Contemporary use of the term 'intension' derives from the traditional logical doctrine that an idea has both an extension and an intension. In this paper we introduce an intensional FOL (First-order-logic) for P2P systems by fusing the Bealer's intensional algebraic FOL with the S5 possible-world semantics of the Montague, we define the intensional equivalence relation for this logic and the weak deductive inference for it. The notion of ontology has become widespread in semantic Web. The meaning of concepts and views defined over some database ontology can be considered as intensional objects which have particular extension in some possible world: for instance in the actual world. Thus, non invasive mapping between completely independent peer databases in a P2P systems can be naturally specified by the set of couples of intensionally equivalent views, which have the same meaning (intension), over two different peers. Such a kind of mapping has very different semantics from the standard view-based mappings based on the material implication commonly used for Data Integration. We show how a P2P database system may be embedded into this intensional modal FOL, and how we are able to obtain a weak non-omniscient inference, which can be effectively implemented. For a query answering we consider non omniscient query agents and we define object-oriented class for them which implements as method the query rewriting algorithm. Finally, we show that this query answering algorithm is sound and complete w.r.t. the weak deduction of the P2P intensional logic.