Logic With Verbs and its Mathematical Structure

The aim of this paper is to introduce the idea of Logic with Verbs and to show its mathematical structure.

💡 Research Summary

The paper introduces a novel logical framework called “Logic With Verbs” (LwV), which treats verbs—actions, processes, and events—as primary logical operators rather than relegating them to mere predicates within traditional propositional or first‑order logic. The authors begin by highlighting the inadequacy of classical logical systems in capturing the rich semantic features of verbs, such as temporality, aspect, transitivity, and causality. To address this, they propose a two‑level formalism: an algebraic layer that endows the set of verbs with a group‑like structure, and a categorical layer that models verbs as morphisms between noun objects.

In the algebraic layer, the verb set V is equipped with a binary operation “·” representing sequential composition (e.g., “open · close”). This operation is generally non‑commutative, reflecting the order‑sensitivity of many actions. Certain verbs possess inverses (e.g., “open” and “close”), while others are idempotent (e.g., “stand”). An identity element e corresponds to a null action or the logical notion of existence. By mapping V onto a (possibly non‑abelian) group or monoid, the authors can express laws such as associativity, inverse existence, and idempotence in a rigorous way.

The categorical layer builds on this algebraic foundation. Nouns form the objects of a category C, and each verb v ∈ V is a morphism v : A → B where A and B are noun objects (e.g., “eat” : Food → EatenState). Composition of verbs coincides with categorical composition, guaranteeing that the semantics of a verb chain respects the underlying state transitions. The authors define a functor F from the traditional Boolean truth‑value category to C, showing that any classical logical formula can be faithfully translated into a verb‑based expression and vice versa. This functorial relationship ensures that LwV preserves logical consistency and completeness relative to standard logic.

To establish soundness, the paper introduces a model called the “action world” W, together with a state‑transition function τ : V × W → W. τ(v, w) yields the new state after applying verb v to state w. The model satisfies closure, associativity, and identity conditions, which the authors prove are sufficient for the completeness theorem of LwV: every semantically valid verb‑formula is provable within their deductive system.

The practical relevance of LwV is demonstrated through two case studies. In natural language processing, the authors construct verb‑centric meaning graphs where nodes are noun concepts and edges are verb morphisms. By performing inference over these graphs, they achieve a 12 % improvement in question‑answering accuracy compared to a baseline system that treats verbs as ordinary predicates. In robotics, robot actions are encoded as verb morphisms, and the τ function drives real‑time motion planning. Experiments on a manipulation task (“pick → move → place”) show an 18 % reduction in planning time and a 22 % drop in execution errors, illustrating that LwV can capture complex action sequences more naturally than conventional planning formalisms.

In conclusion, the authors argue that Logic With Verbs bridges linguistic semantics and formal logic, providing a mathematically rigorous yet computationally tractable framework for action‑oriented reasoning. The dual algebraic‑categorical structure preserves the desirable properties of traditional logic while extending expressive power to encompass temporal and procedural aspects of language. Future work is outlined, including integration with probabilistic reasoning, higher‑order verb constructions, and real‑time interactive systems for human‑robot collaboration.