Are there modern approaches to logic that aren't *grounded* in math?

Question

Are there any modern approaches to logic that are non-mathematical in nature or not grounded in math?

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The introduction to Thoughts, Words and Things, referenced in this answer, contains an interesting passage about the history of logic and how modern readers link logic and the foundations of math.

Readers coming at this material from the point of view of modern logic may be surprised to find very few of what are sometimes called “logical results” — that is, theorems about interesting general logical truths. In fact, you may think what we are doing isn’t really "logic" at all, but more philosophy of language, or even philosophy of mind or epistemology. Why then call it logic? The short answer to this of course is that they called it "logic," and they got there first! A less contentious response would be to point out how much the close connection between logic and the foundations of mathematics in the recent period has shaped our view of what logic is, to the point of making it hard sometimes for us to think of logic in any other terms.

I've seen other comments elsewhere about the success of the mathematical approach to studying logic and how it revolutionized the field, but this passage has a rather different tone. It suggests but does not directly claim that conflating an application of logic and logic itself obscures some important and interesting ideas.

I've really enjoyed learning bits and pieces of traditional syllogistic logic, particularly the pieces that differ radically from mathematical approaches as I understand them, such as not including syllogisms with weaker-than-possible conclusions from the set of valid syllogisms, and not having syllogisms with inconsistent premises.

Answer 1

I would argue mathematics is grounded in logic, not vice versa. There's a great clip of Feynman covering how to explain things to a computer here, which I think makes this clear.

My friend is terrible at mathematics, and brilliant at computer coding - she is relentlessly logical and systematic, but probably has mild discalcula. She strongly argues against the idea facility at maths & coding correlate, or that coding is anything like maths. Code is increasingly looking more fundamental than mathematics.

Digital physics is essentially building physics out of logic, rather than mathematics. Feynman's mentor Wheeler proposed the ultimate version of this, the 'It From Bit' doctrine, that the universe is somehow built from a series of yes & no answers.

The catuskoti or tetralemma is an interesting approach from Indian logic, brought to prominence by the work of Nagarjuna. It does not conform with Greek foundations of logic, the no excluded middle & non-contradiction rules.

I would describe the Zen stories that gave rise to koans, as extremely logical, yet extremely non-mathematical. Lack of context tends to obscure this. Tozan was asked 'What is Buddha?' And replied 'Three pounds of flax.' This is the weight of flax to weave a monks summer robe. But should also be seen against the context of the doctrine of Buddha nature, and other framings of the Buddhist path like Hui Neng's verse.