# What is Validity and Satisfiability in a propositional statement?

I tend to see these words a lot in Discrete Mathematics. I assumed these were just simple words until I bumped into a question.

Is the following proposition Satisfiable? Is it Valid?
$(P \rightarrow Q) \Leftrightarrow (Q \rightarrow R )$

Then I searched in the net but in vain. So I'm asking here. What do you mean by Satisfiable and Valid? Please explain.

A formula is valid if it is true for all values of its terms. Satisfiability refers to the existence of a combination of values to make the expression true. So in short, a proposition is satisfiable if there is at least one true result in its truth table, valid if all values it returns in the truth table are true.

• hey! this was the same stuff that I came across in the net. If you could explain using an example, it would be great. – Vishnu Vivek Dec 14 '12 at 10:36
• The expression (p .AND. q) is satisfiable-- it returns true whenever both p and q are true. On the other hand, (p .OR. .NOT.p) is valid, it always is true regardless of the value of p. – ashley Dec 14 '12 at 22:21
• The expression in your Q is satisfiable but not valid. It returns true for (p, q, r)=(true, true, true) and false for (true, false, true). – ashley Dec 14 '12 at 22:28

Satisfiability -the other way of interpretation
A propositional statement is satisfiable if and only if, its truth table is not contradiction.
Not contradiction means, it could be a tautology also.

Hence, every tautology is also Satisfiable.
However, Satisfiability doesn't imply Tautology.

Another thing to note is, if a propositional statement is Tautology, then its always valid.

Thus, Tautology implies ( Satisfiability + Validity ).

A propositional logic is said to be satisfiable if its either a tautology or contingency. Hence if a logic is a contradiction then it is said to be unsatisfiable. By contingency we mean that logic can be true or false i.e. nothing can be said for sure about the logic.

From Wikipedia,

A well formued formula (wff) is satisfiable if it can be made TRUE by assigning apt logical values to its variables.

On the other hand, a wff is valid only if it's true under every interpretation. In propositional logic, wff's are valid only if they are tautologies.

The wff that you have stated in your question is satisfiable but not valid because its not a tautology which you can easily prove using Truth Tables.