• The Universe contains Things

  • For example, “I”, “Istanbul”, “roses”, ”Turkish books”, “a letter”

• Things have Attributes

  • For example, “large”, “red”, “old”, “which I received yesterday”
  • It is the same as to say that Things have Properties

Based on the book “Symbolic Logic” by Lewis Carroll

One Thing may have many Attributes, and one Attribute may belong to many Things

• The Thing “a rose” may have the Attributes “red,” “scented,” “full-blown,”
• The Attribute “red” may belong to the Things “a rose,” “a brick,” “a ribbon,”

A phrase stating if a Thing has (or hasn’t) an Attribute

• “The rose is red”
• “Istanbul is a small city”

Predicates are either TRUE or FALSE

That is called the truth value of the predicate

Abstracting a little (to make it general), we can say that

• If \(x\) is a Thing and \(A\) is an Attribute
• then \(A(x)\) is a Predicate

When we see \(A(x)\) we say “\(x\) is \(A\)”

Example: \(x\): snow, \(A\) hot, \(A(x)\) is FALSE

We can combine simple predicates to make complex phrases that can be either TRUE or FALSE

To combine predicates, we can use only a few words

• NOT: “The rose is NOT red”
• AND: “The rose is red AND Istanbul is a small city”
• OR: “The rose is red OR Istanbul is a small city”
• IF…THEN: “IF the rose is red THEN Istanbul is a small city”

This rule applies to a single predicate \(P(x)\)

The truth value of the complex predicate “NOT \(P(x)\)” depends on the truth value of \(P(x)\). We can see it in the truth table

The predicate \(P(x)\text{ OR }Q(y)\) is TRUE if any of the parts is TRUE

Notice that this is an Inclusive OR

“XOR” is TRUE if any of \(P(x)\) and \(Q(y)\) is TRUE, but not both

Now the result if FALSE if \(P(x)\) and \(Q(y)\) are TRUE at the same time

\(P(x)\) is EQUIVALENT to \(Q(y)\)

\(P(x)\) is true IF AND ONLY IF \(Q(y)\) is true

(\(P(x)\) is true IF \(Q(y)\) is true) AND (\(P(x)\) is true ONLY IF \(Q(y)\) is true)

(IF \(Q(y)\) is true THEN \(P(x)\) is true) AND (IF \(P(x)\) is true THEN \(Q(y)\) is true)

(\(Q(y)\) IMPLIES \(P(x)\)) AND (\(P(x)\) IMPLIES \(Q(y)\))

\(P(x)\) is NECESSARY AND SUFFICIENT for \(Q(y)\)

When it is true that “IF \(P(x)\) THEN \(Q(y)\)”?

\(P(x)\) IMPLIES \(Q(y)\)

IF \(P(x)\) is true THEN \(Q(y)\) is true

\(Q(y)\) is true IF \(P(x)\) is true

\(Q(y)\) is NECESSARY for \(P(x)\)

\(P(x)\) is SUFFICIENT for \(Q(y)\)

\(P(x)\) is true ONLY IF \(Q(y)\) is true

We can easily combine all the previous operations

• \(P(x)\) AND \(Q(y)\) AND \(R(z)\)
• \(P(x)\) OR NOT \(Q(x)\)

We use parenthesis to avoid ambiguity. For example

• NOT \(P(x)\) AND \(Q(y)\) can be
  • NOT (\(P(x)\) AND \(Q(y)\))
  • (NOT \(P(x)\)) AND \(Q(y)\)

Let’s compare the two interpretations

They are different. Parenthesis are important

Two predicates are equivalent when they have the same truth table

For example

Negation of AND is the OR of negations

Negation of OR is the AND of negations

To make phrases about Attributes we have to speak about the Things having these attributes

So far we saw how to evaluate the truth value of a logical phrase depending on the specific things (\(x\) and \(y\))

Now we care about the truth of the phrase in general

We have two key words, called quantifiers:

• “All” (sometimes written as \(\forall\))
• “Some”, or “Exists” (written as \(\exists\))

“Not all things are natural” means “There are some things that are not natural”

• There exists a thing \(x\) such that NOT \(Natural(x)\)
• \(\exists x,\, \neg Natural(x)\)

“Nothing is too expensive” means “All things are not too expensive”

• For all things \(x\), NOT \(TooExpensive(x)\)
• \(\forall x, \neg TooExpensive(x)\)