Propositional Logic Functions

Propositional logic and the four basic truth functional connectives

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:

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

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 isn't 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:

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's 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 isn't 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:

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.

As we've seen, a conjunction conjoins two separate propositions to form a complex proposition. The conjunction is true if and only if both conjuncts are true. We can represent this information using what is called a truth table. Truth tables represent how the truth value of a complex proposition depends on the truth values of the propositions that compose it. Here is the truth table for 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. (Later on in this chapter we will use truth tables to prove whether an argument is valid or invalid. Understanding that will require more subtlety, but what I have so far introduced is not complicated at all).

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. 

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 didn't," 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 doesn't is temporal information. For example, in the conjunction:

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

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:

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

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:

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:

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:

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:

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

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.

Source: Matthew J. Van Cleave
This work is licensed under a Creative Commons Attribution 4.0 License.

Exercise

Identify which of the following sentences are truth-functional conjunctions. If the sentence is a truth-functional conjunction, identify the two conjuncts (by writing them down).

1. Jack and Jill are coworkers.
   
2. Tom is a fireman and a father.
   
3. Ringo Starr and John Lennon were bandmates.
   
4. Lucy loves steak and onion sandwiches.
   
5. Cameron Dias has had several relationships, although she has never married.
   
6. Bob and Sally kissed.
   
7. A person who plays both mandolin and guitar is a multi- instrumentalist.
   
8. No one has ever contracted rabies and lived.
   
9. Jack and Jill are cowboys.
   
10. Josiah is Amish; nevertheless, he is also a drug dealer.
    
11. The Tigers are the best baseball team in the state, but they are not as good as the Yankees.
    
12. Bob went to the beach to enjoy some rest and relaxation.
    
13. Lauren isn't the fastest runner on the team; still, she is fast enough to have made it to the national championship.
14. The ring is beautiful, but expensive.
    
15. It is sad, but true that many Americans do not know where their next meal will come from.

Answers

1. Not truth functional.
2. Truth functional: Tom is a fireman. Tom is a father.
3. Not truth functional.
4. Not truth functional.
5. Truth functional: Cameron Dias has had several relationships. Cameron
   Dias has never married.
6. Not truth functional.
7. Not truth functional.
8. Not truth functional.
9. Truth functional: Jack is a cowboy. Jill is a cowboy.
10. Truth functional: Josiah is Amish. Josiah is a drug dealer.
11. Truth functional: The Tigers are the best baseball team in the state. The Tigers are not as good as the Yankees.
12. Truth functional: Bob went to the beach to enjoy some rest. Bob went to the beach to enjoy some relaxation.
13. Truth functional: Lauren isn't the fastest runner on the team. Lauren is fast enough to have made it to the national championship.
14. Truth functional: The ring is beautiful. The ring is expensive.
15. Truth functional: It is sad that many Americans to not know where their next meal will come from. It is true that many Americans do not know where their next meal will come from.

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-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:

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

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:

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:

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 doesn't 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).

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:

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:

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

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:

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:

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:

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:

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 your are speeding, or your tail light 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:

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 can't remember which place he was, you know that:

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

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've just pointed out), there is no way to build an inclusive or out of the exclusive or and other truth-functional connectives. We will see how to represent an exclusive or in section 2.5.

Exercise

Translate the following English sentences into our formal language using conjunction (the dot), negation (the tilde), or disjunction (the wedge). Use the suggested constants to stand for the atomic propositions.

1. Either Bob will mop or Tom will mop. (B = Bob will mop; T = Tom will mop)
   
2. It is not sunny today. (S = it is sunny today)
   
3. It is not the case that Bob is a burglar. (B = Bob is a burglar)
   
4. Harry is arriving either tonight or tomorrow night. (A = Harry is arriving tonight; B = Harry is arriving tomorrow night)
   
5. Gareth does not like his name. (G = Gareth likes his name)
   
6. Either it will not rain on Monday or it will not rain on Tuesday. (M = It will rain on Monday; T = It will rain on Tuesday)
   
7. Tom does not like cheesecake. (T = Tom likes cheesecake)
   
8.  Bob would like to have both a large cat and a small dog as a pet. (C = Bob would like to have a large cat as a pet; D = Bob would like to have a small dog as a pet)
   
9. Bob Saget is not actually very funny. (B = Bob Saget is very funny)
   
10. Albert Einstein did not believe in God. (A = Albert Einstein believed in God)
    

Answers

1. B v T
2. ~S
3. ~B
4. A v B
5. ~G
6. ~M v ~T
7. ~T
8. C ⋅ D
9. ~B
10. ~A

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:

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

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:

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's start 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:

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:

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:

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/won't go to class or that he will/won't play video games, but only that he won't 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:

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:

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 ranges over (influences) the whole sentence. In this case, the main operator is the negation, since it influences the truth value of all the rest of the sentence. In contrast, in the previous example (~C ⋅ G), the main operator was the conjunction rather than the negation since it influences both parts of sentence (i.e., both the "~C" and the "G"). 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:

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

What I've 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's start 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:

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:

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:

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:

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:

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's move to a different example. Consider the sentence:

This sentence contains four atomic propositions:

As before, I've 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'd 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:

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:

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's finish this section with one final example. Consider the sentence:

This sentence contains four atomic propositions:

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:

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"):

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:

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.

