Content area
Humans exhibit distinct yet complementary cognitive capacities for logical and relational reasoning, foundational to understanding language processing and meaning representation. These capacities find mathematical expression in two prominent computational frameworks: formal semantics, which maps expressions to truth values through logical structures; distributional semantics, which represents meaning through contextual relationships in vector spaces. This research establishes well-defined mathematical compatibilities between these philosophies of language by constructing structural correspondences. These correspondences: (1) preserve core semantic relationships; (2) respect compositionality and logical dependencies; (3) allow for embeddings of intensional structures (e.g., modality and tense) into continuous representations. Consequently, a reconciliation emerges between `meaning as reference' and `meaning as use' while retaining computational tractability. Five fundamental results follow: (1) a delineation and classification of mathematical linguistics as distinct from yet complementary to computational linguistics, itself distinct yet complementary to natural language processing; (2) a categorical framework organizing extensional and intensional models under a cohesive theoretical structure, such that semantic representation and processing remain modular and order-independent; (3) proofs of structure-preserving homomorphisms between formal and distributional semantics, demonstrating compatibility between symbolic and sub-symbolic approaches while maintaining compositionality and logical dependencies, subject to certain limitations; (4) a generalized vector logic compatible with compositionality that respects the representation of logical operators within distributional spaces; (5) a generalized grounding problem and proposed heuristic for identifying grounding problems, with which we show that neurosymbolic semantics is, indeed, grounded. These results show that logical and relational reasoning, though distinct cognitive processes, are mathematically compatible, albeit at the cost of linearity. This research establishes a linguistically grounded and cognitively plausible foundation for investigating reasoning and semantic representation in human and machine language processing, framed as a possible solution to the symbol grounding problem in neurosymbolic artificial intelligence.
