Harmony via Reductions and Expansions

Abstract

The notion of harmony arises by considerations about the notion of assertion in the theory of meaning. When applied to rules in the format of natural deduction, 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 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 [11, 12]. 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, [11]).¹ 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: Informal statement 1

The consequences that can be drawn from the assertion of a proposition can be neither more nor less than those that are guaranteed by the satisfaction of its assertibility conditions.

To clarify this idea, Dummett (see [11], p. 454) discusses an example aimed at showing that the violation of harmony induces irrational elements in linguistic practices. The example considered is ‘boche’, a derogatory term with which the Anglo-Americans referred to German people during the First World War. The conditions for applying the predicate to a person (and thus for asserting his being a boche) are that the person is of German nationality, while the consequences that can be drawn from the assertion of ‘He is a boche.’ are the barbarism and cruelty of the subject of the proposition. The disharmony between the two aspects of statements of this type shows the non-rationality of the use of the term ‘boche’ and the need to modify the linguistic practices that involve it.

Harmony therefore has not only a descriptive value, but also a normative component, i.e. if the assertibility conditions of a proposition and the consequences that can be drawn from its assertion do not coincide, then the linguistic community ought to change its practices.²

As Dummett himself admits, the universal validity of the principle of harmony is a very strong demand, and it is doubtful that the two aspects of any possible assertion in a given language are in perfect harmony. Nonetheless, given that we are willing to concede that the linguistic practices in which we are involved as speakers are rational (at least for the most part), it seems natural to expect a theory of meaning to satisfy certain general conditions that guarantee that the practices it describes are (at least in principle) harmonious.

2 Harmony and Natural Deduction

Some of these conditions depend on another essential aspect of language, namely that a competent speaker of a language is able to produce and understand a potentially infinite set of distinct utterances, starting from his understanding of a finite set of minimal linguistic units (words) endowed with meaning. This aspect of language, perhaps the only one distinguishing human language from other forms of animal language, is made possible by the existence of expressions which can be used to build complex expressions starting from simpler expressions. One class of these expressions is that of logical constants, which in particular allow the formation of complex propositions starting from one or more simpler propositions.

As on the syntactic level complex propositions are obtained by composing simpler propositions, on the semantic level the meaning of complex propositions will depend on the meaning of their components and on the way in which they are composed. This principle, called compositionality, is embodied in a theory of meaning of the type outlined by Dummett by rules that specify the assertibility conditions (respectively the consequences of the assertions) of complex propositions in terms of the asseribility conditions (resp. the consequences of the assertions) of their components.

Table 1.1 The natural deduction calculus $\texttt{NI}$

 The paradigm for these rules are those of the calculus of natural deduction for intuitionistic logic $\texttt{NI}$ [20, 65] . In this calculus (whose rules are depicted in Table 1.1), two types of rules are associated with each logical constant: introduction rules and elimination rules. The introduction rules for a logical constant $\dagger$ are those which allow one to infer a complex proposition having $\dagger$ as main operator, and thus they specify the assertibility conditions of such propositions; in the elimination rules for $\dagger$, a complex proposition having $\dagger$ as main operator acts as the main premise³ of the rules, and these rules thus specify which consequences can be drawn from its assertion. 

In the case of conjunction, the introduction rule $\wedge$I allows one to infer from two propositions their conjunction,⁴ and thus expresses the fact that the assertibility conditions of a conjunction are satisfied when those of both conjuncts are. The two elimination rules $\wedge$E$_1$ and $\wedge$E$_2$ allow one to infer from a conjunction each of the two conjuncts (respectively), and thus express the fact that the consequences that can be drawn from the assertion of a conjunction are all those that can be obtained from the two conjuncts.

In order to guarantee the harmony between the two aspects of assertion, the rules of introduction and elimination that govern the logical constants cannot be chosen arbitrarily. In particular, an inappropriate choice of introduction and elimination rules for a connective may result in a situation analogous to that of ‘boche’. Exemplary in this sense is the binary connective ${\texttt{tonk}}$ introduced by Prior [76], governed by the following pair of introduction and elimination rules:

As in the case of ‘boche’, the consequences that can be drawn from the assertion of a complex proposition governed by ${\texttt{tonk}}$ do not coincide with what is warranted by the fulfillment of its assertibility conditions, and the disharmony between the two aspects of the practice of assertion deprives this of rationality: given the rules of ${\texttt{tonk}}$ every proposition can be inferred from any other.

As (and even more than) in the case of boche, the strong intuition that ${\texttt{tonk}}$ is “semantically defective” shows that not every collection of introduction and elimination rules for a connective is apt to determine the meaning of complex propositions in which the connective is the main operator.

Hence, in natural deduction, the requirement of harmony as applying to logically complex propositions becomes a condition that should be satisfied by the collections of introduction and elimination rules. We can informally state it as follows⁵:

Harmony: Informal statement 2

What can be inferred from a logically complex proposition by means of the elimination rules for its main connective is no more and no less than what has to be established in order to infer that very logically complex proposition using the introduction rules for its main connective.

The rules of ${\texttt{tonk}}$ display no match between what can be inferred using the elimination rule and what is needed to establish the premise of the elimination rule using the introduction rule.⁶ Thus, even if A and B are meaningful statements, their “contonktion” $A \mathbin {\texttt{tonk}}B$ is nonsense since the rule governing $\mathbin {\texttt{tonk}}$ are ill-formed.⁷ In contrast to the rules of ${\texttt{tonk}}$, the rules for conjunction of $\texttt{NI}$ display a perfect match. 

