## study five: truth tables

The phrase "truth tables" could mislead; the phrase is ambiguous. A good definition should solve this minor difficulty, but not all difficulties associated with truth tables disappear thereby. As with other schemes, Euler circles among them, the correspondence between the characteristics of the forms and the properties of truth tables is peculiar enough to raise suspicious questions about the correspondence. In any case, the use of truth tables in this chapter is a heuristic method for the analysis of propositional forms and arguments. In fewer words, truth tables are schemata for the analysis of forms and relations among them.

As such, the use of truth tables requires some additional information about logical connectives; the construction of truth tables; and the various uses of truth tables. With the truth table method, we can describe the meaning of each of the symbols that stand for the connectives of propositional forms. We then provide explicit instructions for constructing the appropriate truth table followed by a number of applications.

Some logical connectives were introduced as far back as Chapter Two. The connectives join simple statements to form compound statements. In Chapter Two, and since then, "and," "or," "implies," and "not" have been used to connect simple statements into compounds of propositional forms. We shall now express their meanings by means of truth tables. It should be emphasized that the laws of logic (the Law of Identity, the Law of Excluded Middle, and the Law of Contradiction) are the basis for all truth tables as shall be shown.

The law that states that every proposition is either true or false is expressed in truth table fashion as follows:

p p'

_____

T F

F T

It means that when a proposition is true, its denial is false; when a proposition is false, its denial is true. Of course, there are only two rows in this truth table since there are only two possibilities: a proposition is either true or false.

Next, we show truth table descriptions of conjunction, disjunction, and implication. These truth tables consist of four rows, since there are four possibilities given two values (True and False) and two propositions.

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 and every one of its conjuncts is true. Otherwise, it is false as the following truth table shows. The first row depicts the meaning of true conjunction; the other three rows, when a conjunction is false.

p q (p q)

_____________

T T T

T F F

F T F

F F F

An inclusive disjunction is false in one and only one set of circumstances: 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. These characteristics are described by the following truth table.

p q (p + q)

________________

T T T

T F T

F T T

F F F

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

One combination of values is fatal to an implication: when the antecedent is true and the consequent false. Otherwise, an implication is true as this truth table shows.

p q (p < q)

______________

T T T

T F F

F T T

F F T

Row #2 describes the one combination of truth values in which an implication is false. If rows 3 and 4 are a source of perplexity, then substitute the following statements for the variables that satisfy the conditions of these rows to convince yourself that the implications are indeed true as depicted in the truth table.

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).

In both cases, the implication "p < q" is true.

As such, the use of truth tables requires some additional information about logical connectives; the construction of truth tables; and the various uses of truth tables. With the truth table method, we can describe the meaning of each of the symbols that stand for the connectives of propositional forms. We then provide explicit instructions for constructing the appropriate truth table followed by a number of applications.

**Logical Connectives**Some logical connectives were introduced as far back as Chapter Two. The connectives join simple statements to form compound statements. In Chapter Two, and since then, "and," "or," "implies," and "not" have been used to connect simple statements into compounds of propositional forms. We shall now express their meanings by means of truth tables. It should be emphasized that the laws of logic (the Law of Identity, the Law of Excluded Middle, and the Law of Contradiction) are the basis for all truth tables as shall be shown.

The law that states that every proposition is either true or false is expressed in truth table fashion as follows:

**Negation**p p'

_____

T F

F T

It means that when a proposition is true, its denial is false; when a proposition is false, its denial is true. Of course, there are only two rows in this truth table since there are only two possibilities: a proposition is either true or false.

Next, we show truth table descriptions of conjunction, disjunction, and implication. These truth tables consist of four rows, since 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 and every one of its conjuncts is true. Otherwise, it is false as the following truth table shows. The first row depicts the meaning of true conjunction; the other three rows, when a conjunction is false.

p q (p q)

_____________

T T T

T F F

F T F

F F F

**Disjunction**An inclusive disjunction is false in one and only one set of circumstances: 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. These characteristics are described by the following truth table.

p q (p + q)

________________

T T T

T F T

F T T

F F F

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

**Implication**One combination of values is fatal to an implication: when the antecedent is true and the consequent false. Otherwise, an implication is true as this truth table shows.

p q (p < q)

______________

T T T

T F F

F T T

F F T

Row #2 describes the one combination of truth values in which an implication is false. If rows 3 and 4 are a source of perplexity, then substitute the following statements for the variables that satisfy the conditions of these rows to convince yourself that the implications are indeed true as depicted in the truth table.

