Propositional Logic

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

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.

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

**List of Identities:**

- P (P P) ----- idempotence of
- P (P P) ----- idempotence of
- (P Q) (Q P) ----- commutativity of
- (P Q) (Q P) ----- commutativity of
- [(P Q) R] [P (Q R)] ----- associativity of
- [(P Q) R] [P (Q R)] ----- associativity of
- (P Q) ( P Q) ----- DeMorgan's Law
- (P Q) ( P Q) ----- DeMorgan's Law
- [P (Q R] [(P Q) (P R)] ----- distributivity of over
- [P (Q R] [(P Q) (P R)] ----- distributivity of over
- (P True) True
- (P False) False
- (P False) P
- (P True) P
- (P P) True
- (P P) False
- P ( P) ----- double negation
- (P Q) ( P Q) ----- implication
- (P Q) [(P Q) (Q P)]----- equivalence
- [(P Q) R] [P (Q R)] ----- exportation
- [(P Q) (P Q)] P ----- absurdity
- (P Q) (Q P) ----- contrapositive

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.

Let

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] .

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