A question on the truth-functional connectives Hi guys I'm new to sentential logic and here is the problem: 
Suppose that there is a truth-functional connective "#", and we only know that (P # (P # P)) is a tautology. Now we are asked to say something, if any, about the truth-table of "#". 
The answer given is : The truth-value of (P # P) must be T. I cannot understand this thoroughly. What I have learned about the truth-functional connective is only its definition, which states that the truth-value of the new sentence constructed by it depends only on the truth-value of its components sentence. Then I know if we can write down the truth table for a connectives like "$\neg$", it would be a truth-functional connective since the value of "$\neg$P" depends only on P. 
This structure (P # (P # P)) looks a bit complicated for me, and can anyone help me to explain its meaning? 
 A: Once you know the definition of $\#$, you evaluate something like $(P \# (P \# P))$ just like other mathematical statements: first evaluate the inside $(P \# P)$, and once you know what that is (say, $(P\#P)$  evaluates to $Q$), then evaluate $(P \# Q)$.
For example, suppose # is defined by the following truth table:
\begin{array}{c|c|c}
 P & Q & (P \# Q) \\ 
\hline
 T & T & F \\  
 T & F & T \\
 F & T & T \\  
 F & F & T \\
\end{array}
Then we can work out $(P \# (P \# P))$ as follows:
\begin{array}{c|c|c}
 P & (P \# P) & (P \# (P \# P))\\ 
\hline
 T & F & T \\  
 F & T & T \\
\end{array}
And so now we see that $(P \# (P \# P))$ is a tautology ... meaning that $(P \# (P \# P))$ can be a tautology without $(P \# P)$ being true. Indeed, I just defined the $\#$ as the NAND which was already pointed out to be a counterexample to the claim the book apparently makes.
OK, so is there maybe something else that we can say about the definition of $\#$? Let's consider all of the 16 possible definitions of $\#$:
\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}
 P & Q & \#_1 & \#_2 & \#_3 & \#_4 & \#_5 & \#_6 & \#_7 & \#_8 & \#_9 & \#_{10} & \#_{11} & \#_{12} & \#_{13} & \#_{14} & \#_{15} & \#_{16}\\ 
\hline
 T & T & T & T & T & T & T & T & T & T & F & F & F & F & F & F & F & F\\  
 T & F & T & T & T & T & F & F & F & F & T & T & T & T & F & F & F & F\\
 F & T & T & T & F & F & T & T & F & F & T & T & F & F & T & T & F & F\\  
 F & F & T & F & T & F & T & F & T & F & T & F & T & F & T & F & T & F\\
\end{array}
And now let's evaluate $\varphi_i=(P \#_i (P \#_i P))$ for each of these 16 possible definitions (I'll flip the table so I have some more room):
\begin{array}{c|c|c}
\# & P = T & P = F\\
\#_1 & (T \#_1 (T \#_1 T)) = (T \#_1 T) = T & (F \#_1 (F \#_1 F)) = (F \#_1 T) = T\\
\#_2 & (T \#_2 (T \#_2 T)) = (T \#_2 T) = T & (F \#_2 (F \#_2 F)) = (F \#_2 F) = F\\
\#_3 & (T \#_3 (T \#_3 T)) = (T \#_3 T) = T & (F \#_3 (F \#_3 F)) = (F \#_3 T) = F\\
\#_4 & (T \#_4 (T \#_4 T)) = (T \#_4 T) = T & (F \#_4 (F \#_4 F)) = (F \#_4 F) = F\\
\#_5 & (T \#_5 (T \#_5 T)) = (T \#_5 T) = T & (F \#_5 (F \#_5 F)) = (F \#_5 T) = T\\
\#_6 & (T \#_6 (T \#_6 T)) = (T \#_6 T) = T & (F \#_6 (F \#_6 F)) = (F \#_6 F) = F\\
\#_7 & (T \#_7 (T \#_7 T)) = (T \#_7 T) = T & (F \#_7 (F \#_7 F)) = (F \#_7 T) = F\\
\#_8 & (T \#_8 (T \#_8 T)) = (T \#_8 T) = T & (F \#_8 (F \#_8 F)) = (F \#_8 F) = F\\
\#_9 & (T \#_9 (T \#_9 T)) = (T \#_9 F) = T & (F \#_9 (F \#_9 F)) = (F \#_9 T) = T\\
\#_{10} & (T \#_{10} (T \#_{10} T)) = (T \#_{10} F) = T & (F \#_{10} (F \#_{10} F)) = (F \#_{10} F) = F\\
\#_{11} & (T \#_{11} (T \#_{11} T)) = (T \#_{11} F) = T & (F \#_{11} (F \#_{11} F)) = (F \#_{11} T) = F\\
\#_{12} & (T \#_{12} (T \#_{12} T)) = (T \#_{12} F) = T & (F \#_{12} (F \#_{12} F)) = (F \#_{12} F) = F\\
\#_{13} & (T \#_{13} (T \#_{13} T)) = (T \#_{13} F) = F & (F \#_{13} (F \#_{13} F)) = (F \#_{13} T) = T\\
\#_{14} & (T \#_{14} (T \#_{14} T)) = (T \#_{14} F) = F & (F \#_{14} (F \#_{14} F)) = (F \#_{14} F) = F\\
\#_{15} & (T \#_{15} (T \#_{15} T)) = (T \#_{15} F) = F & (F \#_{15} (F \#_{15} F)) = (F \#_{15} T) = F\\
\#_{16} & (T \#_{16} (T \#_{16} T)) = (T \#_{16} F) = F & (F \#_{16} (F \#_{16} F)) = (F \#_{16} F) = F\\
\end{array}
So, we see that $(P \# (P \# P))$ is a tautology for $\#_1$, $\#_5$, and $\#_9$. What do these definitions have in common? Well, one thing to note is that for all of them it holds that $(P \# Q)$ is true whenever $P$ is false. And that, I think, is pretty much the most interesting thing you can really point out for this problem.
A: To say that $P\#(P\#P)$ is a tautology means that $F\#(F\#F)=T$ and $T\#(T\#T)=T.$
Note that $F\#F=F$ implies $F\#(F\#F)=F\#F=F.$ Thus, in order for $F\#(F\#F)=T$ to hold, we must have $F\#F=T$ and $F\#T=T.$
$T\#(T\#T)=T$ holds if either $T\#T=T$ (and we don't care about $T\#F$) or else $T\#T=F$ and $T\#F=T.$
So there are three possibilities for the connective $\#$:
$$P\#Q=T$$
or
$$P\#Q=P\rightarrow Q$$
or
$$P\#Q=\neg(P\land Q).$$
