Zeroth order logic

Sometimes known as Aristotelian or propositional logic. It deals with simple propositions. The canonical one being:

Socrates is a man
All men are mortal
Therefore Socrates is mortal

An argument of this form with two premises is called a syllogism.

All arguments of the type above can be formalized as:

\begin{equation} \begin{prooftree} \AxiomC{$p$} \AxiomC{$p\rightarrow q$} \BinaryInfC{$q$} \end{prooftree} \end{equation}

where the meaning of $p$ and $q$ is left completely arbitrary.