Try using these steps to create your own translations of the sentences in exercise 10 below.

Exercise

Translate the following English sentences into our symbolic language using any of the three truth functional operators (i.e., conjunction, negation, and disjunction). Use the constants at the end of each sentence to represent the atomic propositions they are obviously meant for. After you have translated the sentence, identify which truth- functional connective is the main operator of the sentence. (Note: not every sentence requires parentheses; a sentence requires parentheses only if it contains more than two atomic propositions).

1. Bob does not know how to fly an airplane or pilot a ship, but he does know how to ride a motorcycle. (A, S, M)
   
2. Tom does not know how to swim or how to ride a horse. (S, H)
   
3. Theresa writes poems, not novels. (P, N)
   
4. Bob does not like Sally or Felicia, but he does like Alice. (S, F, A)
   
5. Cricket is not widely played in the United States, but both football and baseball are. (C, F, B)
   
6. Tom and Linda are friends, but Tom and Susan aren't - although Linda and Susan are. (T, S, L)
7. Lansing is east of Grand Rapids but west of Detroit. (E, W)
   
8. Either Tom or Linda brought David home after his surgery; but it wasn't Steve. (T, L, S)
   
9. Next year, Steve will be living in either Boulder or Flagstaff, but not Phoenix or Denver. (B, F, P, D)
   
10. Henry VII of England was married to Anne Boleyn and Jane Seymour, but he only executed Anne Boleyn. (A, J, E)
    
11. Henry VII of England executed either Anne Boleyn and Jane Boleyn or Thomas Cromwell and Thomas More. (A, J, C, M)
    
12. Children should be seen, but not heard. (S, H)
    

Answers

1. (~A ⋅ ~S) ⋅ M (The main operator is the second dot - in this case it doesn't actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator).
2. ~S ⋅ ~H (The main operator is the conjunction).
   
3. P ⋅ ~N (The main operator is the conjunction).
   
4. (~S ⋅ ~F) ⋅ A (The main operator is the second dot - in this case it doesn't actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator).
5. ~C ⋅ (F ⋅ B) (The main operator is the first dot - in this case it doesn't actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator).
6. (T ⋅ ~S) ⋅ L (The main operator is the second dot - in this case it doesn't actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator).
7. E ⋅ W (There is only one truth functional operator, the conjunction. So that is by default the main operator!)
8. (T v L) ⋅ ~S (The main operator is the conjunction).
9. (B v F) ⋅ ~(P v D) (The main operator is the conjunction).
10. 10.(A ⋅ J) ⋅ E (The main operator is the second dot - in this case it doesn't actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator).
11. (A ⋅ J) v (C ⋅ M) (The wedge is the main operator).
12. S ⋅ ~H (The main operator is the conjunction).

"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's look at an example of each.

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's 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 every one except the first scenario: 

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):

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

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:

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:

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:

and

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:

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

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:

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:

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

Let's move on to the English locution "neither...nor" as in:

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:

Exercise

For each of the following, write out what atomic proposition each constant stands for. Then translate the sentences using the constants you have defined. Finally, after you have translated the sentence, identify which truth-functional connective is the main operator of the sentence.

1. Coral is not both a plant and an animal. (P, A)
   
2. Although protozoa and chimpanzees are both eukaryotes, they are not both animals. (There are four atomic propositions here; just use A, B, C, and D for each different proposition).
   
3. Neither chimpanzees nor protozoa are prokaryotes. (C, P)
   
4. China has not signed the Kyoto Protocol and neither has the United States. (C, U)
   
5. Either Chevrolet or McDonald's will support the Olympic team, but they won't both support it. (C, M)
   
6. Peter Jennings is either a liar or has a really bad memory. (L, M)
   
7. Peter Jennings is neither a liar nor has a really bad memory. (L, M)
   
8. Peter Jennings is both a liar and has a really bad memory. (L, M)
   
9. Peter Jennings is not both a liar and a person with a really bad memory. (L, M)
   
10. Chevrolet won't support the Olympic team this year, and McDonald's won't either. (C, M)
    
11. Mother Theresa may be a saint. Even so, she has not been canonized yet by the Catholic Church. (S, C)
    
12. The best distance runner of the last two decades is either Paul Tergat or Haile Gebrselassie, but it certainly isn't Jim Ryun. (T, G, R)
    
