Conditionals, quantification, and strong mathematical induction
The paper deals with the formalization of the principle of strong mathematical induction: (SMI) If a number has a property whenever all of its predecessors do, then all numbers have that property. According to the author, a formalization of (SMI) using material implication \(\supset\) as the only conditional connective is heuristically problematic, since it (i) results in a nested conditional in the antecedent of a conditional, (ii) fails to make a distinction between on the one hand conditionals that are involved in universal generalizations and on the other hand implications, and (iii) allows it to do without a separate base case. It is suggested to translate (SMI) into a language with two kinds of conditional connectives, viz. the so-called `conditional assertion' / (for the conditional that comes with a universal quantifier) and relevant implication \(\to\). Moreover, a base case covering 0 is introduced. The paper does not offer much as far as (SMI) itself is concerned. The motivation for / has nothing to do with (SMI) in particular; however, the use of / in addition to \(\to\) of course provides a remedy for (i) and (ii). No motivation is given for the use of relevant implication. Clearly the intended implication for (SMI) is \(\supset\). If \(\supset\) is to be replaced by \(\to\), then the addition of a base case indeed ``is obvious, if not altogether forced'' (p. 323).
principle of strong mathematical induction
scientific article; zbMATH DE number 20838