4 CHAPTER 4

Formal Methods Of Evaluating Arguments

In chapter 1 we introduced the concept of validity and the informal test of validity. According to that test, in order to determine whether an argument is valid we ask whether we can imagine a scenario where the premises are true and yet the conclusion is false. If we can, then the argument is invalid; if we cannot then the argument is valid. The informal test relies on our ability to imagine certain kinds of scenarios as well as our understanding of the statements involved in the argument. However, because not everyone has the same powers of imagination or the same understanding, this informal test of validity is neither precise nor objective. For example, while one person may be able to imagine a scenario in which the premises of an argument are true while the conclusion is false, another person may be unable to imagine such a scenario. As a result, the argument will be classified as invalid by the first individual, but valid by the second individual.

That is a problem because we would like our standard of evaluation of arguments (i.e., validity) to be as precise and objective as possible, and it seems that our informal test of validity is neither. It is not precise because the concept of being able to imagine x is not precise—what counts as imagining x is not something that can be clearly specified. What are the precise success conditions for having imagined a scenario where the premises are true, and the conclusion is false? But the informal test of validity also is not objective since it is possible that two different people who applied the imagination test correctly could come to two different conclusions about whether the argument is valid.

The goal of a formal method of evaluation is to eliminate any imprecision or lack of objectivity in evaluating arguments. As we will see by the end of this chapter, logicians have devised a number of formal techniques that accomplish this goal for certain classes of arguments. What all of these formal techniques have in common is that you can apply them without really having to understand the meanings of the concepts used in the argument. Furthermore, you can apply the formal techniques without having to utilize imagination at all. Thus, the formal techniques we will survey in this chapter help address the lack of precision and objectivity inherent in the informal test of validity. In general, a formal method of evaluation is a method of evaluation of arguments that does not require one to understand the meaning of the statements involved in the argument. Although at this point this may sound like gibberish, after we have introduced the formal methods, you will understand what it means to evaluate an argument without knowing what the statements of the argument mean. By the end of this chapter, if not before, you will understand what it means to evaluate an argument by its form, rather than its content.

However, I will give you a sense of what a formal method of evaluation is in a very simple case right now, to give you a foretaste of what we will be doing in this chapter. Suppose I tell you:

It is sunny and warm today.

This statement is a conjunction because it is a complex statement that is asserting two things:

It is sunny today. AND It is warm today.

These two statements are conjoined with an “and.” So the conjunction is really two statements that are conjoined by the “and.” Thus, if I have told you that it is both sunny and warm today, it follows logically that it is sunny today. Here is that simple argument in standard form:

It is sunny today and it is warm today.

Therefore, it is sunny today. (from 1)

This is a valid inference that passes the informal test of validity. But we can also see that the form of the inference is perfectly general because it would work equally well for any conjunction, not just this one. This inference has a particular form that we could state using placeholders for the statements, “it is sunny today” and “it is warm today”:

A and B

Therefore, A

We can see that any argument that had this form would be a valid argument. For example, consider the statement:

Kant was a deontologist and a Pietist.

That statement is a conjunction of two statements that we can capture explicitly in the first premise of the following argument:

Kant was a deontologist and Kant was a Pietist.

Therefore, Kant was a deontologist. (from 1)

Regardless of whether you know what the statements in the first premise mean, we can still see that the inference is valid because the inference has the same form that I just pointed out above. Thus, you may not know what “Kant” is (one of the most famous German philosophers of the Enlightenment) or what a “deontologist” or “Pietist” is, but you can still see that since these are statements that form a conjunction, and since the inference made has a particular form that is valid, this particular inference is valid. That is what it means for an argument to be valid in virtue of its form.

4.1 Propositional Logic And The Four Basic Truth Functional Connectives

Watch and Learn

Learn the basics of Propositional Logic by watching A Crash Course in Formal Logic Pt 7a: Propositional Logic, Symbols, and Functions.

Propositional logic (also called “sentential logic”) is the area of formal logic that deals with the logical relationships between propositions. A proposition is simply what I called in section 1.1 a statement. Some examples of propositions are:

Snow is white

Snow is cold

Tom is an astronaut

The floor has been mopped

The dishes have been washed

We can also connect propositions together using certain English words, such as “and” like this:

The floor has been mopped and the dishes have been washed.

This proposition is called a complex proposition because it contains the connective, “and” which connects two separate propositions. In contrast, “the floor has been mopped” and “the dishes have been washed” are what are called atomic propositions. Atomic propositions are those that do not contain any truth-functional connectives. The word “and” in this complex proposition is a truth-functional connective. A truth-functional connective is a way of connecting propositions such that the truth value of the resulting complex proposition can be determined by the truth value of the propositions that compose it. Suppose that the floor has not been mopped but the dishes have been washed. In that case, if I assert the conjunction, “the floor has been mopped and the dishes have been washed,” then I have asserted something that is false. The reason is that a conjunction, like the one above, is only true when each conjunct (i.e., each statement that is conjoined by the “and”) is true. If either one of the conjuncts is false, then the whole conjunction is false. This should be pretty obvious. If Bob and Sally split chores and Bob’s chore was to both vacuum and dust whereas Sally’s chore was to both mop and do the dishes, then if Sally said she mopped the floor and did the dishes when in reality she only did the dishes (but did not mop the floor), then Bob could rightly complain that it is not true that Sally both mopped the floor and did the dishes! What this shows is that conjunctions are true only if both conjuncts are true. This is true of all conjunctions. The conjunction above has a certain form—the same form as any conjunction. We can represent that form using placeholders—lowercase letters like p and q to stand for any statement whatsoever. Thus, we represent the form of a conjunction like this:

p and q

Any conjunction has this same form. For example, the complex proposition, “it is sunny and hot today,” has this same form which we can see by writing the conjunction this way:

It is sunny today and it is hot today.

