Probabilistic characterisation of models of first order theories

We study probabilistic characterisation of a random model of a finite set of first order axioms. Given a set of first order axioms and a structure which we only know is a model of , we are interested in the probability that would satisfy a sentence ψ. Answering this question for all sentences in the language will give a probability distribution over the set of sentences which can be regarded as the probabilistic characterisation of the model . We investigate defining these probabilistic characterisations as the limit of probability functions imposed on the set of finite models of . We show how a symmetry axiom can uniquely specify the probability function over finite models and will study the existence of the limit in terms of the quantifier complexity of .