There are many logical systems, which relatively few people actually understand, but they all rest upon certain axioms or assumptions. However, these axioms can never be logically validated, which means that there is no logical foundation to logic. This point is not new, and it’s really only a more formal way of understanding endless child-like inquisitiveness, where we continually ask ‘Why?’ in response to any answer until we simply give up. (For a brilliantly humorous portrayal of this point, see this Louis CK clip – just be ready for some funny but wild language.) Being thoroughly logical necessarily leads to the logical conclusion that logic has no logical foundation. It is the proverbial snake eating its tail, and then its head.
Atheists, for example, may point out the circular reasoning of claiming that the Bible is the Word of God because the Bible says it is, and we know we can trust the Bible because we know it is the Word of God. Atheists may also point out the logical difficulties of the First-Cause argument, which basically says that in order to avoid an infinite regress in the cosmic causal chain of events, there must be an Uncaused First-Cause, generally called God. However, such atheists end up in very similar logical difficulties as the Christians they inveigh against, for any logical defense of logic is either infinitely circular (like the “Bible is the Word of God” argument) or an infinite regress (like the “First-Cause” argument). To the degree that atheists are correct when they argue against these theistic views, to the same degree they have undermined any logicality in their own position. (In reality, both groups need to reach much deeper understanding.)
The process of trying to find a logical foundation to logic ultimately leads to:
a) infinite circularity; or,
b) infinite regress.
And both options, (a) or (b), ultimately lead to:
c) sub-logical devolution; or,
d) trans-logical evolution.
For example, by saying that an axiom, A1, for system S1 is validated by S1, we have entered perpetual circularity, which is to say that A1 is justified by S1, and S1 is justified by A1, and, again, A1 is justified by S1, and so on. In other words, A1 –> S1 –> A1 –>∞, which is to say that A1 validates S1 and S1 validates A1, a process that has no logical end point.
However, if we validate A1 by appealing to another system, S2, then we will have to validate each axiom, such as A2, that underpins S2. That means that we will then have to appeal to a further system, S3, to validate A2, and so forth, leading to perpetual regress. In other words, A1 –> S2 –> A2 –> S3 –> A3 –> S4–>∞, which is simply to say that the process of appealing to further systems has no logical end point.
Once we realize that there really is no logical foundation to logic, whether through the necessity of infinite circularity or infinite regress, then we can either devolve to a sub-logical level, or we can evolve to a trans-logical level. For example, just because there is ultimately no logical foundation to logic, it does not mean that we thereby have free license to be completely illogical. However, by understanding the inherent limitations of logic, we can both rely upon logic so far as possible, and then transcend it when appropriate, which is to enter into the intuitive or mystical realm. Many academics quickly run from any talk of mysticism, but they are simply being hypocrites when they do so, for the fact is that the foundations of logic are a very profound mystery, a mystery that forever transcends the inherent limitations of logical thought.