  A fundamental issue for Church is when two names, lambda terms, have the same sense. Three alternatives are considered. Common to all three alternatives are the assumptions that sense is unchanged under the renaming of bound variables (with the usual conditions of freeness), and under \(\beta\) reduction. Beyond these, Alternative 0 is somewhat technical and is only briefly mentioned, Alternative 1 is fine grained, making senses distinct as far as possible, while Alternative 2 makes two terms have the same sense whenever equality between them is a logical validity. The proper definition of the alternatives is axiomatic, and altogether various combinations of some 53 axiom schemes are introduced, with none examined in detail. Clearly Church was proposing an investigation, rather than presenting full results.