$$\def\mathblue#1{\colorbox{blue}{$#1$}} \def\mathgreen#1{\colorbox{green}{$#1$}} \def\mathred#1{\colorbox{red}{$#1$}} \def\mathhl#1{\colorbox{Orchid}{$#1$}} \def\lequiv{\fallingdotseq}$$ $\def\lequiv{\fallingdotseq}$ 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 $x \in A$,

$$\def\infer#1#2{\dfrac{#2}{#1}}$$ 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 $$\infer{Q}{P_1 \quad \ldots \quad P_n}$$ where $Q,P_i$ are propositional schemata. Here, $Q$ is called the conclusion and $P_i$ are called the premises.