# Invalidity of propositional formula

It may be a silly doubt but I am really confused about valid and invalid terms.

From whatever I read, I concluded that:

Validity and invalidity apply only to arguments, not to propositional statements. For a (simple)statement either it can be true or false. For arguments- It is valid if all the premises are true, then the conclusion must be true. Invalid when it is possible that all the premises are true and the conclusion is false.

For a propositional formula (compound propositions) - a tautology, contingency, contradiction, satisfiable, unsatisfiable, valid terms make sense.

Now my confusion is:

1. Can we use the $Invalid$ term with a compound proposition (propositional formula)? If yes, Then what is the condition for its invalidity?

It certainly makes sense to have terminology to distinguish truth/falsity of propositional statements from validity/invalidity of arguments that purport to be logical reasoning. On the other hand it is very common for authors to use valid to describe a compound formula with assignments of truth and falsity to its atomic propositions when it results in the overall formula being true (resp. invalid when an assignment gives a false result).

Good authors will be careful in the use of terminology (and in the definitions), but informal usage will vary. Ultimately if you want to use the terms valid/invalid for compound propositions, you should provide consistent definitions that alert readers or listeners to your meaning.

• Thank you. So usually authors use $valid$ terminology when a compound formula is a tautology and $invalid$ terminology when the truth table of formula contains $at least$ one $false$ ? – user9014873 Jul 20 '18 at 14:39
• I'd say careful authors use the term tautology when every truth-table assignment gives true and satisfiable when some truth-table assignment gives true. Even though the idea of truth-table assignments is clear and simple, from a computational complexity point of view this gives a non-polynomial time algorithm. However (for Boolean logic) the respective problems of deciding TAUT and SAT are convertible in polynomial time. I point this out mainly to motivate you to be articulate about your own definitions when using the term valid to refer to a compound formula. – hardmath Jul 20 '18 at 14:45

Let’s first clear a terminological point.

Validity as in “this argument is valid” belongs to classic formal logic.

Validity as in “a tautology is a valid proposition” belongs to model theory.

That’s two meaning of valid. In mathematical logic we can talk about a valid proposition (in the 2nd sense of “valid”). Roughly it means true under any possible interpretation. A contradiction is the negation of a valid proposition. A contingent proposition is a neither valid nor a contradiction. Invalid means not valid, that is contradictory or contingent.