3 Harmony, Reductions and Expansions

Both informal characterizations of harmony given above make clear that harmony is two-fold condition, and we will refer to its two components as the ‘no more’ and ‘no less’ aspect of harmony.⁸ The two aspects of harmony are closely connected with two different kinds of deductive patterns.

Patterns of the first kind are those giving rise to maximal formulas occurrences, sometimes referred to as ‘local peaks’ [12] or ‘hillocks’ (used by von Plato to translate Gentzen’s original ‘Hügel’ [64]). These are formula occurrences which are the major premise of an application of an elimination rule and that are the consequence of an application of one of the introduction rules.

When the rules for a connective are in harmony, configurations of this kind are clearly redundant. In particular, the possibility of “leveling” these local peaks shows that harmonious elimination rules allow one to infer no more than what has to be established to infer their major premise by introduction.

Prawitz [65] defined certain operations on derivations called reductions that, when applied to a derivation, transform it into another one by getting rid of a single maximal formula occurrence.⁹

In the case of conjunction, there are two patterns of this kind, of which one can get rid as follows:

  Patterns of the other kind are those in which the premises of applications of introduction rules have been obtained applying the corresponding elimination rules.¹⁰ These patterns could be described as local valleys since they result when one infers a complex proposition from itself by first eliminating and then reintroducing its main connective. Prawitz [66] defined operations that are, in a sense, the dual of reductions, called (immediate) expansions to introduce such valleys within a derivation. The possibility of “expanding” a derivation via a local valley amounts to the fact that harmonious elimination rules allow one to infer no less than what is needed to infer their major premise by introduction.¹¹ In the case of conjunction, the expansion is the following:

  In the case of implication, we have the following reduction and expansion:¹²$^,$¹³

  Most of the literature on harmony has focused on logical constants. However, there seems to be no principled reason to restrict the account of harmony just sketched to these expressions only. For any inductively definable n-ary predicate P, it is possible to formulate introduction and elimination rules for atomic propositions $P(t_1,\ldots , t_n)$, so that an introduction rule for a primitive n-ary predicate P yields a derivation having $P(t_1,\ldots , t_n)$ as conclusion (where $t_1,\ldots ,t_n$ are singular terms), while an elimination rule for a primitive n-ary predicate P is one that, applied to a derivation having $P(t_1,\ldots , t_n)$ as conclusion and, possibly, other derivations, yields a derivation of some other proposition. For example, the following are the introduction and elimination rules for the unary predicate $\texttt{Nat}$ expressing the property of being a natural number  (in the rules, we use S as a unary function symbol for the successor function and with A(t/x) we indicate capture-avoiding substitution of t for x in A):¹⁴

Whenever the major premise $\texttt{Nat}\,t$ is the consequence of an application of one of the two introduction rules for $\texttt{Nat}$, a reduction is readily defined, and an expansion can be defined as well, by applying the elimination rule taking A to be $\texttt{Nat}\,x$.¹⁵

In the following we will (with a few exceptions) restrict ourselves to rules governing propositional connectives, thereby disregarding the exact nature of the non-logical vocabulary. The application of the ideas presented in the next chapters to specific languages, such as the one of arithmetic, represents an interesting challenge, but goes beyond the scope of the present work.

4 Some Formal Definitions

We will henceforth write $\mathscr {D}_1{\mathop {\triangleright }\limits ^{1\supset \beta }}\mathscr {D}_2$ (respectively $\mathscr {D}_1{\mathop {\triangleleft }\limits ^{1\supset \eta }}\mathscr {D}_2$) when $\mathscr {D}_2$ is obtained by one application of the reduction (respectively expansion) for implication from $\mathscr {D}_1$.¹⁶ This is to be understood to mean that

• either $\mathscr {D}_1$ and $\mathscr {D}_2$ are of the form depicted to the left-hand and right-hand side of the reduction (resp. expansion) above;

• or that $\mathscr {D}_2$ is obtained by replacing in $\mathscr {D}_1$ one of its subderivations having the form depicted on the left-hand side of the reduction (resp. expansion) with a derivation having the form depicted on the right-hand side of the reduction (resp. expansion). (This latter case will be referred to as the congruence condition for $\supset$-reduction (resp. expansion).)

We will refer to these relations as one-step $\mathbin {\supset }\beta$-reduction and one-step $\mathbin {\supset }\eta$-expansion.

We will use a similar notation for one-step reductions and expansions of conjunction and of the other connectives introduced below. For connectives with more than one introduction (resp. elimination) rule we use subscripts (e.g. $\mathscr {D}_1{\mathop {\triangleright }\limits ^{1\wedge \beta _1}}\mathscr {D}_2$ and $\mathscr {D}_1{\mathop {\triangleright }\limits ^{1\wedge \beta _2}}\mathscr {D}_2$ in the case of conjunction) to distinguish between the relations induced by the reductions getting rid of maximal formula occurrences which are the consequences (resp. major premises) of different introduction (resp. elimination) rules.

Sometimes, we will omit the subscripts and/or the indication of the connective (thus writing e.g. $\mathscr {D}{\mathop {\triangleright }\limits ^{1\beta }}\mathscr {D}'$ and $\mathscr {D}{\mathop {\triangleleft }\limits ^{1\eta }}\mathscr {D}'$) where the omission of the connective (and possibly of the subscript) indicates that $\mathscr {D}_2$ can be obtained from $\mathscr {D}_1$ using some reduction (resp. expansion) for some connective.

If a $\beta$-reduction is available to get rid of a maximal formula occurrence whose main connective is $\dagger$, the latter will be referred to as a $\dagger \beta$-redex (contraction of reducible expression).

