From Explosion to Trivialisation: Rethinking Logical Standards
Explosion is the logical rule stating that from any contradiction, anything follows. This principle turns contradictions into potentially catastrophic events because if encountered, they render every proposition true. The only logical system where everything is true is known as the Trivial theory. If this theory holds, inquiry ceases, because every statement you make is true. For example, both "There is a Sun" and "There is no Sun" would be true. The implications stretch to any declaration: "The world is a fiction" and "The world is real" would simultaneously hold true, as would the notion that a sudden billion-dollar increase in your bank account could justify quitting your job. Clearly, triviality is bad.
Debates among logicians about the true nature of logic, its rules, and whether multiple logics exist are ongoing, but all agree that triviality is unacceptable. The rule of explosion is generally applied negatively in logical practice; its primary use is to prevent the derivation of contradictions because if one does arise—boom—everything becomes true.
In mathematics, the rule of explosion is pragmatic. Mathematical research often involves setting foundational principles and deriving results without contradiction. However, as I discuss in Logic in the Wild, specifically in the section "Logic in the Weird," with explosion logic becomes ineffective in scenarios that are weird or absurd. Even the concept of truth can appear contradictory, and simple logical arguments can demonstrate this paradox. To counteract these issues, logicians have concocted theoretical monsters—complex constructs they claim enhance our understanding of truth, though often in ways that are ad hoc and arbitrary.
A more progressive approach is to abandon the rule of explosion and accept some contradictions as inevitable. This leads to the notion of "trivialisation," a broader principle I propose, which infers that everything follows from incoherence. If logic, serving as the guardian of coherence, fails, then incoherence ensues, which is detrimental. While twentieth-century logicians obsessed on reducing trivialisation to explosion, the new millennium is seeing a shift towards a more tolerant perspective.
One challenging aspect remains: how do we measure incoherence if inconsistency alone is insufficient? Ensuring coherence without enforcing strict consistency is a complex but essential task, particularly within community settings where overly stringent logical standards can trivialise diverse viewpoints. This issue underscores the need for a reevaluation of how we apply logical principles to ensure they enhance rather than stifle discourse.
