Sunday, January 1, 2012

Philosophical Digression on Paraconsistent Logic

While reading about Godel's Incompleteness Theorems before writing my post on it, I encountered a strange idea: The notion of "paraconsistent logic". As I explain in my Godel post, the Principle of Explosion states that "from a contradiction, all else follows"; if a formal theory is capable of expressing a contradiction, then both a statement and its negation can be used for proofs. This often leads to all statements being true in a theory. Such a theory is termed inconsistent.

But certainly not all contradictions lead to the principle of explosion! If I simultaneously believe, for instance, that an apple is both green and red, this will not lead me to the contradiction that a number can be both odd and even. Therefore, the Principle of Explosion applies only within the scope of what objects the contradiction speaks of (directly or indirectly). The study of paraconsistent logic considers the implications of having rigorous logic theories in which some contradictions are possible.

This initially seemed to me to be an academic endeavor at best and absurd at worst. But the idea stuck with me. Is it possible that the human mind is a paraconsistent deductive system? Presumably, our mind works on the basis of some logic (whether it be consistent or not). And yet, peoples' beliefs vary widely. Some find the existence of a god to be a clear, almost self-evident truth, while others find such an assertion to be nothing short of absurd. Similar situations are found throughout politics. Could it be possible that the ability for two people to follow progressions of reasoning which reach wildly different and contradictory conclusions, which to each of them seem utterly and completely proven correct, is indicative of a not entirely consistent system of thought? And yet, people do not generally believe in everything at once, as would likely be the case in an inconsistent system. Perhaps the concept of a "paraconsistent" logic is the one which most closely models the human psyche. Perhaps the notion of a "belief" is, at its core, simply a true logical statement which requires a paraconsistent theorem in order to be proven.

Just some thoughts. I admittedly haven't read much of anything on formal paraconsistent logic yet; I had never even heard of it until very recently. Any arguments for or against the viewpoint are welcome, and if I have in some way mischaracterized any concepts or terminology here, please let me know. For the time being, I definitely want to find some good introductory material to read up on about this exotic form of logic.

One last note...I'll be changing the name of the blog, as I have heard from a few people that they feel uncomfortable re-posting it due to the name such as it is. "Math, etc." sounds good to me.