Circular reasoning
Circular reasoning (Latin: circulus in probando, circle in proving) is a form of logical fallacy in which one assumes the fact that one is trying to prove. Begging the question is a form of circular reasoning.
An analysis
Usually, if one is trying to prove something, one must start with something else already accepted as valid. Thus the simplest argument would use the Law of Detachment and run this way:
 If P, then Q.
 P.
 Therefore, Q.
For a slightly more complex example, one uses the Law of the Syllogism to establish an intermediate fact between the alreadyaccepted fact and the fact one wishes to prove. Thus:
 If P, then Q.
 If Q, then R.
 Therefore, if P, then R.
 P.
 Therefore, Q and R.
Finally, one can use contraposition to prove that a thing is not true:
 If P, then Q.
 Not Q.
 Therefore, not P.
Notice that all these proofs must start with a proposition already verified or denied.
Now examine this line of reasoning:
 If P, then Q.
 If Q, then R.
 If R, then P.
 P.
 Therefore, Q.
 Therefore, R.
 Therefore, P.
Each of these three conditional statements might be part of a valid argument if one uses it alone. But when one uses them all together, and then tries to assert P, one finds that the assertion is a necessary step in the proof. The line of reasoning is thus not a line at all, but a circle. One might as well assert "P; therefore P" and expect that to standand thus commit the related fallacy of proof by assertion.
For this reason, every logical system must begin with a set of generally accepted assumptions called postulates or axioms (from the Greek αξιος or axios worthy or deserving). (Similarly, any set of definitions must start with a set of fundamental terms that need no definition.) But an axiom, by definition, must be a fundamental property of nature upon which all reasonable people can agree. To attempt to prove an axiom by asserting a condition that depends upon it, is unavailing and unacceptable.
An example
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Circular reasoning would include using any scientific model that assumed a particular fact of nature in order to infer that fact later on. The classic example of this is the use of carbon14 dating to infer a great age for the earth. According to this model, cosmic rays strike atoms of nitrogen (specifically, ^{14}N) and convert them to ^{14}C. This ^{14}C then combines with oxygen in the air to form radioactive carbon dioxide. Plants take this up and form radioactive glucose, and animals that consume these plants consume the sugars and thus take in ^{14}C into their bodily systems. As long as the animals are alive, the ^{14}C in their systems is in dynamic equilibrium with the level of ^{14}C in the atmosphere. When they die, the ^{14}C begins to decay and is no longer subject to replacement. Thus, by knowing the level of ^{14}C in the atmosphere and comparing it to the level of ^{14}C in the animal's remains, one can determine how long ago the animal died.
The problem is that the model assumes that the level of ^{14}C in the atmosphere is itself in dynamic equilibrium. That in turn requires the earth to be very old. If the earth is young, then the level of ^{14}C would not have been at equilibrium in the earliest years of the earth (and still might not be), and thus all computed "ages" for organic fossils would be exaggerated, some greatly.
See Also
