TITLE: Is the disjunction of these two false statements true?
QUESTION [8 upvotes]: Let $\mathcal{L}$ be the formal system of statement calculus. Let $A$ be a formula of $\mathcal{L}$. Then each of the three formulas 
\begin{gather*}
A \vee (\sim A) \\
A \implies [(\sim A) \implies A] \\
(\sim A) \implies [A \implies (\sim A)] \\
\end{gather*}
is a theorem of $\mathcal{L}$. 
By case analysis, 
$$[(\sim A) \implies A] \vee [A \implies (\sim A)]$$
is a theorem of $\mathcal{L}$. But this seems intuitively incorrect. 
For example, each of the statements
\begin{gather*}
(\forall x \in \mathbb{N})[(x < 3) \implies (x \ge 3)]\\
(\forall x \in \mathbb{N})[(x \ge 3) \implies (x < 3)]
\end{gather*}
is false. However, the disjunction of these two statements is true, by the argument above.
Edit. I guess the example was not very good. 
Suppose you flip a coin. It is a fair coin and it cannot stand up on its side. It lands at your feet on a solid, flat surface. 
Let $A$ be the statement that the coin shows heads. Then $(\sim A)$ is the statement that the coin shows tails. The statement
\begin{gather*}
\text{"if the coin shows heads, then it shows tails,}\\ 
\text{or}\\ 
\text{if the coin shows tails, then it shows heads"}
\end{gather*} 
is true, even though both disjuncts are false. 
Maybe this question no longer belongs on math.stackexchange.com.

REPLY [8 votes]: Perhaps what is counter-intuitive is the fact that whenever $A$ is false, then $A\Rightarrow B$ is true, based on how logical implication is defined:
$$
\begin{array}{|c|c|c|}
\hline
A & B & A\Rightarrow B \\
\hline
{\bf F} & F & {\bf T}\\
{\bf F} & T & {\bf T}\\
T & F & F\\
T & T & T\\
\hline
\end{array}
$$
Applied to the example of the coin:
if the coin shows heads, then it shows tails
                     or
if the coin shows tails, then it shows heads

Let's say a coin is flipped, and it shows heads.
Then the following statement is false:
the coin shows tails

and thus the second half of the disjunction is true:
if the coin shows tails, then it shows heads

therefore the whole disjunction is true.
Similarly if tails were shown.