Propositional-Logic

Propositional logic, also known as sentential logic and statement logic, is the branch of logic that deals

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Propositional Logic

• Propositional logic is the simplest form of logic.

• In it, facts of the real world are represented by sentences without arguments.

• A proposition will never be both true and false at the same time.

• Principles that must always be obeyed:

  • Principle of identity: A true proposition is true; a false proposition is false.
  • Principle of non-contradiction: No proposition can be both true and false at the same time.
  • Principle of the excluded middle: Every proposition is either true or false, with no other logical state for it.

The logical value of a proposition is:

• Truth if the proposition is true.

• Falsehood if the proposition is false.

• V(p) = Logical value of P = V (True)

• V(q) = Logical value of Q = F (False)

Which of the sentences below are propositions:

1. The moon is the only satellite of planet Earth. V(l) = F
2. Stop!
3. Would you like a cup of coffee?
4. The city of Salvador is the capital of the state of Amazonas. V(s) = F
5. The number 712 is odd. V(n) = F
6. I'm not quite sure if I like this color.
7. Fifteen is less than seven. V(q) = F
8. How are you?
9. There is life on other planets in the universe. V(u) = V
10. He is very talented. V(e) = V

Answer.

A proposition is a statement that can be either true or false. Let's analyze each sentence:

The moon is the only satellite of planet Earth. V(l) = F This is a proposition, but it is false. (The moon is not the only satellite of Earth.)

Stop! This is not a proposition. It is a command.

Would you like a cup of coffee? This is not a proposition. It is a question.

The city of Salvador is the capital of the state of Amazonas. V(s) = F This is a proposition, but it is false. (Salvador is the capital of Bahia, not Amazonas.)

The number 712 is odd. V(n) = F This is a proposition, and it is false. (712 is an even number.)

I'm not quite sure if I like this color. This is not a proposition. It expresses uncertainty.

Fifteen is less than seven. V(q) = F This is a proposition, but it is false. (Fifteen is greater than seven.)

How are you? This is not a proposition. It is a question.

There is life on other planets in the universe. V(u) = V This is a proposition, and its truth value is true, according to the notation.

He is very talented. V(e) = V This is a proposition, and its truth value is true, according to the notation.

Simple Propositions

Simple Propositions: A simple proposition or atomic proposition is one that does not contain any other proposition as an integral part of itself.

• Notation: They are generally designated by lowercase letters such as p, q, r, s,..., to represent propositions.
  • Example: p: John is intelligent.

Compound Propositions

Compound Propositions: A compound proposition or molecular proposition is one formed by the combination of two or more propositions.

• Notation: They are represented by uppercase letters P, Q, R, S,...

  • P(p, q): Felines are furry and birds have feathers.
  • R(r, s): I will buy a car or a bicycle.

Simple Exercise

Compound Propositions: If we combine, for example, the following statements: "Elephants are large" and "Basketballs are round", using the "and" connective, we get the proposition: "Elephants are large and basketballs are round".

What is the formal notation, and the logical value, of the simple propositions and the compound proposition?

We have "Elephants are large" as formal E = V and "Basketballs are round" as formal B = V. So we have E and B = V. The formal notation for the compound proposition is: E and B.

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