Cesare Cozzo

“Identità” qui si intende nel senso già precisato da Aristotele di “identità numerica”, che si ha “quando i nomi sono parecchi, ma la cosa è una sola” e, non nel senso di “identità specifica”, che si ha invece “quando gli oggetti, pur essendo parecchi, non rivelano differenze quanto alla specie”.1 In questo articolo intendo fornire al lettore indicazioni introduttive (non certo esaustive) sul posto che la nozione di identità numerica occupa nella logica contemporanea e nell’area di riflessione filosofica del Novecento più strettamente legata alla logica, la filosofia analitica. Nella terza parte proporrò qualche spunto per un’indagine originale.

Is a rational dispute over the validity of a fundamental logical law possible? In his lecture ‘Logics and Metalogics’, Timothy Williamson criticizes Dummett’s approach to this problem and maintains that a semantic theory does not provide a way of settling disputes over the validity of fundamental logical laws. I argue that Dummett’s view is different from the view criticized by Williamson. Dummett does not think that a semantic theory alone can settle a dispute over the validity of a fundamental logical law. Such disputes, according to Dummett, should be settled by the theory of meaning. A semantic theory in the sense that is usual among logicians is a theory of logical consequence. A meaning theory is a theory of understanding.

In spite of some objections voiced by Cora Diamond, the author agrees with Michael Dummett, who detects in Wittgenstein’s Remarks on the Foundations of Mathematics a denial of the objectivity of proof: Wittgenstein identifies being-a-proof with being-treated-as-a-proof. The denial of the objectivity of proof is implausible. But it seems to be a consequence of the rule-following considerations. After examining interpretations of the rule-following considerations advanced by Saul Kripke, Crispin Wright and John McDowell, an argument is exhibited, which starts from Wright’s interpretation and leads to the denial of the objectivity of proof. It is shown, however, that the argument contains at least two problematic steps and should be rejected. The result is that one can consistently adopt the view that meaning is «plastic» or «open» which stems from the rule-following considerations) without renouncing the objectivity of proof.

The concept of “necessity of thought” plays a central role in Dag Prawitz’s essay “Logical Consequence from a Constructivist Point of View” (Prawitz 2005). The theme is later developed in various articles devoted to the notion of valid inference (Prawitz, 2009, forthcoming a, forthcoming b). In section 1 I explain how the notion of necessity of thought emerges from Prawitz’s analysis of logical consequence. I try to expound Prawitz’s views concerning the necessity of thought in sections 2, 3 and 4. In sections 5 and 6 I discuss some problems arising with regard to Prawitz’s views.