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
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’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:
~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/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:
~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’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:
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’s 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’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:
(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’s finish 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.
Try using these steps to create your own translations of the sentences in the exercise below.
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)