Article: Using parentheses (Dr. Van Cleave)

By Matthew Van Cleave

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:

            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:

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:

            (~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:

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

Do Quiz 8a before moving further in the course.