Truth functions and truth table I am trouble with the following question:
"The connective “unless” can be ambiguous, and this exercise will pinpoint
the ambiguity.
We awake at dawn, and we are told
We will have a picnic today unless it is raining at 10 A.M.
Let $P$ be "We will have a picnic today" and Q be "it is raining at 10 A.M." $P$u $Q$ denote “P unless Q.” (This is not a standard notation.) Complete
as much of a truth table as possible for PuQ, and discuss any ambiguous
lines.
My attempt 
$$\begin{array} {|c|}
\hline
P & Q & PuQ& \\ \hline
F & T & T&  \\ \hline
T & F & T \\ \hline
T & T & F \\ \hline
\end{array}$$
 A: See the discussion in this post and the extract from Stephen Cole Kleene, Mathematical logic (1967 - Dover ed  2002). 
According to his proposal [page 64] :


$A \lor B$ is $A$ unless $B$ [usually] and is $A$ except when $B$ [usually].


An "informal" argument in support of this "translation" is the following.
We can rewrite our "P unless Q" :

"We will have a picnic today" unless "it is raining at 10 A.M." 

as :

if "it is not raining at 10 A.M.", then "we will have a picnic today".

But this is "if not Q, then P", which in the truth-functional model is equivalent to :


$Q \lor P$.


But there is an ambiguity in "translating" from the natural language into the truth-functional model, as referred in the question.
If we decide for XOR, we have to leave the "equivalence" of "P unless Q" with "if not Q, then P". 
I agree with StumpyLeg's answer; three case are "clear-cut". The doubt is with the $T-T$ case: here lies the choice between OR and XOR.  
A: $P$, $Q$ both false is clear-cut: $P u Q$ is false. (If there's no rain and no picnic, the announcement was erroneous.) If there's any ambiguity, it's the T/T case (last line in the OP's table). It comes down to whether "unless" should be read as OR (then the statement is true) or as XOR (then it's false). That seems to be the intent of the example, since rain would normally nix a picnic but the announcement doesn't explicitly commit to cancellation in case of rain.
A: One might be tempted to say that "$P$ unless $Q$" is the same as "if not $Q$, then $P$". In fact this is also the semantic used e.g. in PERL, which seems to be the only programming language that allows if $condition then functioncall(); to be written as functioncall() unless !$condition;.
But in natural language the use may vary. For example, "We'll go to the movies unless it rains" may be interpreted differenetly than "If it doesn't rain, we'll go to the movies". The former seems to suggest much more strongly that also "If it rains then we do not go to the movies" holds.
A: unless = if not
P unless Q = P if not Q
I think?