Although we could write the conjunction that way, it is more natural in English to conjoin the adjectives “sunny” and “hot” to get “it is sunny and hot today.” Nevertheless, these two sentences mean the same thing (it is just that one sounds more natural in English than the other). In any case, we can see that “it is sunny today” is the proposition in the “p” place of the form of the conjunction, whereas “it is hot today” is the proposition in the “q” place of the form of the conjunction. As before, this conjunction is true only if both conjuncts are true. For example, suppose that it is a sunny but bitterly cold winter’s day. In that case, while it is true that it is sunny today, it is false that it is hot today—in which case the conjunction is false. If someone were to assert that it is sunny and hot today in those circumstances, you would tell them that is not true. Conversely, if it were a cloudy but hot and humid summer’s day, the conjunction would still be false. The only way the statement would be true is if both conjuncts were true.

In the formal language that we are developing in this chapter, we will represent conjunctions using a symbol called the “dot,” which looks like this: “” Using this symbol, here is how we will represent a conjunction in symbolic notation:

p q

In the following sections we will introduce four basic truth-functional connectives, each of which have their own symbol and meaning. The four-basic truth-functional connectives are: conjunction, disjunction, negation, and conditional. In the remainder of this section, we will discuss only conjunction.

Here is how to understand this truth table. The header row lists the atomic propositions, p and q, that the conjunction is composed of, as well as the conjunction itself, p q. Each of the following four rows represents a possible scenario regarding the truth of each conjunct, and there are only four possible scenarios: either p and q could both be true (as in row 1), p and q could both be false (as in row 4), p could be true while q is false (row 2), or p could be false while q is true (row 3). The final column (the truth values under the conjunction, p q) represents how the truth value of the conjunction depends on the truth value of each conjunct (p and q). As we have seen, a conjunction is true if and only if both conjuncts are true. This is what the truth table represents. Since there is only one row (one possible scenario) in which both p and q are true (i.e., row 1), that is the only circumstance in which the conjunction is true. Since in every other row at least one of the conjuncts is false, the conjunction is false in the remaining three scenarios.

At this point, some students will start to lose a handle on what we are doing with truth tables. Often, this is because one thinks the concept is much more complicated than it actually is. (For some, this may stem, in part, from a math phobia that is triggered by the use of symbolic notation.) But a truth table is actually a very simple idea: it is simply a representation of the meaning of a truth-functional operator. When I say that a conjunction is true only if both conjuncts are true, that is just what the table is representing. There is nothing more to it than that.

Watch and Learn

Learn how to use Truth Tables by watching A Crash Course in Formal Logic Pt 7b: Truth Tables for Propositions

There is more than one way to represent conjunctions in English besides the English word “and.” Below are some common English words and phrases that commonly function as truth-functional conjunctions.

but	yet	also	although
however	moreover	nevertheless	still

It is important to point out that many times English conjunctions carry more information than simply that the two propositions are true (which is the only information carried by our symbolic connective, the dot). We can see this with English conjunctions like “but” and “however” which have a contrastive sense. If I were to say, “Bob voted, but Caroline did not,” then I am contrasting what Bob and Caroline did. Nevertheless, I am still asserting two independent propositions. Another kind of information that English conjunctions represent but the dot connective does not is temporal information. For example, in the conjunction:

Bob brushed his teeth and got into bed

There is clearly a temporal implication that Bob brushed his teeth first and then got into bed. It might sound strange to say:

Bob got into bed and brushed his teeth

since this would seem to imply that Bob brushed his teeth while in bed. But each of these conjunctions would be represented in the same way by our dot connective, since the dot connective does not care about the temporal aspects of things. If we were to represent “Bob got into bed” with the capital letter A and “Bob brushed his teeth” with the capital letter B, then both of these propositions would be represented exactly the same, namely, like this:

A B

Sometimes a conjunction can be represented in English with just a comma or semicolon, like this:

While Bob vacuumed the floor, Sally washed the dishes.

Bob vacuumed the floor; Sally washed the dishes.

Both of these are conjunctions that are represented in the same way. You should see that both of them have the form, p q.

Not every conjunction is a truth-function conjunction. We can see this by considering a proposition like the following:

Maya and Alice are married.

If this were a truth-functional proposition, then we should be able to identify the two, independent propositions involved. But we cannot. What would those propositions be? You might think two propositions would be these:

Maya is married

Alice is married

But that cannot be right since the fact that Maya is married, and that Alice is married is not the same as saying that Maya and Alice are married to each other, which is clearly the implication of the original sentence. Furthermore, if you tried to add “to each other” to each proposition, it would no longer make sense:

Maya is married to each other

Alice is married to each other

Perhaps we could say that the two conjuncts are “Maya is married to Alice” and “Alice is married to Maya,” but the truth values of those two conjuncts are not independent of each other since if Maya is married to Alice it must also be true that Alice is married to Maya. In contrast, the following is an example of a truth-functional conjunction:

Maya and Alice are women.

Unlike the previous example, in this case we can clearly identify two propositions whose truth values are independent of each other:

Maya is a woman

Alice is a woman

Whether or not Maya is a woman is an issue that is totally independent of whether Alice is a woman (and vice versa). That is, the fact that Maya is a woman tells us nothing about whether Alice is a woman. In contrast, the fact that Maya is married to Alice implies that Alice is married to Maya. So the way to determine whether or not a conjunction is truth-functional is to ask whether it is formed from two propositions whose truth is independent of each other. If there are two propositions whose truth is independent of each other, then the conjunction is truth-functional; if there are not two propositions whose truth is independent of each other, the conjunction is not truth-functional.

