Argument in Discrete Mathematics

Alternatively, is there a package just for this?

Yes, I believe ebproof really fits the bill.

\documentclass{article}
\usepackage{ebproof}
\usepackage{mathtools}
\usepackage{amssymb}
\begin{document}
\begin{prooftree}
    \hypo{&P(1)}
    \infer[no rule]1{&P(2)}
    \infer[no rule]1{&\forall n \in \mathbf{N} ((P(n) \land P(n+1) \implies P(n+2))}
    \infer1{& \therefore \forall n \in \mathbf{N} (P(n))}
\end{prooftree}
\end{document}

The & character is here to tune the alignment, there is no need to explicitly declare the math. environment, and the package is flexible enough to accommodate multiple variations.

As a side note, you can get the therefore symbol using other techniques.

Since all material is in math mode, you should use an array environment, with a single column of type l, instead of a tabular environment.

\documentclass{article}
\usepackage{amssymb,booktabs}
\let\implies\Rightarrow % ? -- maybe '\rightarrow'?
\newcommand\N{\mathbb{N}}

\begin{document}
\noindent
$\begin{array}{@{}l@{}} % "@{}" suppresses whitespace padding
P(1) \\
P(2) \\
\forall n \in \N \bigl((P(n) \land P(n+1) \implies P(n+2)\bigr) \\
\midrule % for a well-spaced horizontal rule
\therefore\forall n \in \N (P(n)) 
\end{array}$
\end{document}

Almost the same solution, but in display mode. I suggest using booktabs for horizontal rules, so you have a better vertical spacing:

\documentclass{article}
\usepackage{mathtools, amssymb}%
\usepackage{booktabs}
\newcommand\NN{\mathbf{N}}

\begin{document}

\[ \begin{array}{l}
  P(1) \\
  P(1) \\
  \forall n \in \NN,\bigl(P(n) \land P(n+1) \implies P(n+2)\bigr) \\
\midrule
  \therefore\forall n \in \NN (P(n))
\end{array} \]

\end{document}