Logic studies the process of argumentation.

A proposition is any statement that can be judged as either true or false. Thus, there is always a binary value attributed to it: its truth value, which can be either true (T) or false (F), with no third option. This property is called the Principle of the Excluded Middle. Together with the Principle of Identity (which states that every being is identical to itself) and the Principle of Non-Contradiction (which states that no proposition can be both true and false simultaneously), they form the foundation of Aristotle's classical logic.

A logical argument is a sequence of propositions, such that the initial ones are the premises and the last one is the conclusion.

First premise: All humans are mortal.

Second premise: I am human.

Conclusion: Therefore, I am mortal.

An argument also consists of connectives, which are the structures responsible for combining one or more propositions to produce a new, more complex one.

Negation (¬)

It has the function of changing the meaning of a proposition to the opposite meaning. For example, given the proposition p: “I like camping”, the proposition ¬p will be “I do not like camping”.

The propositions p and ¬p always have different logical values. That is, when p is true, ¬p is false and vice versa.

Conjunction (∧) and disjunction (∨)

It is used to combining two propositions p and q. While the conjunction represents the sense of complementarity, the disjunction represents the sense of simultaneity. For example, for the sentence “The sky is blue or the sea is red” to be true, it is enough that one of the sentences: “The sky is blue” or “The sea is red” is true. On the other hand, for the sentence “The sky is blue and the sea is red” to be true, it is necessary that both sentences are true.

Conditional (→)

It is used to formulate a proposition that conveys an idea of cause and effect. An example of a sentence that has this meaning is “If it breaks, then she fix it”. Symbolically, this sentence is represented as p → q, where p: “It breaks” and q: “She fix it”.

In natural language, it is said that p is a sufficient condition for q, and that q is a necessary condition for p.

Associated conditional propositions

• Contrapositive (¬q → ¬p): If she doesn't fix it, then it doesn't break.

• Reciprocal (q → p): If she fixes it, then it breaks.

• Inverse (¬p → ¬q): If it doesn't break, then she doesn't fix it.

Biconditional (↔)

It indicates what is the equivalence relation between p and q in the following sense: p ↔ q will be true if p and q are both true or both false; and it will be false otherwise.

In natural language, it is said that p is a necessary and sufficient condition for q, or that p if and only if q.

Cards

This is an adaptation of an experiment proposed by Wason that sought to understand whether people knew how to intuitively use classical logic.

The five cards 2, 3, A, E, M are on a table, and each has a number on one side and a letter on the other. We must decide whether the following proposition is true:

"If a card has a vowel on one side, then it has an even number on the other side."

What is the smallest number of cards we need to turn over to decide correctly?

First, we have to turn over cards A and E, which are vowels. In fact, if on the other side of them there is an odd number, the proposition will be false. Note that we do not need to turn over card M (which is a consonant), because, regardless of the number on the other side, the proposition will never be invalidated. We have to turn over card 3, because if we have a vowel on the other side, the proposition will be false. Finally, it is not necessary to turn over card 2, since regardless of the letter on the other side, the sentence cannot be invalidated. Therefore, we have to turn over three cards: 3, A, E.

Gods

This is a more challenging puzzle, published in The Harvard Review of Philosophy in 1996.

Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which.

This puzzle is very tricky, mainly because of the Random god, who gives meaningless answers, and the language barrier. So let's first consider two simpler versions before moving on to the general case.

The Puzzle Without the Language Barrier

Imagine for now that we understand the gods' language and can translate their answers into simple yes or no. So, we have:

• If you know that a god is either True or False (not Random), you can figure out which one is which by simply asking them something you already know the answer to. For example, by asking "Is 1 + 1 = 2?", True will say yes, and False will say no.

• If you know that a god is True, you can figure out who the other two are by pointing to one of the other gods and asking "Is this Random?".

  • If the answer is yes, that god is Random, and the remaining one must be False.
  • If the answer is no, that god is False, and the remaining one is Random.

• If you know that a god is False, you can figure out the other two in the same way as you would in (2).

From this, we see that the key is to ask one question to identify a god who isn't Random. Once you find that god, you can figure out if they're True or False by asking them, "Is 1 + 1 = 2?" Then you can use the process above to identify the others.

Finding a Non-Random God

To identify a god who isn't Random, you can ask any of the gods: "Are you True if and only if this other god is Random?" pointing to one of the other two.

• If you're asking True, they'll say yes if you're pointing at Random, and no if you're pointing at False.

• If you're asking False, they'll lie: they'll say yes if you're pointing at Random, and no if you're pointing at True.

• If you're asking Random, their answer is useless, but the god you're pointing at and the remaining god are definitely not Random.

So, if the answer is yes, the third god (neither the one you're pointing at nor the one you're questioning) is not Random. If the answer is no the god you're pointing at is not Random.

The General Case

Now let's return to the original puzzle, where the gods answer da and ja. The same strategy works, but with a twist.

Before asking a question, you can start by saying, "The word da means yes if and only if…" and then follow it with the question as before. This way:

• If the god would have answered yes in English, they'll say da.

• If the god would have answered no in English, they'll say ja.

This adjustment lets you translate the gods' answers back into yes or no and solve the puzzle the same way as before.