[reposted April 7, 2001]

HE TRUTH TABLE for a valid argument schema has no line on which the premises can be true while the conclusion is false; the truth table for an invalid argument, on the other hand, does have at least one such line. When we construct a truth table for an argument schema, it is the absence or presence of such a line that identifies the schema as valid or invalid.

The object of the procedure of "truth value assignment" is to discover without actually constructing the whole truth table, whether the truth table for a schema would contain such a line; and to specify one such line if the schema is invalid, or to prove that there would be no such line if the schema is valid.

Normally when we test an argument schema, we do not yet know whether it is valid or invalid. Therefore we must be systematic, for otherwise our failure to specify a line that sketches a counter example will not prove that there is no such line. It is important to note that when we use "truth value assignment," we are engaged in a sequence of inferences: we begin by assuming that on some line of the argument schema's truth table, the premises are true and that the conclusion is false; we then infer what values the letters must have for that assumption to hold. If we succeed in specifying values of the sentence letters which result in all the premises' being true while the conclusion is false, we will have sketched a counter example to the schema: it is therefore invalid. On the other hand, if we find that the sentence letters cannot be assigned values necessary for the assumption to hold, then the assumption must be rejected: the schema is then valid.

In order to be systematic in following the procedure of truth value assignment, we will observe these rules:

1. Lay the schema out horizontally on your sheet of paper.

2. Assign numbers in sequence to the steps in which you first assign truth values to the various sentence letters, or to the steps in which you are able to narrow the possible options for the values which letters can have (e.g., if you determine that two letters must have opposite values).

3. When you first assign a truth value to one occurrence of a letter, place a circle around that assignment (placing the appropriate number next to it); then write that value--without circles--under every other occurrence of that letter in the argument schema. (An uncircled truth value shows that the assignment was originally determined elsewhere in the schema.)

4. Each time you make a new assignment, go through the premises and conclusion and show to what values they "resolve"--using underlining to indicate a schema whose resolution is found below the line. When a premise resolves to "T" or the conclusion resolves to "F," place a checkmark prominently below it: you will not need to deal with that sentence schema further.

5. If you find a sentence letter or schema which must be assigned two opposite values in order to make the premises true and the conclusion false, place an "X" nearby: you have discovered that the original assumption has produced a contradiction and therefore must be false; the argument schema is valid.

6. If you reach a point in your reasoning where there are several options for possible assignments, and no option is dictated by the assumption that the premises are true while the conclusion is false, then draw an horizontal line: to the side indicate that you will now assume that a certain sentence letter has a specified value, e.g., "assume P = 'T'," and then proceed as before. If you now find a line sketching a counter example, you have succeeded in showing that the argument schema is invalid. On the other hand, if you still have not found a line sketching a counter example, you must draw another horizontal line, repeating the original schema below the line, and indicating the new (and opposite) assumption, e.g., "assume P = 'F'." If both the assumption that P = "T" and P = "F" fail to produce a line which sketches a counter example, you have shown the argument schema to be valid.