If we search for the term biconditional in the dictionary of the Royal Spanish Academy (RAE), we won’t find it. The concept, however, is used frequently in the field of philosophy and of the logic.
A biconditional is a proposition which has one double conditionality, fixed by the formulas that it relates in a binary way. In colloquial language, the idea is associated with the expression “If and only if”: the biconditional is true if the terms it relates share the truth value (that is, if both formulas are true or if both formulas are false). On the other hand, when the formulas have different truth values (since one is false and the other is true), the biconditional is false.
In other words, a biconditional implies that R is a condition sufficient and necessary to S. It may also be stated that “If R, then S” So what “If S, then R”.
Let’s take the example of the following proposition: “A human being belongs biologically to the male gender if he has male reproductive organs”. Leaving aside cultural and identity questions, it can be affirmed that a human being is part of the masculine gender “If and only if” has male reproductive organs.
Returning to the formulas mentioned above: “If a human being belongs biologically to the male gender, then he has male reproductive organs”. This can also be expressed backwards: “If a human being has male reproductive organs, then biologically he belongs to the male gender”. As you can see, we are facing a biconditional proposition: requires both terms to have the same value really to be true.
In addition to the “particles” or “links” that we have mentioned that are essential in the biconditional, we cannot ignore other elements that, in the same way, are used in it. We are referring, for example, to “is necessary and sufficient for” or “is equivalent to”.
In the same way, we cannot ignore other really important aspects of the biconditional. We are referring, for example, to the fact that it is also used strongly within the field of mathematics. In this case, it must be stated that the symbols used to affect the aforementioned biconditional are the double-headed arrows, one in each direction.
Likewise, it must be taken into account that, with the advancement of technology, we also come across the fact that it is also important within what is known as digital logic. In this case, the biconditional operator to be used is XNOR.
In addition to what is indicated, in order to summarize certain ideas, we have to start from the fact that the biconditional proposition has several forms of translation, among which we can highlight the following:
-P is a necessary and sufficient condition for q.
-P yes and only yes q. An example would be: “P = A triangle is right. Q = A triangle has a right angle “, which would result in a triangle being a right angle and only if it has a right angle”.
-If p then q and reciprocally.
-Q is a necessary and sufficient condition for p.
-Q yes and only yes p.
-If q then p and reciprocally.