We indicate with ${\mathop {\triangleright }\limits ^{\beta }}$ the relation of $\beta$-reduction which is the reflexive and transitive closure of the relation of one-step $\beta$-reduction. That is, we write $\mathscr {D}{\mathop {\triangleright }\limits ^{\beta }}\mathscr {D}'$ when for some $n\ge 1$ there is a sequence of n-derivation $\mathscr {D}_1, \ldots , \mathscr {D}_n$ (to which we will refer to as a $\beta$-reduction sequence for $\mathscr {D}$) such that $\mathscr {D}_1=\mathscr {D}$, $\mathscr {D}_n=\mathscr {D}'$ and $\mathscr {D}_{i-1}{\mathop {\triangleright }\limits ^{1\beta }}\mathscr {D}_{i}$ for each $1<i\le n$. Similar notation and terminology will be adopted for $\eta$-expansion as well.

Sometimes it will be useful to refer to the inverses of these relations as well, that we will indicate with ${\mathop {\triangleleft }\limits ^{(1)\mathbin {\supset }\beta }}$ (respectively ${\mathop {\triangleright }\limits ^{(1)\mathbin {\supset }\eta }}$). Clearly, also these relations may be dubbed (one-step) expansions and reductions respectively, and hence ‘$\mathbin {\supset }$-reduction’ is actually ambiguous between ${\mathop {\triangleright }\limits ^{\mathbin {\supset }\beta }}$ and ${\mathop {\triangleright }\limits ^{\supset \eta }}$ (and ‘$\supset$-expansion’ between ${\mathop {\triangleleft }\limits ^{\supset \eta }}$ and ${\mathop {\triangleleft }\limits ^{\supset \beta }}$). When precision is required we will speak of $\mathord {\supset }\beta$-reduction and $\mathord {\supset }\eta$-reduction (and $\mathbin {\supset }\eta$-expansion and $\mathbin {\supset }\beta$-expansion), but as we already did until now, we will however use ‘$\supset$-reduction’ (resp. ‘$\supset$-expansion’) to indicate the relation ${\mathop {\triangleright }\limits ^{\mathbin {\supset }\beta }}$ (resp. ${\mathop {\triangleleft }\limits ^{\mathbin {\supset }\eta }}$).

We observe that we will often speak of “applications” of $\mathord {\supset }\beta$-reduction (and similarly for other reductions/expansion), thereby treating $\mathord {\supset }\beta$ as a function that given a derivation $\mathscr {D}$ and a particular maximal formula occurrence of the form $A\mathbin {\supset }B$ in $\mathscr {D}$ yields the result of reducing that maximal formula occurrence. We take this notion of reduction as function as clear enough and we omit its precise definition.

We will use the term conversion to refer to $\beta$-reductions, $\beta$-expansions, $\eta$-reductions, $\eta$-expansions, as well as to further transformations on derivations to be introduced in the following chapters.

5 Some Formal Results

A derivation $\mathscr {D}$ is called $\beta$-normal iff it is not possible to $\beta$-reduce it any further (i.e. iff $\mathscr {D}{\mathop {\triangleright }\limits ^{\beta }}\mathscr {D}'$ implies $\mathscr {D}'=\mathscr {D}$). In the $\{\supset ,\wedge \}$-fragment of $\texttt{NI}$ (we will indicate this fragment as $\texttt{NI}^{\wedge \mathbin {\supset }}$) Prawitz [65] showed how any given derivation $\mathscr {D}$ can be transformed into a $\beta$-normal one by successive applications of the $\mathord {\supset }\beta$- and $\mathord {\wedge }\beta$-reductions.

The proof is non-trivial, since an application of $\mathord {\supset }\beta$-reduction to a given derivation may yield a derivation containing the same number of maximal formula occurrences, or even more. (For an example consider the result of $\beta$-reducing the encircled occurrence of $A\mathbin {\supset }B$ in the derivation of Table 1.2 below.) However, it is always possible to find a maximal formula occurrence such that, by $\beta$-reducing it, the number of maximal formula occurrences of maximum degree in the resulting derivation is lower than in the original derivation, where the degree of a maximal formula occurrence is the number of logical constants it contains.¹⁷ Therefore, for every derivation $\mathscr {D}$ in $\texttt{NI}^{\wedge \mathbin {\supset }}$ there is a $\beta$-reduction sequence starting with $\mathscr {D}$ and ending with a $\beta$-normal derivation.

Table 1.2 A derivation illustrating the non-triviality of normalization

This result, known as the weak normalization theorem for $\beta$-reduction in $\texttt{NI}^{\wedge \mathbin {\supset }}$, has been strengthened using a method introduced by Tait [107] to what is nowadays called the strong normalization theorem for $\beta$-reduction in $\texttt{NI}^{\wedge \mathbin {\supset }}$, namely that in $\texttt{NI}^{\wedge \mathbin {\supset }}$ there are no infinite $\beta$-reduction sequences (that is, in the process of $\beta$-reducing a derivation, no matter which maximal formula occurrence is chosen at any step, if one keeps on $\beta$-reducing one will always reach a $\beta$-normal derivation).

Given their significance for the following, we discuss some properties of $\beta$-normal derivations in $\texttt{NI}^{\wedge \mathbin {\supset }}$.

