Wffs Versus Valid Formulas in Inferences
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:
Famous quotes containing the words inferences, formulas and/or valid:
“The modern world needs people with a complex identity who are intellectually autonomous and prepared to cope with uncertainty; who are able to tolerate ambiguity and not be driven by fear into a rigid, single-solution approach to problems, who are rational, foresightful and who look for facts; who can draw inferences and can control their behavior in the light of foreseen consequences, who are altruistic and enjoy doing for others, and who understand social forces and trends.”
—Robert Havighurst (20th century)
“You treat world history as a mathematician does mathematics, in which nothing but laws and formulas exist, no reality, no good and evil, no time, no yesterday, no tomorrow, nothing but an eternal, shallow, mathematical present.”
—Hermann Hesse (18771962)
“It is not enough that France should be regarded as a country which enjoys the remains of a freedom acquired long ago. If she is still to count in the worldand if she does not intend to, she may as well perishshe must be seen by her own citizens and by all men as an ever-flowing source of liberty. There must not be a single genuine lover of freedom in the whole world who can have a valid reason for hating France.”
—Simone Weil (19091943)