It instantiates a variable with an empty world path. 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. For simplicitys sake it is usually assumed that the agents are homogeneous, i. Proof theory and modal logic sonia marin supervised by. 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. W e introduce the completeness problem for modal logic and examine its complexity. Kripke completeness revisited university of helsinki. A modala word that expresses a modalityqualifies a statement.
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. You need only check that the axioms and the rule of modus ponens is. Frank pfenning, rowan davies, a judgemental reconstruction of modal logic, mathematical structures in comp. Categorical and kripke semantics for constructive modal logics. Modal logic is, strictly speaking, the study of the deductive behavior of the expressions it is necessary that and it is possible that. This chapter discusses semantical analysis of modal logic ii and nonnormal modal propositional calculi. Reflexivity is thus an intuitively natural require ment. To do so we must be very explicit about the following important distinction. I will not explicitly state that this set is in the axiom set for any logic up for study, but merely give the modal axioms. The task is to prove the correspondence between the socalled t axiom.
This book presupposes that readers know the attractions and power of this approach, including the notions of logical syntax, semantics, proof, and metatheory of formal systems. Boldface letters represent longer worldpaths, whereas normal letters represent single terms. Modal logic basics suppose now that we have an n modal logic l and want to introduce in it the universal modalities with their intended interpretation. Introduction to modal logic carnegie mellon university. Firstorder logic of proofs city university of new york. Chellas provides a systematic introduction to the principal ideas and results in contemporary treatments of modality, including theorems on completeness and decidability.
In the hellenistic period, the logicians diodorus cronus, philo the dialectician and the stoic chrysippus each developed a modal system that accounted for the interdefinability of possibility and necessity, accepted axiom t see below, and combined elements of modal logic and temporal logic in attempts to solve the notorious master argument. Modal logic is the study of modal propositions and the logical relationships that they bear to one another. As 0 and should behave like the s5 box and diamond, the most natural way to do this is to take the n f 1 modal logic lu l axioms of s5 for and 0 0po qipo i 1 n in the language. In basic modal logic we have two new sentential operators. Instead, we obtain a modal logic with both normal and nonnormal modalities, together with additional axiom that speci. A new s4 classical modal logic in natural deduction medeiros, maria da paz n. First we define a formal semantic framework for modal logic, namely, kripke semantics. Modal logic axiom s4, transitive and reflexive frame. 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. A modal epistemic logic for agents is obtained by joining together modal logics, one for each agent.
A semantic perspective 3 chapters in this handbook. The reflexivity axiom provides a formal definition. Indeed, the satisfiability problem is npcomplete for any modal logic that includes the negative introspection axiom. Modal logic is meant to capture seeming entailments between such alethic and deontic notions. Introduction to modal logic cmu math carnegie mellon university. I will not explicitly state that this set is in the axiomset for any logic up for study, but merely give the modal axioms. We develop a probabilistic semantics for modal logic, which was introduced in recent years by dana scott. We may impose additional requirements, corresponding. Partiality and adjointness in modal logic uc berkeley philosophy. To model negation one may use possibleworlds models in the style of kripke that have an accessibility relation r n peculiar to negation. We also consider the logic and semantics in classical setting. Interestingly, the classical version of our logic is not a normal modal logic, even in the case of relational semantics.
It is formed with propositional calculus formulas and tautologies, and inference apparatus with substitution and modus ponens, but extending the. Reflexivity axiom for inferring functional dependencies. The decomposition and separation rules are sufficient for modal logics k and kd. S4 axiom is a class of transitive and reflexive frames. The possibility of unicorns and modal logic lee walters michael dummett argues, against saul kripke, that there could have been unicorns. Together, these axioms partially axiomatize the modal system k, including all the usual axioms, but not the dual axiom. But should reflexivity of equality be taken as an axiom of logic or as a theorem of set theory. W, we define what does it mean for a formula to be true satisfied in a world w of a model. Inspire your inbox sign up for daily fun facts about this day in history, updates, and special offers. The strong modal operator is symbolized by the box.
In this tutorial, we give examples of the axioms, consider some rules of inference and in particular, the derived rule. Basic concepts in modal logic1 stanford university. Completeness and definability in the logic of noncontingency evgeni e. 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. Natural deduction based upon strict implication for normal modal logics cerrato, claudio, notre dame journal of formal logic, 1994.
Normal modal propositional calculi a modal propositional calculus mpc is given by a denumerably infinite list of. This seems trivial but it cannot be deduced from the augmentation axiom and transitivity axiom. So if you have a relation with attributes a,b,c you can deduce functional dependencies like. Independence of the dual axiom in modal k with primitive. A modal is an expression like necessarily or possibly that is used to qualify the truth of a judgement. Canty, john thomas, notre dame journal of formal logic, 1964. Modal logic is a textbook on modal logic, intended for readers already acquainted with the elements of formal logic. It is a normal modal logic, and one of the oldest systems of modal logic of any kind. Examples for convenience, we reproduce the item logicmodal logic of principia metaphysica in which the modal logic is defined. To get a semantics for modal logic, we introduce the notion of a normal model structure.
This is a summary of some things that can be said about negation understood as an impossibility operator. Marcos 2005 provided a sound and complete axiom atization of the minimal reflexiveinsensitive logic. This semantics is intimately related to an older, topological semantics for modal logic developed by tarski in the 1940s. Humberstone 17 has proposed a semantics for modal logics based on partial. The proof of sufficiency, which is omitted by many, proceeds by constructing a normal characteristic matrix by lindenbaums method. Department of mathematical logic faculty of mathematics and mechanics moscow state university, 119899 moscow, russia abstract hilbertstyle axiomatic systems are presented for versions of the modal logics k. Before moving on, i give a few definitions that will be important as we move on, and list a. D, 4, 5, with noncontingency as the sole modal primitive. Knowing all this, i have implemented a solver for propositional modal logic s4 and it also terminates with a finite model. This is a method for finding the relevant axiom, but it remains to be proved that the axiom is sufficient as well as necessary.
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