Harmony and Paradox: Intensional Aspects of Proof-Theoretic Semantics by Luca Tranchini

Part I 
Harmony

Chapter 1 
Harmony via Reductions and Expansions 
Abstract The notion of harmony arises by considerations about the notion of asser-
tion in the theory of meaning. When applied to rules in the format of natural deduc-
tion, harmony can be explained by making reference to certain transformations on 
derivations, called reductions and expansions. Reductions are a key ingredient of the 
proof of normalization for the calculus of natural deduction for intuitionistic logic. 
In this calculus, normal derivations that are closed (i.e. such that their conclusions 
depend on no assumption) end with an introduction rule, a fact which is referred to 
as the canonicity of closed normal derivations. The condition for the canonicity of 
normal derivations in a more general setting are discussed. The chapter ends with a 
brief discussion of other accounts of harmony

1.1 Meaning Theory and Harmony 
A theory of meaning for a given language is a description of what a subject needs 
to know to qualify as a competent speaker of that language. In spite of widespread 
agreement on this general characterization, the question of what does ‘to know’ mean 
in this context has received very different answers. 
One of the most influential answers to this question is that of Dummett. 
His starting point is the observation that the competent speakers of a language are 
able to interact with each other using a wide range of speech acts such as questions, 
commands, and—most importantly—assertions (see, e.g. p. 417,).The key 
aspects of the practice of assertion are the abilities of speakers to make assertions 
under appropriate conditions and to react appropriately to assertions made by other 
speakers. Thus, an essential task of a theory of meaning is that of accounting, on the 
one hand, of the knowledge of the conditions under which a proposition is correctly 
asserted, i.e. the assertibility conditions of the proposition; and, on the other hand, 
of the knowledge of the consequences that can be drawn from the assertion of a 
proposition. 
Since the practice of assertion is rational, there must be a close connection between 
these two aspects of assertion, a connection to which Dummett refers as the principle 
of harmony:

Harmony and Paradox: Intensional Aspects of Proof-Theoretic Semantics by Luca Tranchini