MODULE: Predicate logic

TUTORIAL Q01: Singular terms and open sentences

To begin our study of predicate logic, we start with the concept of a singular term. A singular term in a natural language is a linguistic expression that has the function of referring to or naming a particular object or thing. For our present purpose we shall take singular terms to include :

proper names : "Peter", "Pakistan", and certain abstract nouns such as "patience", as in "patience is very rare" ...
singular definite descriptions : "the man on the balcony", "the movie", ...
demonstratives : "that cat", "this button here" ...

Exercises

Are these singular terms?

"The most violent animal in the whole universe" [Show answer]
Yes.
"every happy person" [Show answer]
No. Not about a particular thing.
"A bird with red feathers" [Show answer]
No. Does not refer to any particular bird.
"snow is white" [Show answer]
No. This is a sentence.
"Alexander the Great" [Show answer]
Yes. A particular person.
"Beautiful dresses" [Show answer]
No. Not a single thing.

A singular statement is any simple statement with a singular term as the subject, and which ascribes some property to the referent of the singular term. The following statements are all singular statements:

The tallest man in the world is over two meters tall.
That insect on the window is a grasshoper.
Tom Cruise is an actor.
2 is my lucky number.

Names in PL are used to symbolize singular terms. A name is any non-italized letter such as "a", "b", "c". A name is said to refer to some particular object, and the object to which the name refers is called its referent (sometimes also called the extension of the name). For example, the name "Albert Einstein" refers to a famous scientist.

In PL we assume that every name succeeds in referring to some existing object. This is certainly not true in natural languages. For example, the singular term "Santa Claus" presumably does not refer to any actual person. Such singular terms which fail to refer to anything real are said to be empty. In the branch of formal logic known as free logic, there is no assumption that all names refer, but we shall not discuss that approach here.

Exercises

Are these statements correct?

Beethoven is a singular term. [Show answer]
No. Beethoven is a person!
"Beethoven" refers to a person. [Show answer]
Yes.
Beethoven does not refer to a person. [Show answer]
Yes. Beethoven is a person, and a person is not a name, so it does not refer to anything.
"Beethoven" refers to Beethoven. [Show answer]
Yes.
Beethoven refers to "Beethoven". [Show answer]
No.

Variables and open sentences

We shall use small italic letters "x", "y", "z", ... as variables. Grammatically they function similarly as singular terms, even though they do not refer to any particular object. You can think of them as similar to a pronoun like "it".

Given a complete sentence from a natural language, the result of substituting or replacing one or more singular term by a variable is called an open sentence.

For example, replacing the numeral "5" in the sentence

5 is smaller than 7.

by the variable "x" we end up with the open sentence

x is smaller than 7.

Notice that although the original sentence is true, the open sentence that is produced is neither true nor false, because variables do not refer to any particular thing. Compare : if we don't know what the pronoun "it" refers to in "it is expensive", we would not be able to determine the truth or falsity of the sentence.

The use of open sentences provides a way to describe a common feature between statements such as :

5 is smaller than 7.
4 is smaller than 7.
10 is smaller than 7.
21215 is smaller than 7.

All of these statements can be constructed from the same open sentence "x is smaller than 7" by replacing the variable "x" with the appropriate singular term.

Predicates

A predicate in PL is any capital letter such as "A", "B", "C", "P", "Q", "R", etc. They are equivalent to the open sentences introduced earlier. So for example we might use the letter "Cx" to translate the open sentence "x is a city in Asia". Predicates can combine with names to form WFFs in PL. So for example, suppose we use the following translation scheme:

h : Hong Kong
o : Oxford
Cx : x is a city in Asia.

"Ch" would mean "Hong Kong is a city in Asia", which is true. "Co" would mean "Oxford is a city in Asia", and so is a false sentence.

Exercises

Consider these WFFs:

Le, Ls, Ss, Se
Translate them into English using the following translation scheme:

e : The Earth
s : The Sun
Lx : x is larger than the earth.
Sx : x is smaller than the moon.

Which of the WFFs are true, if any? [Show answer]
Only the second one is true.

More complicated WFFs

In PL, those wffs made up of predicates and names can combine to form longer wffs just as in SL. So we can have wffs like "~Le", "(Ss&Se)", "(Le→Le)", etc.. "~Le" is of course true if and only if "Le" is false, and "(Ss&Se)" is true if and only if "Ss" and "Se" are both true. This is just a matter of applying the same rules in SL. "(Le→Le)" is of course a logical truth.

Exercises

Look at the animation on the right carefully.

Suppose we use "a" to name the lightbulb on the left, and "b" to name the lightbulb on the right, "Lx" to mean x is switched on and "Ox" to mean "x is switched off". Determine whether the following wffs are always true :