Watch and Learn

Review the information you have learned about Conjunctions by watching the Critical Thinking Academy’s video, Propositional Logic: Conjunctions

4.2. Negation And Disjunction

In this section we will introduce the second and third truth-functional connectives: negation and disjunction. We will start with negation, since it is the easier of the two to grasp. Negation is the truth-What mood functional operator that switches the truth value of a proposition from false to true or from true to false. For example, if the statement “dogs are mammals” is true (which it is), then we can make that statement false by adding a negation. In English, the negation is most naturally added just before the noun phrase that follows the linking verb like this:

Dogs are not mammals.

But another way of adding the negation is with the phrase, “it is not the case that” like this:

It is not the case that dogs are mammals.

Either of these English sentences expresses the same proposition, which is simply the negation of the atomic proposition, “dogs are mammals.” Of course, that proposition is false since it is true that dogs are mammals. Just as we can make a true statement false by negating it, we can also make a false statement true by adding a negation. For example, the statement, “Cincinnati is the capital of Ohio” is false. But we can make that statement true by adding a negation

Cincinnati is not the capital of Ohio

There are many different ways of expressing negations in English. Here are a few ways of expressing the previous proposition in different ways in English:

Cincinnati is not the capital of Ohio

It is not true that Cincinnati is the capital of Ohio

It is not the case that Cincinnati is the capital of Ohio

Each of these English sentences express the same true proposition, which is simply the negation of the atomic proposition, “Cincinnati is the capital of Ohio.” Since that statement is false, its negation is true.

There is one respect in which negation differs from the other three truth-functional connectives that we will introduce in this chapter. Unlike the other three, negation does not connect two different propositions. Nonetheless, we call it a truth-functional connective because although it does not actually connect two different propositions, it does change the truth value of propositions in a truth-functional way. That is, if we know the truth value of the proposition we are negating, then we know the truth value of the resulting negated proposition. We can represent this information in the truth table for negation. In the following table, the symbol we will use to represent negation is called the “tilde” (~). (You can find the tilde on the upper left-hand side of your keyboard.)

p	~p
T	F
F	T

This truth table represents the meaning of the truth-functional connective, negation, which is represented by the tilde in our symbolic language. The header row of the table represents some proposition p (which could be any proposition) and the negation of that proposition, ~p. What the table says is simply that if a proposition is true, then the negation of that proposition is false (as in the first row of the table); and if a proposition is false, then the negation of that proposition is true (as in the second row of the table).

As we have seen, it is easy to form sentences in our symbolic language using the tilde. All we have to do is add a tilde to left-hand side of an existing sentence. For example, we could represent the statement “Cincinnati is the capital of Ohio” using the capital letter C, which is called a constant. In propositional logic, a constant is a capital letter that represents an atomic proposition. In that case, we could represent the statement “Cincinnati is not the capital of Ohio” like this:

~C

Likewise, we could represent the statement “Toledo is the capital of Ohio” using the constant T. In that case, we could represent the statement “Toledo is not the capital of Ohio” like this:

~T

We could also create a sentence that is a conjunction of these two negations, like this:

~C ~T

Can you figure out what this complex proposition says? (Think about it; you should be able to figure it out given your understanding of the truth-functional connectives, negation and conjunction.) The propositions says (literally): “Cincinnati is not the capital of Ohio and Toledo is not the capital of Ohio.” In later sections we will learn how to form complex propositions using various combinations of each of the four truth-functional connectives. Before we can do that, however, we need to introduce our next truth-functional connective, disjunction.

The English word that most commonly functions as disjunction is the word “or.” It is also common that the “or” is preceded by an “either” earlier in the sentence, like this:

Either Charlie or Violet tracked mud through the house.

What this sentence asserts is that one or the other (and possibly both) of these individuals tracked mud through the house. Thus, it is composed out of the following two atomic propositions:

Charlie tracked mud through the house

Violet tracked mud through the house

If the fact is that Charlie tracked mud through the house, the statement is true. If the fact is that Violet tracked mud through the house, the statement is also true. This statement is only false if in fact neither Charlie nor Violet tracked mud through the house. This statement would also be true even if it was both Charlie and Violet who tracked mud through the house. Another example of a disjunction that has this same pattern can be seen in the “click it or ticket” campaign of the National Highway Traffic Safety Administration. Think about what the slogan means. What the campaign slogan is saying is:

Either buckle your seatbelt or get a ticket

This is a kind of warning: buckle your seatbelt or you’ll get a ticket. Think about the conditions under which this statement would be true. There are only four different scenarios:

Your seatbelt is buckled	You do not get a ticket	True
Your seatbelt is not buckled	You get a ticket	True
Your seatbelt is buckled	You get a ticket	True
Your seatbelt is not buckled	You do not get a ticket	False

The first and second scenarios (rows 1 and 2) are pretty straightforwardly true, according to the “click it or ticket” statement. But suppose that your seatbelt is buckled, is it still possible to get a ticket (as in the third scenario—row 3)? Of course, it is! That is, the statement allows that it could both be true that your seatbelt is buckled and true that you get a ticket. How so? (Think about it for a second and you’ll probably realize the answer.) Suppose that your seatbelt is buckled but you are speeding, or your taillight is out, or you are driving under the influence of alcohol. In any of those cases, you would get a ticket even if you were wearing your seatbelt. So, the disjunction, click it or ticket, clearly allows the statement to be true even when both of the disjuncts (the statements that form the disjunction) are true. The only way the disjunction would be shown to be false is if (when pulled over) you were not wearing your seatbelt and yet did not get a ticket. Thus, the only way for the disjunction to be false is when both of the disjuncts are false.

