A Peer-to-Peer (P2P) system is composed of peer processes (peers) which are interconnected in overlay networks. We dicuss two types of trustworthiness, subjective and objective trustworthiness. The subjective trustworthiness STij of an acquaintance peer pj for a peer pi is obtained by directly communicating with the acquaintance peer pj. The more number of correct replies the acquaintance peer pj sends to the peer pi, the more trustworthy the peer pi recognizes the acquaintance peer pj. The objective trustworthiness of an acquaintance pj shows reputation of the acquaintance peer pj, which is an average value of subjective trustworthiness of the peer pj. We discuss the subjective and objective types of trustworthiness of an acquaintance peer by using the Fuzzy logic.