Symbolish

Symbolish is a notation system designed for expressing mathematical, logical, and computational ideas using a compact, semantically explicit set of symbols. It seeks to balance readability for humans with precision for machines, reducing ambiguity in formal reasoning and proofs. The name blends “symbol” with a suffix implying similarity or closeness to traditional notation, signaling its goal of integration rather than replacement.

Core to Symbolish is a small, well-defined grammar. Expressions are built from atoms, such as identifiers or constants; operators that define how atoms combine; and transformations that express equivalence, inference, or rewriting rules. The syntax emphasizes locality: the meaning of a subexpression is determined by its immediate structure and operator labels, enabling reliable parsing and straightforward translation to machine logic.

Symbolish emerged in the 2010s as a collaborative effort among mathematicians, computer scientists, and educators to improve formal reasoning pedagogy and automated verification. A formal specification and reference implementation were released by the Symbolish Consortium, which promotes adapting the notation for proof assistants and educational tools while maintaining backward compatibility with core algebraic ideas.

Usage examples illustrate its approach. A simple implication might be written as [A ⇒ B], where the brackets delimit a constructive rule; a membership claim as [x ∈ S]. The system supports embedding programming-style constructs, such as [f(x) ↦ y], to denote function application and transformation.

Symbolish is designed to interoperate with existing tools, offering automatic translation to LaTeX for human-readable documents and to formal backends for proof checking. While not universally adopted, it is used in some university courses and a handful of proof-assistant projects aiming to streamline notation without sacrificing rigor.