Truth tables may be defined as schemata for the analysis of forms and relations among them. Their use requires additional information about logical connectives and the construction of truth tables.

Logical Connectives

The logical connectives or logical constants for "and," for "or," for "implies," and for "not"" are used to connect propositional variables to form compounds of propositional forms. The letters p, q, r, etc. stand for any propositions whatsoever. The symbol usually assigned for "not" is the ~ (tilde). We have used the prime symbol, " ' " attached to the variable letter (or logical expression) that is negated. Conjunction is sometimes assigned a centered dot ( · ) or the ampersand (&). Our notation simply places one propositional variable next to another to show conjunction. The customary symbol for a disjunction is the wedge (v), but we use the + sign to join disjuncts. Implication is most often indicated by a horseshoe symbol É but we shall continue to use the "< " sign to stand for "if ... then" relation. These logical constants are said to be truth-functional connectives because by means of their truth table definitions, we calculate the truth or falsity of compound propositional expressions. We first define negation, then show truth table definitions of conjunction, disjunction, and implication.

Negation

A proposition is either true or false. When a proposition is true, its denial is false; when a proposition is false, its denial is true. Since there are only two possibilities, there are only two rows in this truth table. The next three truth table definitions consist of four rows each because in each case there are four possibilities given two values (True and False) and two propositions.

Conjunction

A conjunction is true if and only if both conjuncts are true; or, if a conjunction consists of more than two, then it is true only if each conjunct is true. This condition is met only in the first row of the truth table; the other three rows fail to meet the condition for a true conjunction.

Disjunction

An inclusive disjunction is false in one and only one set of circumstances where both disjuncts are false. If the disjunction consists of more than two disjuncts, then a disjunction is false only when each and every one of the disjuncts is false. Otherwise, the disjunction is true, as the following truth table shows.

The fourth row depicts the meaning of a false disjunction. The first three rows, where one or the other or both disjuncts are true, complete the full meaning of inclusive disjunction.

Implication

One combination of values is fatal to an implication and that is the condition displayed in the second row of the next truth table. There the antecedent is true and the consequent false. The other rows show that an implication is true. Thus, an implication is false if and only if the its antecedent is true and its consequent is false.

Ordinarily the combination of truth values in the first two rows causes no perplexity. The truth value combinations of the last two rows, however, sometimes raise mild objections. Can a false statement imply a true one? Yes it can. Indeed, a false statement can imply another false statement. Try this. Using the expression "if x is less than 2, then x is less than 4" substitute values for "x" to obtain the truth values of rows 3 and 4 of the truth table. In both cases, the implication "p < q" is true.

Row 3: let p = "3 is less than 2," (false antecedent) and q = "3 is less than 4" (true consequent).

Row 4: let p = "4 is less than 2," (false antecedent) and q = "4 is less than 4" (false consequent).

Truth Table Construction

The instructions for a truth table construction are easy to follow. Symbolize the argument using letter variables to stand for particular propositions. The conjunction of the premises constitute the antecedent of an implication, and the conclusion becomes the consequent. Now, count the number of distinct propositional variables. Note, "p" and "p' " are not "distinct variables; "p" and "q" are distinct, in our sense of the word. If an implication contains two distinct propositional variables, then the number of rows is four. A single proposition can be true or false, two truth-values, but a compound proposition of two simple propositions has four possibilities: both can be true; the first true and the second false; the first false and the second true; and both can be false. If an expression contains three distinct propositional variables, then the number of rows is eight. The formula for calculating the number of rows is R = 2n, where R stands for rows, and n stands for the number of distinct variables. For 3 distinct propositional variables, the number of rows is 23, or 2 raised to the third power: 2 x 2 x 2 = 8 rows.

The arrangement of the two values, true and false, is governed by two practical concerns: (1) Does the truth table contain all possible combinations of true and false? (2) Does the arrangement of truth values, "T" and "F," depict, in consistent and identical fashion, a truth table that all can use without confusion. Follow these steps:

As a simple example, let us analyze this expression: (p q) + r. It reads, "either p and q, or r." It is a disjunction, and the first disjunct is a conjunction. (The order of the variables p, q, and r do not matter.)

The numbers in parentheses indicate the order of operations. Column (1), the simplest, is identical to column (iii); its the identical variable. Next, enter the T's and F's for the conjunction (p q) in column (2). Last, enter the values for the disjunction in column (3). When is a disjunction false? When all of its disjuncts are false, and that obtains in Row 8.

