• The Universe contains Things

  • For example, “I,” “London,” “roses,” “redness,” “old English 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

From “Symbolic Logic” by Lewis Carroll

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

• Thus, the Thing “a rose” may have the Attributes “red,” “scented,” “full-blown,” etc.; and the Attribute “red” may belong to the Things “a rose,” “a brick,” “a ribbon,” etc.

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

• “The rose is red”
• “London 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

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

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

Let \(P(x)\) and \(Q(y)\) be two predicates. The complex predicate \(P(x)\text{ AND }Q(y)\) has a truth value depending on the truth values of \(P(x)\) and \(Q(y)\). We can see it in a truth table

Notice that this is an Inclusive OR

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

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

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 \(Q(y)\)

Two predicates are equivalent when they have the same truth table

For AND and OR, the order is not important

Let’s compare the two interpretations

They are different, so parenthesis are important

Negation of AND is the OR of negations

Negation of OR is the AND of negations

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

From the list of questions produced in Homework 2 (and any new questions you have)

• Which ones are Scientific Questions?
• Which ones are interesting to other people?

The question “why?” can have two types of answers

• What is the purpose of something
• What is the mechanisms that causes something