7 Translation

A Translation Method

We build on the translation method we used for propositional logic. That is, we will first indicate how natural language sentences are constructed from simpler sentences by means of the usual propositional connectives. The difference is that we will now look for further structure into the sentences we would have formalized with the help of a propositional variable in propositional logic.

Predication

We use predications to translate the most basic parts of the target sentence.

Consider the sentence:

The translation method for propositional logic generates:

In a further step, we now make use of predicates and constants to translate the simplest sentences:

⊕Translation key
$a$: Alice
$b$: The King
$c$: The Queen
$R$ _ _ : _ talked to _

Quantifier phrases require more discussion. We will look at the case of simple quantifier expressions such as ‘something’ and ‘everything’ and we will then move to more complex quantifier phrases such as ‘some apple’ and ‘every computer’, which require separate discussion.

Quantifiers

We paraphrase the relevant sentences to make clear how they may be built from simpler constituents with the help of quantifiers such as ‘something’ and ‘everything’.

One may rephrase ‘Something is heavy’ as ‘Some thing is such that it is heavy’. The pronoun ‘it’ will rendered as a variable, which will be bound by the existential quantifier that corresponds to ‘something’. Consider the sentence:

We rephrase it with the help of a pronoun, which will eventually be replaced with a variable:

⊕Translation key
$P$ _ : _ is heavy.

Similarly for the sentence:

We now rephrase it with the help of a pronoun, which will eventually be replaced with a variable:

⊕Translation key
$Q$ _ : _ is a material object.

Similarly for predicates with more than one argument place:

⊕Translation key
$a$: San Francisco
$R$ _ _ : _ is south of _

Common quantifier phrases

Common quantifier phrases combine a quantifier such as ‘some’ or ‘a’ and ‘all’ or ‘every’ with a monadic predicate such as ‘apple’ or ‘student’ as in ‘some apples’ or ‘every city’.

There are two main types of natural language sentence to consider:

When the quantifier phrase combines ‘some’ with a monadic predicate, then we proceed as follows:

⊕Translation key
$P$ _ : _ is an apple

Consider the sentence:

We now rephrase it with the help of a pronoun, which will eventually be replaced with a variable:

⊕Translation key
$P$ _ : _ is an apple
$Q$ _ : _ is delicious

Similarly for the sentence:

⊕Translation key
$a$: San Francisco
$P$ _ : _ is a city
$R$ _ _ : _ is south of _

When the quantifier phrase combines ‘every’ or ‘all’ with a monadic predicate, then we proceed as follows:

⊕Translation key
$P$ _ : _ is a computer

Consider the sentence:

We now rephrase it with the help of a pronoun, which will eventually be replaced with a variable:

⊕Translation key
$P$ _ : _ is a computer
$Q$ _ : _ is expensive

Similarly for the sentence:

⊕Translation key
$P$ _ : _ is a computer
$R$ _ _ : _ is connected to _
$a$: The network

Multiple Generality

Some predicates with multiple argument places may combine with more than one quantifier phrase:

Consider the sentence:

We now paraphrase the sentence:

⊕Translation key
$R$ _ _ : _ is south of _

Consider now the sentence:

We now paraphrase the sentence:

⊕Translation key
$P$ _ : _ is a city
$R$ _ _ : _ is south of _

For yet another example, consider the sentence:

We now paraphrase the sentence:

⊕Translation key
$P$ _ : _ is a city
$R$ _ _ : _ is south of _

Some sentences combine a universal and existential quantifier phrases.

Consider the sentence:

We now paraphrase the sentence in stages:

⊕Translation key
$P$ _ : _ is a student
$Q$ _ : _ is a course
$R$ _ _ : _ takes _

Consider now the sentence:

We now paraphrase the sentence in stages:

⊕Translation key
$P$ _ : _ is a student
$Q$ _ : _ is a course
$R$ _ _ : _ takes _

For another example, consider the sentence:

We now paraphrase the sentence in stages:

⊕Translation key
$P$ _ : _ is a student
$Q$ _ : _ is a course
$R$ _ _ : _ takes _

Issues with Translation

Translation into quantificational logic is subject to its own special difficulties. We distinguish at least two families of problems.

