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: 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

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:

1. $$¬$$ tautology $$=$$ $$¬$$ valid $$=$$ contradiction $$=$$ unsatisfiable
2. $$¬$$ contradiction $$=$$ $$¬$$ unsatisfiable $$=$$ tautology $$=$$ valid
3. Not a tautology $$=$$ not a valid $$=$$ invalid $$=$$ can be either contradiction or contingent
4. Not a contradiction $$=$$ not an unsatisfiable $$=$$ satisfiable $$=$$ can be either tautology or contingent
5. $$¬$$ satisfiable $$≠$$ (unsatisfiable $$=$$ contradiction) $$=$$ invalid $$=$$ can be contradiction or contingent
6. $$¬$$ invalid $$≠$$ (tautology $$=$$ valid) $$=$$ satisfiable $$=$$ can be tautology or contingent
7. Not an invalid $$=$$ valid $$=$$ tautology
8. 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?

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.

• I believe your first conclusion is my point 3 in doubt section of question. The only thing I missed to state "explicitly" in point 3 is that it can also be satisfiable (even thought I have specified contingent in point 3, which is also satisfiable). Same is case with your conclusion 2, which I feel is exactly same as my point 1 in doubt section of question. Right? – anir Jan 21 at 12:17
• @anir - contingent is an "old" term : "neither true under every possible valuation (i.e. tautology) nor false under every possible valuation (i.e. contradiction)." Thus, according to this def, a contingent proposition cannot be a contradiction, and thus cannot be unsatisfiable. – Mauro ALLEGRANZA Jan 21 at 12:31
• Regarding your points, the gist of my long comemnt is : it is not necessary (and it can be misleading) to consider all the above cases, when we have only three basic terms in place : tautology/satisfiable/ contradiction. Then we can easily apply them to a formula starting with a negation sign. – Mauro ALLEGRANZA Jan 21 at 12:34
• Hi Mauro, sorry to grab your attention like this. I needed to know exhaustively how all terms relate to each other, especially because those terms are used in exams. Thats why I also included contingent, valid and invalid in the discussion. I have edited new question to put all relations I can think of exhaustively. Can you please have a look at it and reopen the question if appropriate. 4 people have already voted to reopen. – anir Sep 9 at 6:10