First, each derivation in $\texttt{NI}^{\wedge \mathbin {\supset }}$ has a unique $\beta$-normal form, i.e. no matter how a derivation is reduced, one will always end up with the same $\beta$-normal derivation. This property is an immediate consequence of the confluence property (sometimes Church-Rosser property) of the relation of $\beta$-reduction, that is the fact that if $\mathscr {D}{\mathop {\triangleright }\limits ^{\beta }}\mathscr {D}_1$ and $\mathscr {D}{\mathop {\triangleright }\limits ^{\beta }}\mathscr {D}_2$, then there is $\mathscr {D}'$ such that both $\mathscr {D}_1{\mathop {\triangleright }\limits ^{\beta }}\mathscr {D}'$ and $\mathscr {D}_2{\mathop {\triangleright }\limits ^{\beta }}\mathscr {D}'$. (See Table 1.3 for an example of two $\beta$-reduction sequences for the same derivation ending with the same $\beta$-normal derivation. In the table, the target of each arrow is a derivation obtained by $\beta$-reducing one of the maximal formula occurrences in the derivation which is the source of that arrow.)

Table 1.3 An example of confluence

Second, $\beta$-normal derivations in $\texttt{NI}^{\wedge \mathbin {\supset }}$ enjoy the subformula property, that is every formula occurring in a $\beta$-normal derivation is either a subformula of the conclusion or of one of the undischarged assumptions of the derivation.¹⁸ The subformula property is an immediate consequence of the peculiar form of $\beta$-normal derivations in $\texttt{NI}^{\wedge \mathbin {\supset }}$. This can be described using the notion of track, where a track is a sequence of formula occurrences in a derivation such that (i) the first is an assumption of the derivation; (ii) all other members of the sequence are the consequence of an application of an inference rule of which the previous member is one of the premises; (iii) none of them is the minor premise of an application of $\mathbin {\supset }$E.

In each track of a $\beta$-normal derivation in $\texttt{NI}^{\wedge \mathbin {\supset }}$, all eliminations precede the introductions (otherwise the track, and hence the derivation would contain a maximal formula occurrence). The two parts (either of which is possibly empty) of a track are separated by a minimal part. This is a formula occurrence which is both the consequence of an elimination and the premise of an introduction. Furthermore, each formula occurrence in the elimination part is a subformula of the preceding formula occurrence in the track, and each formula occurrence in the introduction part is a subformula of the next formula occurrence in the track (since the premises of introduction rules are of lower complexity than the consequences, and the consequences of elimination rules are of lower complexity than the (major) premise).

A bit more formally, we have the following:

Fact 1

(The form of tracks) Each track $A_1\ldots A_{i-1}, A_i, A_{i+1},\ldots A_n$ in a $\beta$-normal derivation in $\texttt{NI}^{\wedge \mathbin {\supset }}$ contains a minimal formula $A_i$ such that

• If $i>1$ then $A_j$ (for all $1\le j< i$) is the major premise of an application of an elimination rule of which $A_{j+1}$ is the consequence and thereby $A_{j+1}$ is a subformula of $A_j$.

• If $n>i$ then $A_j$ (for all $i\le j< n$) is the premise of an application of an introduction rule of which $A_{j+1}$ is the consequence and thereby $A_j$ is a subformula of $A_{j+1}$.

Proof

For a derivation to be $\beta$-normal, all applications of elimination rules must precede all applications of introduction rules in all of its tracks: This warrants the existence of a minimal formula in each track. Since a track ends whenever it “encounters” the minor premise of an application of $\supset$E, the subformula relationships between the members of a track hold (as it can be easily verified by checking the shape of the rules of $\texttt{NI}^{\wedge \mathbin {\supset }}$).   $\square$

From this it follows (almost) immediately that $\beta$-normal derivations in $\texttt{NI}^{\wedge \mathbin {\supset }}$ enjoy the subformula property: each formula in a $\beta$-normal derivation is the subformula either of the conclusion or of one of the undischarged assumptions of the derivation.

Fact 2

(Subformula property) All formulas in a $\beta$-normal derivation in $\texttt{NI}^{\wedge \mathbin {\supset }}$ are subformulas either of the conclusion or of some undischarged assumption.

Proof

The proof of the theorem is by induction on the order of tracks, where the order of a track is defined as follows: The tracks to which the conclusion belong are of order 0. A track is of order n if its last formula is the minor premise of an application of $\mathbin {\supset }$E whose major premise belong to a track of order $n-1$ (see for details [65], Chap. III, Sect. 2).   $\square$

In the calculus $\texttt{NI}^{\wedge \mathbin {\supset }}$, strong normalization and confluence hold for $\eta$-reduction as well.¹⁹ These results hold moreover for the relation of $\beta \eta$-reduction (notation ${\mathop {\triangleright }\limits ^{\beta \eta }}$), that is defined  as the reflexive and transitive closure of one-step $\beta \eta$-reduction (notation ${\mathop {\triangleright }\limits ^{1\beta \eta }}$), which is the union of ${\mathop {\triangleright }\limits ^{1\beta }}$ and ${\mathop {\triangleright }\limits ^{1\eta }}$.

For the relation resulting by putting together $\beta$-reduction and $\eta$-expansion, Prawitz [66] established a weak normalization theorem by showing that by successively applying expansions it is possible to transform any given a $\beta$-normal derivation in $\texttt{NI}^{\wedge \mathbin {\supset }}$ into one in which all minimal formula occurrences of all tracks are atomic. This relation is also confluent, but not strongly normalizing (due to the possibility of constructing looping infinite chains of $\eta$-expansions followed by $\beta$-reductions). However, in the case of the purely implicational fragment of $\texttt{NI}$ (we refer to it as $\texttt{NI}^{\mathbin {\supset }}$) Mints [49] has shown that strong normalization can be recovered by disallowing $\eta$-expansions of formulas of the form $A\mathbin {\supset }B$ which are the consequence of $\mathbin {\supset }$I or the major premise of $\mathbin {\supset }$E (see also [35] for a discussion).

