.
Comment: Recall that
a sequent can have no premises. Such a sequent is thus of the form .
Comment: By the definition
of validity, a sequent is valid if and only if, on every truth value assignment
for the sequent on which all the premises are true is also one on which
the conclusion is true. But it is trivial that every truth value
assignment makes all the premises of a sequent
with no premises true. So by this definition, for such a sequent to be
valid, its conclusion 
must
be true on all truth value assignments.
Comment: Equivalently,
is valid if and only if the truth table for
is true on all rows.
Example of a valid sequent
without premises
Example of an invalid
sequent without premises
Definition:
A sentence
is a tautology (or is tautologous ) if and only if the sequent 
is valid.
Equivalently:
Alternative Definition:
A sentence
is a tautology (or is tautologous ) if and only if it is
true on every row of its truth table.
Definition:
A sentence
is inconsistent iff it is true on no truth value assignments.
Equivalently:
Alternative Definition:
A sentence
is inconsistent iff it is false on every row of its truth table.
Example of an inconsistent
sentence
Definition:
A sentence
is inconsistent iff it is true on no truth value assignments.
Equivalently:
Alternative Definition:
A sentence
is inconsistent iff it is true on no rows of its truth table.
Definition:
A sentence
is contingent iff it is neither tautologous nor inconsistent.
Equivalently:
Theorem:
A sentence
is contingent iff it is true on at least one row of its truth table and
false on at least one row of its truth table.