Paradox

OCKHAM AND THE INSOLUBILIA 195was long confused with Duns Scotus). Under the traditionaldefinition, an argument is valid if and only if it is impossiblefor the premises to be true and the conclusion false. In acommentary on Aristotle’s Prior Analytics, Pseudo-Scotuspresents an apparent counterexample: “God exists, therefore,this argument is invalid.” Pseudo-Scotus is presupposingthat “God exists” is a necessary truth. Less-devoutreaders are free to substitute any necessary truth, say, “Allequilateral triangles are equiangular.” If the argument isvalid, then since the premise is true, the conclusion must betrue. But then the argument is both valid and invalid.Contradiction. Therefore, the argument is not valid. Butwait! If the argument is invalid, then it is possible for thepremise to be true and conclusion false. Since the premiseis a necessary truth, the argument is invalid only if theconclusion is false. But then the argument is valid! Contradictionagain!Pseudo-Scotus suggests that the counterexample can beavoided by adding an extra requirement for validity: theconclusion must not deny its own validity. Here he is applyingOckham’s proposal that the insolubility be stopped by banningself-reference.Ockham realized that a ban on all self-reference wouldmean the loss of innocent sentences such as “This sentence isin English.” He was willing to accept such casualties ascollateral damage. When a gardener poisons aphids, heforesees that he will poison innocent insects along with thepests. There might be more specialized poisons but thegardener does not feel under an obligation to avoid all sidedestruction.The medievals are interested in eliminating theglitch but are not expecting the solution to be deeply illuminating.They just want to de-bug their logical systems.