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).

In both cases, the implication "p < q" is true.

**Truth Table Construction**

The number of rows in a truth table depends on the number of distinct propositions represented by letter-variables in the argument expressed as an implication. The letter-variables are better known as "propositional variables." If an implication contains two distinct propositional variables, then the number of rows is four. Why? Because 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.

Note, "p" and "p'" are not "distinct variables;" "p" and "q" are distinct, in our sense of the word.

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.

With these concerns in mind, the construction of a truth table follows these steps:

STEP 1: Count the number of distinct variables in the expression to be analyzed. Suppose you count 3 distinct variables.

STEP 2: Determine the number of rows required for the truth table using the formula R = 2n. R = 23 = 8 rows.

STEP 3: Divide the first column in half and place T's in the first half, F's in the second half. See the first column of the figure below.

STEP 4: Divide the second column into two's place alternate 2 T's and 2 F's in the rows as shown in the second column of the figure.

STEP 5: The third column has alternate T's and F's for the full number of rows in the truth table, as shown in the third column of the figure.

p q r

________

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

________

I ii iii

STEP 6: Once you have entered all possible permutations of truth-values into your truth table rows and columns, then deal row by row with any compound statements that require analysis, assigning T's and F's under each compound, according to the previously defined meanings of contradiction, conjunction, disjunction, and implication.

For example, assume that part of an implication to be analyzed by truth table methods is a disjunction symbolized as: [(p q) + r)]. First enter the T's and F's for the conjunction (p q) in column iv, then for the other disjunct (r) in column vi. (Column vi truth-values are identical to those in column iii; identical variable.) The truth-values of column v below are true, except for Row #8. Why? Because a disjunction (inclusive or) is false only when all of its disjuncts are false; and this occurs only in Row #8: the disjuncts (p g) and (r) are both false.

p q r [(p q) + r] ROWS

____________________________

T T T T T T 1

T T F T T F 2

T F T F T T 3

T F F F T F 4

F T T F T T 5

F T F F T F 6

F F T F T T 7

F F F F F* F 8

____________________________

i ii iii iv v vi

STEP 7: Answer questions by inspection of the rows of your truth table with T's and F's. For example, does the truth of (p q) depend on the truth of (r)? You will have to examine the truth table to realize that it does not matter. (Hint: Find rows where (p q) is true.)

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. (Recall, the use of parentheses or brackets are intended as punctuation devices to indicate the sense of the expression accurately.) 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.

p q (p < q) (p q')' (p' + q)

_______________________________

1 T T T T T

2 T F F F F

3 F T T T T

4 F F T T T

________________________________

i ii iii iv v

The truth values (T's and F's), beyond the first two columns were assigned according to the definitions of the logical connectives. Now the question: What can be inferred?

If we had started with English sentences, the inferences would prove to be more interesting. Notice columns iii, iv, and 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 relations between implication, conjunction, and disjunction, described in Chapter Four. The truth table displays their interdefinability. But there is more as further development will show.

**Symbolizing Implications**

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

p only if q p < q

p thus q p < q

p therefore q p < q

p hence q p < q

p if q q < p

p since q q < p

p because q q < p

p for q q < p

p when q q < p

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 accurate symbolization of the 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?" 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 transformation of the intended sense of a proposition.

**Two Examples**

Next, we provide two examples for application of truth table methods. The first is one from Gordon Clark's Logic, modified slightly to guarantee univocal meaning; the second is from the Workbook that accompanies his book. Our 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 D.L. Moody was a successful evangelist or Billy Sunday was a failure. If Billy Sunday was not a failure, Billy Graham is. Either D.L. Moody was not a [successful evangelist] or Billy Graham is not [a failure]. If Billy Sunday was a failure, then Billy Graham is not. (p. 111)*

Let m stand for "Moody was a successful evangelist." Let s stand for "Sunday was a failure." Let g stand for "Graham is a failure. Each of the variables stands for a simple, that is to say, single proposition. The compound propositions that follow contain simple propositions connected by logical connectives according to the sense of the propositions above.

