Tiered logic for contextualizing logics

Towards the later part of the 1980s, the subject of context became an issue in computer science especially in the field of Artificial Intelligence (AI). It came into the spotlight primarily because axioms in logic systems are obviously true only in a certain context. With a small effort, one can easily show that a logical statement is false by by citing the "context" wherein it obviously will not hold. Since then, advances have been made in the process of understanding the role of context in dealing with logical statements and the subject is now also important to the field of linguistics and pragmatics. Mostly it finds an application in the field of reconciling disparate knowledge bases(KBs) or heterogenous ontologies which are described by knowledge representation languages (KRL). In this work, we will show a simple technique for contextualizing logics that is easy to use and avoids complicated machinery which is characteristic of most approaches. In another sense, the technique presented here may be seen also as a technique for combining logics for the purpose of contextualizing them. The approaches some have taken in contextualizing logics require the use of complex logic machineries such as elaborate modal semantics or highly sophisticated abstractions dependent on category theory. However it is not certain if the "nice" properties of a logic such as soundness, completeness and decidability can be achieved by the existing approaches. In the field of contextualizing ontologies, they are gained by paying the high price of complicated presentation. Moreover, in order for the recent approaches to be used in the field, changes or modifications to the internal workings of present reasoning engines are required. Such costs may be excessive and discourage practical usage opportunities. Hence the issue is how such contextualization can take which is less complicated, uses existing reasoning engines for practical applications and requires as much as possible no (or almost no) internal modifications to them. Such ideas have been the consideration of this work. At the time of writing, there are about five logical theories of context that may be used as foundation for contextualizing logics. Of all of these, there are two theories that are worth mentioning. The first is the technique known as the Local Model Semantics (LMS) which has been popularly used as a basis for contextualizing ontology languages. IX The other one, which is often compared to the first, is the Propositional Logic of Context (PLC). This latter theory has not been given much attention by practitioners in the field of contextualizing ontology languages even though historically it came first. As far as we know, there is no work that has used the latter for such a problem. Though this thesis takes a few ideas from LMS, it takes more inspiration from PLC and thus serves as a counter argument for and on behalf of PLC, that it is a promising foundational basis for contextualizing logics. Knowing how vital a role context plays in logic theories, Tiered Logic is a method (TLM), or a framework, for contextualizing logics. It is also a method for combining logics. This method follows a layered approach in combining logics, in other words the outcome of this process is a layered logic (in this case two layered) wherein the logic in the bottom layer called tier-0, is of a certain logic species while the top layer, called tier-1, is of another logic species which in general is different from the logic that is at the bottom. The value of this process is that when combining or contextualizing logics, the resulting combination is a lot easier to understand and that the "nice" properties of the logics may be transferred over to the resulting combined logic. This is specially true when tier-0 happens to be a decidable logic and tier-1 is another decidable logic. Specifically, TLM is tailor fitted for the situation where one has a proliferation of incompatible ontologies, i.e., they are heterogenous languages or differing vocabulary bases and the problem then is how to reconcile or navigate them. As a process, TLM can only be grasped by going through the motion or task of layering itself, hence the work shows several cases of combining or contextualizing logics by using a number of examples. After giving the Introductory and Background chapters, we start the first example in Chapter 3, where we treat, that is to say, at tier-0 we have Propositional Logic (PL) and also at tier-1. In Chapter 4, we do the same for First Order Logic (FOL), we treat. In Chapter 5 we do the same but treatwhere we use The Common Algebraic Specification Language (CASL) at tier-0 and PL in tier-1. We continue in the same spirit and treatin Chapter 6 where DL stands for Description Logic and +PL stands for the Positive Propositional Logic. In Chapter 7, we apply the results ofto practical issues and show how the logic solves some defects found in present approaches. In Chapter 8 we point the way on how an ontology algebra may be achieved. In the last, Chapter 9, we give the benefits and limitations and the future work where the research may be moved further and forward. In all of these logic systems we prove soundness, completeness and decidability results for the created contextualized system and note how simple and smoothly these pleasant properties have been achieved through TLM. These examples show the flexibility of TLM in that it can work with popular logics in tier-0 and accommodates even those that do not have proof systems but us the tableau. Finally, we gives thoughts to ponder on what happens when in tier-1 the logic is more expressive than PL, such as FOL; what would it be like and are there practical benefits for this, etc