March 27th, 2017

Logic Arguments

Speaking the Truth

A brief introduction to Logic

  • 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

Attributes

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.

Simple Predicates

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

Complex Predicates

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”

AND rule

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

\(P(x)\) \(Q(y)\) \(P(x)\) AND \(Q(y)\)
TRUE TRUE TRUE
TRUE FALSE FALSE
FALSE TRUE FALSE
FALSE FALSE FALSE

OR rule

\(P(x)\) \(Q(y)\) \(P(x)\) OR \(Q(y)\)
TRUE TRUE TRUE
TRUE FALSE TRUE
FALSE TRUE TRUE
FALSE FALSE FALSE

Notice that this is an Inclusive OR

Exclusive OR rule

\(P(x)\) \(Q(y)\) \(P(x)\) XOR \(Q(y)\)
TRUE TRUE FALSE
TRUE FALSE TRUE
FALSE TRUE TRUE
FALSE FALSE FALSE

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

NOT rule

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

\(P(x)\) NOT \(P(x)\)
TRUE FALSE
FALSE TRUE

Combining predicates

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 \(Q(y)\)

Logical equivalence

Two predicates are equivalent when they have the same truth table

\(P(x)\) \(Q(y)\) \(P(x)\) AND \(Q(y)\) \(Q(y)\) AND \(P(x)\)
TRUE TRUE TRUE TRUE
TRUE FALSE FALSE FALSE
FALSE TRUE FALSE FALSE
FALSE FALSE FALSE FALSE

For AND and OR, the order is not important

NOT \(P(x)\) AND \(Q(y)\)

Let’s compare the two interpretations

\(P(x)\) \(Q(y)\) NOT (\(P(x)\) AND \(Q(y)\)) (NOT \(P(x)\)) AND \(Q(y)\)
TRUE TRUE FALSE FALSE
TRUE FALSE TRUE FALSE
FALSE TRUE TRUE TRUE
FALSE FALSE TRUE FALSE

They are different, so parenthesis are important

An important rule

De Morgan’s law

\(P(x)\) \(Q(y)\) NOT (\(P(x)\) AND \(Q(y)\)) (NOT \(P(x)\)) OR (NOT \(Q(y)\))
TRUE TRUE FALSE FALSE
TRUE FALSE TRUE TRUE
FALSE TRUE TRUE TRUE
FALSE FALSE TRUE TRUE

Negation of AND is the OR of negations

Another De Morgan’s law

\(P(x)\) \(Q(y)\) NOT (\(P(x)\) OR \(Q(y)\)) (NOT \(P(x)\)) AND (NOT \(Q(y)\))
TRUE TRUE FALSE FALSE
TRUE FALSE FALSE FALSE
FALSE TRUE FALSE FALSE
FALSE FALSE TRUE TRUE

Negation of OR is the AND of negations

Implication

The most important one

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

\(P(x)\) \(Q(y)\) IF \(P(x)\) THEN \(Q(y)\)
TRUE TRUE TRUE
TRUE FALSE FALSE
FALSE TRUE TRUE
FALSE FALSE TRUE

Homework

Homework 3

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?

Two types of “Why”

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

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