These examples reveal a pattern: a disjunction is a truth-functional statement that is true in every instance except where both of the disjuncts are false. In our symbolic language, the symbol we will use to represent a disjunction is called a “wedge” (v). (You can simply use a lowercase “v” to write the wedge.) Here is the truth table for disjunction:

p	q	p v q
T	T	T
T	F	T
F	T	T
F	F	F

As before, the header of this truth table represents two propositions (first two columns) and their disjunction (last column). The following four rows represent the conditions under which the disjunction is true. As we have seen, the disjunction is true when at least one of its disjuncts is true, including when they are both true (the first three rows). A disjunction is false only if both disjuncts are false (last row).

As we have defined it, the wedge (v) is what is called an “inclusive or.” An inclusive or is a disjunction that is true even when both disjuncts are true. However, sometimes a disjunction clearly implies that the statement is true only if either one or the other of the disjuncts is true, but not both. For example, suppose that you know that Bob placed either first or second in the race because you remember seeing a picture of him in the paper where he was standing on a podium (and you know that only the top two runners in the race get to stand on the podium). Although you cannot remember which place he was, you know that:

Bob placed either first or second in the race.

This is a disjunction that is built out of two different atomic propositions:

Bob placed first in the race

Bob placed second in the race

Although it sounds awkward to write it this way in English, we could simply connect each atomic statement with an “or”:

Bob placed first in the race or Bob placed second in the race.

That sentence makes explicit the fact that this statement is a disjunction of two separate statements. However, it is also clear that in this case the disjunction would not be true if all the disjuncts were true, because it is not possible for all the disjuncts to be true since Bob cannot have placed both first and second. Thus, it is clear in a case such as this, that the “or” is meant as what is called an “exclusive or.” An exclusive or is a disjunction that is true only if one or the other, but not both, of its disjuncts is true. When you believe the best interpretation of a disjunction is as an exclusive or, there are ways to represent that using a combination of the disjunction, conjunction and negation. The reason we interpret the wedge as an inclusive or rather than an exclusive or is that while we can build an exclusive or out of a combination of an inclusive or and other truth-functional connectives (as I have just pointed out), there is no way to build an inclusive or out of the exclusive or and other truth-functional connectives.

Watch and Learn

Review the information you have learned about Disjunctions by watching the Critical Thinking Academy’s video, Propositional Logic: Disjunctions

4.3. Using Parentheses To Translate Complex Sentences

We have seen how to translate certain simple sentences into our symbolic language using the dot, wedge, and tilde. The process of translation starts with determining what the atomic propositions of the sentence are and then using the truth functional connectives to form the compound proposition. Sometimes this will be fairly straightforward and easy to figure out—especially if there is only one truth-functional operator used in the English sentence. However, many sentences will contain more than one truth-functional operator. Here is an example:

Bob will not go to class but will play video games.

What are the atomic propositions contained in this English sentence? Clearly, the sentence is asserting two things:

Bob will not go to class. AND Bob will play video games

The first statement is not an atomic proposition, since it contains a negation, “not.” But the second statement is atomic since it does not contain any truth-functional connectives. So, if the first statement is a negation, what is the non-negated, atomic statement? It is this:

Bob will go to class

I will use the constant C to represent this atomic proposition and G to represent the proposition, “Bob will play video games.” Now that we have identified our two atomic propositions, how can we build our complex sentence using only those atomic propositions and the truth-functional connectives? Let usstart with the statement “Bob will not go to class.” Since we have defined the constant “C” as “Bob will go to class” then we can easily represent the statement “Bob will not go to class” using a negation, like this:

~C

The original sentence asserts that, but it is also asserts that Bob will play video games. That is, it is asserting both of these statements. That means we will be connecting “~C” with “G” with the dot operator. Since we have already assigned “G” to the statement “Bob will play video games,” the resulting translation should look like this:

~C G

Although sometimes we can translate sentences into our symbolic language without the use of parentheses (as we did in the previous example), many times a translation will require the use of parentheses. For example:

Bob will not both go to class and play video games.

Notice that whereas the earlier sentence asserted that Bob will not go to class, this sentence does not. Rather, it asserts that Bob will not do both things (i.e., go to class and play video games), but only one or the other (and possibly neither). That is, this sentence does not tell us for sure that Bob will/will not go to class or that he will/will not play video games, but only that he will not do both of these things. Using the same translations as before, how would we translate this sentence? It should be clear that we cannot use the same translation as before since these two sentences are not saying the same thing. Thus, we cannot use the translation:

~C G

since that translation says for sure that Bob will not go to class and that he will play video games. Thus, our translation must be different. Here is how to translate the sentence:

~(C G)

I have here introduced some new symbols, the parentheses. Parentheses are using in formal logic to show groupings. In this case, the parentheses represent that the conjunction, “C G,” is grouped together and the negation ranges over that whole conjunction rather than just the first conjuct (as was the case with the previous translation). When using multiple operators, you must learn to distinguish which operator is the main operator. The main operator of a sentence is the one that connects the main groupings of the sentence. In this case, the “connector” is the negation, since it “connects” the only grouping in this sentence. In contrast, in the previous example (~C G), the main operator was the conjunction rather than the negation. We can see the need for parentheses in distinguishing these two different translations. Without the use of parentheses, we would have no way to distinguish these two sentences, which clearly have different meanings.

