We return with another new content of the course of propositional logic, in this occasion I will dedicate myself to develop an interesting logical connective, that is, the logical disjunction or simply disjunction.
I will dedicate myself to explaining with some examples where we will see a small problem with disjunctive reasoning and how to solve this problem by defining two types of statements, that is, the inclusive statement and the exclusive statement.
What is a logical disjunction?
It is also known as logical sum, this type of statement gives us the alternative or possibility of choosing the validity of one or more of its component statements, I mean the logical disjunction.
Among all known logical connectives, disjunction has a double meaning, but in mathematics, it is necessary to differentiate it symbolically, they can be differentiated as inclusive and exclusive disjunction.
Let’s see an example to understand what is the logical disjunction and its variants, although subtle but identifiable.
Example
The disjunctive statement of the type “Samantha is male or female” is a selective statement, because we can select which simple statement is true. The proposition can be broken down as follows:
- Samantha is a woman
- Samantha is a man
It is known and also has a historical record that Samantha is not a unisex name, that we are dealing with a person of the female sex.
But since the connective “or” give us the choice between one of the two, we choose “Samantha is a woman.” We then say the following:
- Samantha is a woman (true statement).
- Samantha is a man (false statement).
Therefore “Samantha is male or female” is a true statement by a matter of choice.
In the previous example, we saw a compound statement where it was possible to choose any of the simple statements with at least one true validity so that the entire statement is true, that is, only one option could be chosen from the two available options.
The following example will deal with another type of disjunction where we have the possibility of choosing both at the same time without any contradiction, if we have the possibility \( \mathrm{A} \) and another \( \mathrm{B} \), any of them can be chosen and even choose simultaneously both, let’s see:
Example:
The statement “My cat is a feline or is it an animal” is a statement in which either of the two alternatives can be selected, we break down the proposition.
- My cat is a feline.
- My cat is an animal.
What option can we choose to determine that your compound statement is true? As we can see, both simple statements are true.
Then we can choose both, and with this concludes that our statement “My cat is a feline or is it an animal” is also true.
These types of statements that despite being similar, have some differences, we will explain them in the following sections.
Inclusive disjunction
This type of disjunction refers to illustrative example 2 and has the property of being able to choose any true statement that composes it (if it exists) to determine that our proposition that forms it is valid, here is its definition:
Definition:
The inclusive disjunction with symbol \( \vee \) is a logical connective that joins two statements \( p \) and \( q \) forming a new statement \( p \vee q \) in such a way that its truth value is false if the statements \( p \) and \( q \) turn out to be false, otherwise, it turns out to be true if at least one of its component statements is true.
In general, the inclusive disjunction is also called a logical disjunction, from now on any statement formed hierarchically by an inclusive disjunction will be called an inclusive statement.
According to the definition that we have just proposed for the meaning of the symbol \ (\ vee ) (which is literally written with the word “or”), an inclusive statement must have 3 possible choices to indicate that the statement is true and one that we can be given as false. Let’s look at some examples.
Example:
- The number 2 is real or integer.
- Cats have four legs or have tails.
- I read a book wearing a cap or sitting down.
For any of these examples, it is possible that any of the simple statements of the inclusive statements can be made simultaneously or also choose only one of them.
Truth table of the inclusive disjunction
Based on these examples, the truth table for this logical connective is:
\[ \begin{array}{ c | c | c } p & q & p \vee q \\ \hline V & V & V \\ V & F & V \\ F & V & V \\ F & F & F \end{array} \]
Some logical laws of inclusive disjunction
In the same way as the logical conjunction, the inclusive disjunction also possesses some important logical properties and laws, we list it here.
Let the propositions \( p \), \( q \) and \( r \) have:
- Associative law: \( ( p \vee q ) \vee r = p \vee ( q \wedge r ) \)
- Existence of the neutral element: \( \mathrm{V} (p) \vee F = \mathrm{V} (p) \)
- Commutative law: \( p \vee q = q \vee p \)
- Distributive laws of inclusive disjunction and conjunction:
\[ p \vee (q \wedge r) = ( p \vee q ) \wedge ( p \vee r ) \\ p \wedge ( q \vee r ) = ( p \wedge q ) \vee ( p \wedge r ) \] - Existence of the complementary element: \( \mathrm{V} ( \sim p \vee p ) = V \)
- The negation of disjunction results in conjunction: \( \sim ( p \vee q ) = \sim p \wedge \sim q \)
Inclusive disjunction and the relationship with the union.
