Fundamentals of Logic and Reasoning

Posted on Dec 8, 2024 in English

Key Thinkers

• Leibniz (1646-1716)
• George Boole (1815-1864)
• Aristotle (384-322 BC)

Classification of Logic

• Inductive: For example, it is common for Corinthians to win the matches they play. As the Corinthians are playing, then they will win.
• Deductive: For example, every man is mortal. Dunga is a man. Therefore, Dunga is mortal.

Propositions: Arguments, Premises, Conclusion

Premise: Every even number is divisible by 2. Premise: The number 8 is divisible by 2. Conclusion: Therefore, the number 8 is an even number.

Premise: Every student loves math. Premise: Philip is a student. Conclusion: Therefore, Philip loves math.

Premise: If it does not rain, I’ll play football. Premise: It rained. Conclusion: So, I’ll play football. (Note: This conclusion does not logically follow from the premises.)

Fundamental Principles of Propositions

• Identity: A true proposition is true. A false proposition is false.
• Non-Contradiction: A proposition cannot be true and false simultaneously.
• Excluded Middle: Every proposition is either true or false.

Complex Formulas

Let:

• P: It’s cold.
• Q: It’s raining.

Then:

• ~P ^ ~Q: It is not cold and it is not raining.
• (P ^ ~Q) -> P: If it is cold and it is not raining, then it is cold.
• P v ~Q: It is cold or it is not raining.
• P -> ~Q: If it is cold, then it is not raining.
• P <-> ~Q: It is cold if and only if it is not raining.

Reverse Engineering Propositions

Let:

• P: Gisele is tall.
• Q: Gisele is elegant.

Then:

• Gisele is tall and elegant: P ^ Q
• Gisele is tall, but is not elegant: P ^ ~Q
• It is not true that Gisele is short and elegant: ~(~P ^ Q)
• It is true that Gisele is short or that she is not elegant: ~P v ~Q

Order of Precedence

1. ~ (Negation)
2. ^ or v (Conjunction or Disjunction)
3. -> (Conditional)
4. <-> (Biconditional)

Therefore, the biconditional is the strongest connective, and negation is the weakest.

Expressions in Portuguese

• (^): e (and), mas (but), também (also), além disso (moreover)
• (v): ou (or)
• (->): se p então q (if p then q), p implica q (p implies q), p, então q (p, then q), p somente se q (p only if q), q segue-se a p (q follows p)
• (<->): p se e somente se q (p if and only if q), p é uma condição necessária e suficiente para q (p is a necessary and sufficient condition for q)
• (~): não p (not p), é falso que p (it is false that p), não é verdade que p (it is not true that p)

Logic Puzzles and Solutions

Problem 1

Five girls are sitting in the front row of the classroom. They are: Maria, Mariana, Marina, Marisa, and Matilde. Marisa is at one end and Marina is on the other. Mariana sits next to Marina, and Matilde sits next to Marisa. Now answer:

1. How many are between Marina and Marisa? 3
2. Who is in the middle? Maria
3. Who is between Matilde and Mariana? Maria
4. Who is between Marina and Maria? Mariana
5. How many are between Marisa and Mariana? 2

Problem 2

What is the missing number in the square below?

Problem 3

Four students withdrew books from the school library. Each is from a different grade, and their names are: John, Philip, Paula, and Luiza. They are reading different literary genres: suspense, humor, adventure, and romance. One of them is on page 8, another on page 34, the third on page 67, and the last, almost finishing the book, is on page 108. You will discover the name of each child, their grade, the kind of book they are reading, and what page each one is on. To do this, pay attention to the clues below:

• John is on page 67.
• The girl in the 1st grade is at the very beginning of the book of humor.
• Philip, who is in the 4th grade, does not read books of suspense or romance.
• Paula is in the 2nd grade but is not on page 108.
• The boy in the 3rd grade is reading a novel.

John / 3rd grade / Romance / Page 67

Luiza / 1st grade / Humor / Page 8

Philip / 4th grade / Adventure / Page 108

Paula / 2nd grade / Suspense / Page 34

Problem 4

There were 30 people at a party, and $30.00 was spent on tickets, as follows:

• Men pay $2.00
• Women pay $0.50
• Children pay $0.10

How many men, women, and children joined the party?

14 Men, 1 Woman, 15 Children

Tautology and Contradiction

Tautology is a proposition whose values in its truth table are always true.

Contradiction is a proposition whose logical values in its truth table are always false.

A proposition that is neither a Tautology nor a Contradiction is called a Contingency.

Logical Equivalence

Logical equivalence is used in statements of valid arguments, sets, and their properties. Because they have characteristics similar to arithmetic with numbers, such properties are known as “Algebras of Propositions.”

Problem: Movie Night

Three friends decided to go to the cinema to see a movie, but at that time, the only option was “All is a Matter of Logic.” Then a discussion arose, as some claimed to have seen the movie:

• Tuca: If Joca did not watch, then Kika also did not attend.
• Joca: Tuca did not watch the movie, but Kika did.
• Kika: I watched the movie, or Joca did not attend.

Propositions are:

• P: Tuca watched the movie.
• Q: Joca watched the movie.
• R: Kika watched the movie.

Using the truth table, answer the following questions:

1. If all watched the movie, who is lying?
2. If everyone is telling the truth, who has not watched the movie?

Resolution

Testimony of Tuca: ~Q -> ~R

Testimony of Joca: ~P ^ R

Testimony of Kika: R v ~Q

General Truth Table:

Answers

1. If all watched the movie (P: V, Q: V, R: V), we are in the first line. Then we see that the liar is Joca.
2. If everyone is telling the truth, we are in the last line. So nobody saw the movie.