What is Logic?
It is our ability to reason.
It is the foundation of sound thinking.

Where do we start?
I start with the question:
What is a sentence ?
Every good sentence needs a noun or subject, an action word or a verb and an object or the completion of the idea of the sentence.

How is a mathematical sentence different from an grammatical sentence?
A mathematical sentence has a certain truth value. It is called a closed sentence .
A sentence whose truth value is uncertain is called an open sentence .

The negation of a sentence changes the truth value of the original sentence.
It is represented by the symbol ~.
If p represents my sentence then ~p represents it’s negation.

In the space provided write a sentence in words.
Then write the negation of that sentence in words.

A conjunction connects two simple sentences using the word “ and “ to make a compound sentence.
Given the two simple sentences
Sara goes swimming.
Tom is a life guard.
The conjunction would be
Sara goes swimming and Tom is a life guard

The word And is represented by the symbol

Example
Let p represent: Sara goes swimming.
Let q represent: Tom is a life guard.
Sara goes swimming and Tom is a life guard.
Would be represented by: p
 q

The compound sentence p
 q is only true when both individual sentences are true.
T
F
F p
T
Truth Table
F
T
F q
T p
 q
T
F
F
F

A disjunction connects two simple sentences using the word “ or “ to make a compound sentence.
Given the two simple sentences
Sara will study
Sara goes to the movies
The compound sentence would be
Sara will study or Sara goes to the movies

The connector “ or ” is represented by the symbol
“  .”
Example:
Let p represent: Sara will study
Let q represent: Sara goes to the movies
Sara will study or Sara will go to the movies. represents: p
 q

The disjunction p
 q is only false when both p and q are false.
p
T
T
F
F
Truth Table q p
 q
T
F
T
T
T
F
T
F

Let p represent: Dinner is a Meal
Let q represent: Winter is a season a) Write a complete sentence for ~p V q b) What is the truth value of the sentence?

A conditional is a compound sentence usually formed by the words if….then
to combine 2 simple sentences. The
 is used to represent if…then symbolically.
Given the simple sentences
John studies
John gets good grades
The compound sentence would be:
If John studies then John gets good grades.

A conditional statement is also called an implication p implies q.
Let p represent: John studies
Let q represent: John gets good grades
If John studies then John gets good grades.
Represent by: p
 q

The Definition of the Conditional
In the conditional p
 q p : is the antecedent q : is the consequent antecedent  consequent
( hypothesis ) ( conclusion )

The conditional statement is only false when a true hypothesis implies a false conclusion .
T
F
F p
T
Truth Table q p
 q
T T
F
T
F
F
T
T

[( p
q ) ^~p ]
 q p q p V q ~p ( p V q ) ^~p [( p V q ) ^~p ]
 q
T T T F F T
T F
F T
T
T
F
T
F
T
T
T
F F F T F T
Only false when antecedent is true and the consequent is false .

A tautology is a compound statement that is always true.
To determine if a compound statement is a tautology construct a truth table .
If the last column is always true then the statement is a tautology.
Conditional statements that are tautologies are called implications .

↔ translates into “if and only if” p
↔ q means: p
 q and q
 p
Or symbolically p
 q ^ q
 p

p
 q
q
 p
p
T q
T p
 q q
 p ( p
 q ) ^ ( q
 p )
T
F
T
T
T
F T F
F T T F F
F F T T T
Only false when antecedent is true and the consequent is false .

Converse, Inverse, and
Contrapositive
From a conditional we can also create additional statements referred to as related conditionals . These include the converse , the inverse , and the contrapositive .

Converse, Inverse, and
Contrapositive
Statement Formed by Symbols
Conditional Given an hypothesis and conclusion
Converse
Inverse
Switch the hypothesis and conclusion
Negate both the hypothesis and the conclusion
Contrapositive Negate and Switch both p q q p
~p ~q
~q ~p

Write the converse, inverse, and contrapositive of the statement:
If it is cloudy outside then it will rain.
Converse:
Inverse
Contrapositive

Write the converse, inverse, and contrapositive of the statement:
If you live in New York, then the capital is Albany
Converse:
Inverse
Contrapositive