Implication for Conditional Reasoning
Implication meaning is not something that can be easily explained. In Mathematics, implication essentially deals with conditional statements. Conditional statements are those that state that if one ‘condition’ is met or if one thing is true, then the related fact or idea will also be true. A typical conditional statement has an antecedent and a precedent, where if the antecedent is true, then the precedent must also hold. Normally, the antecedent is synonymous with the hypothesis, where the precedent is the conclusion or the result of the hypothesis.
The implication symbol is denoted by \[p \to q\]. Here, p is the precedent, or the hypothesis, whereas q is the antecedent, also known as the conclusion or the consequence, concerning logical implication. The implied meaning will be better understood with the help of the following examples.
If you work hard, then you will succeed.
If I save up money, then I shall be able to buy the car.
If it rains, then we must stay indoors.
As you can see, most of these statements are ‘if-then’ clauses. This is the very essence of logical implication.
Propositions
Propositions are an integral part of implication logic, where the declarative or assertive statement can either be true or false. However, the same statement cannot be true and false at the same time. This means that it is either of the two, but not both. To understand this better, we will take the following example.
Example: Here, let p and q be the propositions.
The conditional statement represents the proposition, where the given conditional statement holds false only when p is true and q is false. In all other cases, the statement is false. The implication logic truth table gives the values of the stated implications.
Now, certain specific terminologies are used to express \[p\to q\]. They are as follows:
If p, then q
If p,q
q if p
q when p
p implies q
p only if q
q follows from p
p is a sufficient condition for q
Converse, Contrapositive and Inverse
Converse, contrapositive and inverse are different variations of a conditional statement. The converse statement of $p\to q$ is the reverse of the statement, and gives the proposition $q\to p$. The contrapositive of the same statement will be $\sim q\to \sim p$, and the inverse proposition would logically be $\sim p\to \sim q$.
According to logical implication, the following truth table will help you figure out the values, corresponding to the converse, contrapositive, and inverse propositions of the conditional statement $p\to q$.
Bi-Conditional Statements or Equivalence
Bi-conditional statements are also termed double implication or equivalence. In such cases, the propositions are combined so that both the propositions have the same truth value.
Let us assume that p and q are the two propositions that we are going to use here. Then, the implication symbol or representation, $p\leftrightarrow q$, is the proposition for the statement p, if and only if q. This means that the double implication or the bi-implication of $p\leftrightarrow q$ will only be true if both p, and q hold, and will be false for all other conditions.
The expression, $p\leftrightarrow q$, means either p is necessary and sufficient for q, or if p, then q, and the converse of that. To understand the concept better, follow the implication symbol that is provided in the table below.
Examples on Logical Implication
Example 1: Prove that the proposition $q\to p$ and $\sim p\to \sim q$ are not the same as $p\to q$
Solution:
To prove that the propositions $q\to p$ and $\sim p\to \sim q$ are not the same as $p\to q$, we take the following double implication truth table.
Therefore, from the above-mentioned truth table, we can see that the propositions $q\to p$ and $\sim p\to \sim q$ are not the same as $\sim p\to \sim q$.
Example 2: What will be the converse, contrapositive, and inverse of the implication statement “if you smile, you look pretty”?
Solutions:
As you already know, if p, q is a method of expressing the conditional statement $p\to q$, the converse, contrapositive, and inverse of the given statement are as follows:
Converse: If you look pretty, then you are smiling.
Contrapositive: If you are not looking pretty, then you are not smiling.
Inverse: If you are not smiling, then you are not looking pretty.
Example 3: What will be the converse and contrapositive of the logical implication statement “if it gets cloudy, then it will rain”?
Solution:
As you already know, if p, q is a method of expressing the conditional statement $p\to q$, the converse, contrapositive, and inverse of the given statement are as follows:
Converse: If it is raining, then it is cloudy.
Contrapositive: If it is not raining, then it is not cloudy.
Conclusion
Mathematical implication or logical implication essentially deals with conditional statements. Such statements are termed conditional statements since there are two separate parts of the statement, where if one condition is met, the other holds true.
In Mathematics, a proposition is an integral part of a conditional statement, and the implication expression is represented by $p\to q$. Here, p is the precedent or the hypothesis, whereas q is the antecedent, which is also termed the conclusion or the consequence of the hypothesis.
FAQs on Implications and Conditional Statements for JEE
1. How many different types of propositional logic are there, and what are they?
There are six different types of propositional logical statements. They are shown in the following table.
Sr. No. | Type of Statement | Implication Symbol |
1 | Simple | p |
2 | Negation | $\sim p$ |
3 | Junction | $p \land q$ |
4 | Disjunction | $p\,\text{v}\,\text{q}$ |
5 | Conditional | $p\to q$ |
6 | Bi-conditional | $p\leftrightarrow q$ |
2. What do you mean by the term implication?
A logical implication essentially refers to the compound statement ‘if p, then q.’ The implication symbol for the above-quoted statement is $p\to q$, and is read as p implies q. The logic statement is false only when p holds true and q is false. Apart from that, the conditional statement is always true. An implication is any conditional statement where one is the required clause or situation for the other to occur. Implications are extensively used in many different fields like artificial intelligence and machine learning, apart from Mathematics and Applied Maths.
3. How are logical implications and conditional statements represented in Boolean logic?
Boolean logic is a branch of Mathematics, algebra in particular where there can be only outputs or results. According to boolean algebra either the outcome will be true or false. Therefore implications too will have only their truth values that get represented. In Boolean logic, the statement p implies q, which is mathematically represented as $p\to q$. This represents the statement “if p is true, then q must also be true”. Conversely, this mathematical expression also represents “if p is not true, then neither will q hold true”.