I have a confusion regarding how to check whether a wff is satisfiable, unsatisfiable and valid.
As far as I understood, valid means the truth table must be a tautology, otherwise it is not a valid wff. For eg, p or (not p) is valid since it is a tautology, whereas p and (not p)
is a contradiction. So it is not valid.
Unsatisfiable means the truth table is a contradiction. ie. A wff which is not valid is unsatisfiable. Eg. p and (not p)
Satisfiable can be a tautology or even if there is one True value in the truth table. Eg. p or (not p) is tautology and not(p implies q) has one True. So, they are satisfiable.
Am I correct about the terms? If not, please explain with simple examples like what I've used. Thanks in advance.
UPDATED:
Please check out [satisfiability][1] In that there is a definition about satisfiability which confuses me.
It says, a formula 'A' is satisfiable if and only if 'not(A)' is not a tautology
Best Answer
The following may seem rather strange, but I hope it'll help you think about these three concepts.
Let $\Bbb B$ denote the boolean set $\{ T, F \}.$ Now, think of the WFF containing $n$ propositions as a function: $\Bbb{B}^n \to \Bbb{B}.$ I will use the following WFF as my running example: $p \land \lnot q.$ That's a "function" $\mathbb{B}^2 \to \Bbb{B}.$ It's a function that takes two boolean inputs (each being true or false) and produces a boolean value $f(p, q) = p \land \lnot q.$ The truth table is just a tabulation of this "function" for all (four) possible values of $p$ and $q.$
This "function" could be true for some values of $p, q$. It could also be false for some other values of $p, q.$ Some functions are always (constant) true or always (constant) false. This is exactly the distinction between satisfiable, valid, and unsatisfiable, respectively.
Now, a WFF is valid if $f(p, q) = T$ for all values of $p, q.$ It's unsatisfiable if $f(p, q) = F$ for all values of $p, q.$ The former two cases resemble constant functions. A WFF is satisfiable if $f(p, q) = T$ for some values of $p, q.$ It could be true/false for all other values; that doesn't matter. What matter is that it's possible to find values of $p, q$ to satisfy the WFF (satisfy a WFF = making the WFF true).