Let us assume a logical system with only 3 axioms (laws of thought):
Axiom 1: A statement that is true will remain true till a change is made in the system.
Axiom 2: A statement may not be true and not true.
Axiom 3: A statement must either be true or not true.
Basic rules are that statements refer to the truth of other statements and can use boolean operators AND, OR and NOT to tell if-then statements.
Then, we state a statement, G:
G can not be proven as true.
Now, let us fill in this table (with ticks and crosses):
G is Provable true Provable false Not provable true/false
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True ?1 ?2 ?3
False ?4 ?5 ?6
Obviously, blank 2 and 4 cannot be, and can be crossed.
If G is true, then it states that it can not be proven true. Therefore, blank 1 is crossed out.
If G is false, then it can be proven, either as true or as false. Under our current assumption (of it being false), we can only prove it as being false.
Is this enough to conclude that G is false?