### II.3 Examples Propositional Logic

#### Propositional Logic Examples [Slide 23]

Example 1.

 true ¬P true ∧ P false ∨¬P false ⊃ P P ≡ (P ∧ R)

#### Propositional Logic Examples [Slide 24]

Example 2.

Let P denote the proposition “The moon is made of cheese”

Let Q denote the proposition “The moon is red”

The sentence “If the moon is red, it is not made of cheese” is translated as propositional formula Q (¬P)

#### Truth Table: And [Slide 25]

 P Q P ∧ Q false false false false true false true false false true true true

true iﬀ (if and only if) both operands are true

#### Truth Table: Or [Slide 26]

 P Q P ∨ Q false false false false true true true false true true true true

true iﬀ (if and only if) either operands are true

#### Truth Table: Not [Slide 27]

 P ¬P false true true false

true iﬀ operand is false.

#### Truth Table: Implication [Slide 28]

 P Q P ⊃ Q false false true false true true true false false true true true

true iﬀ ﬁrst is true and second is true or the ﬁrst is false.

#### Truth Table: Equivalence [Slide 29]

 P Q P ≡ Q false false true false true false true false false true true true

true iﬀ both operands have the same value.

#### Semantic Reasoning: Truth tables [Slide 30]

• From truth table take into account ALL its constituents. The formula P ((¬P) Q) has the following constituents ¬P, (¬P) Q as well as P and Q.

 P Q ¬P (¬P) ⊃ Q P ∧ ((¬P) ⊃ Q) false false false true true false true true
• The values in the last column determine the value of the proposition:
• if some values are true, the proposition is known to be satisﬁable
• if the values are all true, the proposition is valid
• if all are false, the proposition is not satisﬁable (contradiction).

#### Reasoning with Truth table [Slide 31]

• Works ﬁne when there are 2 variables: 4 lines in the table.
• three variables: 8 lines in the table.
• 20 variables is deﬁnitely out of hand: 220 lines in the table. You don’t want to look at a million lines, and if you do, you will make mistakes!
• Hundreds of variables - not in a million years.
• There are other ways to reason, but outside the scope of this course.

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