1. The Purpose of a Truth Table
The purpose of the truth table is to determine all the conditions of truth and falsity for the given logical statement. The truth table allows us to determine the complete set of logical possibilities for any given statement, first by inputting all possible combinations of truth-values for the atomic sentences involved in the statement, and second by determining the output, the overall truth value of the logical statement, for all possible combinations of inputs.
Given the rigid and determined structure of the logical system we are currently studying, and given that atomic sentences within a logical statement can either be true or false, the truth table represents the complete set of logical possibilities for the statement as a whole given all possible combinations of truth values for the atomic sentences involved in the statement.
Importantly, the function of the truth table is not to tell us when a particular statement is true in our world. Rather, the truth table represents all the ways that the particular logical statement may be true or false in possible worlds. The truth table can be understood as a kind of elaborate catalogue for a given logic statement, where the goal of the table is to provide a complete record of all of the ways that the logical statement may ‘appear’ according to the rules of logic. As we complete our truth table for a logical statement, we determine under what conditions (under what set of possible circumstances) the logical statement turns out to be true or false. Since the truth table represents all of the logical possibilities for a given logical statement, we are able to read off from the truth table under what conditions is the statement as a whole true or false. In other words, the table reveals the ‘complete’ set of logical properties of the given logical statement according to the core principles of the logical system.
For our purposes, the main function of the truth table is to determine whether the given argument is valid or invalid. Remember, the definition of validity that we are working with is:
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.
Put differently, an argument is invalid if and only if there is a particular combination of truth values for the atomic sentences found in the argument (represented on the truth table) that produces the following two logical results: (1) the overall truth value for each and every logical statement that is a premise is/are TRUE, and (2) the overall truth value for the logical statement that is the conclusion is FALSE.
More practically, an argument is invalid is and only if there is a row on the truth table where the overall truth value for ALL of the premises is TRUE, AND on that same row, the overall truth value for the conclusion is FALSE.
Since the truth table represents all the logical possibilities for any given argument, the truth table is always able to reveal whether an argument is valid or invalid. A valid argument simply means that there exists no logically possible world, no possible combination of truth values for atomic sentence, where the premises of the argument are true and the conclusion of the argument is false. An invalid argument, then, simply means that there exists a possible combination of truth values for the atomic sentences of an argument to produce the problematic result where true premises support a false conclusion. To be sure, we are not here concerned with whether the given argument is true or false in our real world. Instead, we are concerned with whether, according to the rules of our logical system, the given argument is valid or invalid in principle or in terms of logical possibility.
Later in the text, we will be employing a system of derivation for proving how a given argument is valid in a direct and efficient way.
