Translations

In this section, we briefly discuss how to translate from English to our formal language. It can be tricky to identify the right translation for an English sentence (in some cases, there may be different ways to translate a sentence). The goal is to find a formula that captures the logical form of the sentence and that comes as close as possible to the meaning of the sentence. The main steps to translate an English sentence into the a formal language are:

1. If needed, paraphrase the sentence to identify all the basic sentences and the main connective.
2. Write down a translation key associating each basic sentence with an atomic proposition. Then, replace each basic sentence with the corresponding atomic proposition.
3. Replace the remaining English words with the appropriate Boolean connectives. Be careful to properly group the resulting formula.

Example

Fix the following translation key:

• Ann got an A in PHIL 171. $\rightsquigarrow$ $A$
• Ann got an B in PHIL 171. $\rightsquigarrow$ $B$
• Charles got an A in PHIL 171. $\rightsquigarrow$ $C$

Using this translation key, we can translate the following sentences:

1. Ann didn't get a B in PHIL 171.
   Paraphrase: It is not the case that Ann got a B in PHIL 171.
   Translation: $\neg B$

2. Ann got an A or B in PHIL 171.
   Paraphrase: Ann got an A in PHIL 171 or Ann got a B in PHIL 171.
   Translation: $(A\vee B)$

3. Ann and Charles got an A in PHIL 171.
   Paraphrase: Ann got an A in PHIL 171 and Charles got an A in PHIL 171.
   Translation: $(A\wedge C)$

4. If Ann got an A in PHIL 171, then so did Charles.
   Paraphrase: If Ann got an A in PHIL 171, then Charles got an A in PHIL 171.
   Translation: $(A\rightarrow C)$

Example

Fix the following translation key:

• I will have fish. $\rightsquigarrow$ $F$
• I will have salad. $\rightsquigarrow$ $D$
• I will have soup. $\rightsquigarrow$ $P$

Using this translation key, we can translate the following sentences:

1. Either I will have fish or salad and soup.
   Translation: $(F\vee (D\wedge P))$
   Note that $((F\vee D)\wedge P)$ is not a correct translation (the English sentence is a disjunction, but this formula is a conjunction). The second sentences is the translation of "I will have either fish or salad and I will have soup."

2. I will have fish and salad or soup.
   Translation: $(F\wedge (D\vee P))$

3. I will have fish if I don't have salad.
   Translation: $\neg D\rightarrow F$

4. I will not have fish and salad.
   Translation: $\neg (F\wedge D)$

5. I will not have fish and I will not have salad.
   Translation: $(\neg F\wedge \neg D)$

6. I will have neither fish nor salad.
   Translation: $\neg (F\vee D)$

I conclude with some general comments about finding translations of English sentences.

Logical form

When translating, it is important to capture both the logical form and the meaning. For example, the correct translation of "I will have steak and fish." is $(S \wedge F)$. Although, $(F\wedge S)$ does mean the same thing, it has a different logical form ("I will have fish and steak."). Both sentences are conjunctions, but the order of the conjuncts is different.

Challenge

The sentences "I will have steak and fish" and "I will have fish and steak" are translated to different formulas, but the sentences mean the same thing. Can you think of a sentence that is translated as a conjunction in which the order of the conjuncts matters? That is, re-ordering the conjuncts changes the meaning of the sentence?

One difficulty with translating English sentences to formulas is that there may be multiple ways to translate the same sentence. For example, there are two ways to translate sentences involving "neither...nor". Suppose that $R$ stands for "it is raining" and $S$ stands for "it is sleeting". Then, the translation of "It is neither raining nor sleeting." is either of the following formulas:

1. $\neg (R\vee S)$
2. $(\neg R\wedge\neg S)$

To see why both formulas can be used to translate "It is neither raining nor sleeting.", note that the following are all paraphrases of this sentence:

• It is not true that either it is raining or it is sleeting.
• Neither is it raining nor is it sleeting.
• Neither is it true that it is raining nor is it true that it is sleeting.

Exclusive/inclusive disjunction

Recall that we translated "Ann got an A or B in PHIL 171" as $(A\vee B)$. There is an implicit assumption associated with this sentence: "it is not the case that Ann got both an A and B in PHIL 171". Contrast this with the sentence: Either Ann got an A in PHIL 171 or Charles got an A in PHIL 171. This sentence is translated as $(A\vee C)$. In this case, there is no implicit assumption that Ann and Charles cannot both get an A in PHIL 171.

