General
Considerations about Arguments and Statements in SL

Arguments are composed of premises
and a conclusion. Both the premises and
the conclusion are statements, no matter which function they serve in an
argument. Arguments are either valid or
invalid, and statements are either tautologous, contingent, or contradictory;
premises either are or are not consistent with each other.

If an argument is valid, it is NOT
possible to assign truth values such that it is shown that there can be all
true premises with a false conclusion, i.e., a valid argument is one in which
the falsehood of the conclusion is inconsistent with the truth of the premises. But if an argument is invalid, it is
possible to assign truth values such that the falsehood of the conclusion is
consistent with the truth of the premises.

Statements either are or are not
consistent with each other. If a set of
premises is consistent, this implies that all of those premises can be true at
the same time. In such a case, the
argument in which those premises appear could be either valid or invalid. If an argument with consistent premises is
valid, it may also be a sound argument.
But if a set of premises is inconsistent, the argument in which they
appear is valid simply due to the fact that it would never be possible to have
all the premises true at the same time.
And if all the premises cannot be true at the same time, then it would
be impossible to assign truth values (in a proof of invalidity) such that all
the premises could be true while the conclusion is false.

Every argument, whether valid or
invalid, can be stated as a corresponding conditional. A corresponding conditional statement is
simply a statement that is formed through a consideration of the premises and
conclusion f an argument in which one conjoins all the premises and indicates
that if those premises were true, and if the argument is deductively valid,
they would imply the conclusion. Thus,
a corresponding conditional from an original argument is to be formed by
conjoining all the premises and placing an implication symbol between the
premises and conclusion such that the implication is the major connective of
the resulting statement. When an
argument is valid and is given as a corresponding conditional, the truth table
test of the conditional statement is a tautology. If the argument is invalid, when given as a corresponding
conditional, it will not yield a tautology.

Furthermore, if the conclusion of an
argument is a tautologous statement, the argument in which it appears is
automatically understood to be valid since, again, it would not be possible to
assign values such that the conclusion could be false while the premises are true
SINCE THE CONCLUSION WOULD BE NECESSARILY TRUE. But a statement with a contradictory conclusion could be
invalid. This is true since a
contradictory statement is necessarily false, and a test of invalidity depends
upon the fact that an argument can be invalid only if when all the premises are
true, the conclusion could be false.

Premises are thus either consistent
or inconsistent. A truth tree test will
show whether a set of premises is consistent or not. If consistent, a proof of consistency can be done. If inconsistent, a proof of inconsistency is
appropriate. When a set of premises is
consistent, the argument in which they appear could be either valid or invalid. If the argument is valid, one or more of the
proof methods can be applied to prove that it is the case, or a truth tree or
table can be used. If an argument has
inconsistent premises, it is known to be valid "by default" and can
be tested in the same ways in which an argument with consistent premises can be
tested or proven.

Corresponding conditionals are used
and tested only with truth tables.
Statements are tested in the same way.
If an argument is invalid, a proof of invalidity will show clearly at
least one possible truth value assignment combination that yields all true
premises with a false conclusion.