## Table of Contents

- What is a Conditional Statement?
- How to Write a Conditional Statement?
- Conditional Statement Examples
- Lesson Summary

Learn about conditional statements. Identify what a conditional statement is, learn how to write a conditional statement, and see conditional statement examples.
Updated: 02/25/2022

- What is a Conditional Statement?
- How to Write a Conditional Statement?
- Conditional Statement Examples
- Lesson Summary

What is a conditional statement? Simply put, a conditional statement is an **if-then** statement, e.g., '"**If** Jane does her homework, **then** Jane will get a good grade."' The conditional statement's definition emphasizes a relationship between two ideas, wherein one idea follows from the other. In the example, Jane getting a good grade follows from the idea of Jane doing her homework. Thus, conditional statements are an important part of mathematical and logical reasoning because it allows one to make deductions in a clear and rigorous way.

The question of how to write a conditional statement is not a difficult one. The first requirement is that there are two independent propositions (complete sentences). Then, the terms '"if"' and '"then"' are used in front of those propositions. Note that the meaning of a conditional statement is determined by the if and then but not their order. The proposition that follows the if is called the **hypothesis** or antecedent, while the proposition that follows the then is called the **conclusion** or consequent.

To take an example, '"If Jane does her homework, then Jane will get a good grade"' is the same as '"Jane will get a good grade if Jane does her homework."' '"Jane does her homework"' is the hypothesis, and '"Jane will get a good grade"' is the conclusion. In second formulation, the '"then"' is implied while the '"if"' remains explicit. The meaning changes when the if precedes the other idea. The claim that '"If Jane will get a good grade then, Jane will do her homework"' has a different, and perhaps unintuitive, meaning. This sentence suggests that Jane doing her homework depends on her getting a good grade.

Conditional statements can be symbolized in order to make it easier to manipulate them in logical analysis. Often, an arrow symbol pointing to the right is used to indicate a conditional relationship with the hypothesis of a conditional statement on the left of the arrow and the conclusion on the right. The hypothesis and conclusion are generally symbolized with letters, which act like variables. For example, '"If Jane does her homework, then Jane will get a good grade,"' can be symbolized, {eq}H \rightarrow G {/eq}. The H represents the entire hypothesis, the G represents the conclusion, and the arrow represents the conditional relationship. Thus, there are three conditional statement symbols.

Conditional statements in logic are very important, since they set up the basis for many **valid argument forms**. A valid argument is one in which one can make an inference from premises with certainty. Or in other words, the inference necessarily follows. One of the most commonly used valid argument forms is called modus ponens. In modus ponens, a conditional statement is given along with an affirmation of the hypothesis. From these two premises, it follows that the conclusion must follow. Note that '"conclusion"' is being used in a different sense, as the proposition that follows from one or premises. Modus ponens is expressed as follows:

- Premise 1: {eq}A \rightarrow B {/eq}

- Premise 2: A
- Conclusion: B

A conditional statement by itself does not state whether the hypothesis or the conclusion is true. It merely claims that the conclusion follows from the premise. Thus, if one wants to know whether the conclusion is true, one must figure out whether the hypothesis is true. Or in other words, a conditional statement does not immediately state anything about material reality. To return to the example, '"If Jane does her homework, then Jane will get a good grade,"' does not state whether Jane will do her homework. It merely states that if she does, then good grades will follow. It may be the case that Jane refuses to do her homework, at which point, it is unknown what her grade will be. If one wants to know what her grade will be, then the hypothesis will need to be asserted separately. An argument predicting her grade, when organized as a modus ponens would be written as follows:

- Premise 1: If Jane does her homework, then Jane will get a good grade.
- Premise 2: Jane has done her homework.
- Conclusion: Jane will get a good grade.

Given that any two propositions can be conditionally related, there are an infinite number of conditional statement examples. The major clue to identifying conditional statements is the if (and usually a then). Consider the following:

- If that is an owl, then it is the weirdest bird I have seen.
- If it is after sundown, then I should get home and feed my cats.
- If a number is even, then it is divisible by two.
- You can clock out early if your work is finished for the day.
- If you give me a hamburger today, then I will gladly pay you on Tuesday.

Conditional statements are statements wherein one proposition is said to follow from another. Conditional statements require, then, two propositions, which are logically independent sentences, as well as the terms '"if"' and usually '"then."' The proposition that follows the if is called the **hypothesis**, while the proposition that follows the then is called the **conclusion**. Note that conditional statements only assert that the conclusion follows from the hypothesis but not the other way around. The meaning of a conditional statement is determined by the location of the if and the then rather than by order that the propositions appear in a sentence. In formal notation, the hypothesis and conclusion are symbolized by different letters, and the conditional relationship is usually symbolized by an arrow, e.g., {eq}A \rightarrow B {/eq}.

By themselves, conditional statements do not assert anything about the actual state of affairs. However, conditional statements are important in **valid argument forms**, which are logical patterns in which a given claim necessarily follows from one or more premises. One of the most important valid argument forms is modus ponens, in which a conditional statement is coupled with an assertion that the hypothesis is independently true. From these two premises, it follows that the conclusion of the conditional statement must be true (as long as the conditional and the hypothesis are in fact true).

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Frequently Asked Questions

Conditional statements are written by connecting two propositions with the words if and then. For example, "if it is winter time, then you will likely hear Christmas carols." is a conditional statement. It can be symbolized by writing the propositions as letters and using an arrow to represent the conditional relationship, A -> B.

One example of a conditional statement is "If the rug is dirty, then the rug should be vacuumed." "The rug is dirty" is the hypothesis, and "the rug should be vacuumed" is the conclusion.

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