February 9, 2026

Paracomplete logic and the Aymara language

Last week my EPFL colleague Martin Rohrmeier (Digital and Cognitive Musicology) pointed out to me the truly fascinating Aymara people and their language, which can directly express 3-valued logic (true, false, uncertain). Aymara is an “existence proof” that the rules of human reasoning are elastic and need not necessarily be dominated exclusively by the classical 2-valued Aristotelian standard. Paracomplete or “gappy” reasoning can be natural and intuitive; it depends on what you're accustomed to.

This dovetails with the intuitive sense I've arrived at in my grounded deduction project on building usable paracomplete formal systems. At the cost of slightly-more-complex deduction rules, we get far greater expressiveness: namely unconstrained recursive functions and predicates, including traditionally paradoxical ones like the Liar (“this sentence is false”), without inconsistency. We even appear to escape the “fundamental” limits that Gödel's incompleteness theorems are commonly said to impose on “all formal systems” (implicitly meaning all classical formal systems, but what else is there?). Basic grounded arithmetic (BGA), a minimal Peano-style formulation of grounded deduction, formally modeled in Isabelle/HOL, is provably consistent, semantically complete, and powerful enough to express not only arithmetic but arbitrary Turing-complete computation, a combination that Gödel proved impossible in classical reasoning.

Personally I think that grounded deduction is even usable, natural, and “makes sense” too – even more than classical logic, once you're immersed deeply enough. I doubt many others will agree at the moment; that's fine. But I wonder if anyone who grew up speaking and thinking in Aymara might find grounded deduction less foreign? If you are or know of someone who could weigh in on this, please get in touch!