Here is a different example where we must utilize parentheses:

Noelle will either feed the dogs or clean her room, but she will not do the dishes.

Can you tell how many atomic propositions this sentence contains? It contains three atomic propositions which are:

Noelle will feed the dogs (F)

Noelle will clean her room (C)

Noelle will do the dishes (D)

What I have written in parentheses to the right of the statement is the constant that I’ll use to represent these atomic statements in my symbolic translation. Notice that the sentence is definitely not asserting that each of these statements is true. Rather, what we have to do is use these atomic propositions to capture the meaning of the original English sentence using only our truth-functional operators. In this sentence we will actually use all three truth-functional operators (disjunction, conjunction, negation). Let usstart with negation, as that one is relatively easy. Given how we have represented the atomic proposition, D, to say that Noelle will not do the dishes is simply the negation of D:

~D

Now consider the first part of the sentence: Noelle will either feed the dogs or clean her room. You should see the “either…or” there and recognize it as a disjunction, which we represent with the wedge, like this:

F v C

Now, how are these two compound propositions, “~D” and “F v C” themselves connected? There is one word in the sentence that tips you off—the “but.” As we saw earlier, “but” is a common way of representing a conjunction in English. Thus, we have to conjoin the disjunction (F v C) and the negation (~D). You might think that we could simply conjoin the two propositions like this:

F v C ~D

However, that translation would not be correct, because it is not what we call a well-formed formula. A well-formed formula is a sentence in our symbolic language that has exactly one interpretation or meaning. However, the translation we have given is ambiguous between two different meanings. It could mean that (Noelle will feed the dogs) or (Noelle will clean her room and not do the dishes). That statement would be true if Noelle fed the dogs and also did the dishes. We can represent this possibility symbolically, using parentheses like this:

F v (C ~D)

The point of the parentheses is to group the main parts of the sentence together. In this case, we are grouping the “C ~D” together and leaving the “F” by itself. The result is that those groupings are connected by a disjunction, which is the main operator of the sentence. In this case, there are only two groupings: “F” on the one hand, and “C ~D” on the other hand.

But the original sentence could also mean that (Noelle will feed the dogs or clean her room) and (Noelle will not wash the dishes). In contrast with our earlier interpretation, this interpretation would be false if Noelle fed the dogs and did the dishes, since this interpretation asserts that Noelle will not do the dishes (as part of a conjunction). Here is how we would represent this interpretation symbolically:

(F v C) ~D

Notice that this interpretation, unlike the last one, groups the “F v C” together and leaves the “~D” by itself. These two grouping are then connected by a conjunction, which is the main operator of this complex sentence.

The fact that our initial attempt at the translation (without using parentheses) yielded an ambiguous sentence shows the need for parentheses to disambiguate the different possibilities. Since our formal language aims at eliminating all ambiguity, we must choose one of the two groupings as the translation of our original English sentence. So, which grouping accurately captures the original sentence? It is the second translation that accurately captures the meaning of the original English sentence. That sentence clearly asserts that Noelle will not do the dishes and that is what our second translation says. In contrast, the first translation is a sentence that could be true even if Noelle did do the dishes. Given our understanding of the original English sentence, it should not be true under those circumstances since it clearly asserts that Noelle will not do the dishes.

Let us move to a different example. Consider the sentence:

Either both Bob and Karen are washing the dishes or Sally and Tom are.

This sentence contains four atomic propositions:

Bob is washing the dishes (B)

Karen is washing the dishes (K)

Sally is washing the dishes (S)

Tom is washing the dishes (T)

As before, I have written the constants than I’ll use to stand for each atomic proposition to the right of each atomic proposition. You can use any letter you would like when coming up with your own translations, as long as each atomic proposition uses a different capital letter. (I typically try to pick letters that are distinctive of each sentence, such as picking “B” for “Bob”.) So how can we use the truth functional operators to connect these atomic propositions together to yield a sentence that captures the meaning of the original English sentence? Clearly B and K are being grouped together with the conjunction “and” and S and T are also being grouped together with the conjunction “and” as well:

(B K)

(S T)

Furthermore, the main operator of the sentence is a disjunction, which you should be tipped off to by the phrase “either…or.” Thus, the correct translation of the sentence is:

(B K) v (S T)

The main operator of this sentence is the disjunction (the wedge). Again, it is the main operator because it groups together the two main sentence groupings.

Let usfinish this section with one final example. Consider the sentence:

Tom will not wash the dishes and will not help prepare dinner; however, he will vacuum the floor or cut the grass.

This sentence contains four atomic propositions:

Tom will wash the dishes (W)

Tom will help prepare dinner (P)

Tom will vacuum the floor (V)

Tom will cut the grass (C)

It is clear from the English (because of the “not”) that we need to negate both W and P. It is also clear from the English (because of the “and”) that W and P are grouped together. Thus, the first part of the translation should be:

(~W ~P)

It is also clear that the last part of the sentence (following the semicolon) is a grouping of V and C and that those two propositions are connected by a disjunction (because of the word “or”):

(V v C)

Finally, these two grouping are connected by a conjunction (because of the “however,” which is a word the often functions as a conjunction). Thus, the correct translation of the sentence is:

(~W ~P) (V v C)

As we have seen in this section, translating sentences from English into our symbolic language is a process that can be captured as a series of steps:

Step 1: Determine what the atomic propositions are.

Step 2: Pick a unique constant to stand for each atomic proposition.

Step 3: If the sentence contains more than two atomic propositions, determine which atomic propositions are grouped together and which truth-functional operator connects them.

