[PL02] Variables and predicates

§1. 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.

§2. 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. (Recall our discussion in sentential logic: WFF = well formed formula. i.e. a grammatical sentence defined by the rules of the language of the formal system.) 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.

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?

§3. 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.

Look at the circle and star animation carefully.

Suppose we use "a" to name the circle, "b" to name the star, the right, "Bx" to mean x is black and "Ox" to mean "x is orange". Determine whether the following wffs are always true :

  1. ( Oa ∨ Ob )
  2. ( ~Ba → Bb )
  3. ~ ( Ba ↔ Bb )
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