This connective is very easy to understand, its development is simple and simplified, the logical bicionditional has premises that can be interchanged and its truth value remains unchanged.
It is the last logical connective that we will study, its mathematical concept is double-edged, those where two statements always go hand in hand, one depends on the other and vice versa, it is an intuitive concept that we will deal with shortly.
It is also related to the conditional, conjunction, logical disjunction and negation, the combination of which can be used as alternative definitions and is what we will see next.
Definition of biconditional
The bicionditional is a logical connective denoted by \( \leftrightarrow \) that connects two statements \( p \) and \( q \) forming a new statement \( p \leftrightarrow q \) such that its validity is true if its component statements have the same truth value and false if they have opposite truth values.
For two statements \( p \) and \( q \) connected by a biconditional can mutually depend on each other. Here \( p \) can be the antecedent of \( q \) as well as \( q \) can be the antecedent of \( p \).
Since two statements under this connector are mutually dependent, there is no hierarchy between the two.
From now on, when we refer to a statement hierarchically formed with this connective, we will call it a biconditional statement.
Another point that we must take into account is that this connector is commutative, the following equivalence is fulfilled \( p \leftrightarrow q \equiv q \leftrightarrow p \).
The literal meaning of the logical biconditional between two open statements or sentences is “If and only if“, in this case, the statement \( p \leftrightarrow q \) is read \( p \) if and only if \( q \).
Not always two statements can be literally connected in this way to make sense of a new proposition, although at a symbolic level they can, as \( p \leftrightarrow q \), let’s see an example to explain this detail.
Example
Case 1: We have the statement:
- I will leave the house if and only if it gets dark.
Not every statement can be a biconditional since the above statement can be written as follows:
- It gets dark if and only if I leave the house.
It is impossible for it to magically get dark because we simply leave the house, therefore, this statement is unfeasible, let’s look at another case.
Case 2: Let the following statement be true:
- I will leave the house as long as my mother buys a chocolate.
This statement can be written in inverted form as follows:
- My mother will buy a chocolate as long as I leave the house.
As we can see, this type of statements has more coherence than case 1, however, in propositional logic the meaning of the arguments will not be taken into account, restricting only and only to their truth values, which we will see in a truth table shortly.
Relationship with the conditional and logical conjunction
As I said before, the biconditional is simple to understand, there is not much magic in its explanation. There is a connection between the material conditional and the logical conjunction that can be related to the biconditional. We already saw this with the previous example of case 2 where the statements were commutative (interchangeable).
Two statements that mutually depend on each other means that any of them can be the antecedent of the other and vice versa, which is fulfilled that \( p \leftrightarrow q \) as well as \( q \leftrightarrow p \) and as any of these combinations is true it follows that the biconditional is also true, therefore, we achieve a new relation between biconditional with the logical conjunction and material conditional in the following way:
\[ ( p \rightarrow q ) \wedge ( q \rightarrow p ) \equiv p \leftrightarrow q \]
Truth Table of The Biconditional Logic
This table tells us that both the antecedent and the consequent must be either true or false for the biconditional statement to be true. The following truth table for the biconditional shows this characteristic.
\[ \begin{array}{ c | c | c } p & q & p \leftrightarrow q \\ \hline V & V & V \\ V & F & F \\ F & V & F \\ F & F & V \end{array} \]
This means that the proposition \( q \) is a sufficient and necessary condition for \( p \), but we will see these points with the concept of logical equivalence and that many times they are confused with the biconditional.
Relationship between exclusive disjunction
In the section on logical disjunction, we mentioned that the exclusive disjunction is opposite to the biconditional. Let us first look at the truth table of the exclusive disjunction:
\[ \begin{array}{ c | c | c } p & q & p \bigtriangleup q \\ \hline V & V & F \\ V & F & V \\ F & V & V \\ F & F & F \end{array} \]
According to these truth value tables we can write a relationship between the logical biconditional and the exclusive disjunction as follows:
\[ p \leftrightarrow q \equiv \sim ( p \bigtriangleup q ) \]
Although we could also have written it this way:
\[ p \leftrightarrow q \equiv \sim ( p \nleftrightarrow q ) \]
Where the symbol \( \nleftrightarrow \) also represents the exclusive disjunction.
Some logical laws of biconditionality
This has already been discussed above but I will repeat it again. For the case of the biconditional, if we have two statements \( p \) and \( q \) implies that \( p \) is antecedent of the consequent \( q \) and symmetrically also \( q \) is antecedent of the consequent \( p \), symbolically two relations must be fulfilled in this way:
- \( p \rightarrow q \)
- \( q \rightarrow p \)
These two conditional statements must be fulfilled simultaneously to be mutually dependent, therefore it requires a logical conjunction between them, then the logical biconditional of \( p \) and \( q \) would be:
\[ ( p \leftrightarrow q ) \equiv ( p \rightarrow q ) \wedge ( q \rightarrow p ) \]
Two other properties are:
\[ p \leftrightarrow q \equiv ( p \wedge q ) \vee ( \sim p \wedge \sim q ) \\ p \leftrightarrow q \equiv \sim ( p \bigtriangleup q ) \]
Differences between biconditional and logical equivalence
These differences are initially not entirely distinguishable, but we can say that the biconditional is a logical operator like the four mathematical operations and the logical equivalence is analogous to the equals sign ( \( = \) ), it is not precisely a logical operator, what it does is to relate two statements. Here are its symbols:
- Biconditional material, symbol: \( \leftrightarrow \)
- Logical equivalence, symbol: \( \equiv \)
The differences that we can find between these two are:
| Biconditional material | Logical Equivalence |
|---|---|
| The biconditional between two statements is another statement. A biconditional statement is not always true. The biconditional of two statements \( p \) and \( q \) can be expressed as an identity of the type \( ( p \rightarrow q ) \wedge ( q \rightarrow p ) \). | Logical equivalence is the equality between two affirmative statements. The logical equivalence between two statements is always true. Logical equivalence not only cannot be expressed as \( ( p \rightarrow q ) \wedge ( q \rightarrow p ) \), it does not allow it either because it is not a statement. |
This section has come to an end, we have seen that there is not much to explain with respect to this connective, there are no contradictions or paradoxes in its analysis and there is not much to say.
For the next entry we will finally focus on the truth table of each of the logical connectives and that’s all friends, see you in the next post, see you soon.