## Identities

### Subjects to be Learned

• Identities(tautologies) of propositional logic
• Dual of proposition

### Contents

From the definitions(meaning) of connectives, a number of relations between propositions which are useful in reasoning can be derived. Below some of the often encountered pairs of logically equivalent propositions, also called identities, are listed.
These identities are used in logical reasoning. In fact we use them in our daily life, often more than one at a time, without realizing it.
If two propositions are logically equivalent, one can be substituted for the other in any proposition in which they occur without changing the logical value of the proposition.

Below corresponds to and it means that the equivalence is always true (a tautology), while means the equivalence may be false in some cases, that is in general a contingency.

That these equivalences hold can be verified by constructing truth tables for them. Click here for more detailed discussions about it.

First the identities are listed, then examples are given to illustrate them.

List of Identities:

1. P (P P) ----- idempotence of
2. P (P P) ----- idempotence of
3. (P Q) (Q P) ----- commutativity of
4. (P Q) (Q P) ----- commutativity of
5. [(P Q) R] [P (Q R)] ----- associativity of
6. [(P Q) R] [P (Q R)] ----- associativity of
7. (P Q) ( P Q) ----- DeMorgan's Law
8. (P Q) ( P Q) ----- DeMorgan's Law
9. [P (Q R] [(P Q) (P R)] ----- distributivity of over
10. [P (Q R] [(P Q) (P R)] ----- distributivity of over
11. (P True) True
12. (P False) False
13. (P False) P
14. (P True) P
15. (P P) True
16. (P P) False
17. P ( P) ----- double negation
18. (P Q) ( P Q) ----- implication
19. (P Q) [(P Q) (Q P)]----- equivalence
20. [(P Q) R] [P (Q R)] ----- exportation
21. [(P Q) (P Q)] P ----- absurdity
22. (P Q) (Q P) ----- contrapositive
Let us see some example statements in English that illustrate these identities.

Examples:

1. P (P P) ----- idempotence of
What this says is, for example, that "Tom is happy." is equivalent to "Tom is happy or Tom is happy". This and the next identity are rarely used, if ever, in everyday life. However, these are useful when manipulating propositions in reasoning in symbolic form.

2. P (P P) ----- idempotence of
Similar to 1. above.

3. (P Q) (Q P) ----- commutativity of
What this says is, for example, that "Tom is rich or (Tom is) famous." is equivalent to "Tom is famous or (Tom is) rich".

4. (P Q) (Q P) ----- commutativity of
What this says is, for example, that "Tom is rich and (Tom is) famous." is equivalent to "Tom is famous and (Tom is) rich".

5. [(P Q) R] [P (Q R)] ----- associativity of
What this says is, for example, that "Tom is rich or (Tom is) famous, or he is also happy." is equivalent to "Tom is rich, or he is also famous or (he is) happy".

6. [(P Q) R] [P (Q R)] ----- associativity of
Similar to 5. above.

7. (P Q) ( P Q) ----- DeMorgan's Law
For example, "It is not the case that Tom is rich or famous." is true if and only if "Tom is not rich and he is not famous."

8. (P Q) ( P Q) ----- DeMorgan's Law
For example, "It is not the case that Tom is rich and famous." is true if and only if "Tom is not rich or he is not famous."

9. [P (Q R] [(P Q) (P R)] ----- distributivity of over
What this says is, for example, that "Tom is rich, and he is famous or (he is) happy." is equivalent to "Tom is rich and (he is) famous, or Tom is rich and (he is) happy".

10. [P (Q R] [(P Q) (P R)] ----- distributivity of over
Similarly to 9. above, what this says is, for example, that "Tom is rich, or he is famous and (he is) happy." is equivalent to "Tom is rich or (he is) famous, and Tom is rich or (he is) happy".

11. (P True) True.   Here True is a proposition that is always true. Thus the proposition (P True) is always true regardless of what P is.
This and the next three identities, like identities 1 and 2, are rarely used, if ever, in everyday life. However, these are useful when manipulating propositions in reasoning in symbolic form.

12. (P False) False

13. (P False) P

14. (P True) P

15. (P P) True
What this says is that a statement such as "Tom is 6 foot tall or he is not 6 foot tall." is always true.

16. (P P) False
What this says is that a statement such as "Tom is 6 foot tall and he is not 6 foot tall." is always false.

17. P ( P) ----- double negation
What this says is, for example, that "It is not the case that Tom is not 6 foot tall." is equivalent to "Tom is 6 foot tall."

18. (P Q) ( P Q) ----- implication
For example, the statement "If I win the lottery, I will give you a million dollars." is not true, that is, I am lying, if I win the lottery and don't give you a million dollars. It is true in all the other cases. Similarly, the statement "I don't win the lottery or I give you a million dollars." is false, if I win the lottery and don't give you a million dollars. It is true in all the other cases. Thus these two statements are logically equivalent.

19. (P Q) [(P Q) (Q P)]----- equivalence
What this says is, for example, that "Tom is happy if and only if he is healthy." is logically equivalent to ""if Tom is happy then he is healthy, and if Tom is healthy he is happy."

20. [(P Q) R] [P (Q R)] ----- exportation
For example, "If Tom is healthy, then if he is rich, then he is happy." is logically equivalent to "If Tom is healthy and rich, then he is happy."

21. [(P Q) (P Q)] P ----- absurdity
For example, if "If Tom is guilty then he must have been in that room." and "If Tom is guilty then he could not have been in that room." are both true, then there must be something wrong about the assumption that Tom is guilty.

22. (P Q) (Q P) ----- contrapositive
For example, "If Tom is healthy, then he is happy." is logically equivalent to "If Tom is not happy, he is not healthy."

The identities 1 ~ 16 listed above can be paired by duality relation, which is defined below, as 1 and 2, 3 and 4, ..., 15 and 16. That is 1 and 2 are dual to each other, 3 and 4 are dual to each other, .... Thus if you know one of a pair, you can obtain the other of the pair by using the duality.

Dual of Proposition
Let X be a proposition involving only , , and as a connective. Let X* be the proposition obtained from X by replacing with , with , T with F, and F with T. Then X* is called the dual of X.

For example, the dual of [P Q ] P is [P Q ] P, and the dual of [ P Q] [ T R] is [ P Q] [ F R] .

Property of Dual: If two propositions P and Q involving only , , and as connectives are equivalent, then their duals P* and Q* are also equivalent.

### Test Your Understanding of Identities

Indicate which of the following statements are correct and which are not.

Click Yes or No , then Submit. There are two sets of questions.

Next -- Examples of Use of Identities

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