Step 4: Determine what the main operator of the sentence is (i.e., which truth functional operator connects the groups of atomic statements together).

Step 5: Once your translation is complete, read it back and see if it accurately captures what the original English sentence conveys. If not, see if another way of grouping the parts together better captures what the original sentence conveys.

4.4. “Not Both” And “Neither Nor”

Two common English phrases that can sometimes cause confusion are “not both” and “neither nor.” These two phrases have different meanings and thus are translated with different symbolic logic sentences. Let us look at an example of each.

Carla will not have both cake and ice cream.

Carla will have neither cake nor ice cream.

The first sentence uses the phrase “not both” and the second “neither nor.” One way of figuring out what a sentence means (and thus how to translate it) is by asking the question: What scenarios does this sentence rule out? Let us apply this to the “not both” statement (which we first saw back in the beginning of section

2.4). There are four possible scenarios, and the statement would be true in everyone except the first scenario:

Carla has cake	Carla has ice cream	False
Carla has cake	Carla does not have ice cream	True
Carla does not have cake	Carla has ice cream	True
Carla does not have cake	Carla does not have ice cream	True

To say that Carla will not have both cake and ice cream allows that she can have one or the other (just not both). It also allows that she can have neither (as in the fourth scenario). So, the way to think about the “not both” locution is as a negation of a conjunction, since the conjunction is the only scenario that cannot be true if the statement is true. If we use the constant “C” to represent the atomic sentence, “Carla has cake,” and “I” to represent “Carla has ice cream,” then the resulting symbolic translation would be:

~(C I)

Thus, in general, statements of the form “not both p and q” will be translated as the negation of a conjunction:

~(p q)

Note that the main operator of the statement is the negation. The negation applies to everything inside the parentheses—i.e., to the conjunction. This is very different from the following sentence (without parentheses):

~p q

The main operator of this statement is the conjunction and the left conjunct of the conjunction is a negation. In contrast with the “not both” form, this statement asserts that p is not true, while q is true. For example, using our previous example of Carla and the cake, the sentence

~C I

would assert that Carla will not have cake and will have ice cream. This is a very different statement from ~(C I) which, as we have seen, allows the possibility that Carla will have cake but not ice cream. Thus, again we see the importance of parentheses in our symbolic language.

Earlier (in section 2.3) we made the distinction between what I called an “exclusive or” and an “inclusive or” and I claimed that although we interpret the wedge (v) as an inclusive or, we can represent the exclusive or symbolically as well. Since we now know how to translate the “not both,” I can show you how to translate a statement that contains an exclusive or. Recall our example

Bob placed either first or second in the race.

As we saw, this disjunction contains the two disjuncts, “Bob placed first in the race” (F) and “Bob placed second in the race” (S). Using the wedge, we get:

F v S

However, since the wedge is interpreted as an inclusive or, this statement would allow that Bob got both first and second in the race, which is not possible. So, we need to be able to say that although Bob placed either first or second, he did not place both first and second. But that is just the “not both” locution. So, to be absolutely clear, we are asserting two things:

Bob placed either first or second. AND Bob did not place both first and second.

We have already seen that the first sentence is translated: “F v S.” The second sentence is simply a “not both F and S” statement:

~(F S)

Now all we have to do is conjoin the two sentences using the dot:

(F v S) ~(F S)

That is the correct translation of an exclusive or. Notice that when conjoining the “F v S” to the “~(F S)” I needed to put parentheses around the “F v S” to show that it was grouped together. Thus, it would have been incorrect to write:

F v S ~(F S)

since that is not a well-formed formula. The problem, as before, is that this sentence is ambiguous between two sentences that have different meanings:

F v (S ~(F S)

(F v S) ~(F S)

While both of these sentences are well-formed, only the latter is the correct translation of the exclusive or.

Let us move on to the English locution “neither…nor” as in:

Carla will eat neither cake nor ice cream.

This statement might be true if, for example, Carla was on a diet (and was sticking to her diet). Using the same method I introduced earlier, we can ask under what conditions the statement would be true or false. As before, there are only four possibilities, which I represent symbolically this time:

There is only one circumstance in which this statement is true and that is the one in which it is false that Carla eats cake and false that Carla eats ice cream. That should be obvious from the meaning of the “neither nor” locution. Thus, the correct translation of a “neither nor” statement is as a conjunction of two negations:

~C ~I

The main operator of this statement is the dot, which is conjoining the ~C with the ~I. Thus, the form of any “neither nor” statement can always be translated as a conjunction of two negations:

~p ~q

As we will see in a later section (where we will prove it), this statement is also equivalent to a negation of a disjunction:

~(p v q)

Thus, the English locution “neither nor” can also be translated using this statement form.

Watch and Learn

Review the information you have learned about Negations by watching the Critical Thinking Academy’s short videos: Propositional Logic: Double Negation Propositional Logic: not–(A and B) Propositional Logic: not–(A or B)

4.5 The Truth Table Test Of Validity

So far, we have learned how to translate certain English sentences into our symbolic language, which consists of a set of constants (i.e., the capital letters that we use to represent different atomic propositions) and the truth-functional connectives. But what is the payoff of doing so? In this section we will learn what the payoff is. In short, the payoff will be that we will have a purely formal method of determining the validity of a certain class of arguments—namely, those arguments whose validity depends on the functioning of the truth-functional connectives. This is what logicians call “propositional logic” or “sentential logic.”

