Signalling in independence-friendly logic

Independence-Friendly (IF) logic is an extension of first-order logic that embodies some aspects of extensive games of imperfect information. Acts of signalling (communication between players), which are absent in game-theoretic presentations of first-order logic, play a fundamental role in the evaluation of IF sentences. Perhaps surprisingly, signalling effects keep playing an important role even if we extend our semantics to a more general strategic and probabilistic approach; in this article we shall analyse some manifestations of this phenomenon. First we show how David Lewis' signalling games used for the analysis of conventions can be expressed in IF logic. Then we use the Lewis' signalling defining IF sentence to give a very different proof of the fact (already stated in [15]) that there is an IF sentence which expresses all rational numbers. More precisely, we prove that for every rational q in the interval [0,1], there is a finite model such that the probabilistic value of the Lewis' signalling sentence on this model is q. Next, by means of a variant of the Lewis' signalling sentence we also prove a result concerning the impact of single relations of independence on the spectrum of probabilistic values of an IF sentence. Our theorem gives a partial solution to the following problem. Removing a relation of independence, the probabilistic value p of a formula changes to some greater value q; are all such couples (p,q) realized on some structure? In an appendix, we show that the results proved for our probabilistic semantics can be transferred to the equilibrium semantics already defined in the literature ([18]).