Couple of days back I asked this question. And after reading comments and answer there, even though I knew the definitions of the different terms (tautology, contradiction, contingent, satisfiable, unsatisfiable, valid and invalid) involved, I was quick to realize that I dont know a lot of subtleties, especially connection between these terms. I didnt find one place discussing all these connections. So I spent some time reading comments and answers on that and other linked post and come up with some understanding which I have put below and want your confirmation for it.
First some quick definitions (you may skip this as this super basic. I have put it to avoid any possible ambiguity):
- Tautology: a formula that is true in every possible interpretation
- Contradiction: a formula that is false in every possible interpretation
- Satisifiable: a formula that is true under at least one interpretation
- Unsatisfiable: a formula is unsatisifable if it is contradiction
- Valid: a formula is valid if and only if it is true under every interpretation
- Invalid: a formula is that is false under at least one interpretation
Now to get hold of the connection between all these terms. I prepared below figure:
Next as stated in this comment, "Not a typeX" has two meanings: (1) "The negation of a typeX" and "Not a statement that can be categorized as a typeX".
So, with this in mind, I came up with following relations:
- $¬$ tautology $=$ $¬$ valid $=$ contradiction $=$ unsatisfiable
- $¬$ contradiction $=$ $¬$ unsatisfiable $=$ tautology $=$ valid
- Not a tautology $=$ not a valid $=$ invalid $=$ can be either contradiction or contingent
- Not a contradiction $=$ not an unsatisfiable $=$ satisfiable $=$ can be either tautology or contingent
- $¬$ satisfiable $≠$ (unsatisfiable $=$ contradiction) $=$ invalid $=$ can be contradiction or contingent
- $¬$ invalid $≠$ (tautology $=$ valid) $=$ satisfiable $=$ can be tautology or contingent
- Not an invalid $=$ valid $=$ tautology
- Not a satisfiable $=$ unsatisfiable $=$ contradiction
- $\neg$ means "negation" (or I dont know if it will be more correct or will add more sense if I say a "Boolean negation")
- "Not a TypeX" means statement "cannot be categorized as TypeX"
- $=$ simply means "is"
- $\neq$ simply means "is not"
I know these are pretty much basic stuff but was source of a lot of confusion for me and I wanted all these related things noted down at one place in correct way. Can anyone confirm if above bullet points and figure are correct?