These observations suggest that it is important to distinguish between an exclusive disjunction and an inclusive disjunction. An inclusive disjunction is a disjunction in which both disjuncts could be true at the same time, while an exclusive disjunction excludes the possibility that both disjuncts are true. When defining the connectives and our formal language, we did not specify whether "$\vee$" is intended to express an inclusive or exclusive disjunction. Following the standard convention in logic texts, we will assume that "$\vee$" is an inclusive disjunction. When it is important to be explicit about a disjunction being exclusive rather than inclusive, you should add the implicit conjunct. For instance, an extended translation of "Ann got an A or B in PHIL 171" is

$$(A\vee B)\wedge\neg(A\wedge B).$$

Non-standard uses of connectives

One complication that can arise is non-standard uses of connectives. For example, sometimes sentences involving "and" should be translated as an atomic sentence rather than a conjunction. Consider the sentence "Ann and Bob are dancing". This does not mean the same thing as "Ann is dancing and Bob is dancing". So, the first sentence should be translated as an atomic sentence.

Another non-standard use of "and" is illustrated with the following sentence:
Cheat on the exam and you will fail the class.

One may be tempted to translate this as $(C\wedge F)$ (where $C$ means "You will cheat on the exam" and $F$ means "You will fail the class"). However, "You cheated on the exam and you will fail the class" does not mean the same thing as the original sentence. The correct translation is $(C\rightarrow F)$: If you cheat on the exam, then you will fail this class.

Expressing the connectives

There are many different ways of expressing the Boolean connectives in English:

Symbol	English expressions
$\vee$	or, unless, either ... or ...
$\wedge$	and, yet, but, however, both, also, although, nevertheless, still, also, although, moreover, additionally, furthermore
$\neg$	not, it is not the case that, it is false that
$\rightarrow$	if ... then ..., only if, given that, in case, provided that, on condition that, sufficient condition, necessary condition, unless

For example, each of the following sentences is a conjunction:

• The cat is napping and the dog is chasing his tail.
• The cat is napping, but the dog is chasing his tail.
• Although the cat is cat is napping, the dog is chasing his tail.
• The cat is napping, though the dog is chasing his tail.
• The cat is napping; however, the dog is chasing his tail.
• The cat is napping, whereas the dog is chasing his tail.

Each of these sentences is translated as $C\wedge D$ (where $C$ means "the cat is napping" and $D$ means "the dog is chasing his tail").

There are also many ways to express a conditional:

• If John ran, then Mary laughed.
• If John ran, Mary laughed.
• Mary laughed, provided that John ran.
• Given that John ran, Mary laughed.
• Mary laughed if John ran.
• John ran only if Mary laughed.

Each of these sentences is translated as $J\rightarrow M$ (where $J$ means "John ran" and $M$ means "Mary laughed").

Unless

Consider the following sentence:

Ann will tutor Bob in logic, unless Carla does.

How should we translate this into our formal language? Although we do not have a connective that directly symbolizes "unless", we can find a formula that expresses the above sentence. Start by fixing the following translation key:

• Ann will tutor Bob in logic. $\rightsquigarrow$ $A$
• Carla will tutor Bob in logic. $\rightsquigarrow$ $C$

The following sentence means the same thing as the above sentence:

     If Carla does not tutor Bob in logic, then Ann will tutor Bob in logic.

The translation of this sentence is $\neg C\rightarrow A$.

Another way of paraphrasing the original sentence is:

     If Ann does not tutor Bob in logic, then Carla will tutor Bob in logic.

The translation of this sentence is $\neg A\rightarrow C$.

Either of the above two formulas can be used as a translation of the original sentence. Arguably, the best translation of the original sentences is $A\vee C$: Either Ann will tutor Bob in logic or Carla will tutor Bob in logic.

Further reading

A very good discussion of translating English sentences into formulas (including many examples) is Chapter 4 of Symbolic Logic: A First Course by G. Hardegree (note that this text uses "$\&$" rather than "$\wedge$" to symbolize conjunction). See also Chapter 2 of the online textbook carnap.io by G. Leach-Krouse and J. Ehrlich.

Practice Questions

Fix the following translation key:

• Ann will go to the party. $\rightsquigarrow$ $A$
• Bob will go to the party. $\rightsquigarrow$ $B$
• Carla will go to the party. $\rightsquigarrow$ $C$

Using this translation key, translate the following sentences. For these translations, you should assume that the only people that could possibly attend the party are Ann, Bob or Carla (no one else is invited).

1. Ann will go to the party if neither Bob nor Carla goes.

2. Ann will go to the party only if Bob doesn't.

3. Ann will go to the party if Bob doesn't.

4. Ann will go to the party, but Bob won't.

5. If Ann will go to the party, then Bob and Carla both won't go to the party.

6. No one is going to the party. (None of Ann, Bob or Carla is going to the party).

7. Bob and Carla are not both going to the party.

8. Neither Bob nor Carla is going to the party.

9. Someone is going to the party.

10. Exactly one person is going to the party.