Review Question

Interpret the following sentences.

• (Smoke ⇒ Fire) ⇒ (¬Smoke ⇒ ¬Fire)
• [(Smoke ∧ Heat) ⇒ Fire] ∧ [(Smoke ⇒ Fire) ∨ (Heat ⇒ Fire)]
• (P ⇒ Q) ⇒ [(R v P) ⇒ (R v Q)]

Interpretation of Sentences:

1. (Smoke ⇒ Fire) ⇒ (¬Smoke ⇒ ¬Fire):
   • If where there is smoke, there is fire, then it logically follows that where there is no smoke, there is no fire. This sentence expresses a form of the contrapositive relationship.
2. [(Smoke ∧ Heat) ⇒ Fire] ∧ [(Smoke ⇒ Fire) ∨ (Heat ⇒ Fire)]:
   • The first part asserts that if there is both smoke and heat, there is fire. The second part presents a disjunction, stating that either where there is smoke, there is fire, or where there is heat, there is fire. The entire sentence combines these conditions.
3. (P ⇒ Q) ⇒ [(R v P) ⇒ (R v Q)]:

• If whenever P implies Q, then the statement "If either R or P, then either R or Q" is also true. This sentence involves implications and disjunctions based on the given variables P, Q, and R.

The Potential of First-order Logic

First-order logic (FOL) models the world in terms of Objects, Properties, Relations, and Functions.

Examples:

• Objects: Students, lectures, companies, cars, etc.
• Relations: Brother-of, bigger-than, outside, part-of, has-color, owns, etc.
• Properties: blue, oval, even, large, etc.
• Functions: father-of, best friend, second-half, more-than, etc.

Quantifiers

Sentences