$$$$ $$ What mathematics describes is a long standing issue in philosophy. It is hard to break into this subject, but I hope that my intent can be shared. Since I am a mathemtics student who is interested in formal systems, I can’t help to feel nervous when my professor says $A$ is a subset of $B$ if for all $x$, $x\in B$ whenever $x\in A$,

$$$$ Rule systems are seen in almost every programming languages paper. They are a tool for conveying very precise notions of computation. In general, is not easy to give a succinct definition of a rule system, but one can be described easily by adopting a specific representation. A rule system consists of a set of statements of the form $${\displaystyle \frac{P_1\dots P_n}{Q}}$$ where $Q,P_i$ are propositional schemata. Here, $Q$ is called the conclusion and $P_i$ are called the premises.