13. Jim Ryun was the best high school miler of all time, but he ran a slower time than Alan Webb. (R, W)
    
14. Neither Paul Tergat nor Haile Gebrselassie knows how to play hockey, but they both know how to play soccer. (A, B, C,
15. Ethiopians are neither good bobsledders nor tennis players, but they are excellent distance runners. (B, T,
16. Before Helen Keller met Annie Sullivan, she could neither speak, read, nor communicate. (S, R, C)
    
17. Although Helen Keller learned to communicate, she never learned to play soccer or baseball. (C, S, B)
18. Tom is allowed to play football or soccer, but not both. (F, S)
    
19. Tom will major in either engineering and physics, or business and sociology. (E, P, B, S)
    
20. Cartman is both xenophobic and racist, but he isn't a murderer or a thief. (X, R, M, T)

Answers

1. 
   
   • a. P = Coral is a plant; A = Coral is an animal
   • b. ~(P ⋅ A)
   • c. Main operator is the negation
2. 
   
   • a. A = Protozoa are eukaryotes; B = Chimpanzees are eukaryotes; C
   • = Protozoa are animals; D = Chimpanzees are animals
   • b. (A ⋅ B) ⋅ ~(C ⋅ D)
   • c. Main operator is the second dot
3. 
   
   • a. C = Chimpanzees are prokaryotes; P = Protozoa are prokaryotes
   • b. ~(C v P)
   • c. Main operator is the negation
4. 
   
   • a. C = China has not signed the Kyoto Protocol; U = The United States has not signed the Kyoto Protocol
   • b. ~(C v U)
   • c. Main operator is the negation
5. 
   
   • a. C = Chevrolet will support the Olympic team; M = McDonald’s will support the Olympic team
   • b. (C v M) ⋅ ~(C ⋅ M)
   • c. Main operator is the first dot
6. 
   
   • a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad memory
   • b. L v M
   • c. Main (and only) operator is the wedge
7. 
   
   • a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad memory
   • b. ~(L v M)
   • c. Main operator is the negation
8. 
   
   • a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad memory
   • b. L ⋅ M
   • c. Main (and only) operator is the dot
9. 
   
   • a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad memory
   • b. ~(L ⋅ M)
   • c. Main operator is the negation
10. 
    
    • a. C = Chevrolet will support the Olympic team this year; M = McDonald’s will support the Olympic team this year
    • b. ~(C v M)
    • c. Main operator is the negation
11. 
    
    • a. S = Mother Theresa is a saint; C = Mother Theresa has been canonized by the Catholic Church
    • b. S ⋅ ~C
    • c. Main operator is the dot
12. 
    
    • a. T = Paul Tergat was the best distance runner of the last two decades; G = Haile Gebrselassie was the best distance runner of the last two decades; R = Jim Ryun was the best distance runner of the last two decades
    • b. (T v G) ⋅ ~R
    • c. Main operator is the dot
13. 
    
    • a. R = Jim Ryun was the best high school miler of all time; W = Jim Ryun ran a slower time than Alan Webb
    • b. R ⋅ W
    • c. Main (and only) operator is the dot
14. 
    
    • a. A = Paul Tergat knows how to play hockey; B = Haile Gebrselassie knows how to play hockey; C = Paul Tergat knows how to play soccer; D = Haile Gebrselassie knows how to play soccer
    • b. ~(A v B) ⋅ (C ⋅ D)
    • c. Main operator is the first dot
15. 
    
    • a. B = Ethiopians are good bobsledders; T = Ethiopians are good tennis players; D = Ethiopians are good distance runners
    • b. ~(B v T) ⋅ D
    • c. Main operator is the dot
16. 
    
    • a. S = Before Helen Keller met Annie Sullivan she could speak; R = Before Helen Keller met Annie Sullivan she could read; C = Before Helen Keller met Annie Sullivan she could communicate
    • b. (~S ⋅ ~R) ⋅ ~C
    • c. Main operator is the second dot
17. 
    
    • a. C = Helen Keller learned to communicate; S = Helen Keller learned to play soccer; B = Helen Keller learned to play baseball
    • b. C ⋅ (~S ⋅ ~B)
    • c. Main operator is the first dot
18. 
    
    • a. F = Tom is allowed to play football; S = Tom is allowed to play soccer
    • b. (F v S) ⋅ ~(F ⋅ S)
    • c. Main operator is the first dot
19. 
    
    • a. E = Tom will major in engineering; P = Tom will major in physics; B = Tom will major in business; S = Tom will major in sociology
    • b. (E ⋅ P) v (B ⋅ S)
    • c. Main operator is the wedge
20. 
    
    • a. X = Cartman is xenophobic; R = Cartman is a racist; M = Cartman is a murderer; T = Cartman is the thief
    • b. (X ⋅ R) ⋅ (~M ⋅ ~T)
    • c. Main operator is the second dot