Remember, we take the disjunction over the elements of a clause, then the conjunction over the entire clause set. So if the clause set is empty, then we have an empty conjunction. If the clause itself is empty, then we have an empty disjunction.
What does it mean to take an empty conjunction or empty disjunction? Let's consider a similar situation. Over the real numbers, what is an empty sum, or an empty product? I claim that an empty sum should be 0; an empty product should be 1. Why is this? Clearly, we have:
sum(2,3,4)+sum(5,6,7) = sum(2,3,4,5,6,7)
sum(2,3,4)+sum(5,6) = sum(2,3,4,5,6)
sum(2,3,4)+sum(5) = sum(2,3,4,5)
Now make the second sum empty:
sum(2,3,4)+sum() = sum(2,3,4)
So sum() should be 0. In the same way, product() must be 1. (Replace "sum" by "product" and "+" by "*" in the lines above.)
In general, a commutative, associative binary operation applied on an empty set should be the identity element for that operation.
Now back to your original example. Since the identity for conjunction is "true", and the identity for disjunction is "false", that is why an empty clause set is true, but empty clause is false.
A formula $A$ is a tautology if it is true with every assignment.
A formula $A$ is satisfiable if there is at least an assignemnt $v$ such that $A$ is true for $v$.
If $A$ is true for the assignment $v$, then its negation, $¬A$, is false for that assignment.
A formula $A$ is a tautology iff its negation, $¬A$, is not satisfiable.
The complement of a decision problem :
is the decision problem resulting from reversing the yes and no answers.
Thus, in a nutshell, if the answer to the problem "is $A$ in TAUT ?" is NO, then $¬A$ is in SAT.
More precisely, the problem of determining if some formula $A$ is not a tautology is thus equivalent to the problem of determining if the negation of the formula, $¬A$, is satisfiable.
It seems to me that it is only a terminological issue. Compare with :
Now we define some additional complexity classes related to $\text {P}$ and $\text {NP}$.
If $L ⊆ \{ 0, 1 \}^∗$ is a language, then we denote by $\overline L$ the complement of $L$.
We make the following definition: $\text {coNP} = \{ L \mid \overline L ∈ \text {NP} \}$.
$\text {coNP}$ is not the complement of the class $\text {NP}$. The following
is an example of a $\text {coNP}$ language: $\overline {\text {SAT} } = \{ \varphi \mid \varphi \text { is not satisfiable} \}$.
The decision problems (or languages) are complementary : not the corresponding classes of formulae.
Best Answer
Yes, that's right: contingencies are sentences which are satisfiable but which also have satisfiable negations. That is, a sentence is a contingency iff it is neither a tautology nor a contradiction.
Incidentally, tautologies are also often called validities, so we could rephrase this as "contingent = satisfiable but not valid." (It's also worth noting that I've almost never seen "contingent" on the more mathematical side of logic - "satisfiable" is almost certainly the term you want to pay more attention to.)