On intuitionistic modal and tense logics and their. A modala word that expresses a modalityqualifies a statement. Im reading an interesting on modal logic by timothy surendonk 1, where he defines the concept of canonical logic as the following. Finite models constructed from canonical formulas indiana. Definability, canonical models, compactness for finitary. For philosophers, modal logic is a powerful tool for semantics.
Filtrations for kripke models were rst introduced and studied by john lemmon 9 and in a general form by krister segerberg 11, 12. Completeness and canonical models open logic project. Can you generalise this to cover lconsistency for an arbitrary normal modal logic l. This is the standard propositional logic with negations, conjunctions and disjunctions of formulas, augmented with a knowledge modality for each. A view of its evolution 5 was a variable neither always true nor always false. Compactness for coalgebraic modal logic is more complicated than for standard modal logic. The logic k has the nite model property the logic k has the nite model property. In standard modal logic, compactness is inherited from the compactness of. A canonical model construction for iterationfree pdl with.
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. Given a normal modal logic l, it is a familiar step to consider the lattice hl. Can we import resultsideas from model theory for modal logic with respect to kripke semanticstopological semantics. 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. The semantics for modal logic are usually given as follows. Technical report 7807, department of mathematics, university of amsterdam, 1978. Many concepts in philosophy of language can be formalized in modal logic. Modal logic is meant to capture seeming entailments between such alethic and deontic notions. First we define a frame, which consists of a nonempty set, g, whose members are generally called possible worlds, and a binary relation, r, that holds or not between the possible worlds of g. The individual modules can then be considered endpoints on. A canonical model is a design pattern used to communicate between different data formats. Pdf denability, canonical models, compactness for finitary.
In studies in logic and the foundations of mathematics, 2003. Let k denote the minimal modal logic and mean is derivable in k. On canonical modal logics that are not elementarily determined. This proves completeness with respect to socalled modalstructures.
T is compared to the canonical model as known from modal logic. The canonical model notes for lecture 6 eric pacuit march, 2012 notation. On intuitionistic modal and tense logics and their classical. The logical model is specific to a class of databases, such as relational databases. This book is an introduction to logic for students of contemporary philosophy.
Monomodal examples are also constructed that extend k4 and are related to the proof of noncanonicity of the mckinsey axiom. The choice of logical connectives depends on the development of propositional logic one wants to follow. Compactness for coalgebraic modal logic is more complicated than for stan. However, a common implementation of a canonical is more of an overall enterprise scope, and is then partitioned or subclassed to represent functional areas of the business and common data concepts such as items, customers, invoices, orders, reference. Tools and techniques in modal logic marcus kracht ii. An introduction to modal logic 2009 formosan summer school on logic, language, and computation 29 june10 july, 2009. Most recently, modal symbolism and model theory have been put to use in computer science, to formalise reasoning about the way programs behave and to express dynamical properties of transitions between states. Canonical formulas for modal and superintuitionistic logics. Basic concepts in modal logic1 stanford university. Model checking and temporal logic are very hot research areas in computer science which use modal logics extensively. The canonical model for any system is the model defined by the procedure used in our competeness proof for the. These special kripke frames were introduced in the mid1960s by lemmon and scott 25, and independently by cresswell 2 and makinson 27, as an extension of the method of completeness proof due to.
We then uniformly constructfor each modal predicate logic l a canonical model whose theory is exactly l. The strong modal operator is symbolized by the box, while the weak modal operator is symbolized by the diamond. Given a world win an lmodel m, show that the set of formulas f. Conceptual model vs logical model vs canonical model. Algebraic tools for modal logic mai gehrke yde venema esslli01 august 17, 2001 helsinki, finland. A modal logic l is called canonical if it is valid in the canonical frame f l whose points are the maximally lconsistent sets of formulas. The stable model semantics is defined for a logic program. To show the converse, we observe that in the canonical model for cpdl, s5 w the s5accessibility relation commutes with each of the atomic transition relations. The canonical model for t is built on a re exive frame, were going to show.
Showing a system s to be complete relative to a class c. The canonical data model cdm is a data model that covers all data from connecting systems andor partners. Pdl is also related to logics used in knowledge representation, it is basically the same as the description logic. It covers i basic approaches to logic, including proof theory and especially model theory, ii extensions of standard logic such as modal logic that are important in philosophy, and iii some elementary philosophy of logic. Over a period of three decades or so from the early 1930s there evolved two kinds of mathematical semantics for modal logic.
James bean, in soa and web services interface design, 2010. More on completeness and the canonical model 1 a show that the normal modal logic kd. On the polynomial finite model property of modal and intermediate logics. Canonical formulas for modal and superintuitionistic. The polytheistic approach to modal logics alethic modal logic. This is an advanced 2001 textbook on modal logic, a field which caught the attention of computer scientists in the late 1970s.
In this lecture we consider a logic as the set of its tautologies. Instead, we obtain a modal logic with both normal and nonnormal modalities, together with additional axiom that speci. Denability, canonical models, compactness for finitary coalgebraic modal logic article pdf available in electronic notes in theoretical computer science 651 september 2002 with 22 reads. The concepual model is database neutral, and results from data analysis. The way the data is modelled will be different from the connected data models, but still the cdm is able to contain all the data from the connecting data models. We also consider the logic and semantics in classical setting. These notes are meant to present the basic facts about modal logic and so to provide a common. Canonical kripke models play a role similar to the lindenbaumtarski algebra construction in algebraic semantics.
On the degree of incompleteness in modal logic and the covering relations in the lattice of modal logics. Show that the canonical modal for the modal logic s4. In first approximation, modal logic im using the term loosely can be understood as an interesting fragment of firstorder logic for simplicity i ignore e. A canonical model for a modal logic is typically a kripke structure whose worlds are all the maximally consistent sets of the underlying logic whith respect to some notion of provability such a set. Another issue which arises when we focus on nitary logics is compactness. In basic modal logic we have two new sentential operators.
In mathematical logic, mathematical linguistics and algorithm theory, pages 7583, kalinin state university, kalinin, 1983. The choice of logical connectives depends on the development of proposi. Interestingly, the classical version of our logic is not a normal modal logic, even in the case of relational semantics. The chellas text in uenced me the most, though the order of presentation is inspired more by goldblatt. Modal logic is a simplified form of the first order predicate logic. To capture those notions, we need the concept of an open model, an model, analogous to a canonical model in modal logic, in which all possibilities or, at least, all possibilities consistent with certain constraints are realized. The completeness proof strategy is basically the same as for intuitionistic logic. It covers i basic approaches to logic, including proof theory and especially model theory, ii extensions of standard logic such as modal logic that are important in philosophy, and. Lecture notes on completeness and canonical models citeseerx. There exist modal logics that are validated by their canonical frames but are not sound and complete for any elementary class of frames. A model for the basic modal language is a pair m f.
Surendonk, timothy j does ek 4 have the finite model property. This logic is defined by combining positive free logic with equality pfl. Sometimes one can still show completeness in such cases by first using the canonical model method to prove completeness for the system relative to a larger. 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. This does not mean the cdm is just a merge of all the data models. The canonical model notes for lecture 6 stanford ai lab. Chapter 1 modal logics of space institute for logic.
274 557 917 947 1486 629 1217 1513 362 256 10 1290 1409 1519 1511 878 587 1178 222 1374 841 1019 368 897 813 1309 555 845 1418 609 81 287 1270 763 998 633