In the previous video, you learned about different types of basic logical connectives that helps in creating compound prepositions. These connectives are join two or more atomic statements to form compound preposition which also have a truth value of its own.
We have discuss negation, conjunction, and disjunction connectives so far, the remaining two connectives are discussed in this article.
You can also watch our YouTube video on implication and biconditional.
Implication ( $\implies$)
The implication connective takes one or more variables that represent atomic statements and make a compound statement. The implication is denoted with the $\implies$ symbol which means “implies” or “if…then..”.
In programming world, if-then is popularly known as conditional statement.
For example, Let $ p$ and $q$ be two atomic statements.
\begin{aligned} &p: "Today\hspace{3px} is\hspace{3px} Sunday."\\\\ &q: "I\hspace{3px} don't \hspace{3px} go\hspace{3px} to\hspace{3px} school."\\\\ &Then,\hspace{3px} p \implies q \hspace{3px} is \hspace{3px} "if \hspace{3px} p,\hspace{3px} then\hspace{3px} q" \hspace{3px} or \hspace{3px} "p \hspace{3px} implies \hspace{3px} q", \\\\ &p \implies q: \hspace{3px} "If\hspace{3px} today\hspace{3px} is \hspace{3px} Sunday, \hspace{3px} then \hspace{3px} I \hspace{3px} don't \hspace{3px} go \hspace{3px} to \hspace{3px} school" \end{aligned}
The statement
Other English Phrases That Represent If…Then..
There are several other ways to write implication in English, some of them are
- q, if p
- q, provided that p
- q unless p
- q follows from p
- p sufficient condition or q
- q necessary condition for p
- because p, q
- p therefore q
Translating English sentences to logic statements with variables may be tricky because there could be several meaning to same statement and we are not asserting a particular truth value to statement
It means that we are not saying that
This could be understood with the help of truth table of implication.
Suppose
Let say
For the row 3 and row 4, we could not use
Suppose
The only case when connective is false is when
Read about “Paradoxes of Material Implication”.
Simplification of implication
Another way to write implication is to simplify it. The if-then
Biconditional Operator ( )
The biconditional connective also takes one of more atomic statements and create a compound statement that has a truth value of its own. The biconditional is an “if and only if” or “iff” statement. The biconditional operator is denoted by
For example if and are two logical atomic statements.
: "I am hungry"
: "I worked very hard this morning"
Then
: "I am hungry if and only if I worked very hard this morning"
Here is the truth table for biconditional connective.
From the truth table it is evident that when both the variables have same truth values – true or false. The compound preposition with a biconditional connective is true.
For example, let
The biconditional is implication that true from
: "A polygon is a triangle"
: "A triangle has 3 sides"
Using the biconditional we can make following statement.
"A polygon is a triangle if and only if it has 3 sides"
Condition for above statement are
- If polygon is triangle then it has 3 sides (
) - If polygon has 3 sides, then it is a triangle (
)
Note that the both conditions we get and equivalent preposition for biconditional.
Let us verify the claim using a truth table.
From the truth table above, clearly, both expression are logically equal. Whenever you are asked to simplify a compound statements involving biconditionals, replace it with above expression for convenience.