8. Testing for Validity

An Introduction to Truth Tables:  Testing for Validity

We have focused on constructing truth tables for single logical expressions, where these expressions can range in complexity.  For a given logical expression, we can determine under what conditions is that statement true or false overall.  But we have yet to consider perhaps the most important function of the truth table:  to determine whether an argument is valid.

An argument is, by definition, composed of multiple statements.  Every argument must include one conclusion and at least one premise.  Validity refers to the logical relationship between the statements that make up an argument.

An argument is valid if and only if it is impossible for ALL of the premises of an argument to be true while the conclusion is false. 

To test for validity, we need to determine whether a particular relationship between the statements that make up an argument is possible.  We have already determined by the governing connective column of a logical expression represents the ‘overall’ truth properties of that expression.  Since validity asks us to make an assessment about the relationship between logical expressions, what this amounts to in practical terms is an assessment of the governing connective columns found within the expressions that make up the argument.

A truth table is a catalogue of inputs and outputs for a logical expression.  The table ‘tracks’ what happens when we consider, for example, P to the TRUE and Q to the TRUE, P to be TRUE and Q to the FALSE, and so forth.  In terms of validity, we can use a truth table to determine if there are conditions where a given set of inputs for the atomic sentences employed in the argument turn out to produce a situation where the argument is valid.  Put differently, the truth table can be used to determine whether there is a row (a particular set of inputs) where the governing connective column for each premise is TRUE and, on that same row, the governing connective column for the conclusion is FALSE.

Testing for the validity of an argument is making a comparison among the governing connective columns for the statements that make up an argument.  So, in order to make such a comparison, we must construct a truth table that includes all of the statements that make up the argument – the premise(s) and the conclusion.

We begin our test for validity in a familiar fashion.  On one large truth table, we complete its truth table for each individual logical statement.  Since our table is a bit more complicated, we need to introduce some new symbols to help make sure that we mark out the premises and the conclusion.  We include these symbols in the header row of our large truth table.

Symbol:

, (Comma):  In arguments involving multiple premises, commas are used to separate the premises from each other.

|- (Therefore):  Since every argument must have one conclusion and at least one premise, the ‘|-‘ symbol indicates which statement is the conclusion of the argument.

Example:

P & Q, R -> ~(P v Q), ~P |- P <-> ~Q

This is an argument that has three premises that are separated from each other using commas; the conclusion of this argument is P <-> ~Q, given that this is the statement that follows the ‘|-‘ symbol.

Once we have completed the truth table for each logical expression (treating each expression as independent from all other expressions found on the table), we then focus our attention on the governing connective columns.  We assess whether, given a set of inputs for the atomic sentences, the argument is capable of producing a possible ‘logical world’ where all of the premises are TRUE overall and yet the conclusion is FALSE overall.  If the argument is capable of producing such a scenario, then the argument must be invalid.  An invalid argument is one where it is possible for true premises to lead to a false conclusion.  If, however, there are no inputs within the argument that produce a scenario where the premises are TRUE and the conclusion is FALSE, then the argument overall must be considered valid.

In practical terms, testing for validity amounts to completing one large table that includes all of the statements that make up the argument, focusing our attention on the governing connective columns of the statements, and then looking to see if a particular pattern emerges if any of the rows of the truth table.

Importantly, at this stage of our test, we are only concerned with the governing connective columns for each logical statement.  Since an invalid argument is one where it is possible for the conclusion to be false even when all the premises are true, we are looking for a row on the truth table where the governing connective columns for all the premises are TRUE and, on that same row, the governing connective column for the conclusion is FALSE.

For an argument with one premise, the pattern we are looking for is:

T |- F

For an argument with two premises, the pattern we are looking for is:

T, T |-F

For an argument with three premises,

T, T, T |- F

And so forth.  If we cannot find a row where, given the length of the argument, the appropriate pattern emerges, then it must be the case that there is no possible set of inputs (no possible logical world) that would result in true premises leading to a false conclusion.  If we do find at least one row where the pattern emerges, then the argument is invalid, regardless of how many other rows in the table ‘pass the test.’

In the next chapter, we will run through an example of using truth tables to determine whether an argument is valid.