Paraconsistent logics find structure in our inconsistent world

By | January 11, 2022

Zach Weber writes:

Here is a dilemma you may find familiar. On the one hand, a life well lived requires security, safety and regularity. That might mean a family, a partner, a steady job. On the other hand, a life well lived requires new experiences, risk and authentic independence, in ways incompatible with a family or partner or job. Day to day, it can seem not just challenging to balance these demands, but outright impossible. That’s because, we sense, the demands of a good life are not merely difficult; sometimes, the demands of a good life actually contradict. ‘Human experience,’ wrote the novelist George Eliot in 1876, ‘is usually paradoxical.’

One aim of philosophy is to help us make sense of our lives, and one way philosophy has tried to help in this regard is through logic. Formal logic is a perhaps overly literal approach, where ‘making sense’ is cashed out in austere mathematical symbolism. But sometimes our lives don’t make sense, not even when we think very hard and carefully about them. Where is logic then? What if, sometimes, the world truly is senseless? What if there are problems that simply cannot be resolved consistently?

Formal logic as we know it today grew out of a project during the 17th-century Enlightenment: the rationalist plan to make sense of the world in mathematical terms. The foundational assumption of this plan is that the world does make sense, and can be made sense of: there are intelligible reasons for things, and our capacity to reason will reveal these to us. In his book La Géométrie (1637), René Descartes assumed that the world could be covered by a fine-mesh grid so precise as to reduce geometry to analysis; in his Ethics (1677), Baruch Spinoza proposed a view of Nature and our place in it so precise as to be rendered in proofs; and in a series of essays written around 1679, G W Leibniz envisioned a formal language capable of expressing every possible thought in structure-preserving, crystalline symbols – a characteristica universalis – that obeys precise algebraic rules, allowing us to use it to find answers – a calculus ratiocinator.

Rationalism dreams big. But dreams are cheap. The startling thing about this episode is that, by the turn of the 20th century, Leibniz’s aspirations seemed close to coming true due to galvanic advances across the sciences, so much so that the influential mathematician David Hilbert was proposing something plausible when in 1930 he made the rationalist assumption a credo: ‘We must know, we will know.’ [Continue reading…]

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