6 Canonicity

It is not the case that $\beta$-normal derivations have the subformula property in every calculus of natural deduction consisting of harmonious rules.²⁰ Consider for example the calculus $\texttt{NI}^{2\wedge \mathbin {\supset }}$, the extension of $\texttt{NI}^{\wedge \mathbin {\supset }}$ with quantification over propositions governed by the following rules (we indicate with A(B/X) the result of substituting the free occurrences of X in A with B):²¹

The rules are in harmony as testified by the possibility of defining the following reduction and expansion (in the reduction, $\mathscr {D}(B/X)$ indicates the derivation that results by uniformly substituting all free occurrences of X in $\mathscr {D}$ with B):

In contrast to the other elimination rules so far encountered, the consequence of an application of $\forall ^2$E might be of higher complexity than its premise, due to the fact that the complexity of the formula B (called the witness of the rule application) can be arbitrary. This features complicates significantly the proof of normalization of $\beta$-reduction in $\texttt{NI}^{2\wedge \mathbin {\supset }}$ (strong normalization of $\beta$-reduction in a calculus akin to $\texttt{NI}^{2\wedge \mathbin {\supset }}$ was first established by Girard [22] using a generalization of the method of Tait mentioned above). Moreover, $\beta$-normal derivations in $\texttt{NI}^{2\wedge \mathbin {\supset }}$ do not enjoy the subformula property, as shown by the following $\beta$-normal derivation (in the example, the witness of the application of $\forall ^2$E is $\forall X. X\mathbin {\supset }C$ which is also the premise of the rule application):

There is however another, though much weaker, property which is warranted by the harmonious setup of the rules of the calculus. This property, to which we will refer to as canonicity, is not enjoyed by any $\beta$-normal derivation, but only by those $\beta$-normal derivations which are also closed (i.e. such that all their assumptions are discharged): In harmonious calculi, any such derivation ends with an introduction rule.

Canonicity holds for any arbitrary natural deduction calculus provided it satisfies the following two conditions:

Definition 1.1

(Harmonious calculus) A natural deduction calculus²² is said to be harmonious iff:

• 1. All rules of the calculus are either introduction or elimination rules. We stress that

  • No restriction is imposed on introduction rules (in particular no complexity condition has to be satisfied for the result to hold);

  • only a mild requirement is imposed on elimination rules, namely that no elimination rule is such that its applications can discharge assumptions in the derivation of the major premise.

• 2. For every maximal formula there is a $\beta$-reduction to get rid of it (i.e. every maximal formula is a $\beta$-redex).

Fact 3

(Canonicity) In an harmonious natural deduction calculus, every closed and $\beta$-normal derivation ends with an application of an introduction rule.

Proof

We check by induction on the number of inference rules applied in a derivation that either the antecedent of the theorem is false or the consequent is true. If the derivation consists only of an assumption it is not closed. The inductive case of a derivation consisting of $n+1$ rule applications falls into two sub-cases. The derivation ends either with an introduction rule or with an elimination rule. In the former case, we are done. In the latter case, we have to show that the derivation is either open or it is not $\beta$-normal. We apply the induction hypothesis to the subderivation of the major premise and we distinguish two sub-cases: either the subderivation is open or it is not $\beta$-normal and then so is the whole derivation (because of the mild requirement on elimination rules in condition 1 above); or the subderivation ends with an introduction rule, but then the whole derivation is not $\beta$-normal as the major premise of the application of the elimination rule yielding the conclusion of the derivation is obtained by introduction.   $\square$

Henceforth, we will refer to derivations ending with an introduction rule as canonical derivations.²³

7 Normalization, Subformula Property, Canonicity and Harmony

  In every harmonious calculus, $\beta$-normal derivations can be equivalently characterized as those containing no maximal formula occurrence.

In calculi which are not harmonious, however, this is not in general the case. Consider the extension of $\texttt{NI}^{\wedge \mathbin {\supset }}$ with the rules for ${\texttt{tonk}}$, which we will refer to as $\texttt{NI}^{\wedge \mathbin {\supset }\texttt{tonk}}$. The calculus is not harmonious in that it fails to satisfy the condition 2 of Definition 1.1: In such a calculus, it is not possible to devise a reduction to get rid of any maximal formula occurrence whose main connective is ${\texttt{tonk}}$, as the following example shows (in the example A is an arbitrary proposition and p an atomic one):

The occurrence of $(A\mathbin {\supset }A)\; {\texttt{tonk}}\; p$ in T is maximal. Nonetheless, there is no way to further $\beta$-reduce the derivation, which therefore qualifies as $\beta$-normal.

To consider the above derivation as $\beta$-normal may appear counterintuitive at first. The reason is that, on the one hand, the notion of normal derivation is usually presented as meant to capture the intuitive idea of a derivation containing no redundancy; and, on the other hand, maximal formula occurrences are usually taken to constitute a kind of conceptual redundancy within derivations. Sticking to these intuitions, one may expect a necessary condition for a derivation of any calculus (and not just of an harmonious one) to qualify as normal to be that the derivation does not contain any maximal formula occurrences.