Suppose we wanted to display the relations between (p < q), (pq' )', and (p' + q). The first reads: if p then q; the second: it is not the case that p and not-q; the third: either not-p or q. There are 2 distinct variables; using R = 2n, the number of rows is 4. The first column will contain 2 T's and 2 F's. The second column will consist of T's and F's, for 4 rows as shown in the truth table below. In every case, the use of parentheses or brackets are intended as punctuation devices to indicate accurately the sense of the expression.

The truth values (T's and F's), beyond the first two columns were assigned according to the definitions of the logical connectives. What can we conclude? If we had started with English sentences, the inferences would prove to be more interesting. But for now, examine column iii through v. The expressions have identical truth values in these columns which means that they are logically equivalent. If one is true, the others are true also; and if any one is false, the others are false too. This truth table shows the interdefinability of implication, conjunction, and disjunction described in the previous Study.

Symbolizing Implications

Students sometimes encounter difficulty in symbolizing more complicated expressions of implications. The following list contains some of the more common expressions of implications.

Other ways of expressing implications may not have the "If ..., then" formulation. We have used the word "implies" in "p implies q;" also, "q is implied by p." Another example: "Saving faith means belief in an understood proposition" is an implication made plain as, "If you possess saving faith, then you possess belief in the understood propositions of the Gospel." The key word is the verb, means. Thus, x means y is a formula for an implication: if x, then y. The "if" introduces the antecedent of an implication, but "only if" introduces the consequent as in: "You are saved, only if you believe the Good News of the Bible." Careful attention to the sense of a proposition is required for accurately symbolizing a proposition.

Other Symbolizing Difficulties

Difficulties in symbolizing conjunctions may occur when the word "and" is absent but implied. Other conjunction words used instead of "and" are: "but," "yet," "however," "although," "whereas," "nevertheless," and sometimes "plus," or only a comma or semicolon. Is there a difference between "not both p and q" and "both not-p and not-q?" Yes, there is! The first is a denial of a conjunction, (pq)'; the second is a conjunction of denials, (p' q'). To complicate matters, sometimes "and" is used but the proposition is not a conjunction as in "1 and 1 is equal to 2," or "Peter and Paul were contemporaries."

Symbolizing disjunctions proves difficult when it is not clear which sense of or is the intended sense. Using the phrase "and/or" distinguishes the inclusive sense from the others; the phrase "but not both" signals the exclusive sense. The trouble is that these phrases are often implied, not explicitly stated. Of course, "+" stands for the inclusive sense; we have no symbol for the exclusive sense having determined that the inclusive serves our purposes well. Nevertheless, suppose the exclusive sense is intended as in "Either you are regenerate or you are forever lost." One or the other, but not both. Symbolized, it is: (r + l) (rl)'. Another minor difficulty: "Neither p, nor q" is not (p' + q'); the correct symbolization is (p + q)', a denial of the disjunction. A less difficult case is the use of "unless" in "Unless you study logic, you will believe propaganda." This proposition means "Either you study logic, or you'll believe propaganda." Again, careful attention is required to achieve the correct translation of the intended sense of a proposition.

A Demonstration

This demonstration will serve as a summary of truth table construction and application. The purpose is not only to show the advantages of symbolizing propositions, but to indicate how truth table methods may assist in understanding relations among propositions.

Either Planck was a successful physicist or Einstein was in some respects a failure. If Einstein was in some respects a failure, then Hawking is a failure. Either Planck was not a successful physicist or Hawking is not a failure. If Einstein was a failure, then Hawking is not.

Let p stand for "Planck was a successful physicist." Let e stand for "Einstein was a failure in some respects." Let h stand for "Hawking is a failure. Each of the variables, in this case, stands for a simple, that is to say, single proposition. Symbolizing the compound propositions, we have:

*Rows 2 and 6 are the only ones that have values of true for columns iv-vii, inclusive. Notice the contradictory values for p column i, rows 2 and 6; nothing can be said about Planck. But e is true in rows 2 and 6, and h is false in the same rows. So it is true that Einstein was a failure, but false that Hawking is a failure, according to this truth table analysis. The compound propositions are not offered in support of a position; so, no argument is involved and questions about validity do not apply.