Special Quantifier Phrases

Some quantificational structures pose special problems.

A

A whale lives underwater

Quantifier phrases of the form ‘a $P$’ may on occasion be translated with the help of a universal quantifier and they may on other occasions be translated with the help of an existential quantifier. Consider the contrast between typical utterances of the sentences below:

The first sentence is generally used to express a universal claim:

⊕Translation key
$P$ _ : _ is a whale
$Q$ _ : _ lives underwater

Translation key:

On the other hand, the second sentence is generally used to express an existential claim:

⊕Translation key
$P$ _ : _ is a student
$R$ _ _ : _ aces _
$a$: The Midterm

Any

Any philosopher thinks.

The quantifier phrase ‘any’ may be translated with the help of a universal or an existential quantifier in different cases.

Compare the sentences:

The first sentence invites the use of a universal quantifier.

⊕Translation key
$P$ _ : _ is a philosophert
$Q$ _ : _ thinks

On the other hand, the second sentence suggests an existential quantifier:

⊕Translation key
$P$ _ : _ is a philosophert
$Q$ _ : _ thinks
$a$: Hegel

Only

Only seniors are eligible for the prize

How should we paraphrase this sentence? Here is a suggestion:

⊕Translation key
$P$ _ : _ is a senior
$Q$ _ : _ is eligible for the prize

The interaction between quantification and negation requires some care as well:

Consider the sentence:

There is a choice between two equivalent interpretations. Consider, first, the sentence:

⊕Translation key
$P$ _ : _ is a senior
$Q$ _ : _ is eligible for the prize

For an alternative interpretation of the original sentence, consider:

Structural Ambiguity

A sentence may be structurally ambiguous.

Consider the sentence:

There is a choice between two non-equivalent interpretations:

⊕Translation key
$R$ _ : _ is a cause of _

The alternative interpretation proceeds as follows:

⊕Translation key
$R$ _ : _ is a cause of _

The distinction is specially important when it comes to the evaluation of an argument such as the following:

1. Everything has a cause.
2. If everything has a cause, then God is the cause of everything.
3. God is the cause of everything.

The crucial observation is that premise two is plausible on one but not the other interpretation of the first premise.

Consider the sentence:

There is a choice between two non-equivalent interpretations:

⊕Translation key
$P$ _ : _ is a student
$Q$ _ : _ is a course
$R$ _ _ : _ registers for _

The alternative interpretation is this:

⊕Translation key
$P$ _ : _ is a student
$Q$ _ : _ is a course
$R$ _ _ : _ registers for _

We are in a position to combine the skills we have gained in order to determine whether an argumnet formulated in English exemplifies a valid quantificational form.

Exercises

1. Translate the following sentences into the language of quantificational logic.

   1. Some Trojans fear Achilles, but Achilles fears every Trojan.

   2. All Trojans are courageous.

   3. Only Trojans come from Troy.

   4. No Greek comes from Troy.

   5. Achilles fears Paris, but Paris fears no one.

   6. Some Greeks fears some Trojans, and some Trojans fear some Greeks.

2. Assess the following arguments for quantificational validity.

   1. All poisonous frogs are amphibians. Kermit is a posisonous amphibian. Therefore, Kermit is a frog.

   2. Aristotle admired anyone who taught for free. Socrates taught for free, but no sophist did ever teach for free. So, Aristotle did not admire a sophist.

   3. No weekday comes after Saturday. Monday is a weekday, but Sunday is not. Therefore, Monday does not come after Sunday.

   4. Some cube is green all over and some cube is red all over. No cube is both green all over and red all over. Some sphere is green all over. So, some sphere is not red all over.

   5. No Trojan fears Achilles, but Achilles fears some Trojans. Paris is a Trojan. So, Achilles fears Paris but Paris does not fear Achilles.

   6. Some stadium is close to University Park. The Coliseum is a stadium. Therefore, the Coliseum is close to University Park.

   7. Los Angeles is connected by direct flight to London. London is connected by direct flight to every major European city. Barcelona is a major European city. Therefore, Los Angeles is connected by direct flight to Barcelona.

   8. All Greeks feared Hector or Paris. Paris feared no Greek who feared Hector. Therefore, some Greeks feared Paris.