It is however doubtful that maximal formula occurrences should always count as constituting a redundancy. This is certainly the case in $\texttt{NI}^{\wedge \mathbin {\supset }}$, where consecutive applications of $\mathbin {\supset }$I and $\mathbin {\supset }$E, or of $\wedge$I and either $\wedge$E$_1$ or $\wedge$E$_2$ do constitute a conceptual detour. But what about a calculus containing the rules for $\mathbin {\texttt{tonk}}$? The rules for $\mathbin {\texttt{tonk}}$ are clearly not in harmony. This is tantamount to denying that we had already a derivation of the consequence of an application of the elimination rule, provided that the premise had been established by introduction. In other words, when we establish something passing through a complex formula governed by $\mathbin {\texttt{tonk}}$, we are not making an unnecessary detour. The fact that the rules for $\mathbin {\texttt{tonk}}$ are not in harmony means exactly that in some (actually most) cases it is only by appealing to its rules that we can establish a deductive connection between two propositions not involving $\mathbin {\texttt{tonk}}$. This is the opposite of the claim that maximal formula occurrences having $\mathbin {\texttt{tonk}}$ as main connective constitute a redundancy. Rather, they are the most essential ingredient for establishing a wide range of derivability claims. For example, in the derivation ${\textbf {T}}$, the maximal formula occurrence $(A\mathbin {\supset }A){\texttt{tonk}}\,p$ is in no way redundant: without passing through it, it would have been impossible to establish the conclusion p.²⁴

Hence, we do not take the fact that $\beta$-normal derivations in a calculus like $\texttt{NI}^{\wedge \mathbin {\supset }\texttt{tonk}}$ might contain maximal formula occurrence as showing that there is something amiss with the definition of normality.

Of course this is not to deny that there is something amiss with $\texttt{NI}^{\wedge \mathbin {\supset }\texttt{tonk}}$ and in fact the notion of $\beta$-normal derivation can be used to make clear what is amiss with this calculus. Although $\beta$-reduction is normalizing in $\texttt{NI}^{\wedge \mathbin {\supset }\texttt{tonk}}$ (this can be easily proved in the same way as it was done for $\texttt{NI}^{\wedge \mathbin {\supset }}$), neither do $\beta$-normal derivations have the sub-formula property, nor are all closed $\beta$-normal derivations canonical.

In the next chapters, we will show that the canonicity of closed $\beta$-normal derivations of harmonious calculi plays a crucial role for their (proof-theoretic) semantic interpretation. The fact that not every $\beta$-normal derivation in a calculus like $\texttt{NI}^{\wedge \mathbin {\supset }\texttt{tonk}}$ is canonical thus shows its semantic defectiveness.

It is worth stressing that among harmonious calculi we find not only calculi such as $\texttt{NI}^{2\wedge \mathbin {\supset }}$—in which $\beta$-normal derivations do not possess the subformula property—but also calculi in which $\beta$-reduction is not (weakly) normalizing. Like the failure of the subformula property for $\beta$-normal derivation in $\texttt{NI}^{2\wedge \mathbin {\supset }}$, the failure of normalization of $\beta$-reduction does not invalidate Fact 3, i.e. in every harmonious calculus, even those in which not every derivation can be reduced to a $\beta$-normal derivation, those derivations which are both closed and $\beta$-normal are canonical as well.

We conclude by observing that all (standard) natural deduction calculi for classical logic are obtained by the addition of one or more rules to $\texttt{NI}$ which are neither introduction nor elimination rules. Hence these calculi do not comply with the definition of ‘harmonious calculus’ given above. It is by now commonplace that classical logic can be given an harmonious presentation by either abandoning natural deduction (typically in favor of sequent calculus) or by enriching natural deduction in different ways (typically, by allowing derivations to have multiple conclusions, or by considering “refutation” rules along side standard “proof” rules). Although we do not exclude the possibility of systematically applying the ideas to be developed in the next chapters (possibly in modified form) to these other formal settings, in the present work we will restrict our attention to standard natural deduction calculi, and therefore leave classical logic out of the picture.

8 A Quick Comparison with Other Approaches

The account of harmony sketched in Sect. 1.3 differs from the account of harmony stemming from Belnap [2] , who cashed out the no more and no less aspects of the informal definition of harmony in terms of conservativity and uniqueness respectively.²⁵

Following Dummett (who refers to conservativity as ‘global’ harmony and to the availability of reductions as ‘intrinsic’ harmony), for some authors (see e.g. [94], pp. 1204–5) the distinctive feature of Belnap’s conditions is their being “global”, in contrast with other “local” ways of rendering the informal definition of harmony, such as the one in terms of reductions and expansions.²⁶

In our opinion, however, what crucially distinguishes the account of harmony sketched in the previous section from the one of Belnap—as well as from those of other authors, such as Tennant’s (see, e.g. [108])—is something else: Both conservativity and uniqueness are defined in terms of derivability (i.e. of what can be derived by means of the rules for a connective) and not in terms of properties involving the internal structure of derivations (i.e. of how something can be derived). We propose to refer to accounts of harmony based on derivability as extensional, while those making explicit reference to the internal structure of derivations will be referred to as intensional.

The choice of this terminology will become clear in the course of the next chapter, in which it shown how starting from reductions and expansions one naturally arrives at a notion of identity of proofs and of formula isomorphism.

Notes to This Chapter

• 1. For a recent critical discussion of the thesis that assertion plays a distinguished role among speech acts, see [81].

• 2. The exact meaning of harmony is however open to different interpretations. In particular, in the subsequent literature, there is no agreement on whether harmony should be considered as a descriptive or a normative criterion; nor whether it should be considered as a criterion of “significance” or of “logicality” (that is, if expressions governed by rules that are not in harmony should be considered as meaningless; or as meaningful but not belonging to the logical vocabulary) or of something else. In the present chapter, we will stick to the reading of harmony as meaningfulness. A more specific characterization of the significance of harmony will be given in Sect. 3.6. See also Note 7 below.

