On the Nature of Logic: Illogically Logical and Logically Illogical

The world of logic can often seem to stand on its head, especially when considering phrases like 'illogically logical' and 'logically illogical.' These concepts highlight the profound influence that context and perspective have on our understanding of rationality and reasoning. Let's delve into these ideas and explore the complexities that arise.

The Illusion of Classical Logic

It is remarkably easy to forget that our current understanding of logic is but one perspective, shaped by human reasoning and advancements in technology. Consider microwave ovens, an invention that within the scope of classical logic might be deemed illogical or even psychotic. However, as Keith Devlin points out, the logic of smoking may seem illogical from a health standpoint but perfectly logical from the perspective of socialization. This dual perspective challenges our assumptions about what is logical.

Logical Frameworks in Context

The term 'illogically logical' can describe phenomena that appear to conform to logical reasoning but are fundamentally flawed or based on incorrect premises. A fallacy within an argument might seem logical at first glance but collapse under deeper scrutiny. Similarly, 'logically illogical' describes scenarios that defy classical logic yet make sense within a different framework. In certain contexts, such as quantum mechanics, outcomes can defy classical logic but are still logical within the quantum theory framework.

Just as the structure of matter and the dynamics of electricity are theoretical, current technology is still very primitive. The concept that Logic is a veridical golden book with unchanging rules is a misconception. Logic itself is a human construct, defined and operated upon through various formal systems. Modern logic includes non-classical and paraconsistent logics that challenge and expand our understanding of what constitutes logical reasoning.

The Foundations of Logic

One of the key elements in classical logic is the modus ponens, a rule of inference that states: if p implies q and p is true, then q must also be true. This may seem simple and uncontentious, but it serves as a prime example of circular reasoning cloaked in logical terms. The criticism lies in the notion that the truth of the statement "if you can conceive of Zeus, then Greek mythology is real" is derived from the mere conceivability of Zeus. There is no defined material implication that clarifies the relationship between conceiving of Zeus and the truth of Greek mythology.

Challenges in Defining Logic

Attempts to define logic more precisely, like those made by Nuel Belnap and Anderson, involve using numbered annotations to boil down the scheme to a more concrete framework. However, these efforts do not resolve the inherent circularity in foundational definitions. It is akin to the game 'vish,' where defining a term using other terms circularly leads to an endless loop of references. Thus, many contemporary thinkers argue that logic, as a human construct, is not entirely veridical.

Expanding the Frontiers of Logic

External to classical logic, there are fields like ordinal arithmetic that deal with non-standard arithmetic structures. Concepts like the behavior of aleph-null, the cardinality of the continuum, can lead to surprising conclusions. For example, according to certain definitions, aleph-null plus another number might still equal aleph-null. This can be seen in the concept of logical systems that break down in hypothetical or other-worldly dimensions, where classical logic may not apply.

Consider a scenario inspired by sages and legends: in another dimension, the laws of nature might permit absurdities like a horse running through a wall or a fruit being multiplied. From an outsider's perspective, this would be illogical, as it breaks the patterns we are accustomed to. However, from within that dimension, these phenomena would be perfectly logical.

Conclusion

The argument that logic is a fixed, veridical truth system that applies in all circumstances and dimensions is contentious. While our classical logic is a powerful tool, it is not the only system or the ultimate truth. The very nature of logic suggests that it is a contextual and human-defined framework. As we continue to explore the frontiers of knowledge, our understanding of logic will undoubtedly evolve, with new challenges and questions arising.