Propositional Logic in Artificial Intelligence

Learning Outcome:

At the end of the class, students should be able to:

• Define primitive statements
  
• Form a compound statements
  
• define negation, conjunction, disjunction, conditional, biconditional, tautology, contradiction and contingency statements.
  
• construct truth table.
  

LOGICAL CONNECTIVES

Main components of symbolic logic are:

• Proposition/Statements
• Connectives

PROPOSITION/STATEMENTS

• A proposition is a declarative sentence that is either true or false.
• The truth or falsify of a statement is called truth value.
• Usually denoted by letters p, q, r, s and so on. Example:
  

✔ You will be late to school if you miss the bus;
✔ Ms. Nina will have a broader audience next month;
✔ I did not join the competition;
✔ Today is Monday;
✔ I love reading thriller books;
✔ The number 3 is an odd integer;

Logical connectives and compound statements

Logical Connective	Compound statement	Symbolic compound statement
∧	p and q (conjunction)	p ∧ q
∨	p or q (disjunction)	p ∨ q
→	if p then q (conditional/implication)	p → q
↔	p if and only if q (biconditional)	p ↔ q

• If the compound statement has two variables (p and q) the truth table must be constructed with 4 rows.
• If 3 variables (p, q and r), then 8 rows.

Two primitive statements are defined as follows,

s: Your handbag is stylish
c: I like its colour.

Connectives join primitives statements into more complex statement:

Your handbag is stylish and I like its colour.

In symbolic: s ∧ c

Connective “but” has an identical role as “and”, thus, use same symbol ∧

Your handbag is stylish but I like its colour

In symbolic: s ∧ c

EXERCISE

Answer :

1. Boston is the capital of Massachusetts. Truth value: True.
2. Miami is the capital of Florida. Truth value: False. (The capital of Florida is Tallahassee)
3. 2 + 3 = 5. Truth value: True.
4. 5 + 7 = 10. Truth value: False.
5. x + 2 = 11

x + 2 = 11" not a proposition because it contains a variable (x) and is not a statement that can be determined as true or false without additional information about the value of x.

TRUTH VALUES AND TRUTH TABLES(NOT)

p	¬q
True	False
False	True

• The operation “not” or ¬ turns single statement into negation and it is not a connective.

For example, the negation of I like you is I don’t like you.

p: I like you.
¬q: I don't like you.

TRUTH VALUES AND TRUTH TABLES(CONJUNCTION /AND)

Combine primary statements by the word “and”denoted by p ∧ q (if 2 statements)

If p is true and q is true, then p ∧ q is true. Otherwise, p ∧ q is false.

p	q	p ∧ q
False	False	False
False	True	False
True	False	False
True	True	True

p: I play the piano (false)
q: I study logic (true)

Thus,

p ∧ q: I play the piano and study logic is a false statement

TRUTH VALUES AND TRUTH TABLES(DISJUNCTION/OR)

Combine primary statements by the word “or”. Denoted by p ∨ q (if 2 statements)

If p is true or q is true or both p and q are true, then p ∨ q is true. Otherwise, p ∨ q is false.

p	q	p ∨ q
False	False	False
False	True	True
True	False	True
True	True	True

p: 2+3=6 (false)
q: 3>2 (true)

Thus,

p ∨ q: (2+3=6) or (3>2) is a true statement

TRUTH VALUES AND TRUTH TABLES (CONDITIONAL STATEMENTS)

A compound statement of the form “If p then q”, p → q In p → q, p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence).

If p is true and q is false, then the conditional p → q (p implies q) is false. Otherwise, p → q is true.

p	q	p → q
False	False	True
False	True	True
True	False	False
True	True	True

If Amy is a human being, then Amy has feeling. Defined primitive statement:

p : Amy is a human being (is the antecedent/hypothesis).
q : Amy has feeling (is the consequent).

Thus, the only way the statement is false , that is:

p → q: If Amy is a human being and she doesn’t have feeling.

TRUTH VALUES AND TRUTH TABLES (BICONDITIONAL STATEMENTS)

If p and q have the same truth value, then p ↔ q is true. If p and q have opposite truth value, then p ↔ q is false.

p	q	p ↔ q
False	False	True
False	True	False
True	False	False
True	True	True