Module: Predicate logic
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We have discussed the limitations of SL in an earlier tutorial. For example, SL cannot be used to show that the following argument is valid:
All hackers are nerds.
Mitch is a hacker.
So Mitch is a nerd.
Why can't SL show that this argument is valid? When this argument is translated into SL, some information is lost. When this argument is translated into SL, each statement is translated by a different sentence letter. The result is an invalid sequent:
P, Q ⊧ R.
Some information about the relation between the statements in this argument has been lost in translation. For one thing, the premise, "Mitch is a hacker", and the conclusion, "Mitch is a nerd", have something in common; both statements contain the name "Mitch". But in the translation, "Q" and "R" don't have anything in common.
The relation between the statements in an argument is important to its validity. Indeed, the relation between the parts of statements in an argument is important to its validity. To see this, notice what happens if we change the name "Mitch" in the conclusion to the name "Henry". The result is an argument which is not valid:
All hackers are nerds.
Mitch is a hacker.
So Henry is a nerd.
In order to preserve relations between statements, we need a formal system which can formalize the parts of statements. Predicate logic (PL) does just that.
Some statements are very simple. Some statements, like "John eats" and "Archibald runs" have two parts. Traditionally, these parts have been called the subject and the predicate. For example, in "John eats", "John" is the subject and "eats" is the predicate. In "Archibald runs", "Archibald" is the subject and "runs" is the predicate. Similarly, in "Archibald is hungry", "Archibald" is the subject and "is hungry" is the predicate.
What are the subject and predicate of "Mitch is a hacker"?
In many cases, the subject of a sentence is 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 :
A singular statement is any simple statement with a singular term as the subject, followed by a predicate, which ascribes some property to the referent of the singular term. The following statements are all singular statements:
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.
Are these singular terms?
Are these statements correct?