Kai Wehmeier

Identity, we're told, is the binary relation that every object bears to itself, and to itself only. But how can a relation be binary if it never relates two objects? This puzzled Russell and led Wittgenstein to declare that identity is not a relation between objects. The now standard view is that Wittgenstein's position is untenable, and that worries regarding the relational status of identity are the result of confusion. I argue that the rejection of identity as a binary relation is perfectly tenable. To this end, I outline and defend a logical framework that is not committed to an objectual identity relation but is nevertheless expressively equivalent to first-order logic with identity. After it has thus been shown that there is no indispensability argument for objectual identity, I argue that we have good reasons for doubting the existence of such a relation, and rebut a number of attempts at discrediting these reasons.

In the first part of this paper we investigate the intuitionistic version $iI\!\Sigma_1$ of $I\!\Sigma_1$ (in the language of $PRA$ ), using Kleene's recursive realizability techniques. Our treatment closely parallels the usual one for $HA$ and establishes a number of nice properties for $iI\!\Sigma_1$ , e.g. existence of primitive recursive choice functions (this is established by different means also in [D94]). We then sharpen an unpublished theorem of Visser's to the effect that quantifier alternation alone is much less powerful intuitionistically than classically: $iI\!\Sigma_1$ together with induction over arbitrary prenex formulas is $\Pi_2$ -conservative over $iI\!\Pi_2$ . In the second part of the article we study the relation of $iI\!\Sigma_1$ to $iI\!\Pi_1$ (in the usual arithmetical language). The situation here is markedly different from the classical case in that $iI\!\Pi_1$ and $iI\!\Sigma_1$ are mutually incomparable, while $iI\!\Sigma_1$ is significantly stronger than $iI\!\Pi_1$ as far as provably recursive functions are concerned: All primitive recursive functions can be proved total in $iI\!\Sigma_1$ whereas the provably recursive functions of $iI\!\Pi_1$ are all majorized by polynomials over ${\Bbb N}$ . 0 $iI\!\Pi_1$ is unusual also in that it lacks closure under Markov's Rule $\mbox{MR}_{PR}$