Is this formula satisfiable?

Question

I am confused whether or not my explanation for whether or not this formula is satisfiable is correct. Note that the question state it should be Brief and it should not be necessary to write down a full truth table.

I asked someone and they said my explanation was not ok and one was only giving an example of the formula not being satisfiable. And that the formula was satisfiable...

Here is the formula: (p→q)→(¬p→¬q)

And here is my explanation why I think its not satisfiable

• A→B is satisfiable as long as you don't have both A= true and B= false.
• When p = false, and q = true
• A or (p→q)= true , and B or (¬p→¬q) = false
• Therefore is not satisfiable.

Could somone improve or confirm if my explanation is ok

Answer 1

The formula is satisfiable. To make an implication true, it is enough to make the first term false.

And it is easy to make $p\to q$ false. Make $p$ true and $q$ false.

Answer 2

You seem to be confusing satisfiable propositions with tautologies.

• A proposition is satisfiable iff there is at least one assignment to each variable that makes the proposition evaluate to be true.
• A proposition is a tautology iff every assignment to the variables will make the proposition evaluate to be true.

Notice that the propsition evaluates to be true if, for example, we assign $p$ to be true and $q$ to be true. So the proposition is indeed satisfiable.

Answer 3

Possibly you are mixed up between satisfiable and tautology.

A statement is a tautology if it is always true. You have shown that the statement may be false, so it is not a tautology. But this is not what was asked.

A statement is satisfiable if it is sometimes true. So you need to find one example of truth values for $p,q$ such that the statement is true. This should not be too hard by trial and error (or see other people's answers).

Also worth noting: a statement is satisfiable if its negation is sometimes false, that is, if its negation is not a tautology.