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

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.

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    It is hard to understand what "non-mathematical" approaches, as you understand them, are. I thought I understood it (what used to be called "logic" and is now renamed into "epistemology") until the last paragraph. Excluding weaker-than-possible conclusions or inconsistent premises are studies of interesting subsystems of a grand system, a perfectly mathematical enterprise. There are relational extensions of the syllogistic, for example, radically different from the predicate calculus (referenced here), would that be "non-mathematical"? – Conifold Jan 21 at 4:21
  • @Conifold the last paragraph was meant to indicate an example of properties that can be accounted for when viewed through the lens of mathematics, but suggest, at least to me, a different heritage. – Gregory Nisbet Jan 21 at 4:42
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    Perhaps the different heritage is that traditional logicians tried to stay closer in form and motivation to natural language and reasoning rather than giving them up for more technical benefits. If that's what it is perhaps what you are looking for goes under "natural logic" and "semantics of natural language". References I linked are of this sort. – Conifold Jan 21 at 4:50
  • Logic is not "grounded" in mathematics: but, as well as many other fields of inquiry (physics, economy) the mathematical tool-set greatly improved the study of logic. – Mauro ALLEGRANZA Jan 21 at 9:08
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    Mathematics is an enormously great source for representing various definitive concepts when studying logic and similar ideas. It's more a tool with which you can express abstract quantities, draw implications, identify functions, generalise relationships, plot data, extrapolate results, rigorously prove conjectures, organise and communicate information with strong clarity, etc. To me, the question asked is analogous to "Are there approaches to language that is not grounded in writing?" – Mr Pie Jan 21 at 19:00

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.

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    I think the idea that coding is non-mathematical reflects an overly-narrow view of mathematics. I certainly consider coding to be mathematical. – Noah Schweber Jan 21 at 19:28
  • @NoahSchweber: Where do you stand on maths as a kind of logic, or logic as a kind of maths? I have to admit I think narrow definitions and trying to force hard lines between disciplines is not useful, but if maths is a kind of logic it's fair to say coding as applied logic need not be considered maths. What do you see asserting coding as maths achieves? – CriglCragl Jan 21 at 19:45

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