Symbolizing the compound propositions, we have:

- m + s Either Moody was a successful evangelist or Sunday was a failure.
- s' < g If Sunday was not a failure, Graham is a failure.
- m' + g' Either Moody was not a successful evangelist or Graham is not a failure.
- s < g' If Sunday was a failure, Graham is not a failure.

m s g (m + s) (s' < g) (m'+ g') (s < g')

_____________________________________________

1 T T T T T F F

2 T T F T T T T**

3 T F T T T F T

4 T F F T F T T

5 F T T T T T F

6 F T F T T T T**

7 F F T F T T T

8 F F F F F T T

_____________________________________________

i ii iii iv v vi vii

**Rows 2 and 6 are the only ones that have values of true for columns iv-vii, inclusive. Notice the contradictory values for m in column i, rows 2 and 6; nothing can be said about Moody. But s is true in rows 2 and 3, and g is false in the same rows. So it is true that Sunday was a failure, but false that Graham 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.

*The second example:*

Either there is a God or there is not. If you bet on God and win, you win infinitely. If you lose, you lose nothing. Therefore, bet on God.

Either there is a God or there is not. If you bet on God and win, you win infinitely. If you lose, you lose nothing. Therefore, bet on God.

Let g stand for "God exists." Let w stand for "you win." Finally, let l stand for "you lose nothing."

Now, symbolizing the compound propositions, we have the following:

- g + g' Either God exists or he does not.
- g < w If God exists, you win.
- g' < l If God doesn't exist, you lose nothing.
*[w + l Either you win or you lose nothing.]**

g w l (g + g') (g < w) (g' < l) [w + l]

_________________________________________

1 T T T T T T T**

2 T T F T T T T**

3 T F T T F T T

4 T F F T F T F

5 F T T T T T T**

6 F T F T T F T

7 F F T T T T T**

8 F F F T T F F

__________________________________________

i ii iii iv v vi vii

*

*Added*

**Rows 1, 2, 5 and 7 have true values in columns iv through vii. Notice the values of g, w, and l; only in row 1 are these all true. Nothing definite can be said about g, w, or l based on this truth table analysis, since there is no agreement in truth-value for a single variable of columns i, ii, or iii in Rows 1, 2, 5 and 7, where all of the compound propositions are true. These compound propositions do not constitute an argument, or at least not a complete one. If the unstated conclusion is "bet on God (a command, not a proposition)," this has not been established by this truth table analysis. If the conclusion is "you win, or you lose nothing," it would have the formulation of a constructive dilemma. It is valid if and only if each of the implications is valid and the disjunction is a complete disjunction.

**Modus Ponens Revisited**

Suppose we submit Modus Ponens to truth table analysis. In this formulation: [(p < q) (p)] < (q), we indicate that the expression is an implication, with the last " < " as the major logical connective; the premises are a conjunction within the brackets and constitute the antecedent of the implication. "(q)," of course, is the conclusion or the consequent of the implication. Can a truth table analysis reveal that Modus Ponens is a valid argument form?

#1 #2 Conclusion

p q (p < q) p

**<**q

____________________________

1 T T T T T T *

2 T F F T T F

3 F T T F T T

4 F F T F T F

____________________________

i ii iii iv v vi

Columns (iv) and (vi) are identical to columns (i) and (ii), respectively, being the identical variables. 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. 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). 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 contrast this argument form with an associated fallacy.

**Fallacy of Affirming the Consequent Revisited**

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

#1 #2 Conclusion

p q (p < q) q

**<**p

____________________________

1 T T T T T T

2 T F F F T T

3 F T T T F F *

4 F F T F T F

_____________________________

i ii iii iv v vi

Columns (ii) and (iv) are identical; columns (i) and (vi) are also identical. 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. 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. 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

**Summary**

Truth tables are schemata for analyzing the relations between different propositions, simple or compound. In this chapter, the meanings of the logical connectives for conjunction, disjunction, implication, and negation were elucidated using truth table methods. Thereafter, we set down instructions for constructing truth tables. Two examples for the implementation of truth table methods served to illustrate what can be inferred by these methods. 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 birds are singing or the baby is crying.
- If the baby is not crying, then the wind is blowing.
- Either the birds are not singing or the wind is not blowing.
- [If the baby is crying, then the wind is not blowing.]*

*Are the birds singing? Is the baby crying? Is the wind blowing?*(Gordon H. Clark. Logic, p. 111).

S = birds are singing; C = baby is crying; B = wind is blowing.

S C B (S + C) (C' < B) (S' + B') (C < B')

___________________________________________

1 T T T

___________________________________________

2 T T F

___________________________________________

3 T F T

___________________________________________

4 T F F

___________________________________________

5 F T T

___________________________________________

6 F T F

___________________________________________

7 F F T

___________________________________________

8 F F F

___________________________________________

i ii iii iv v vi vii

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