Wednesday, January 11, 2012

Theory, Prediction, Fallacy, Negation and Proof

Direct prediction is in fact no proof of a theory:

After all, "A implies B; B: therefore A" is the logical fallacy of affirming the consequent (since some other subject, say "A' ", may imply "B", too).

This would seem to imply that there can be no predictive proofs of scientific theories, only disproofs, although as a practical matter one can calmly accept that the more, and the more different, and the more unexpected, predictions that a theory affords, the more likely it is that it is true (this resembles Hume's later position, say in "Of Miracles", after his original and highly unrealistic idee fixe that "causation is custom").

But if you can formulate the negation of the theory "A", "~A" (easy enough with a simple theory consisting of a single elementary or atomic proposition, and using the tilde "~" to mean logical or Boolean "NOT"), then if you can show "~A" implies a false proposition, it is false, and therefore (since the negation of a falsehood is true) the original theory "A" is true": that is, " ~A implies C; ~C: therefore ~(~A), = A" (a double negation of course canceling each other).

As for more complex theories, if the theory to be proved is "A1 && A2 && A3" (using the double ampersand "&&" to mean logical or Boolean "AND"), then its negation is "~A1 || ~A2 || ~A3" (using the double stroke "||" to mean logical or Boolean "OR"), and the aforementioned proof then hinges on whether that disjunction implies a falsehood, which means that if any one of "~A1", "~A2" or "~A3" implies a falsehood, then the original theory is true.

It seems strange that the original complex theory "A" can be proven on the basis of an assessment of just one of its elements, but, of course, that's not what's happening here: what's being assessed is an element of the NEGATION of that theory.

Note: For those readers unfamiliar with propositional logic due to an incompetent education, "A && B" is equivalent to "~(~A || ~B)", so negating "A && B" gives "~A || ~B".

Keywords: cephalgia, logic, philosophy, prediction, theory


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