# 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. Commented Dec 14, 2012 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. Commented Dec 14, 2012 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). Commented Dec 14, 2012 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.

• Valid: always true regardless of the values of its variables, e.g., P OR NOT(P)..

Slightly more formally: A valid formula is one which is always true, no matter what truth values its variables may have.

• Satisfiable: Can be solved, i.e., true and false values can be assigned to its variables, in a way that the final outcome is true.

Slightly more formally: A satisfiable formula is one which can sometimes be true, i.e., there is some assignment of truth values to its variables that makes it true.

An example of an unsatisfiable formula is P AND NOT(P)..

Why are satisfiability and validity treated together? Because they are related:

To check that G is valid,
we can check that NOT(G) is not satisfiable.


If you want to read more about satisfiability and validity in an accesible way, check the chapter 3 of Mathematics for Computer Science by Lehman, Thomson and Meyer.