How to Argue for and against Abstracta

Realism about abstracta is the view that there are abstract objects, such as properties, sets and numbers, whereas nominalism is the view that there are no abstract objects. The most powerful argument for realism about abstracta is arguably an abductive argument along the lines of (P1-2|C).

(P1) The best total theory with respect to our evidence entails that there are abstract objects

(P2) If the best total theory with respect to our evidence entails that there are abstract objects then we are justified in believing that there are abstract objects

(C) We are justified in believing that there are abstract objects

(P1-2|C) faces the following problem: There are good reasons to think that the argument is only credible if theories are understood to be abstract objects, such as sentence types or propositions. If theories are abstract objects, however, then the argument will be ineffectual against nominalists, since nominalists will reject (P1).

This paper argues that (P1-2|C) can be modified so that it avoids this problem. It also does the same for the related abductive argument for nominalism, which is obtained from (P1-2|C) by replacing ‘there are abstract objects’ with ‘there are no abstract objects’.