# What is this notation style called?

I was reading a paper about implementing the statistical programming language R on the JVM, and encountered this notation I haven't seen before (I'm not formally CS-trained). What is the name of notational style? I've seen it before in other papers--usually as it related to functional programming and the REPL.

• See here1 and Backus-Naur form. Dec 7 '13 at 7:11
• I had seen BNF before (Odersky did it in the Functional Programming Coursera), but usually it was written out into words, so I figured this was something different. Please paste your comment into a full answer and I'll accept it. Dec 7 '13 at 7:56
• I believe that the fraction notation for rules of inference was introduced by [R. Smullyan, Theory of Formal Systems, 1961]. See also en.wikipedia.org/wiki/Formal_system as a starting point. Dec 7 '13 at 11:16
• actually, the fractional notation was used already in Gentzen's paper (1934) on natural deduction and sequent calculus, which I think (but am not 100% sure) is the original source. Dec 7 '13 at 13:52
• Yes, I was wrong, sorry. Smullyan also said so here, in comments: youtube.com/watch?v=P8VBeGl9Law I had rather badly misinterpreted a remark in another paper. Dec 7 '13 at 21:57

The individual statements of the form $A\Rightarrow B$ are sequents and the things that look like fractions are inference rules.