chapter 2: the logic of compound statements 2.1 logical forms and equivalence 12.1 logical forms and...

2.1 Logical Forms and Equivalences 1

Discrete Structures

Chapter 2: The Logic of Compound Statements

2.1 Logical Forms and Equivalence

Logic is a science of the necessary laws of thought, without which no employment of the understanding and the reason takes place.

– Immanuel Kant, 1724 – 1804Foundation for the Metaphysics of Morals, 1785


Logic

• Logic is the study of reasoning; specifically whether reasoning is correct.

Note: We use p, q, and r to represent propositions.

• Logic does:– Assess if an argument is valid/invalid

• Logic does not directly:– Assess the truth of atomic statements


Statements

• Statement or Proposition – any sentence that is true or false but not both


Statements

• Examples of Statements:1. George Washington was the first president of the

United States.

2. Baltimore is the capital of Maryland.

3. Seventeen is an even number.

• The above statements are either true (1) or false (2, 3)


Not Statements

• These are not statements:1. Earth is the only planet in the universe that

contains life.

2. Buy two tickets to the rock concert for Friday

3. Why should we study logic?

• None of the above can be determined to be true or false so they are not statements.


Ambiguity

• It is possible that a sentence is a statement yet we can not determine its truth or falsity because of an ambiguity or lack of qualification.

• Examples:1. Yesterday it was cold.

2. He thinks Philadelphia is a wonderful city.

3. Lucille is a brunette.

For (1) we need to determine what we mean by the word “cold”

For (2) we need to know whose opinion is being considered.

For (3) it depends on which Lucille we are discussing.


Compound Statements

• In speech and writing, we combine propositions using connectives such as and or even or.

• For example, “It is snowing” and “It is cold” can be combined into a single proposition “it is snowing and it is cold.”


• Let:

p = “It is snowing.”

q = “It is cold.”

Compound Statements

Connective Symbol Name Example

Not ~ Negation ~q: It is not the case that it is cold.

And Conjunction pq: It is snowing and it is cold.

Or (inclusive) Disjunction pq: It is snowing or it is cold.

Or (exclusive) Exclusive Or pq: It is snowing or it is cold but not both.

If…then… Conditional pq: If it is snowing, then it is cold.

If and only if (iff) Biconditional pq: It is snowing iff it is cold.


Translating: English to Symbols

• The word but translates the same as and.

“Jim is tall but he is not heavy” translates to

“Jim is tall AND he is not heavy”


Translating: English to Symbols

• The words neither-nor translates the same as not.

“Neither a borrower nor a lender be” translates to

“Do NOT be a borrower and do NOT be a lender”


Truth Values

• If sentences are statements then the must have well-defined truth values meaning the sentences must be either true or false.


Negation

• Definition

If p is a statement variable, the negation of p is “not p” or “it is not the case that p” and is denoted p. It has the opposite truth value from p:

if p is true, p is false

if p is false, p is true


Negation Truth Table

p p

T

F

The truth value for negation are summarized in the truth table on the right.


Conjunction

• Definition

If p and q are statement variables, the conjunction of p and q is “p and q”, denoted p q. It is true when BOTH p and q are true. If either p or q is false, or both are false, p q is false.


Conjunction Truth Table

• The truth value for conjunction are summarized in the truth table on the right.

p q p q

T T

T F

F T

F F


Disjunction (Inclusive Or)

• Definition– If p and q are statement variables, the disjunction

of p and q is “p or q”, denoted p q. It is true when either p is true, q is true, or both p and q are true. If both p and q is false, p q is false.

• Example– You may have cream or sugar with your coffee


Disjunction Truth Table


p q p q

T T

T F

F T

F F


Exclusive Or

• Definition– If p and q are statement variables, the exclusive or

of p and q is “p or q”, denoted pq. It is true when either p is true or when q is true, but not both. If both p and q is false, p q is false. If both p and q is true, p q is false.

• Example– Your meal comes with soup or salad.


Exclusive Or Truth Table


p q p q

T T

T F

F T

F F


Statement Form

• Definition– A statement form is an expression made up of

statement variables such as p, q, and r and logical connectives that becomes a statement when actual statements are substituted for the component statement variables.


Example

• Construct a truth table for the statement form (pq)(pq).

p q p q pq pq (pq)(pq)

T T

T F

F T

F F


Tautology

• Definition– A tautology is a statement form that is always true

regardless of the truth values of the individual statements substituted for its statement variables.


Contradiction

• Definition– A contradiction is a statement form that is always

false regardless of the truth values of the individual statements substituted for its statement variables.


Logically Equivalent

• Definitions– Two statement forms are logically equivalent iff

they have identical truth values for each possible substitution of statements for their statement variables. Logical equivalence of statement forms P and Q is denoted by PQ.


Testing for Logical Equivalence

1. Construct a truth table with one column for the truth values of P and another column for the truth values of Q.

2. Check the combination of truth values of the statement variables.

a. If in each row the truth value of P is the same as the truth value of Q, then PQ.

b. If in each row the truth value of P is not the same as the truth value of Q, then PQ.


Example

Are the statement forms (pq) and pq logically equivalent?

(pq) pq

p q p q pq (pq) (pq)

T T

T F

F T

F F


Theorem 2.1.1 – Logical Equivalence

• Given any statement variables p, q, and r, a tautology t, and a contradiction c, the following logical equivalences hold.

1. Commutative Laws:

2. Associative Laws:

3. Distributive Laws:

p q q p p q q p

p q r p q p p q r p q p

p q r p q p r

4. Identity Laws:

5. Negation Laws: ~ ~

6. Double Ne

p q r p q p r

p p p p

p p p p

t c

t c

gative Law: ~ ~

7. Idempotent Laws:

8. Universal Bound Laws:

9. De Morgan's L

p p

p p p p p p

p p

t t c c

aws: ~ ~ ~ ~ ~ ~

10. Absorption Laws:

11. Negations of and : ~

p q p q p q p q

p p q p p p q p

t c t c ~ c t


De Morgan’s Laws

• The negation of an and statement is logically equivalent to the or statement in which each component is negated.

(p q) p q

• The negation of the or statement is logically equivalent to the and statement in which each component is negated.

(p q) p q


Example – pg. 38 # 33

• Use De Morgan’s Laws to write negations for the statement.-10 < x < 2


Example – pg. 38 # 51

• Use Theorem 2.1.1 to verify the logical equivalences. Supply a reason for each step.

p (q p) p


Next time

• Continue with logical equivalence.• Discuss valid and invalid arguments.

chapter 2: the logic of compound statements 2.1 logical forms and equivalence 12.1 logical forms and...

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