The Role of Explosion in Logic
Explosion is a fundamental rule in logic that stipulates inconsistency leads to triviality. When a contradiction occurs, no matter the context, everything theoretically becomes true. In logical systems that adhere to explosion, the rule isn't so much a tool to be used as a warning to avoid contradictions. For instance, if you want to prove something as absurd as "I'm a donkey," you could start by asserting a contradiction like "the Moon is and isn't made of green jelly." Once you've established this contradiction, technically, everything becomes true—including the statement about being a donkey. Of course, I hope to have not convinced you of such nonsense!
In systems governed by explosion, while such arguments are valid, they aren't sound because the premises (like the Moon’s contradictory composition) cannot all be true. The real utility of explosion is evident in mathematical reasoning, which relies on assumptions and definitions. Here, it’s crucial that first principles not only support the theorems (like the Pythagorean theorem) but also ensure that no contradictions can be proven. Securing this guarantees the reliability of mathematical foundations, a major focus of foundational studies in the 20th century.
However, the rule of explosion is not without its critics. Why reject such a rule? Some contradictions seem inevitable. In Logic in the Wild, I refer to these as "insolubles," a term borrowed from medieval logicians to describe statements that entail their opposites, such as the Liar Paradox ("This sentence is false."). These insolubles present contradictions that could trivialize all logical reasoning if taken to the extreme with explosion.
In response, some modern logicians have moderated their stance, opting to work without the rule of explosion. Contrary to what might seem like an abandonment of rigorous logical foundations, this approach does not end the inquiry but rather complicates it. Bo logician wants triviality, and if all contradictions are true, then everything is true. That would be bad! The challenge lies in distinguishing between good and bad contradictions. For example, a statement like "The Moon is and isn’t made of blue cheese" is nonsensical and clearly problematic. However, the Liar sentence represents a tolerable contradiction because it is an insoluble.
This discussion leads to an intriguing question: if inconsistency isn't the logical measure of triviality, what is? Can we formulate a rule like explosion that accommodates inconsistency yet still flags incoherence as trivializing? In the terms of Logic in the Wild, this would translate to something like "Incoherence entails triviality." But not all cases of incoherence are so severe as to warrant every conceivable conclusion. What are your thoughts on refining our understanding of logical boundaries to better handle the nuances of real-world reasoning?
