# 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. Thanks in advance!

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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.

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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 ).

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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.

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