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Validity of a formula
I am struggling with the definition of validity. In the lecture, we said that a formula is valid, if it is true for all assignments. So, for instance, (A or notA) is valid. But (A and B) is not valid, since setting A or B to false makes the entire formula false.
However, in the solution of the mock exam, it says
It looks like the solution of the mock exam uses another definition for validity. (A => B) is not valid using the definition from the lecture.
My question: How is validity defined in the mock exam? And which definition should we use in the exam on Monday?
This is very unfortunately phrased.
In general, your definition of validity (true for all assignments) is the one that will matter e.g. for the exam.
Now for the mock exam, and why the phrasing makes sense, even if it’s not strictly adherent to that definition:
It is possible (and that’s what happens here) to extend the notion to (more accurately) “valid in some context”, where a context is a set of propositions assumed to be true a priori. For example, an assignment itself induces a context (namely the one that contains A for every variable A that is set to true, and NOT A for every variable A that is set to false). Then the “usual” notion of validity is just “valid in the empty context”.
Alternatively, you can consider A,B and C not to be propositional variables, but instead (variables representing) arbitrary formulas. In that interpretation, “Let A => B be valid” would mean “Let A and B be propositions such that A => B is valid”. For example, if A = B = C, then both A => B and B => C are valid.
This also makes sense, since in general propositional variables can be seen as variables representing whole propositions everywhere outside of the strict (inductive) definition of “formula”.
For example, if a formula F[A,B] in the variables A and B is valid, then F[F1,F2] (where we substitute A by F1 and B by F2) is also valid. So in practice, the distinction between propositional variables and “variables representing arbitrary formulas” is often swept under the rug or ignored entirely.
Thank you for the explanation! Happy to hear that I haven’t completely misunderstood how validity works