International Journal of Theoretical Physics, Vol. 38, No. 12, 1999 How Many Notions of ªSharpº? Gianpiero Cattaneo, 1 Maria Luisa Dalla Chiara, 2 and Roberto Giuntini 3 Received October 13, 1999 1. INTRODUCTION To what extent is ªsharp/unsharpº an unsharp, ambiguous distinction? In the framework of the unsharp approach to quantum theory different definitions of ªsharp physical propertyº can be proposed. One can distinguish two basic kinds of characterizations: (i) Purely algebraic definitions, which only refer to the algebraic structure of the quantum events; (ii) Semantic-statistical definitions, which also refer to the relations between events and states. In the present paper we will only discuss case (i). Case (ii) will be the subject of another paper. Different algebraic environments give different answers to the question, ªWhat is the structure of the quantum events?º A minimal structure is repre- sented by an effect algebra. 4 This is a partial structure with two privileged 1 Dipartimento di Scienze dell’Informazione, UniversitaÁ di Milano, I-20135 Milan, Italy; e-mail: cattang@dsi.unimi.it. 2 Dipartimento di Filosofia, UniversitaÁ di Firenze, I-50139, Florence, Italy; e-mail: dachiara@mailserver.idg.fi.cnr.it. 3 Dipartimento di Scienze Pedagogichee Filosofiche, UniversitaÁ di Cagliari, I-09123 Cagliari, Italy; e-mail: giuntini@idg.fi.cnr.it. 4 Effect algebras have been also called weak ortholagebras [6] and unsharp ortholagebras [2]. Equivalent structures are represented by difference posets ([ 8]). 3153 0020-7748/99/12 00-3153$16.00/ 0 q 1999 Plenum Publishing Corporation