Understanding connection between terms tautology, contradiction, contingent, satisfiable, unsatisfiable, valid and invalid 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[1]: a formula that is true in every possible interpretation



*

*Contradiction: a formula that is false in every possible interpretation

*Satisifiable[1]: a formula that is true under at least one interpretation

*Unsatisfiable: a formula is unsatisifable if it is contradiction

*Valid[2]: 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

Doubt
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:

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*$¬$ 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

*(contradiction $=$ unsatisfiable) $≠$ $¬$ satisfiable $=$ invalid $=$ can be contradiction or contingent

*(tautology $=$ valid) $≠$ $¬$ invalid $=$ satisfiable $=$ can be tautology or contingent

*Not an invalid $=$ valid $=$ tautology

*Not a satisfiable $=$ unsatisfiable $=$ contradiction

Above,

*

*$\neg$ 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?
 A: Long comment
Consider first the negation of the definition of tautology :

A formula $\varphi$ is a tautology  iff is true in every possible interpretation.

We can re-write it as follows : for every interpretation $\mathfrak I$, the formula $\varphi$ is true.
Thus, its negation is : there is an interpretation $\mathfrak I_0$ such that the formula $\varphi$ is not true, i.e. false.
But, to say that the formula is false for some interpretation does not imply that it is false for every interpretation.
Conclusion : 

a formula that is not a tautology can be either unsatisfiable or satisfiable.


Different is the case with the negation of a formula $\varphi$.
If $\varphi$ is a tautology, then it is true for every interpretation.
Thus, its negation $\lnot \varphi$ is false for every interpretation.
Conclusion :

if $\varphi$ is a tautology, then $\lnot \varphi$ is unsatisfiable, i.e. a contradiction.



In this way, we have covered all cases :

tautology (aka : valid), unsatisfiable (aka : contradiction), satisfiable.

