4.3 GE-Harmony
4.3.1 General Elimination Rules
In a series of articles, Read has developed a harmony-as-reduction approach which promises to deliver a series of improvements on the traditional accounts:7
• injective (one-to-one) functionality;
• induction of elim-rules from intro-rules;
• a more accurate analysis of harmony vs normalisation;
• a novel diagnosis of tonkitis.
The resulting notion, called generalised elimination harmony (GE-harmony) by Francez & Dyckhoff (2009), is based on some observations about natural deduc- tion rules first made by Schroeder-Heister (1984a).8 The paper provides schemata for intro- and elim-rules for any n-ary operator which functions as standard tem- plates. Its novelty lies in the power to let sets of intro-rules induce corresponding sets of elim-rules.9 GE-harmony is not only a better analysis of Gentzen’s orig- inal remarks, it is probably the most sophisticated means of understanding the challenge posed by tonk.
7Read (2000,2008b,2008a).
8The extension is originally from Schroeder-Heister (1981). See also Avron (1990) for a connection to sequent calculus.
9Schroeder-Heister also introduces the idea of assuming and dischargingrules (or subderiva- tions) as hypotheses. I am indebted to Allen Hazen for pointing out that this generalisation is anticipated in Fitch (1966). We return to this issue below.
Before we look at the approach more schematically, it is worth revisiting the inversion principle from Section 2.3.3. We are told that if an application of an elim-rule Eλ has a consequence C, then the intro-rules, Iλ, combined with the subderivations of minor premises in Eλ (if any), already yield a derivation of C without the application of Eλ. Inspecting the laundry list of connectives, we see that the moral expressed by Prawitz’s inversion principle is obscured by the fact that elim-rules come in different forms. Whereas conjunction offers the genuine inversion that Weir requested (see Section3.2.4), disjunction has an elim-rule that not only involves subderivations but which ultimately derives an arbitrary formula (which, of course, need not be a subformula of the major premise). If we take the difference seriously, we will notice that what E∨ says is that anything which can be derived from A ∨ B can be derived directly from the grounds for introducing A ∨ B, i.e., from A and B independently.
Thinking of this as a recipe we can give a corresponding rule for ∧:10
Example 4.2. GE∧: A∧B [A, B]u .. .. C C (GE∧)(u)
We then get a corresponding conversion step reminiscent of the ∨ case—the gen- eralised elimination rule for ∧:
Π1 A1 Π2 A2 A1∧A2 [A1, A2]1 Π3 C C (1) Π1 A1 Π2 A2 Π3 C
In the GE form, the ∧-rules follow the style of the standard ∨-rules. The re- duction is simply an explication of the point that anything which can be derived 10This generalisation also owes to the development of linear logic where we distinguish between
multiplicativeandadditiveconjunction (see Girard1987). We leave the details of this discussion to Section4.5.2below.
from the introduced formula can be derived from its grounds directly. In Read’s favoured terminology, we can say that the intro-rules not only expound the suffi- cient grounds for asserting a statement, they also implicitly characterise theneces- sary consequences of asserting the very same statement. If not, there would be no sense in which the elim-rules—which patently concern the necessary conditions— could be induced from the corresponding intro-rules (but see also footnote 41, Chapter 3).
What is the relationship between the GE-rule and the standard ∧rules? If we let C = A we see that the subderivation in GE∧ becomes a trivial derivation from
pA, Bq to A. Assuming something like reflexivity and weakening, i.e., that for any A, B:11
A, B .. .. A
we can disregard the subderivation, and the resulting rule is a standard E∧ rule for A; similarly for B. Conversely, we can derive the generalised rule from the standard rules by observing the following:
A∧B A A∧B B |{z}.. .. C
Here we need two applications of the standard E∧, followed by the subderivation of C from (A, B).12 The result is simply a permutation of the GE-rule.
We might then venture to give a general schema inspired by these examples. First, let us look at some schemata occurring in Francez & Dyckhoff (2009). The
11Compare the axiom
pΓ,A⇒A,∆q in theG3-style sequent calculus. In natural deduction, however, the structural rules are absorbed asdischarge policies. This is a topic that will loom large in what follows (see especially Section5.3.5).
12Again, the derivation tacitly assumes that something likecontraction is possible (more pre- cisely, vacuous discharge).
schemata are slightly modified from their originals: We ignore complications aris- ing from quantifiers, and we use notation which is different in inessential ways. Keep in mind, however, that in Section 4.4 we will discuss some grievances with the particular formulation. We will then proceed to develop an improved version of the GE-template which differs from both Francez & Dyckhoff (2009) and Read (2008a).
The schematic intro-rule for a logical constant λ occuring as principal connective in a formulaϕ is as follows: [Σi]j1,...,jmi ˆ Πi ∆i ϕ (δIj1,...,jm)i
Let us explain the notation: The subscript i (on Σ) indicates that that this is the ith intro-rule for the operator λ in ϕ, where i ∈ {1, ..., n}. [Σi]j1,...,jmi are
(possibly empty) sets of assumptions discharged byδI, and∆i are sets of formulae
(possibly empty).
