\documentclass[12pt]{article}
\usepackage{amsmath,amsthm,amsfonts,amssymb,latexsym, hyperref}

\begin{document}
\begin{center}
\textbf{Logic: Propositional Calculus}
\end{center}

Undefined terms:  Statements and statement variables; a set of logical values $\{T, F\}$; operators $\vee, \wedge, \sim $.

\

Let $\mathbb{S}$ denote the collection of statements and assume in the following that each of $P, Q$ and $R$ is a statement.

\

Axiom L0.  To any statement $P$ a logical value can be assigned.  Two sentences are logically equivalent if they have the same truth values for the same truth values of their clauses.

\

Definition.  The statement $P$ is said to be true if and only if it has truth value $T$; the statement $P$ is said to be false if and only if it has truth value $F$.

\

Interpretations: The statement $P \vee Q$ is interpreted to mean ``$P$ or $Q$''; the statement $P \wedge Q$ is interpreted to mean ``$P$ and $Q$''; and the statement $ \sim P$ is interpreted to mean ``not $P$''.  These interpretations should be consistent with your understanding of the grammar of our (English in our case) language.

\

Axiom L1.

\begin{center}
\begin{tabular}{|c||c|}
\hline
$P$ & $\sim P$ \\
\hline
$T$ & $F$ \\
$F$ & $T$\\
\hline
\end{tabular}
\end{center}

\

Axiom L2.

\begin{center}
\begin{tabular}{|c|c||c|}
\hline
$P$ & $Q$ &  $P \vee Q$ \\
\hline
$T$ & $T$ &  $T$ \\
$T$ & $F$ &  $T$ \\
$F$ & $T$ &  $T$ \\
$F$ & $F$ &  $F$ \\
\hline
\end{tabular}
\end{center}

\newpage

Axiom L3.


\begin{center}
\begin{tabular}{|c|c||c|}
\hline
$P$ & $Q$ &  $P \wedge Q$ \\
\hline
$T$ & $T$ &  $T$ \\
$T$ & $F$ &  $F$ \\
$F$ & $T$ &  $F$ \\
$F$ & $F$ &  $F$ \\
\hline
\end{tabular}
\end{center}

\

\

Theorem 1.1.  If $P$ is a statement then:
$$\begin{array}{ll}
\mbox{i.  } &  P \vee P  =  P \\
\mbox{ii. } &  P \wedge P  =  P \\
\mbox{iii.} &  \sim (\sim P)  =  P
\end{array}$$

\

Theorem 1.2.  The operators $\vee$ and $\wedge$ are commutative: If each of $P$ and $Q$ is a statement then:
$$\begin{array}{ll}
\mbox{i.  } &  P \vee Q  =  Q \vee P \\
\mbox{ii. } &  P \wedge Q  =  Q \wedge P
\end{array}$$

\

Theorem 1.3.  The operators $\vee$ and $\wedge$ are associative: If each of $P, Q$ and $R$ is a statement then:
$$\begin{array}{ll}
\mbox{i.  } &  P \vee (Q \vee R)  =  (P \vee Q) \vee R \\
\mbox{ii. } &  P \wedge (Q \wedge R)  = (P \wedge Q) \wedge R
\end{array}$$

\

Theorem 1.4.  Each of the operators $\vee$ and $\wedge$ distributes over the other: If each of $P, Q$ and $R$ is a statement then:
$$\begin{array}{ll}
\mbox{i.  } &  P \vee (Q \wedge R)  =  (P \vee Q) \wedge (P \vee R) \\
\mbox{ii. } &  P \wedge (Q \vee R)  =  (P \wedge Q) \vee (P \wedge R)
\end{array}$$

\

Theorem 1.5. [De Morgan's Law for logic.] If each of $P$ and $Q$ is a statement then:
$$\begin{array}{ll}
\mbox{i.  } &  \sim (P \vee Q)  =  (\sim P) \wedge (\sim Q) \\
\mbox{ii. } &  \sim (P \wedge Q)  =  (\sim P) \vee (\sim Q)
\end{array}$$

\

Definition.  If each of $P$ and $Q$ is a statement then the statement $P \Rightarrow Q$ [read
``$P$ implies $Q$''] has the following truth values.

\begin{center}
\begin{tabular}{|c|c||c|}
\hline
$P$ & $Q$ &   $P \Rightarrow Q$ \\
\hline
$T$ & $T$ &  $T$ \\
$T$ & $F$ &  $F$ \\
$F$ & $T$ &  $T$ \\
$F$ & $F$ &  $T$ \\
\hline
\end{tabular}
\end{center}

\

Theorem 1.6.  If each of $P$ and $Q$ is a statement then the statement $P \Rightarrow Q$ is equivalent to the statement $(\sim P) \vee Q$.


\end{document}