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Working the vohp contradictionstautologies and contradictions

A.Glen@murdoch.edu.au

https://amyglen.wordpress.com

Chapter 7: Propositional Logic 2

(continued)

Chapter 7: Propositional Logic

Connectives & Truth Tables

3
Dr Amy Glen (Murdoch University) MAS162 – Foundations of Discrete Mathematics Lecture 22 3

More Connectives

There are two other connectives for propositions which are important for our study of logic, called the conditional and biconditional connectives.

p æ q

1
1
0
1

Chapter 7: Propositional Logic
5
Chapter 7: Propositional Logic
6

Consider the conditional proposition:

“If I win the lottery, then I will give everybody a gift.”

Chapter 7: Propositional Logic
7

Example

Dr Amy Glen (Murdoch University) MAS162 – Foundations of Discrete Mathematics Lecture 22 7
I The conditional proposition p æ q is true in every logical possibility except the one in which p is true and q is false.

I That is, it is su�cient to prove that if p is true, then q is true.

I Such a proof is called a direct proof.

Dr Amy Glen (Murdoch University) MAS162 – Foundations of Discrete Mathematics Lecture 22 8

“there are 23 401 words in Chapter 7 of the Unit Notes”
“there are more than 20 000 words in Chapter 7 of the Unit Notes”

The truth of the conditional, p æ q, could be established directly by arguing as follows.

Dr Amy Glen (Murdoch University) MAS162 – Foundations of Discrete Mathematics Lecture 22 9

Four Related Conditionals
Note: p æ q and q æ p are not the same.

q ¬p ¬q

0
1
0
1

1
1
0
0

Dr Amy Glen (Murdoch University) MAS162 – Foundations of Discrete Mathematics Lecture 22 10

Four Related Conditionals . . .

q ¬p ¬q

0
1
0
1

1
1
0
0

You should notice that the truth tables of:
I p æ q and its contrapositive ¬q æ ¬p are the same,
I the converse q æ p and the inverse ¬p æ ¬q are the same.

Chapter 7: Propositional Logic
12

Then

Dr Amy Glen (Murdoch University) MAS162 – Foundations of Discrete Mathematics Lecture 22 12

Dr Amy Glen (Murdoch University) MAS162 – Foundations of Discrete Mathematics Lecture 22 14

So a compound proposition is a tautology if the final column of its truth table consists entirely of 1’s.

Tautologies are very important for our understanding and use of logical reasoning.

Dr Amy Glen (Murdoch University) MAS162 – Foundations of Discrete Mathematics Lecture 22 17

Tautologies and Contradictions . . .

Definition

Example

The proposition (p æ q) ¡ p · ¬q is a contradiction.

Chapter 7: Propositional Logic
19

Example: Recall that p æ q and its contrapositive ¬q æ ¬p have the same truth tables (i.e., the final column is [1, 1, 0, 1]T), so they are logically

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