As an example, take I∧ (Example 4.2): There is only one intro-rule (so i = 1), and Σ is empty as the rule infers A ∧ B as ϕ immediately from ∆ = {A, B}. More interestingly, I→ has Σ = {A}, and discharges the formula after deriving
∆ = B and concluding ϕ = A → B. (A brief warning: As we will see when we later develop GE-harmony for multiple conclusion, the current notation adopted from Francez & Dyckhoff (2009), where∆ as the premises of the intro-rule, is not convenient.)
The suggestion, then, is to (harmoniously) induce the following GE-rule:
Π ϕ Π1 Σ1 ... Πn Σn [∆1]1 Π01 ψ ... [∆n]n Π0n ψ ψ (δGE)(1,...,n)
Call this the GE-induced rule. Notation: ψ is an arbitrary formula. ϕ is the major premise, Σi are (sets of) minor premises corresponding to the assumptions
of intro-rules, ∆i are (sets of) assumptions for subderivations corresponding to
the grounds in intro-rules. Each of the assumption sets ∆ are discharged upon concluding ψ.13 Examples: For our GE-rule for ∧ it is evident that there is only
one set of grounds, ∆1 = {A, B} (but no minor premises Σ). Upon deriving C
from the set, one discharges the assumption set and concludes C from the major premiseϕ =A ∧ B.
What about GE→? Given the description of I→ above we can plot out the induced GE-rule by transferring the content of the intro-rule to the GE-schema. The resulting rule is as follows:
Example 4.3. GE→: A→B A [B]u .. .. C C (GE→)(u)
This rule appears to have been first formulated for natural deduction in Dyckhoff (1987). It is later used for developing an intuitionistic relevant logic in Tennant (1992, 2002), and also for translations between natural deduction and sequent calculus in von Plato (2001).14 As before, the GE-rule is equivalent to the standard
E→, i.e., modus ponendo ponens, given some structural assumptions. To see that it is a special case of the GE-rule, simply let C = B. For the other direction we have the permutation:
A→B A
B.. .. C
13Note that the subscripts tracking the intro-rules and the superscripts tracking the discharges are identical. There is no confusion: We are simply using the same ordering on the subderivations corresponding to intro-rules (left-to-right) as we are on the different discharges.
14A similar typed rule also appeared earlier in Martin-Löf (1984, p. 44). Note that Tennant
However, this is not the rule suggested by Schroeder-Heister, nor is it the rule adopted by Read in his account of GE-harmony (which differ from Francez & Dy- ckhoff2009). Rather, in the case of theGE→rule—and presumably all intro-rules involving subderivations—they employ the idea of the higher-order rule, i.e., an inference rule where subderivations themselves can be assumed and discharged. With higher-order rules, one can schematise GE-rules in a more straightforward manner, albeit with the cost of having variables that range over both formulae and subderivations. Read offers the following template where intro-rules are un- derstood as giving groundsΣi which are ambiguous between direct grounds (as in
I∧) and subderivations involving discharge (as inI→):
ϕ [Σ1] .. .. ψ ... [Σn] .. .. ψ ψ
The critical difference that leads to an output which deviates from the former template is that Σi might stand proxy for a subderivation, written χ ⇒ ξ. As a
consequence, the schema returns the following GE-rule for →:15
Example 4.4. Higher-order rule for →:
A →B [A⇒B]u .. .. C C (GE→ 0 )(u)
We call this type of rules Schroeder-Heister-style rules to distinguish them from the Dyckhoff-style rules. Strictly speaking, the rule is different from the first rule we gave for →. GE→ says that whenever we can derive C from the assumption B, we can derive the conclusion C from the major premiseA → B together with a minor premise A, and subsequently discharge the assumptionB. In contrast, its 15Although the idea of assuming and discharging rules might be philosophically problematic, it has some advantages. Apart from the role it plays in the harmony debate, there is also some evidence that certain connectives can only be axiomatised with such a device. See Hazen (1996) for the example of ((A→B)→B) in intuitionistic logic.
higher-order counterpart says that whenever we can derive C on the assumption that B can be derived from the assumption of A, we can derive the conclusion C from the major premise A → B, and subsequently discharge the assumption A
⇒ B. Yet, even if the rules are not the same—there certainly may be structural environments in which they come apart—they are closely related. Especially, note that GE→0 does not discharge A, it discharges the derivation from A to B. Of course, if it did, the rule would result in a simplified rule much stronger than
modus ponendo ponens.
It hardly needs mentioning that giving all-encompassing schemata for intro- and -elim rules is a tall order. Nonetheless, there is a lot to be said for trying. First, it might assist not only our understanding of harmony, but also the understanding of reduction and normalisation (more on this to follow in Section4.3.3). Second, it is a critical part of givingadequacy (or completeness) results for intuitionistic logic in PTS. Early work was done by both by Prawitz (1978), Zucker & Tragesser (1978), and Schroeder-Heister (1984a). They all rely on developing templates for intro- and elim-rules, and consequently face, either explicitly or implicitly, the difficult question of what is to count as an intro- and elim-rule respectively. Typically, rules like those presented by Milne (see footnote53) do no fit the mold if interpreted as intro-rules. This should not be taken as an argument against such deviant rules, however, but rather as an incentive to improve on the templates. Although, we do not consider rules of Milne’s specific form, we will attempt to achieve some level of generality in what follows. But first we turn to one of the advantages with the first formulation of GE-harmony.