Type of Document Master's Thesis Author Pemmaraju, Sriram V. URN etd-09082012-040214 Title Modal logics of provability Degree Master of Science Department Computer Science and Applications Advisory Committee

Advisor Name Title Roach, John W. Committee Chair Cuda, Thomas V. Committee Member Heath, Lenwood S. Committee Member Keywords

- Logic
- Symbolic and mathematical.
Date of Defense 1989-05-05 Availability restricted AbstractGödel proved his Incompleteness theorems for any theory 'strong' enough to represent recursive functions. In the process he showed that the provability predicate can be represented in such theories. Modal logics of provability are modal logics which attempt to express the concept of 'provability' and 'consistency' using the modal operators '[]' and '<>' respectively. This is achieved by forcing '[]' to behave like the provability predicate. GL is a modal logic which has been shown to be complete and sound with respect to arithmetic theories (theories which can represent all recursive functions), hence results about concepts such as 'consistency,' 'provability' and 'decidabi1ity' in arithmetic theories can be stated and proved in GL. It has also been proved that GL is complete with respect to the class of finite, transitive, reversely well—founded models. This essentially means that the set of theorems of GL is recursive and hence there exists an effective procedure to determine whether a given wff is a theorem of GL or not. We investigate a weaker version of GL called GH and show that GH is not complete with respect to arithmetic theories. We show this by first showing that GH is a proper subset of GL and then showing that the theorems missing from GH are properties of the provability predicate. We finally, show that GH is not complete with respect to the class of transitive, reversely well-founded models and hence not sound and complete with respect to any frame.

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