# Definition:Conditional/Semantics of Conditional

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## Definition

Let $p \implies q$ where $\implies$ denotes the conditional operator.

$p \implies q$ can be stated thus:

*If*$p$ is true*then*$q$ is true.

**$q$ is true***if*$p$ is true.

**(The truth of) $p$***implies*(the truth of) $q$.

**(The truth of) $q$***is implied by*(the truth of) $p$.

**$q$***follows from*$p$.

**$p$ is true***only if*$q$ is true.

The latter one may need some explanation. $p$ can be either true or false, as can $q$. But if $q$ is false, and $p \implies q$, then $p$ can not be true. Therefore, $p$ can be true *only if* $q$ is also true, which leads us to our assertion.

**$p$ is true***therefore*$q$ is true.

**$p$ is true***entails*that $q$ is true.

**$q$ is true***because*$p$ is true.

**$p$***may*be true*unless*$q$ is false.

*Given that*$p$ is true, $q$ is true.

**$q$ is true***whenever*$p$ is true.

**$q$ is true***provided that*$p$ is true.

**$q$ is true***in case*$p$ is true.

**$q$ is true***assuming that*$p$ is true.

**$q$ is true***on the condition that*$p$ is true.

Further colloquial interpretations can often be found in natural language whose meaning can be reduced down **$p$ only if $q$**, for example:

**$p$ is true***as long as*$q$ is true

- Example:
*"Mummy, can I go to the pictures?"**"***As long as**you've done your homework. Have you done your homework? No? Then you cannot go to the pictures."

- In other words:
*"You can go to the pictures***only if**you have done your homework."

- Using the full language of logic:
*"If it is true that you are going to the pictures, it is true that you must have done your homework."*

- Example:

**$p$ is true***as soon as*$q$ is true

*"Are we going to this party, then?"**"*"**As soon as**I've finished putting on my makeup.

- The analysis is the same as for the above example of
**as long as**.

## Examples

The statement:

**If**I pass this course,**then**(it shows that) I have studied hard for it.

may be rephrased as:

*I will pass this course***only if**I have studied hard for it.

*To prove that I have studied hard for this course, it is***sufficient**to know that I passed it.

*For me to pass this course, it is***necessary**for me to study hard for it.

## Sources

- 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 2$ - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.2$: Conditional Statements - 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 1.1$: Introduction