 Assume that (i) verb phrases (VPs) denote functions from individuals into truth values (type \(\langle e,t\rangle\)), (ii) noun phrases denote functions from VP denotations into truth values (type \(\langle\langle e,t\rangle, t\rangle\)), and (iii) sentences denote truth values (type \(t\)). Then, talking about characteristic functions as if they were sets, we have that John denotes the set of all John’s properties, Mary denotes the set of all Mary’s properties, and sang denotes the set of all individuals that sang. Given  (22),  John or Mary denotes the union of John’s properties with Mary’s properties, i.e., the set consisting of exactly those properties that pertain to either John or Mary. So John or Mary sang will come out true just in case this union has as a member the property of singing. This is the case only if either John sang or Mary sang (see  23a).  On the other hand, given  (22),  sang or danced denotes the union of the set of individuals that sang with the set of individuals that danced. So every man sang or danced will come out true just in case this union is a member of the property-set denoted by every man. From this we cannot infer that either every man sang or every man danced (see 23b).