October 5, 2004 1 introduction this short tutorial had its genesis in a coincidence of two events. I was sorting through a pile of notes dating from the time when i was engaged as a professional logician 4. The meaning of set forth in this section is that of epistemic modal. It is easy to prove modal modus ponens, given axiom 1 of modal logic. Dekker and others published possible worlds, belief, and modal logic. Thm joubkal oj symbolic loglc volume 12, number 2, june 1947 the problem of interpreting modal logic w. How does modal logic differ from ordinary predicate logic.
We assume that we possess a denumerably infinite list. There were, and still are, some recurring complaints. 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. On intuitionistic modal and tense logics and their. Algebraic tools for modal logic mai gehrke yde venema esslli01 august 17, 2001 helsinki, finland. A modal epistemic logic for agents is obtained by joining together modal logics, one for each agent. 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. The logic s4, for instance, is a classical modal logic with the axiomsk, t,4and rulesg, modus ponens and all propositional tautologies.
Researchers in areas ranging from economics to computational linguistics have since realised its worth. It is a normal modal logic, and one of the oldest systems of modal logic of any kind. The task is to prove the correspondence between the socalled t axiom. I will not explicitly state that this set is in the axiomset for any logic up for study, but merely give the modal axioms. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. 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.
It is formed with propositional calculus formulas and tautologies, and inference apparatus with substitution and modus ponens, but. So an epistemic logic for agents consists of copies of a. Axiomatic theory modal logic kt axioms 16 40 sg models and formalisms 31. Given any formula, it is straigtforward to make a truth table and determine whether the formula is a tautology, a contradiction, or neither. You are telling us things that we either are not interested in, or else already know well. This algebraic perspective dominated the semantical approach to modal logic until the publication of the groundbreaking works of kripke 30, 31. A new introduction to modal logic is an entirely new work, completely rewritten by the authors. For simplicitys sake it is usually assumed that the agents are homogeneous, i. 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. Propositional modal logic modal logic is the logic of necessity, possibility and other related notions.
Gunther propositional logic our language semantics syntax results modal logic our language semantics. 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. Modal logic gives a frame work for arguing about these distinctions. 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. Or, intuitively, if all possible worlds are accessible from themselves. In our problem encodings we exploit the wellknown correspondences between these. The logic of provability university of california, berkeley. So the presence of axiom m distinguishes logics for necessity from other. The journal of symbolic logic volume 24, number 1, march 1959 a completeness theorem in modal logic saul a. The logic s4, for instance, is a classical modal logic with the axiomsk,t,4and rulesg, modus ponens and all propositional tautologies. 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.
Notice that adding axioms to system k may even result in the inconsistent logic. This is called the law of the excluded middle the statement p q is not a tautology. Additional validities arise as axioms for modal logics with special. The purpose of this paper is to show that such an axiom is indispensable. Philosophy stack exchange is a question and answer site for those interested in the study of the fundamental nature of knowledge, reality, and existence. 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. 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. An overview of applications of modal logic in linguistics can be found in. Axiom t will hold in all pws models just in case the relation r is re. Systems of modal logic department of computing imperial. 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. 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. Chapter 1 modal logics of space institute for logic. This is an advanced 2001 textbook on modal logic, a field which caught the attention of computer scientists in the late 1970s.
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. 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. Independence of the dual axiom in modal k with primitive. Manual of intensional logic van benthem, 1988a extends the canvas.
Distribution axiom of modal logic mathematics stack exchange. 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. Modal logic, philosophy society and department of philosophy, university of uppsala, vol. 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.
Lewissare not intuitively clear until explained in nonmodalterms. 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. There are several ways to formalise a logic as a mathematical object. 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. On the one hand we are guided by the reading of the. Prominent modal logics are constructed from a weak logic called k after saul kripke.
Most accessibility relations in applications of modal logic are re. A modala word that expresses a modalityqualifies a statement. The chellas text in uenced me the most, though the order of presentation is inspired more by goldblatt. So the acceptability of axioms for modal logic depends on which of these. The language of the classical logic is simple, straightforward, and easy to work with. 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. A modal a word that expresses a modalityqualifies a statement. Of particular interest are socalled normal systems of modal logics. Knowing all this, i have implemented a solver for propositional modal logic s4 and it also terminates with a finite model. That is, t will hold just in case rw,w, for all worlds 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. A modal is an expression like necessarily or possibly that is used to qualify the truth of a judgement. Is there any further reduction possible by proving the distribution axiom. Basic concepts in modal logic1 stanford university.
However, the term modal logic may be used more broadly for a family of. The first two are straightforward and are left as an exercise tutorial sheet. A a the accessibility relation is reflexive and symmetric. Pdf a new introduction to modal logic download full. On the role of modal intuition in modal logic 171 walters articulates in words what is presented in the possible worlds diagram above. Axioms are formulas that are considered to be selfevidently true, for which no proof is required. These notes are meant to present the basic facts about modal logic and so to provide a common. S4 axiom is a class of transitive and reflexive frames. Part i james pustejovsky september 27, 2004 1 modal logic so far in this class, we have studied propositional and. All the more important systems have in fact but a single additional rule of necessitation, permitting inference from a to a. Modal logic is, strictly speaking, the study of the deductive behavior of the expressions it is necessary that and it is possible that.1241 605 786 15 1014 1656 719 1346 1156 1595 606 1314 634 1246 779 147 476 1084 1585 269 127 1000 1083 535 798 1591 731 620 1609 402 1508 168 330 1276 365 815 493 1339 414 881 345 708 709 1087 722 998 492 575 836 132