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