of modal logic of any kind. Pdf on sep 1, 2017, antonis achilleos and others published the completeness problem for modal logic find, read and cite all the research you need on researchgate. 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. The first two are straightforward and are left as an exercise tutorial sheet. Most modal logic books accept this as an axiom in addition to others, but i want the fewest axioms possible as i have already reduced them to the distribution axiom, modal semantics and definitions, and some contingent proposition is true. So the acceptability of axioms for modal logic depends on which of these.
Axioms are formulas that are considered to be selfevidently true, for which no proof is required. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. I was sorting through a pile of notes dating from the time when i was engaged as a professional logician 4. Given any formula, it is straigtforward to make a truth table and determine whether the formula is a tautology, a contradiction, or neither. 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. In our problem encodings we exploit the wellknown correspondences between these. Part i james pustejovsky september 27, 2004 1 modal logic so far in this class, we have studied propositional and. Algebraic t o ols for mo dal logic mai gehrke y yde venema general aim there is a long and strong tradition in logic researc h of applying algebraic tec hniques in order to deep en our understanding of logic. Axiomatic theory modal logic kt axioms necessity axiom, called t p p 16 40 sg models and formalisms 32. A modala word that expresses a modalityqualifies a statement. The logic of provability university of california, berkeley. The functions we assume to valuate the truth of a proposition in either system have involved assignment within a model, to a value of either true or false.
S4 axiom is a class of transitive and reflexive frames. Gunther propositional logic our language semantics syntax results modal logic our language semantics. This is called the law of the excluded middle the statement p q is not a tautology. I will not explicitly state that this set is in the axiomset for any logic up for study, but merely give the modal axioms. All the more important systems have in fact but a single additional rule of necessitation, permitting inference from a to a. In logic and philosophy, s5 is one of five systems of modal logic proposed by clarence irving lewis and cooper harold langford in their 1932 book symbolic logic. The logic s4, for instance, is a classical modal logic with the axiomsk,t,4and rulesg, modus ponens and all propositional tautologies. Extending previous answers by chaosandorder and dennis you seem to appreciate why pure logic i take it that you mean classical first order logic is useful in the context of mathematical logic, but you dont see the point in formalizing other modal notions in ordinary language.
A modal a word that expresses a modalityqualifies a statement. Or, intuitively, if all possible worlds are accessible from themselves. Basic concepts in modal logic1 stanford university. His remarks capture the core point of the argument, as well as the open question as to which axiom is at fault for the failure.
How does modal logic differ from ordinary predicate logic. Most accessibility relations in applications of modal logic are re. Propositional modal logic modal logic is the logic of necessity, possibility and other related notions. Coalgebra and modal logic problems, problems i started talking about coalgebra as a successor to work i had been doing with jon barwise on nonwellfounded sets. On the role of modal intuition in modal logic 171 walters articulates in words what is presented in the possible worlds diagram above.
These notes are meant to present the basic facts about modal logic and so to provide a common. Notice that adding axioms to system k may even result in the inconsistent logic. The journal of symbolic logic volume 24, number 1, march 1959 a completeness theorem in modal logic saul a. 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 a the accessibility relation is reflexive and symmetric. Philosophy stack exchange is a question and answer site for those interested in the study of the fundamental nature of knowledge, reality, and existence. Algebraic tools for modal logic mai gehrke yde venema esslli01 august 17, 2001 helsinki, finland. Mathematical logic or symbolic logic is the study of logic and foundations of mathematics as, or via, formal systems theories such as firstorder logic or type theory. However, the term modal logic may be used more broadly for a family of. There are several ways to formalise a logic as a mathematical object. October 5, 2004 1 introduction this short tutorial had its genesis in a coincidence of two events. Pdf a new introduction to modal logic download full. Kripke the present paper attempts to state and prove a completeness theorem for the system s5 of 1, supplemented by firstorder quantifiers and the sign of equality. Independence of the dual axiom in modal k with primitive. Prominent modal logics are constructed from a weak logic called k after saul kripke.
Of particular interest are socalled normal systems of modal logics. Some of the high points are temporality the possible world semantics as given by stig kanger and saul kripke connects formal systems for modal logic and geometrical assumptions about the temporal relation. It is formed with propositional calculus formulas and tautologies, and inference apparatus with substitution and modus ponens, but. Quine there are logicians, myself among them, to \,hom the ideas of modal logic e. That is, t will hold just in case rw,w, for all worlds w. Axiomatic theory modal logic kt axioms 16 40 sg models and formalisms 31. Systems of modal logic department of computing imperial. You are telling us things that we either are not interested in, or else already know well.
So the presence of axiom m distinguishes logics for necessity from other. Distribution axiom of modal logic mathematics stack exchange. The purpose of this paper is to show that such an axiom is indispensable. Modal logic gives a frame work for arguing about these distinctions. The language of the classical logic is simple, straightforward, and easy to work with. The chellas text in uenced me the most, though the order of presentation is inspired more by goldblatt. Manual of intensional logic van benthem, 1988a extends the canvas. Additional validities arise as axioms for modal logics with special. The logic s4, for instance, is a classical modal logic with the axiomsk, t,4and rulesg, modus ponens and all propositional tautologies. Lewissare not intuitively clear until explained in nonmodalterms. The task is to prove the correspondence between the socalled t axiom. On the one hand we are guided by the reading of the.1021 1237 360 516 1614 38 710 1559 1487 1631 1331 1265 796 1098 1136 686 439 1233 388 1153 182 1481 859 530 31 652 221 527 673 98 165 719 718