Explaining Necessary Truths

TL;DR

This work investigates how explanations for necessary truths in deductive reasoning arise from two cognitive mechanisms within a SAT/CNF framework: search-time simplifications (e.g., unit clauses and resolution) and backtracking-driven fictitious contingencies when simplifications are weak. Framed within computational complexity theory, the study notes that counting explanatory structures touches difficult regimes in problems like $k$-SAT, and uses GPT-4o simulations to test hypotheses about which features people (and machines) cite as reasons for a given solution. The authors report that simplifications consistently provide strong, high-frequency explanations (unit clauses and resolution), while backtracking can generate alternative, contingent explanations, especially when simplifications fail, and that high-degree variables (influence) bias explanation choices. They further show that linguistic cues align with the underlying computational roles, discuss connections to conflict-driven clause learning (CDCL) and symmetry challenges, and outline future human studies to validate and extend these findings to mathematical discovery and deductive reasoning.

Abstract

Knowing the truth is rarely enough -- we also seek out reasons why the fact is true. While much is known about how we explain contingent truths, we understand less about how we explain facts, such as those in mathematics, that are true as a matter of logical necessity. We present a framework, based in computational complexity, where explanations for deductive truths co-emerge with discoveries of simplifying steps during the search process. When such structures are missing, we revert, in turn, to error-based reasons, where a (corrected) mistake can serve as fictitious, but explanatory, contingency-cause: not making the mistake serves as a reason why the truth takes the form it does. We simulate human subjects, using GPT-4o, presented with SAT puzzles of varying complexity and reasonableness, validating our theory and showing how its predictions can be tested in future human studies.