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Also known as Begging the Question, a circular argument is one that supposes the proposition (directly or indirectly) within a preceding premise. Or more explicitly, a circular argument makes a conclusion based on something that has already been presumed in the argument.
"The study of physics is beneficial because good physics makes for useful analysis."
Here, we can see that the statement seems quite reasonable until we place it into a more explicit form:
"The study of physics is beneficial because physics makes for a beneficial study."
Where the argument is of the following form for assumption A,
Premise (1): A implies B
Premise (2): Suppose B
Conclusion: Therefore A
Examples that don't follow the circular argument fallacy
Person A: God doesn't exist.
Person B: Here is some historical evidence for the miracles of Jesus.
Person A: Miracles are impossible, therefore God doesn't exist.
Here, this argument doesn't follow the particular "A is B, thus A is B" format. Instead, it appeals more to the "Argument from Personal Incredulity", which attributes that because one personally finds a premise unlikely or absurd, the premise can be supposed to be untrue, or that another favoured but unproven premise is true instead.
Kalam Cosmological argument
A noted example of circular argument is one known as "Kalam Cosmological argument".
Its structure is this:
1. Everything which begins to exist has a cause.
2. The universe began to exist.
3. Therefore, the universe had a cause.
The refutation of the Kalam Cosmological argument is actually rather simple:
The first premise "begins to exist" implies that things that do not begin to exist do not need a cause; they've always existed.
So, the challenge becomes to name some things that do not conform to this qualification, things that have always existed.
"God" is the most ready response, but is there anything else in the category of things that do not begin to exist? Is there any reason to think that there are?
If not, that is if god is the only thing that never began to exist, then the argument MUST read thus:
1. Everything which is not god has a cause.
2. The universe is not god.
3. Therefore, the universe has a cause.
So, the person attempting to use Kalam to prove the existence of god is implicitly putting god into the premise, thereby rendering the argument hopelessly circular.
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