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.

From Coherence to Computers: Logic's Unseen Influence on Technology

If logic serves as the guardian of coherence and sidesteps content or truth, one might wonder about its significance in society. How can a discipline that seemingly detaches itself from reality exert any influence on our daily lives? In "Logic in the Wild," I tackle this question from a practical standpoint. I demonstrate how prioritizing coherence over truth enables us to navigate through complex phenomena when truth is elusive, reconcile new information that contradicts our pre-existing beliefs, or engage constructively with others holding divergent viewpoints without descending into disputes over content or truth.

Moreover, there's an intriguing narrative highlighted by Martin Davis in "The Universal Computer," which credits logic with the inception of the computer. Far from being mere collections of software, computers are, at their core, embodiments of logic. This journey begins, not with the ancients or medieval logicians, but in the 17th century with a young boy. He harbored what he described as one of the "greatest ideas of all times"—the vision of a perfect language capable of articulating all truths. This boy was none other than Gottfried Wilhelm Leibniz, who, as an adult, often boasted about this groundbreaking concept in various letters. Leibniz's vision included not just the perfect language but also the notion of a machine that could operate using this language, enumerating all truths and, by extension, becoming the ultimate repository of knowledge and truth.

Leibniz indeed engineered some simple yet ingenious mechanical devices capable of performing basic arithmetic operations. However, beyond these inventions, his contributions to logic largely consisted of him bragging about his groundbreaking ideas in various letters. The narrative progresses to the 19th century with the mathematician Gottlob Frege, who initiated a renaissance in crafting a perfect logical language. Building on centuries of mathematical advancements, including those by Leibniz, Frege laid the groundwork for what we now recognize as Classical Logic.

It was the English logician Alan Turing who bridged the theoretical with the tangible by devising the concept of the Turing machine, the precursor to modern computers. It took approximately five decades to evolve the technology necessary to actualize this theory, but the culmination of these efforts was the realization of Leibniz’s dream: a perfect machine operating with a perfect language.

So, if the abstract study of logical patterns and the coherence of beliefs and theories seem distant from practical utility, consider the computer as a resounding rebuttal. Through logic, we not only enhance our understanding and interactions but also pave the way for technological marvels that define our contemporary existence.

Unraveling Classical Logic: A Journey from Ancient Roots to Twentieth Century Precision

What is Classical Logic? A 15-Minute Overview

Classical logic, which I prefer to call Twentieth Century logic, is a distinctly European innovation. This discipline is not 'classical' in the traditional sense of deriving directly from ancient philosophies, nor is it similar to classical music, which emerged from the Enlightenment adorned with a veneer of sophistication in churches and courts. Instead, it signifies the rigorous mathematization of an ancient discipline, formalizing thought processes and striving for universal and necessary foundations.

Though not the logic of the Ancients, classical logic inherits and transforms the legacy of those early thinkers over millennia. My narrative, as detailed in Logic in the Wild, suggests that the genesis of logic lies in the necessity to excel in public debates, accompanying the emergence of politics and decision-making through deliberation. This deliberation process—formulating, articulating, and defending ideas—benefited from logical structures that guided coherent argumentation. Logic served as a repository of reliable argument forms, known as syllogistic logic.

During the medieval period, logic was a vibrant field of study, aiming to harmonize Christian dogmas with pagan philosophy and seeking coherence in their amalgamation. Logicians tackled paradoxes such as reconciling God's omniscience with human free will, divine justice with mercy, or creating a rock so heavy that he can't lift it.

The Enlightenment saw a temporary decline in logical pursuits as the burgeoning of new sciences, along with philosophical and mathematical advancements, diverted attention. Syllogistic logic sufficed for then-current needs. However, as mathematicians delved into increasingly complex theories, dealing with the infinitely small and large, they encountered foundational uncertainties. This prompted a reassessment of logical thought, gradually incorporating mathematical operations into logic.

At the dawn of the twentieth century, Frege identified language as a source of uncertainty in mathematical constructs, advocating for new, unambiguous logical languages. These formal languages allowed for the articulation of mathematical principles as necessary truths. With solid axioms and logical rules, mathematics could ensure the integrity of truths from premises to conclusions, providing a stable foundation. That was the dream.

But then, the same paradoxes that preoccupied medieval logicians, as they delved into the extremities of theological notions, resurfaced when twentieth-century mathematicians pushed the boundaries of truth. This historical echo turned twentieth-century logic into a quest for constructing theoretical towers, a narrative thoroughly explored in Logic in the Wild. Consequently, classical logic stands as the culmination of a century's mathematical effort to structure formal languages within these meticulously crafted theoretical structures. Time’s up!