In set theory, inclusive disjunction can be represented by the union between two sets, for example, we have an element that can belong to two different sets, they can be \( x \in \mathrm {A} \) and \( x \in \mathrm {B} \), to represent that the element \( x \) belongs to any of the sets \( \mathrm{A} \) and \( \mathrm {B} \) or both, we write So:
\[ x \in \mathrm{A} \vee x \in \mathrm{B} \]
This is the same as typing:
\[ x \in \mathrm{ A \cup B } \]
It can also be defined as follows:
\[ \mathrm{ A \cup B } = \left \{ x| x \in A \vee x \in B \right \} \]
We can illustrate graphically with Venn diagrams as follows:
This diagram means that the element \( x \) can be in any of these 3 bounded regions.
Exclusive disjunction
This type of disjunction is stricter and refers to illustrative example 1 where a compound statement can’t be true if both are true, at most it is only possible to choose a true statement so that the compound statement is true.
Definition of exclusive disjunction
The exclusive disjunction with symbol \( \bigtriangleup \) is a logical connective that joins two statements \( p \) and \( q \) forming a new statement \( p \bigtriangleup q \) in such a way that its validity is false if the statements \( p \) and \( q \) have the same truth value, otherwise, it turns out to be true if the statements \( p \) and \( q \) have opposite truth values.
As we have seen, there are two types of disjunction, one is the inclusive or weak disjunction and the other is the exclusive or strong disjunction and both use literally the letter “or” but in different ways.
These differences are necessary because there are situations where we can see that not always the same validity of its statements that compose it can give us the same general validity of the main statement, that is, a statement can be true or false with the same truth values of its propositional variables that compose it.
A proposition formed hierarchically by an exclusive disjunction from now on we will call it an exclusive proposition.
Example:
Here are some examples of an exclusive statement.
- Either you are sick or you are healthy.
- Either it is false or it is true.
- Either you are immobile or you are in motion.
These statements have a limit, they are only true if and only if a single propositional variable (simple proposition) that composes it is true.
Truth Table for The Exclusive Disjunction
There is another logical symbolization of this type of disjunction, since it turns out to be opposite to the logical biconditional \( ( \leftrightarrow ) \), therefore, we can also represent it with this symbol \( \nleftrightarrow \), the truth table of the exclusive disjunction is:
\[ \begin{array}{ c | c | c } p & q & p \bigtriangleup q \\ \hline V & V & F \\ V & F & V \\ F & V & V \\ F & F & F \end{array} \]
Some Laws of Exclusive Disjunction
Let’s see some properties of the exclusive disjunction, Let us be the propositions \( p \), \( q \) and \( r \), we have:
- Associative law: \( ( p \bigtriangleup q ) \bigtriangleup r = p \bigtriangleup ( q \bigtriangleup r ) \).
- Commutative law: \( p \bigtriangleup q = q \bigtriangleup p \).
- Distributive law with the logical conjunction: \( p \wedge (q \bigtriangleup r ) = ( p \wedge q ) \bigtriangleup ( p \wedge r ) \)
I hope that with these examples, definitions, properties, and some logical laws you will be able to understand the meaning of disjunction and its only two necessary variants.
Finally, this logical connective in elementary set theory is used to explain the concept of union of two sets and its main properties, let’s see this relationship in the next section.
Exclusive disjunction and disjoint sets
An element can belong to a set \( \mathrm{A} \) or \( \mathrm{B} \) or both, but if such sets have no elements in common, then said element can belong to one and only one of the sets.
The exclusive disjunction can remedy this point, for example, if we have an element \( x \) where it can be contained only in the set \( \mathrm{A} \) or only in the set \( \mathrm{B} \) but not in both, it is represented like this:
\[ x \in \mathrm{A} \bigtriangleup x \in \mathrm{B} \]
Its equivalent would be:
\[ x \in \mathrm{A + B} \]
If we want to merge the two interpretations, it can be expressed as follows:
\[ \mathrm{A + B} = \left \{ x| x \in \mathrm{A} \bigtriangleup x \in \mathrm{B} \right \} \]
Now let’s see how it is represented graphically:
Note that there are no terms in common between the sets \( \mathrm {A} \) and \( \mathrm {B} \), this is represented by the intersection symbol “\( \cap \)”, like this \( \mathrm{A \cap B} = \phi \), the symbol “\( \phi \)” means that there are no elements and is called an empty set.
We come to the end of the topic, I hope it has been very helpful. In the next section I will explain one of the very important logical connectors after the disjunction, I mean the material conditional.
Thank you for getting here, have a good day, and see you soon.