• 3. More precisely, we call the major premise of an application of an elimination rule the one which corresponds, in the rule schema, to the premise in which the connective to be eliminated occurs.

• 4. I am therefore taking for granted that what is established by a proof, and what one can draw inferences from, is a proposition, rather than, say, a judgment that a proposition is true. The distinction between judgment and proposition, on which some authors particularly insist (see, e.g. [106]), will not play any significant role in the present work. The reason for the choice here made is of mere convenience.

• 5.   Terminologically, [12] uses ‘harmony’ sometimes to refer only to the no more aspect of this condition (see, e.g. pp. 247–248, [12]) and sometimes to refer to both (see, e.g. p. 217, [12]). Later on (see, e.g. p. 287, [12]), Dummett introduces the term ‘stability’ to cover both aspects. Here, we will follow Jacinto and Read [34] and use ‘stability’ to refer to the no less aspect of harmony, where ‘harmony’ is understood as covering both aspects. We also remark that sometimes one refers to what has to established in order to infer a proposition A as the (direct, or canonical) grounds for A and harmony is informally stated as the requirement that “Whatever follows from the direct grounds for deriving a proposition must follow from that proposition” (this formulation is due to Negri and von Plato [53], p. 6) to which one may add, ‘(and nothing else).’ in order to stress the two-fold nature of the requirement. Negri and von Plato [53] refer to their informal formulation of harmony as ‘inversion principle’. We will however reserve the term ‘inversion principle’ for functions yielding collections of elimination rules as outputs when applied to collections of introduction rules as inputs. Three distinct inversion principles in our sense will be discussed in Sects. 3.4, 3.7 and 3.9 (on related terminological issues, see also Note 6 to Chap. 3). Finally, we observe that, although Dummett himself stressed that harmony is a two-fold condition, the proof-theoretic semantic literature has been mostly concerned with the no more aspect of it (but see e.g. [7, 8, 52] for notable exceptions) , thus making our informal characterization of harmony, to some extent, non-standard.

• 6. Whereas $\mathbin {\texttt{tonk}}$’s rules fail to meet both the no less and the no more aspect of the informal characterization of harmony, there are connectives which fail to satisfy only one of the two. A connective with the same introduction rule as conjunction and the same elimination rule as implication is an example of a connective failing to satisfy the no less aspect, but satisfying the no more aspect (see pp. 158–159, [52]). For another example, consider the connective whose rules are obtained from those of conjunction by dropping one of the two elimination rules. For a connective satisfying the no less aspect but failing to satisfy the no more aspect one may consider a variant of $\mathbin {\texttt{tonk}}$ with two introduction rules (corresponding to both introduction rules for disjunction). In this case using the elimination rule one would obtain no less than what is needed to introduce the connective again using the second introduction rule.

• 7. A referee objected that from our diagnosis of $A \mathbin {\texttt{tonk}}B$ as nonsense it looks “as if harmony was a criterion for meaningfulness, although perhaps it is best interpreted as a criterion for logicality (in line with Dummett’s own admission that it cannot be reasonably asked for all the expressions of the language).” The objection is fair but a full evaluation of it, though of the utmost importance for the current debates on proof-theoretic semantics, goes beyond the scope of the present work. Here we only remark that: (i) In spite of Dummett’s own admissions, it is undeniable that he is at least strongly sympathetic to the equation between harmony and meaningfulness. (One of) Dummett’s [12] aim(s) is to recast Brouwer’s criticisms of classical mathematics (namely that of being incomprehensible, viz. meaningless) by showing that the rules for the logical constants in classical logic are not harmonious. (We do not thereby want to commit ourselves either to the cogency of Dummett’s arguments, nor to the tenability of Brouwer’s views.) (ii) Even if harmony is not a criterion for meaningfulness, its applicability goes certainly beyond that of logical expressions, as shown by the rules for the predicate ‘x is a natural number’ which we briefly discussed at the end of Sect. 1.3. (We are here implicitly endorsing the view on which the natural number predicate is a non-logical expression. Though widespread, this view has been notoriously challanged by logicist and neo-logicist, see [114].) See also Note 2 above.

• 8.   Schroeder-Heister [94] refers to the two aspects of harmony as the ‘criterion of reduction’ and the ‘criterion of recovery’ respectively. Steinberger [102] refers to collections of rules that fail to meet the no more and no less aspects of harmony as cases of ‘E-strong’ (or equivalently ‘I-weak’) disarmony and cases of ‘E-weak’ (or equivalently ‘I-strong’) disarmony respectively.

• 9. However, new ones may be generated in the process, see below.

• 10. For rules discharging assumptions, we distinguish between their premises and their immediate premises (see Definition A.1 in the appendix). In the case of an instance of $\supset$I with consequence $A\mathbin {\supset }B$, the immediate premise is B, while the premise is the (concrete) rule $A\Rightarrow B$. The distinction is relevant for showing that in the expansion for implication given below, the local valley accords with the general description just given. Through a single application of $\supset$E one obtains a derivation of B from A and $A\mathbin {\supset }B$, that by Definition A.6 (see Sect. A.5 in the appendix) counts as a derivation of the rule $A\Rightarrow B$ (from the assumption $A\mathbin {\supset }B$).

• 11. The idea that expansions express the no less aspect of harmony has been first explicitly formulated by Pfenning and Davies [58].

