Proof-theoretic semantics are usually conceived in opposition to truth-theoretic semantics. In truth-theoretic semantics, truth is considered as a primitive (non-analyzed) notion, and meaning is then explained in terms of it. On the other hand, in proof-theoretic semantics, meaning is explained in terms of (our) inferential abilities, and truth is then explained in terms of proofs. In order to avoid any possible trivialization of proof-theoretic semantics — boiling it down to truth-theoretic semantics — an intensional, rather than an extensional approach, should be adopted. In particular, the semantic value of a sentence A should not be defined in terms of the simple existence of a proof of A, but in terms of the way in which A is proved, i.e. in terms of the inferential structure of proof of A. In order to specify this structure some properties are asked to be satisfied. The most known of them is the property of harmony, which corresponds to the reduction of local complexity peaks (detours) in a proof. However, as Dummett claims, this property is “an excessively modest demand”, and it should be complemented by another property, that of stability. It will be shown that this property can be captured by a more fundamental operation, that of expansion, allowing one to generate local complexity riffs within a proof. Even if this operation could seem very natural to add, it will be shown in fact how it is destructive for the intensional account proper to proof-theoretic semantics. In particular, when this operation of expansion is used in presence of negation and identity, it leads to the collapse of the set of proofs of negative and identity sentences, respectively. Finally, the case of identity is analyzed in details in the framework of Martin-Löf’s type theory. It is shown, in particular, that a possible way of avoiding the collapse between identity proofs can be found in the works of M. Hoffman and T. Streicher, where the operation of expansion is lifted from the level of proof-objects to the level of the sentences which speak about these proof-objects.
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