Dialetheism Alongside Validity as a Social Construct and Logical Nihilism

This week I argued that validity is a social construct and I defended logical nihilism. While I’m at it, I might as well tell you why I’m a dialetheist. To be a dialetheist is to believe in the existence of true contradictions. This might sound perplexing at first. Take, for instance, statements like "I am human and I am not human" or "It’s 23 degrees in Auckland and it isn’t 23 degrees in Auckland." These are clear contradictions, and it would indeed seem absurd to claim they are true. In fact, they are false. This initial sense of absurdity is precisely the challenge that dialetheism addresses.

Historically, in the European tradition, Aristotle is often recognized for formulating the law against contradictions. Rather than definitively defending this law, Aristotle articulated it in various ways, some of which are blatantly false. Despite these inaccuracies, his authority has consistently been used to deny the possibility of contradictions. This complex narrative, including Aristotle's contributions and their implications, is more deeply explored Logic in the Wild.

As I’ve discussed this week, Twentieth-century logic has been obsessively seeking the foundations of mathematics, which are supposed to provide logical underpinnings for all reasoning considered 'valid' (a notion I argue is a social construct). Therefore, asserting the existence of true contradictions in the 21st century directly challenges this long-established tradition, particularly within mathematical logic. To claim that the quest for mathematical foundations was merely a dream, disrupted by the reality of true contradictions, is naturally (and here I empathize with the reaction) perceived as an offense.

However, recognizing true contradictions does not mean we forsake all rationality. Accepting every contradiction would lead to triviality, where everything is considered true - a situation no logician accepts. Even we dialetheists, who acknowledge certain contradictions, staunchly reject this notion of triviality. Our focus lies on what I describe in "Logic in the Wild" as 'insolubles,' statements that, based on first principles, imply each other’s opposites. The Liar paradox is a classic example, asserting “I am a false sentence.” This paradox demonstrates a true contradiction through its self-referential nature: if it’s true, then what it says is true, namely that it’s false, so it’s false. But if it’s false, it’s false that it is false, so it is true (double negation, first principle). So it’s a true contradiction. Weird? Sure, but not trivial.

In the chapter “Logic in the Weird” of Logic in the Wild, you’ll discover how logic guards coherence in the face of the weird, the absurd, the paradoxical, and the insoluble. Writing this book has altered my perspective, especially regarding how logic functions within a community. It has led me to a broader viewpoint that accommodates the practices of twentieth-century mathematical logicians, who sought coherence in mathematics, while also allowing logic to uphold coherence in more perplexing contexts. Recognizing that logic isn't governed by universal laws (logical nihilism) frees it within the community, offering an alternative to full-blown emotional empathy. This, I believe, marks the beginning of logic’s rehabilitation.