Is ~p -> ~q an acceptable conditional statement?

Question

I have a conditional statement that was posted in a chat room that I am trying to parse. I'm going to edit a little to make the question more concise.

The statement is: "No taxation without representation"

I have two possible ways to look at this: either as a conditional statement itself, or as a version of a conditional statement that first needs to be transformed. For example:

Accepting the statement as it is:

• Statement: "No taxation without representation"
• Inverse: "Taxation with representation"
• Converse: "No representation without taxation"
• Contrapositive: "Representation with taxation"

Changing the statement to be free of negatives means that the statement and contrapositive swap and the inverse and converse switch too.

• Statement: "Representation with taxation"
• Inverse: "No representation without taxation"
• Converse: "Taxation with representation"
• Contrapositive: "No taxation without representation"

Now, obviously, this is not a problem for the meaning of the statements, but it has created problems in the conversation because people are confused by the various meanings of the logical statements.

Is there an easier way to deal with the confusion when one person picks the 'original' statement and another picks the contrapositive of that statement (what they might call the 'true original' statement?

This isn't really a reasoning issue, it's more of a communication problem, so I'm interested to hear if there are any language tools or rules that are generally followed when distinguishing in these situations.

For example, I imagine that if there were a name for the inverse/converse and statement/contrapositive pairings then I could just say something like, "The inverse/converse pairing must be proven, only the contrapositive/statement paring is given to be true." But either those words don't exist, or I can't find them (using them in this shown way does not make for simpler communication).

Alternatively, it could be that there is some rule for this that says, "A conditional statement may only have up to one negation." and then we would end up on the same conditional statement, the second one: "Representation with taxation", because "No taxation without representation" is both negating taxation and negating representation (~p -> ~q).

So how do I mind my ps and qs?

Answer 1

Yes, it is.

"No taxation without representation" is :

if not representation, then no taxation,

that is equivalent to :

if taxation, then representation.

If we model the second conditional with : $(Q \to P)$, the first one is its contrapositive : $(\lnot P \to \lnot Q)$.

Another way to read a conditional "if $Q$, then $P$", is :

"$P$ is necessary (condition) for $Q$",

because $Q$ cannot be true unless $P$ is true.

Thus, we have that representation ($P$) is a necessay condition for taxation ($Q$), because if there are no citizens' representatives ($\lnot P$), there are no taxes ($\lnot Q$).