• 12. In actual and schematic derivations,  discharge is indicated with natural numbers placed above the discharged assumptions and in angle brackets to the left of the inference line at which the assumptions are discharged. Sometimes, u, v possibly with subscripts are used in place of numbers. As detailed in Appendix A, according to the “official” definition of derivations all assumptions (and not only those that are discharged) actually carry a numerical label. Which assumptions count as discharged thus depends only on the numerical labels in angle brackets to the left of inference rules. With very few exceptions (see e.g. Footnote 5 to Chap. 2), the labels above undischarged assumptions are irrelevant for the issues described in the present work, and hence they will be mostly omitted. In schematic derivations, a formula in square brackets indicates an arbitrary number ($\ge 0$) of occurrences of that formula, if the formula is in assumption position, or of the whole subderivation having the formula in brackets as conclusion. Square brackets are also used in rule schemata to indicate the form of the assumptions that can be discharged by rule applications.

• 13. That u is fresh for $\mathscr {D}$ means that the application of $\supset$I in the expanded derivation discharges no assumptions of the form A in $\mathscr {D}$.

• 14. That x is an eigenvariable means that x does not occur free in any assumption of the derivation of the minor premise A(Sx/x) other than those discharged by the rule.

• 15. For a general pattern to produce rules for inductively defined predicates covering the identity relation, the predicate ‘being a natural number’ and other more complex notions as special cases, see [42].

• 16. The choice of the notation is motivated by the Curry-Howard correspondence (between derivations in the implicational fragment of $\texttt{NI}$ and terms of the simply typed $\lambda$-calculus), under which the reduction and expansion for implication correspond (respectively) to steps of $\beta$-reduction and $\eta$-expansions on $\lambda$-terms:

  $$ (\lambda x.t) s {\mathop {\rightsquigarrow }\limits ^{\beta }} t[s/x] \qquad t {\mathop {\rightsquigarrow }\limits ^{\eta }} \lambda x.t x $$

• 17. An application of $\mathord {\supset }\beta$-reduction introduces new maximal formula occurrences whose degree is not lower than the one cut away only when: (i) the derivation of the minor premise of the relevant application of $\supset$E contains at least one maximal formula occurrence whose degree is not lower than the one of the maximal formula occurrence cut away by the application of $\mathord {\supset }\beta$; and (ii) the relevant application of $\supset$I discharges more than one assumption. Choose among the maximal formula occurrences in a derivation in $\texttt{NI}$ one of maximal degree which does not fulfill condition (i) above (such a formula occurrence can always be found). Let n be the degree of the chosen formula. By cutting away such a maximal formula occurrence with $\mathord {\supset }\beta$, the number of maximal formula occurrences of degree n necessarily decreases by one.

• 18. For a precise definition of the notion of undischarged assumption of a derivation, see Definition 4 in Appendix A.

• 19. The proof of strong normalization for ${\mathop {\triangleright }\limits ^{\eta }}$ in $\texttt{NI}^{\wedge \mathbin {\supset }}$ can be given by induction on the number of “local valleys”, since there are no complications analogous to those connected with $\mathbin {\supset }\beta$-reduction. The proof of confluence for ${\mathop {\triangleright }\limits ^{\eta }}$ is also immediate since it follows from the confluence of ${\mathop {\triangleright }\limits ^{1\eta }}$ (which is immediate) by a simple induction on the length of $\eta$-reduction sequences. In contrast, the proof of confluence for ${\mathop {\triangleright }\limits ^{\beta }}$ is more involved and it requires the introduction of a relation “between” ${\mathop {\triangleright }\limits ^{1\beta }}$ and ${\mathop {\triangleright }\limits ^{\beta }}$ which can be (almost) immediately shown to be confluent.

• 20. A general definition of what is here understood by a ‘calculus’, although restricted to purely propositional languages (without quantification) is given in the Appendix, see Sect. A.4. At a few points, calculi equipped with rules for first-, or second-order quantification will be mentioned, as in the present section, but no general characterization of them will be provided.

• 21. That X is an eigenvariable here means that X does not occur free in A or in any undischarged assumption on which A depends.

• 22. A precise formulation of what is here understood by ‘introduction rule’ and ‘elimination rule’ is given in the Appendix, see Definition A.12 in Sect. A.9.

• 23. Prawitz refers to derivations ending with an introduction rule as ‘canonical’ starting from [68]. The term ‘canonical’ is used in the same way in Martin-Löf’s constructive type theory, although it does not appear as late as [44] in Martin-Löf’s writings.   Dummett (see [12], pp. 260–261)  defines canonical derivations in a more stringent way, by requiring canonical derivations to consist (roughly said) of introduction rules with the exception of their open subderivations on which no restriction is placed. In particular, the closed derivations in $\texttt{NI}^{\wedge \mathbin {\supset }}$ that qualify as canonical in Dummett’s sense are the $\beta _w$-normal derivations to be discussed in Sect. 2.7 below. On canonical derivations in Dummett’s sense, see also Notes 6 and 14 to Chap. 2.

• 24. In Chap. 6 we will actually provide arguments to reject the assumption that for a derivation to be normal it must be redundancy-free. This is however irrelevant for the present point.

• 25.   The fact that uniqueness is a way of rendering the no less aspect of harmony may not be obvious at first, but see [94], pp. 1204–5. Observe moreover that Belnap’s aim is that of providing conditions that a collection of rules has to satisfy in order to be able to qualify as implicit definitions of a connective, rather than that of defining harmony.

• 26. For a contrasting opinion on the globality of uniqueness, however, see [52], p. 151.