Examples Propositional Logic

Example 1.

true

¬P

true ∧ P

false ∨¬P

false ⊃ P

P ≡ (P ∧ R)

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)

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

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

true iﬀ operand is false.

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

true iﬀ both operands have the same value.

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.

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).

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.

2023-09-12