A basic principle adhered to in most formal logics is the principle of noncontradiction - two contradictory statements cannot both be true (or a statement cannot be both true and false). Should the principle of noncontradiction be assumed in science? How would science operate if it allowed contradictory statements to be considered true? Are there examples found in science as it is practiced (by most scientists) now where contradictory statements are both considered to be true (or where a statement is said to be both true and false)?

Here is a little background on the question:

http://www.math.wichita.edu/history/topics/logic.html

One area of mathematics that has its roots deep in philosophy is the study of logic. Logic is the study of formal reasoning based upon statements or propositions. (Price, Rath, Leschensky, 1992) Logic evolved out of a need to fully understand the details associated with the study of mathematics. At the most fundamental level, mathematics is a language and it is a language of choice and must be communicated with great precision. (Wheeler, 1995) The idea of logic was a major achievement of Aristotle. In his effort to produce correct laws of mathematical reasoning, Aristotle was able to codify and systemize these laws into a separate field of study. The basic principles of logic center on the law of contradiction, which states that a statement cannot be both true and false, and the law of the excluded middle, which stresses that a statement must be either true or false. The key to his reasoning was that Aristotle used mathematical examples taken from contemporary texts of the time to illustrate his principles. Even though the science of logic was derived from mathematics, logic eventually came to be considered as a study independent of mathematics yet applicable to all reasoning. (Kline, 1972)

I only asked about the principle of noncontradiction (law of contradiction or law of noncontradiction), but the same question applies to the law of the excluded middle.