A proposition is a sentence or sentence fragment that has a value of true or false.
Let’s start with a sentence that we know is true.
If x is a positive integer, then X is a counting number.
We can break this sentence into two sentence fragments and the sentence fragments have values of true or false.
- p := x is a positive integer
- q := x is a counting number
The above is read in words in to ways:
- “If p, then q”
- “p implies q”
Consider the following:
We can say this as:
If not q, then not p
If x is not a counting number, then x is not a positive number.
Can you see that this sentence will always be true?
Having p implies q doesn’t give us “q implies p”. However, it does give us “not q implies not p”.
If we have both of them …
- p implies q
- q implies p
then we can say “p if and only if q” and we can write
Now, we can also say
q if and only if p
and write
Rob Sterling 