In the first chapter, we learned the informal test of validity, which required us to try to imagine a scenario in which the premises of the argument were true and yet the conclusion false. We saw that if we can imagine such a scenario, then the argument is invalid. On the other hand, if it is not possible to imagine a scenario in which the premises are true and yet the conclusion is false, then the argument is valid. Consider this argument:

The convict escaped either by crawling through the sewage pipes or by hiding out in the back of the delivery truck.

But the convict did not escape by crawling through the sewage pipes.

Therefore, the convict escaped by hiding out in the back of the delivery truck.

Using the informal test of validity, we can see that if we imagine that the first premise and the second premise are true, then the conclusion must follow. However, we can also prove this argument is valid without having to imagine scenarios and ask whether the conclusion would be true in those scenarios. We can do this by a) translating this sentence into our symbolic language and then b) using a truth table to determine whether the argument is valid. Let usstart with the translation. The first premise contains two atomic propositions. Here are the propositions and the constants that I’ll use to stand for them:

S = The convict escaped through the sewage pipes

D = The convict escaped by hiding out in the back of the delivery van

As we can see, the first premise is a disjunction and so, using the constants indicated above, we can translate that first premise as follows:

S v D

The second premise is simply the negation of S:

~S

Finally, the conclusion is simply the atomic sentence, D. Putting this all together in standard form, we have:

S v D

~S

D

We will use the symbol ““ to denote a conclusion and will read it “therefore.”

The next thing we have to do is to construct a truth table. We have already seen some examples of truth tables when I defined the truth-functional connectives that I have introduced so far (conjunction, disjunction, and negation). A truth table (as we saw in section 2.2) is simply a device we use to represent how the truth value of a complex proposition depends on the truth of the propositions that compose it in every possible scenario. When constructing a truth table, the first thing to ask is how many atomic propositions need to be represented in the truth table. In this case, the answer is “two,” since there are only two atomic propositions contained in this argument (namely, S and D). Given that there are only two atomic propositions, our truth table will contain only four rows—one row for each possible scenario. There will be one row in which both S and D are true, one row in which both S and D are false, one row in which S is true and D is false, and one row in which S is false and D is true.

D	S	S v D	~S	D
T	T
T	F
F	T
F	F

The two furthest left columns are what we call the reference columns of the truth table. Reference columns assign every possible arrangement of truth values to the atomic propositions of the argument (in this case, just D and S). The reference columns capture every logically possible scenario. By doing so, we can replace having to use your imagination to imagine different scenarios (as in the informal test of validity) with a mechanical procedure that does not require us to imagine or even think very much at all. Thus, you can think of each row of the truth table as specifying one of the possible scenarios. That is, each row is one of the possible assignments of truth values to the atomic propositions. For example, row 1 of the truth table (the first row after the header row) is a scenario in which it is true that the convict escaped by hiding out in the back of the delivery van and is also true that the convict escaped by crawling through the sewage pipes. In contrast, row 4 is a scenario in which the convict did neither of these things.

The next thing we need to do is figure out what the truth values of the premises and conclusion are for each row of the truth table. We are able to determine what those truth values are because we understand how the truth value of the compound proposition depends on the truth value of the atomic propositions. Given the meanings of the truth functional connectives (discussed in previous sections), we can fill out our truth table like this:

D	S	S v D	~S	D
T	T	T	F	T
T	F	T	T	T
F	T	T	F	F
F	F	F	T	F

To determine the truth values for the first premise of the argument (“S v D”) we just have to know the truth values of S and D and the meaning of the truth functional connective, the disjunction. The truth table for the disjunction says that a disjunction is true as long as at least one of its disjuncts is true. Thus, every row under the “S v D” column should be true, except for the last row since on the last row both D and S are false (whereas in the first three rows at least one or the other is true). The truth values for the second premise (~S) are easy to determine: we simply look at what we have assigned to “S” in our reference column and then we negate those truth values—the Ts becomes Fs and the Fs becomes Ts. That is just what I have done in the fourth column of the truth table above. Finally, the conclusion in the last column of the truth table will simply repeat what we have assigned to “D” in our reference column, since the last conclusion simply repeats the atomic proposition “D.”

The above truth table is complete. Now the question is: How do we use this completed truth table to determine whether or not the argument is valid? In order to do so, we must apply what I’ll call the “truth table test of validity.” According to the truth table test of validity, an argument is valid if and only if for every assignment of truth values to the atomic propositions, if the premises are true then the conclusion is true. An argument is invalid if there exists an assignment of truth values to the atomic propositions on which the premises are true and yet the conclusion is false. It is imperative that you understand (and not simply memorize) what these definitions mean. You should see that these definitions of validity and invalidity have a similar structure to the informal definitions of validity and invalidity (discussed in chapter 1). The similarity is that we are looking for the possibility that the premises are true and yet the conclusion is false. If this is possible, then the argument is invalid; if it is not possible, then the argument is valid. The difference, as I have noted above, is that with the truth table test of validity, we replace having to use your imagination with a mechanical procedure of assigning truth values to atomic propositions and then determining the truth values of the premises and conclusion for each of those assignments.

Applying these definitions to the above truth table, we can see that the argument is valid because there is no assignment of truth values to the atomic propositions (i.e., no row of our truth table) on which all the premises are true and yet the conclusion is false. Look at the first row. Is that a row in which all the premises are true and yet the conclusion false? No, it is not, because not all the premises are true in that row. In particular, “~S” is false in that row. Look at the second row. Is that a row in which all the premises are true and yet the conclusion false? No, it is not; although both premises are true in that row, the conclusion is also true in that row. Now consider the third row. Is that a row in which all the premises are true and yet the conclusion false? No, because it is not a row in which both the premises are true. Finally, consider the last row. Is that a row in which all the premises are true and yet the conclusion false? Again, the answer is “no” because the premises are not both true in that row. Thus, we can see that there is no row of the truth table in which the premises are all true and yet the conclusion is false. And that means the argument is valid.

