Contradictions Do Not Necessarily Make a Theory Inconsistent

Abstract

Some religious scholars claim that while the corresponding holy texts may be contradictory, they lead to a consistent set of ethical and behavioral recommendations. Is this logically possible? In this paper, somewhat surprisingly, we kind of show that this is indeed possible: namely, we show that if we add, to statements about objects from a certain class, consequences of both contradictory abstract statements, we still retain a consistent theory. A more mundane example of the same phenomenon comes from mathematics: if we have a set-theoretical statement S which is independent from ZF and which is not equivalent to any arithmetic statement, then we can add both arithmetic statements derived from S and arithmetic statements derived from ¬S and still keep the resulting class of arithmetic statements consistent. 1 Can Contradictions Lead to a Consistent Theory? There are seeming logical contradictions in holy books. From the purely logical viewpoint, holy books often contain inconsistent statements. For example, the Bible has two different stories of creation: • in one, Adam was created first and Eve made out of his rib later on – since he felt lonely in the Paradise, while • in the second one, both first humans were created at the same time. How religions treat such contradictions. There are two main approaches to such seeming contradictions. The first approach is to try to re-interpret the text so that the contradictions disappear. Interestingly, there is also a second approach (see, e.g., [7]), that yes, from our viewpoint, this may be perceived as a contradiction, but both contradictory