Well-formed formulas versus valid formulas in inferences

Statement

The notion of valid argument is usually applied to inferences in arguments, but arguments reduce to propositional formulas and can be evaluated the same as any other propositional formula. Here a valid inference means: "The formula that represents the inference evaluates to "truth" beneath its principal connective, no matter what truth-values are assigned to its variables", i.e. the formula is a tautology. Quite possibly a formula will be well-formed but not valid. Another way of saying this is: "Being well-formed is necessary for a formula to be valid but it is not sufficient." The only way to find out if it is both well-formed and valid is to submit it to verification with a truth table or by use of the "laws":

• Example 1: What does one make of the following difficult-to-follow assertion? Is it valid? "If it's sunny, but if the frog is croaking then it's not sunny, then it's the same as saying that the frog isn't croaking." Convert this to a propositional formula as follows:

" IF (a AND (IF b THEN NOT-a) THEN NOT-a" where " a " represents "its sunny" and " b " represents "the frog is croaking":

( ( (a) & ( (b) → ~(a) ) ≡ ~(b) )

This is well-formed, but is it valid? In other words, when evaluated will this yield a tautology (all T) beneath the logical-equivalence symbol ≡ ? The answer is NO, it is not valid. However, if reconstructed as an implication then the argument is valid.

"Saying it's sunny, but if the frog is croaking then it's not sunny, implies that the frog isn't croaking."

Other circumstances may be preventing the frog from croaking: perhaps a crane ate it.

• Example 2 (from Reichenbach via Bertrand Russell):

"If pigs have wings, some winged animals are good to eat. Some winged animals are good to eat, so pigs have wings."

( ((a) → (b)) & (b) → (a) ) is well formed, but an invalid argument as shown by the red evaluation under the principal implication: