Graph-Based Logic and Sketches 1: The General Framework
  Traditional treatments of formal logic provide: 1. A syntax for formulas. 2. An inference relation between sets of formulas. 3. A rule for assigning meaning to formulas (semantics) that is sound with respect to the inference relation.   First order logic, the logic and semantics of programming languages, and the languages that have been formulated for various kinds of categories are all commonly described in this way. The formulas in those logics are strings of symbols that are ultimately modeled on the sentences mathematicians speak and write when proving theorems.   Mathematicians with a categorial orientation frequently state and prove theorems using graphs and diagrams. The theory of sketches provides a formal way to describe mathematical structures and impose requirements (such as equations) on them using graphs, diagrams and similar structures. The graphs, diagrams and other data of a sketch are formal objects that correspond to the graphs and diagrams used by such mathematicians in much the same way as the formulas of traditional logic correspond to the sentences mathematicians use in proofs.   Sketch theory has a well-known and well-developed functorial semantics corresponding to item 3 in the description of logic above. The content of this paper is to propose a structure in sketch theory that corresponds to items 1 and 2 in that description. 
Atish Bagchi, Charles Wells