Tautology, Valid, Contingent, Unsatisfiable, Contradiction: relationship?

Question

I am trying to clear my doubts about various terms: tautology, contradiction, contingent, satisifiable, unsatisfiable, valid and invalid. I have read on them from various sources, and am putting all my understanding below in point form. I don't know whether I am overthinking. I just want exhaustive understanding. Could someone please check whether my understanding below is correct, and where my mistakes are?

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Some of these definitions are straight up from other sources (so they must be correct), and the rest are written from my own understanding:

1. Tautology: a formula or assertion that is true in every possible interpretation (that is, for all assignment of values to its propositional variables). Ref
2. Contradiction: a formula or assertion that is false in every possible interpretation (that is, for all assignment of values to its propositional variables).
3. A formula that is neither a tautology nor a contradiction is said to be logically contingent.
   Such a formula can be either true or false based on the values assigned to its propositional variables.
4. A formula is satisfiable if it is true under at least one interpretation.
   So, it's either contingent or a tautology. Ref
5. If a logic is a contradiction then it is said to be unsatisfiable.
6. A formula is valid only if it is a tautology. Ref
7. A formula is invalid if it is contingent or a contradiction.

Based on these definitions, I prepared a diagram showing how these concepts overlap:

Based on this diagram, I tried to answer questions like, "what is the negation of a tautology?" I felt that it could be either contingent or a contradiction: the above diagram means, "Given an assertion, if it is not a tautology, it can be either contingent or a contradiction." But it seems that I was wrong: the above diagram does not mean, "The negation of a tautology can be contingent or a contradiction." J.G.'s comment pointed out that I was simply negating definitions above, where I should have actually tried investigating how models (sets of values assigned to variables of formulae) satisfying a given formula behave for the negation of that formula. It took a while for me to understand that, and I have now come up with the following relations between any given assertion and its negation (I have given examples in brackets):

Answer 1

1. Following the truth-functional formulation of your Definitions 1 to 3, invalid satisfiable formulae are actually a proper subset of contingent formulae. I have revised your diagram to reflect this.

   As such, in first-order logic, your Definitions 5 to 7 are all incorrect. To wit:

   • $(\exists x\,x\ne x)$ is contingent and unsatisfiable.
   • $(\forall x\:x=x)$ is contingent and valid.
2. • Valid Formulae and Invalid Formulae are complementary sets.
   • Satisfiable Formulae and Unsatisfiable Formulae are complementary sets.
   • Every formula that isn’t valid (non-valid formula) is invalid, and vice versa.
   • Every formula that isn’t satisfiable (non-satisfiable formula) is unsatisfiable, and vice versa.

3. On the other hand,

   • The negation of every valid formula is unsatisfiable, not merely invalid.
   • The negation of every unsatisfiable formula is valid, not merely satisfiable.
   • The negation of every satisfiable formula is invalid, not necessarily unsatisfiable.
   • The negation of every invalid formula is satisfiable, not necessarily valid.

   Examples:

   • $(\forall x\:x=x)$ is valid, while its negation $(\exists x\,x\ne x)$ is unsatisfiable.
   • $(\exists x\,x\ne x)$ is unsatisfiable, while its negation $(\forall x\:x=x)$ is valid.
   • Both $\big(\forall x\,P(x)\big)$ and its negation $\big(\exists x\,¬P(x)\big)$ are satisfiable and invalid.

4. What is the negation of a tautology?

5. Set Complement $(A^\complement=\text{“non-}A\text”.)$ versus Formula Negation $(\lnot \psi=\text{“not }\psi\text”.)$

Answer 2

The notions and their negations will get clearer once you realize the meta-logical quantifications they involve, and how these quantifications systematically behave under negation.

All the definitions you listed quantify over interpretations: A statement is valid iff it is true under all interpretations, satisfiable iff it is true under some interpretation, contradictory iff it is valid under no interpretation, and so on.

In general, we have

• "not all interpretations" = "there exists some interpretation such that not";
• "not some interpretation" = "for all interpretations not";
• "not no interpretation" = "there exists some interpretation such that".

So let's apply these equivalences to each of the definitions*:

-------------------------------------------------------------------------------------------------------------------------    
Notion          Definition                           Negation of definition                  Negation of notion    
-------------------------------------------------------------------------------------------------------------------------
tautological    all i. true (= no i. false)          not all i. true (= some i. false)       contradcictory or contingent
contradictory   all i. false (= no i. true)          not all i. false (= some i. true)       satisfiable
(= unsatisfiable)
contingent      some i. true and some i. false       not (some i. true and some i. false)    contradictory or tautological
                                                     = (not some i. true) or (not some i. false)
                                                     = no i. true or no i. false
                                                     = all i. false or all i. true
satisfiable     some i. true (= not all i. false)    no i. true (= all i. false)             contradictory
unsatsifiable   no i. true (= all i. false)          some i. true (= not all i. false)       satisfiable
(= contradictory)
valid           (see tautological)
(= tautological)
invalid         not valid = not all i. true          all i. true                             tautological

So we have

---------------------------------------------------------------------
Notion              Negation               can but doesn't have to be
---------------------------------------------------------------------
not tautological    = not valid         
                    = invalid
                    = contradictory        unsatisfiable (if contradictory),
                      or contingent        satisfiable (if contingent)
not contradictory   = not unsatisfiable    tautological,
                    = satisfiable          contingent,
                                           invalid
not contingent      = contradictory        unsatisfiable, invalid (if contradictory),
                      or tautological      satisfiable (if tautological)
not satisfiable     = unsatisfiable        invalid (must be)
                    = contradictory
not valid           = invalid

                    = not tautological
                    = contradictory        unsatisfiable (if contradictory),
                      or contingent        satisfiable (if contingent)
not invalid         = valid                satisfiable (must be)
                    = tautological

The problem with the negations your first table is that your negation is to strong: The negation of "all interpretations" is just "not all interpretations", i.e. "there are some interpretations such that not", and not (as you did) "no interpretation". So the negation of "valid" is just "not true under all interpretations", which can be contradictory or contingent, and not "true under no interpretation", which would be contradictory. Likewise, the negation of contradictory (= false under all interpretations) is just "not false under all interpretations", i.e. "true under some interpretations", which satisfiable, and not the stronger statement "true under all interpretations", which would be tautological.

The diagram you made is correct, and explicates the misunderstanding as follows: Negation does not mean opposite. Negating a notion does not get you to the other extreme of the diagram, only to the complementary half, i.e. the entire part not covered by that notion: "not contradictory" gives you everything in the range of "satisfiable", not just the extreme "tautology". "not tautological" just gives you "invalid", not the opposite "contradictory". "Not satisfiable" is "contradiction", not "invalid", "not invalid" is "tautology", not "satisfiable", and lastly, if something is "not contingent" it must be in either "contradiction" or "tautology". That covers all the possible cases.

* Note that in the case of first-order logic, sometimes a distinction is made between a formula that is valid (true in all first-order interpretations) and one that is tautological (a first-order instance of a propositional tautology, i.e. one that has the form of a tautological propositional formula but with predicate logical formulas in the place of propositional variables). All tautological formulas are also valid, e.g. $\forall x P(x) \lor \neg \forall x P(x)$, but not all first-order validities are merely instances of propositional tautologies, e.g. $\forall x (x = x)$.