As discussed in SL10, sentential logic has limitations. 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 does just that. What we will discuss here is a portion of predicate logic, called monadic predicate logic, or MPL.
Question: Why not translate the valid argument above into SL using a valid sequent, such as: "P, Q P"?
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.

  1. What are the subject and predicate of "Mitch is a hacker"?

The subject of a statement is used to refer to an object. The predicate is used to refer to a property which that object is said to have. For example, in the statement "Archibald is hungry", a person, Archibald, is said to have the property of being hungry.

The subject/predicate analysis of sentences traces back at least the Ancient Greek philosopher Plato. In his dialogue "The Sophist", Plato describes and uses this analysis. Contemporary linguists now use more refined ways to analyze the structure of sentences.
Here is how to translate simple statements into MPL: translate the predicate using a capital letter (from "A" to "Z"), translate the subject using a small letter (from "a" to "t"), and then put them together. (The small letters "u", "v","w", "x", "y" and "z" are reserved for other purposes.)

For example, a statement like "Archibald eats" can be translated by "Fa". Here "F" (a predicate letter) translates "eats", and "a" (a constant) translates "Archibald". A statement like "Archibald is happy" could also be translated by "Fa". This time "F" is a predicate letter which translates "is happy" and "a" is a constant which translates "Archibald".

Of course if you are translating an argument that contains both "Archibald is happy" and "Archibald eats" you should choose different predicate letters for "is happy" and "eats". You might use "Ha" for "Archibald is happy" and "Ea" for "Archibald eats". (Do you know why?)

Exercises

Which of the following could be translation of "John eats" into MPL:
  1. Ea
  2. FF
  3. Fa
  4. Fb
  5. Ha
  6. Hx

More Exercises

  1. What are the subject and predicate of "Aaron celebrates"?
  2. Can you guess how might you translate "Aaron eats and Beth sings" into MPL? (Ea & Sb)'); ?>
  3. Can you see how you might translate "Aaron eats or Beth sings" into MPL? (Ea v Sb)'); ?>
  4. Can you see how you might translate this argument into MPL?
    If Archibald eats, Archibald is happy.
    Archibald eats.
    So, Archibald is happy.
    (Ea → Ha), Ea Ha'); ?>
  5. How should you translate "Something eats" into MPL?