Is paraconsistent logic the logic of mathematics?

In classical logic, one contradiction is enough to prove anything. From $A$ and $\neg A$ you get $A \vee B$ for any $B$ you like, and disjunctive syllogism then hands you $B$. Logicians call this the principle of explosion, ex contradictione quodlibet. It is why a single inconsistency in a classical theory is fatal: the theory becomes trivial, because every sentence becomes a theorem.

Paraconsistent logics are built so that this does not happen. My bachelor thesis at ETH Zürich (2021) looks at how they are formalised, what they give up to get there, and whether any of this describes how mathematicians actually work.

What the thesis covers

Explosion and paradox. It starts from classical propositional and first-order logic, derives explosion, and shows where contradictions come from in practice: the liar sentence and Russell’s set $R = \{x \mid x \notin x\}$, for which $R \in R \leftrightarrow R \notin R$.

The Logic of Paradox. Graham Priest’s LP adds a third truth value, paradoxical, which counts as true. I worked through the truth tables and proved two of its basic results:

• LP has exactly the same tautologies as classical logic.
• If $A$ and $B$ share no sentence letters and $B$ can be false, then $B$ does not follow from $A$. That is precisely the property that stops explosion.

What it costs. Disjunctive syllogism, modus ponens and modus tollens are only quasi-valid in LP: they preserve truth as long as nothing paradoxical is involved. I show why the implication cannot simply be patched. With a self-referential sentence (“if this sentence is true, then $A$”) and Tarski’s truth scheme, modus ponens together with absorption proves any $A$ at all, so one of the two has to go.

Paraconsistency applied to mathematics.

• Priest’s inconsistent models of arithmetic: collapsing the natural numbers modulo an equivalence yields a model in which $n = n + m$ and $n \neq n + m$ both hold, without everything else becoming true.
• Chunk and Permeate (Brown and Priest): split an inconsistent theory into consistent chunks and let only selected results flow between them. I follow its application to the Dirac delta, the “function” that is zero everywhere except at the origin and still integrates to one. Both of Dirac’s own arguments become valid reasoning when a nonstandard or smoothed source chunk passes its result to standard analysis.

Philosophy. The last part asks why anyone would want this, from contradictory laws and beliefs to Bohr’s atom and inconsistent data. It then asks which logic mathematicians really use, and whether true contradictions exist at all.

Where I landed

Getting rid of explosion by deleting inference rules is too expensive for mathematics. Most theorems are implications that we apply with something very like modus ponens, and a logic without it cripples ordinary proofs. I found Priest’s other proposal more convincing, his methodological maxim: keep using the classical rules unless there is a specific reason to suspect a paradoxical sentence is involved, and retract the proof if one turns up later. This is close to how mathematics already treats a proof that turns out to have a gap.

Mathematicians also reason more paraconsistently than the textbooks suggest. Many of us think in naive set theory, and Leibniz’s infinitesimals still teach calculus, both of them inconsistent and both useful. What the discipline aims for is consistency, as the rigorous treatments of calculus and of the Dirac delta (distributions) show. The thesis ends with Wittgenstein’s reply to explosion:

“From a contradiction everything would follow.” The reply to that is: well then, don’t draw any conclusions from a contradiction.