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 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