Since the truth table test of validity is a formal method of evaluating an argument’s validity, we can determine whether an argument is valid just in virtue of its form, without even knowing what the argument is about! Here is an example:

(A v B) v C

~A

C

Here is an argument written in our symbolic language. I do not know what A, B, and C mean (i.e., what atomic propositions they stand for), but it does not matter because we can determine whether the argument is valid without having to know what A, B, and C mean. A, B, and C could be any atomic propositions whatsoever. If this argument form is invalid then whatever meaning we give to A, B, and C, the argument will always be invalid. On the other hand, if this argument form is valid, then whatever meaning we give to A, B, and C, the argument will always be valid.

The first thing to recognize about this argument is that there are three atomic propositions, A, B, and C. And that means our truth table will have 8 rows instead of only 4 rows like our last truth table. The reason we need 8 rows is that it takes twice as many rows to represent every logically possible scenario when we are working with three different propositions. Here is a simple formula that you can use to determine how many rows your truth table needs:

2ⁿ (where n is the number of atomic propositions)

You read this formula “two to the n-th power.” So, if you have one atomic proposition (as in the truth table for negation), your truth table will have only two rows. If you have two atomic propositions, it will have four rows. If you have three atomic propositions, it will have 8 rows. The number of rows needed grows exponentially as the number of atomic propositions grows linearly. The table below represents the same relationship that the above formula does:

Number of atomic propositions	Number of rows in the truth table
1	2
2	4
3	8
4	16
5	32

So, our truth table for the above argument needs to have 8 rows. Here is how that truth table looks:

A	B	C	(A v B) v C	~A	C
T	T	T
T	T	F
T	F	T
T	F	F
F	T	T
F	T	F
F	F	T
F	F	F

Here is an important point to note about setting up a truth table. You need to make sure that your reference columns capture each distinct possible assignment of truth values. One way to make sure you do this is by following the same pattern each time you construct a truth table. There is no one right way of doing this, but here is how I do it (and recommend that you do it too). Construct the reference columns so that the atomic propositions are arranged alphabetically, from left to right. Then on the right-most reference column (the C column above), alternate true and false each row, all the way to the bottom. On the reference column to the left of that (the B column above), alternate two rows true, two rows false, all the way to the bottom. On the next column to the left (the A column above), alternate 4 true, 4 false, all the way to the bottom.

The next step is to determine the truth values of the premises and conclusion. Note that our first premise is a more complex sentence that consists of two disjunctions. The main operator is the second disjunction since the two main grouping, denoted by the parentheses, are “A v B” and “C”. Notice, however, that we cannot figure out the truth values of the main operator of the sentence until we figure out the truth values of the left disjunct, “A v B.” So that is where we need to start. Thus, in the truth table below, I have filled out the truth values directly underneath the “A v B” part of the sentence by using the truth values I have assigned to A and B in the reference columns. As you can see in the truth table below, each line is true except for the last two lines, which are false since a disjunction is only false when both of the disjuncts are false. (If you need to review the truth table for disjunction, please see section 2.3.)

A	B	C	(A v B) v C	~A	C
T	T	T	T
T	T	F	T
T	F	T	T
T	F	F	T
F	T	T	T
F	T	F	T
F	F	T	F
F	F	F	F

Now, since we have figured out the truth values of the left disjunct, we can figure out the truth values under the main operator (which I have emphasized in bold in the truth table below). The two columns you are looking at to determine the truth values of the main operator are the “A v B” column that we have just figured out above and the “C” reference column to the left. It is imperative to understand that the truth values under the “A v B” are irrelevant once we have figured out the truth values under the main operator of the sentence. That column was only a means to an end (the end of determining the main operator) and so I have grayed those out to emphasize that we are no longer paying any attention to them. (When you are constructing your own truth tables, you may even want to erase these subsidiary columns once you have determined the truth values of the main operator of the sentence. Or you may simply want to circle the truth values under the main operator to distinguish them from the rest.)

A	B	C	(A v B) v C	~A	C
T	T	T	T T
T	T	F	T T
T	F	T	T T
T	F	F	T T
F	T	T	T T
F	T	F	T T
F	F	T	F T
F	F	F	F F

Finally, we will fill out the remaining two columns, which is very straightforward. All we have to do for the “~A” is negate the truth values that we have assigned to our “A” reference column. And all we have to do for the final column “C” is simply repeat verbatim the truth values that we have assigned to our reference column “C.”

A	B	C	(A v B) v C	~A	C
T	T	T	T T	F	T
T	T	F	T T	F	F
T	F	T	T T	F	T
T	F	F	T T	F	F
F	T	T	T T	T	T
F	T	F	T T	T	F
F	F	T	F T	T	T
F	F	F	F F	T	F

The above truth table is now complete. The next step is to apply the truth table test of validity in order to determine whether the argument is valid or invalid. Remember that what we are looking for is a row in which the premises are true, and the conclusion is false. If we find such a row, the argument is invalid. If we do not find such a row, then the argument is valid. Applying this definition to the above truth table, we can see that the argument is invalid because of the 6^th row of the table (which I have highlighted). Thus, the explanation of why this argument is invalid is that the sixth row of the table shows a scenario in which the premises are both true and yet the conclusion is false.

Chapter Review

Review what you have learned in this chapter about Propositional Logic and Truth Tables by watching A Crash Course in Formal Logic Pt 7c: Truth Tables for Arguments