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.