Circular reasoning is the practice of assuming something, in order to prove the very thing that you assumed. In Logic-speak, you assume that proposition A is true, and use that premise (directly or indirectly) to prove that proposition A is true. This is one of many logical fallacies that routinely get used in heated arguments, and is actually a special case of the fallacy of false assumptions.
Popular examples of Circular Reasoning include 'The Bible must be infallible - this verse says it is the word of God!', and 'The government always obeys the law - this piece of legislation says they must!'.
Prove Anything!
Circular reasoning is very useful because anything at all can be proved with it, including things that are obviously false. This delightful property is easily provable, and is shown how below:
For example, take the statement 'Circular Reasoning can be used to prove anything.'
Now, clearly this statement is part of anything.
Therefore, because 'Circular Reasoning can be used to prove anything', the statement can be proved.
Therefore 'Circular Logic can be used to prove anything.'
This is of course a special case of the logical concept that 'false implies everything'. In other words, if you start with a false premise (like the premise that circular reasoning is a valid form of logical reasoning) then you can 'prove' that any statement is true. If you doubt that a single faulty starting point can be used to construct a towering edifice of nonsense, just look at the Monster Raving Loony Party.
In Practice
Circular reasoning can often be used by stealth to complete proofs that would otherwise be very difficult, or indeed impossible. Don't ever say that you will assume the property you want to prove. Instead, just derive stuff from it and assume the reader knows what you're talking about. Now do several pages of obscure and fiddly maths, preferably using lots of arrows from one section to another. Finally, bring it all back together again, and say that therefore the desired property is true, and do a big 'QED' after it. Success is assured, because by this stage the reader will have forgotten what you started off by assuming, and probably be half asleep to boot.
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