On Agents That Have the Ability to Choose

We demonstrate ways to incorporate nondeterminism in a system designed to formalize the reasoning of agents concerning their abilities and the results of the actions that they may perform. We distinguish between two kinds of nondeterministic choice operators: one that expresses an internal choice, in which the agent decides what action to take, and one that expresses an external choice, which cannot be influenced by the agent. The presence of abilities in our system is the reason why the usual approaches towards nondeterminism cannot be used here. The semantics that we define for nondeterministic actions is based on the idea that composite actions are unravelled in the strings of atomic actions and tests that constitute them. The main notions used in defining this semantics are finite computation sequences and finite computation runs of actions. The results that we obtain meet our intuitions regarding events and abilities in the presence of nondeterminism.     

Elementary Classes in Basic Modal Logic

Dealing with topics of definability, this paper provides some interesting insights into the expressive power of basic modal logic. After some preliminary work it presents an abstract algebraic characterization of the elementary classes of basic modal logic, that is, of the classes of models that are definable by means of (sets of) basic modal formulas. Taking that for a start, the paper further contains characterization results for modal universal classes and modal positive classes.     

Remarks on Gregory's “Actually” Operator

In this note we show that the classical modal technology of Sahlqvist formulas gives quick proofs of the completeness theorems in [8] (D. Gregory, Completeness and decidability results for some propositional modal logics containing actually operators, Journal of Philosophical Logic 30(1): 57–78, 2001) and vastly generalizes them. Moreover, as a corollary, interpolation theorems for the logics considered in [8] are obtained. We then compare Gregory's modal language enriched with an actually operator with the work of Arthur Prior now known under the name of hybrid logic. This analysis relates the actually axioms to standard hybrid axioms, yields the decidability results in [8], and provides a number of complexity results. Finally, we use a bisimulation argument to show that the hybrid language is strictly more expressive than Gregory's language.     

Completeness and Decidability Results for Some Propositional Modal Logics Containing “Actually” Operators

The addition of actually operators to modal languages allows us to capture important inferential behaviours which cannot be adequately captured in logics formulated in simpler languages. Previous work on modal logics containing actually operators has concentrated entirely upon extensions of KT5 and has employed a particular model-theoretic treatment of them. This paper proves completeness and decidability results for a range of normal and nonnormal but quasi-normal propositional modal logics containing actually operators, the weakest of which are conservative extensions of K, using a novel generalisation of the standard semantics.     

Modal Logics of Succession for 2-Dimensional Integral Spacetime

We consider the problem of axiomatizing various natural successor logics for 2-dimensional integral spacetime. We provide axiomatizations in monomodal and multimodal languages, and prove completeness theorems. We also establish that the irreflexive successor logic in the standard modal language (i.e. the language containing and ) is not finitely axiomatizable.     

On Consistent Subsets of Large Sets of Satisfiable Sentences

We extend some results of Adam Kolany to show that large sets of satisfiable sentences generally contain equally large subsets of mutually consistent sentences. In particular, this is always true for sets of uncountable cofinality, and remains true for sets of denumerable cofinality if we put appropriate bounding conditions on the sentences. The results apply to both the propositional and the predicate calculus. To obtain these results, we use delta sets for regular cardinals, and, for singular cardinals, a generalization of delta sets. All of our results are theorems in ZFC.