Modus Ponens Revisited

Once an argument has been symbolized, the premises become a conjunction of premises. This conjunction forms the antecedent of an implication, and the conclusion of the argument becomes the consequent. We thus transform an argument into a truth-functional expression (an implication) suitable for truth table analysis.

Accordingly, Modus Ponens, expressed as an implication is: [(p < q) (p)] < (q). The major logical connective is the second "< ." The premise set, enclosed by brackets, is the antecedent; the conclusion is the consequent of the implication. Will a truth table analysis reveal that Modus Ponens is a valid argument form?

The values of columns (iv) and (vi) are identical to columns (i) and (ii), respectively, being the identical variables. (The numbers in parentheses indicate the order of operations from the first to the fourth.) If the argument is invalid, one would expect to find at least one row in which the premises are both true and the conclusion false. Inspection of Rows #2 and #4 shows "q" is false, but in both cases one of the premises is false. Only in Row #1 are the premises true; and, the conclusion is true also. In a valid argument form it is impossible for the premises to be true and the conclusion false because the form of the conclusion will be true every time the forms of the premises are true. Thus, Modus Ponens is shown to be valid by truth table methods.

The truth table reveals all T's under the major logical connective, "< " of column (v) and the last operation (4). Recall the truth table definition for implication. It is false where the antecedent is true and the consequent false. This condition does not obtain in any of the four rows. Check it out yourself! Under all possible assignments of T's and F's to the distinct variables of this valid argument form, the result reveals all T's under the major logical connective, an additional confirmation that Modus Ponens is a valid argument form.

Let us now analyze the fallacy associated with Modus Tollens.

Fallacy of Affirming the Consequent Revisited

Symbolizing the fallacy as an implication we have: [(p < q) (q)]< (p), again with the second "< " as the major logical connective. As before, the premises form a conjunction within the brackets and constitute the antecedent of the implication. "(p)," of course, is the conclusion or the consequent of the implication.

Columns (ii) and (iv) are identical; columns (i) and (vi) are also identical. (The numbers in parentheses indicate the order of the steps with "(4)" being the last step.) Again, if the argument is invalid, one would expect to find at least one row in which both of the premises are true and the conclusion false. Inspection of Row #3 shows that both of the premises are true and the conclusion false. Notice also that at Row #3, column (v) you have a "F" under the major logical connective because there you have a case of true antecedent and false consequent. Therefore, the argument form is invalid, as we knew it to be.

Don't be confused by the truth values of Row #1: the premises are true and the conclusion is true also. This only indicates the possibility of an invalid argument with true propositions. In a valid argument form, true premises imply a true conclusion --always. "Always" means in each and every row of a truth table. To repeat. The only "F" in column (v), row #3, confirms that we are in the presence of an implication with true premises and a false conclusion: an invalid inference. Thus, the Fallacy of Affirming the Consequent is shown to be an invalid argument form by truth table methods. A similar outcome of invalid would obtain with the formal fallacy associated with Modus Tollens, the Fallacy of Denying the Antecedent. One row in the truth table will reveal true premises and a false conclusion.

Summation

Truth tables are schemata for analyzing the relations between different propositions, simple and compound. In this Study, the definitions of the logical connectives for conjunction, disjunction, implication, and negation were described using truth table methods. Thereafter, we set down instructions for constructing truth tables. A demonstration for the implementation of truth table methods served to illustrate what can be inferred by these techniques. Finally, to further illustrate the usefulness of truth table methods, two argument forms, one valid and one invalid, were subjected to truth table analyses. The results demonstrated that with a valid argument form, expressed as an implication, no single row shows true premises and a false conclusion. On the other hand, the invalid argument form, expressed as an implication, revealed a row with true premises and a false conclusion. As a heuristic method, truth table analyses not only confirm validity and invalidity of argument forms, but provide a practical method for illustrating both.

Review

Either the cat is meowing or the baby is crying. If the baby is not crying, then the wind is blowing. Either the cat is not meowing or the wind is not blowing. If the baby is crying, then the wind is not blowing. (Modified from Gordon H. Clark, Logic, p. 111)

The propositional variables are "c," "b," and "w." c = cat is meowing; b = baby is crying; w = wind is blowing.

Is the cat meowing? Is the baby crying? Is the wind blowing?

© R. Anthony Coletti , 37659, Jonesborough, TN 37659.