To see how the trouble arises in terms of canonical arguments, rather than the relation f-, it is helpful to consider the case of the inference from A → B to B, where A and B are distinct atomic sentences. If there were to be a valid canonical argument for A → B, it would have to enable us to transform any valid canonical argument for A into one for B. Since B is atomic, the only valid canonical argument for B is the one-step argument of taking B as a premise. Hence a valid canonical argument for A → B must have B as an (undischarged) premise; and so it will be transformable into a valid canonical argument for B. The problem, in short, is that there is no way of getting from A to B, except by taking B as premise.

Here, it might be thought, is where Dummett's boundary rules can play a role, since boundary rules license inferences from atomic formulas to atomic formulas. However, three considerations—one technical and two philosophical—show that the problems in the method cannot be avoided by boundary rules as Dummett envisages them.

First, if the counterexamples are to be avoided, there are going to have to be an inordinate number of boundary rules. To forestall the validity of the inference from A → B to B, there must be a rule allowing the inference of B from A (and possibly other premises not including B) for any pair ( A, B) of distinct atomic sentences. To forestall the validity of the inference from no premise to ¬¬ A, there must be a rule allowing the inference of ⊥ from A (again, possibly with other premises). Rules that avoid some anomalies may engender others. For example, if ⊥ can be inferred by boundary rules from premises A and B, and from premises A and C, but not from A and any other premises, then although the inference from no premise to ¬¬ A is no longer valid, the inference from ¬ A to B ∨ C is. It appears, then, that it is unreasonable to expect that boundary rules will avoid the difficulty.

(By the way, it is not clear that a rule allowing the inference of ⊥ from atomic premises should count as a boundary rule at all. Dummett characterizes boundary rules as “rules governing . . . non-logical expressions.” Allowing ⊥ as a conclusion violates this description. After all, a rule allowing the inference of ⊥ from premises A and B is just a rule allowing the inference of ¬B from A, and of ¬ A from B. This significantly weakens the claim that ⊥ is given meaning only by its introduction rule; indeed, it seems to me to weaken the contrast Dummett makes between intuitionistic negation and classical, saying of the latter “there is no way of attaining an understanding of the classical negation operator if one does not have it already” [3,p. 299] Nonetheless, if we are to block the anomalies given by Counterexample 1, we must allow boundary rules with conclusion ⊥.)

Alongside the technical difficulties there are philosophical ones. To use boundary rules in the manner envisioned makes the validity of inferences dependent on which boundary rules there are, and hence, in particular, on empirical claims about the connections of different empirical basic sentences. This is not consistent with the claim that the validity of the logical inferences comes only from the meaning of the logical connectives (as based on the introduction rules).

Finally, even if the latter difficulty is set aside, there is another disturbing consequence, namely, that it becomes impossible to put forth a link between atomic sentences as a supposition, and draw consequences from it. For either the link is taken as a boundary rule, and hence becomes part of the logical framework, usable in any argument anywhere and playing a role in the criterion of validity; or else there is no link, in which case having A → B as a supposition yields B as a valid conclusion, and therefore we can infer from the conditional everything that is yielded by its consequent alone. The irony here is that we have landed in a position akin to Frege's odd-sounding view that “Only true thoughts can be premises of inferences.” [5, p. 335]3

The true nature of the difficulty should be apparent, by now. The intuitionist reading of F → G is, roughly, “from any demonstration of F we can obtain a demonstration of G.” In Brouwer and the early intuitionistic tradition, the notion of demonstration here is taken to be open-ended, identified not with any particular formal system, indeed, not with the entirety of means of demonstration we currently have at our disposal, but as anything that we might come to accept as a demonstration. In later studies, particularly those inspired by Kreisel's work of the 1950s, the generality in talking of “any demonstration” is expressed by speaking of the intuitionist

→ as being “impredicative”: F → G implicitly quantifies over all demonstrations, including those that may contain the very demonstration of F → G. Dummett, in contrast, wants to read “any demonstration” here as meaning “any valid canonical argument”, where this notion is defined in an inductive and hence purely predicative way. It is this restriction that gives rise to the difficulties above, both in the case without boundary rules, and the peculiarities of trying to use a fixed set of boundary rules to block those difficulties.

It is I think far more natural to use the notion of boundary rule in a way not envisaged by Dummett, and in fact inconsistent with Dummett's aim. The definition of “valid” can be revised so that what counts as a valid inference is one that was valid in the old sense given any assumption of boundary rules.4 This revision avoids both of the philosophical difficulties just canvassed. It does not restrict allowable arguments to a fixed set of accepted ones, but rather allows any collection of possible arguments from atomic sentences to atomic sentences. Since all sets of boundary rules are considered, there is no need for empirical input to determine which boundary rules should be adopted.

Technically, the consideration of all sets of boundary rules amounts to the consideration of different model-theoretic structures. There are two equivalent ways of formulating this. Given a set S of boundary rules, the relation f--relative-to-S, or f-S as we shall write it, can be defined by appropriate changes in clauses (4) and (5), keeping clauses (1)–(3) as is. Alternatively, (1)–(5) can be kept as is, and the domain of sets altered to contain all and only sets α that are closed under all the boundary rules in S and do not contain ⊥. For our purposes, the latter procedure is more convenient. For any set α of atomic sentences, let clS (α) be the closure of α under the rules in S, that is, the smallest set β such that α ⊆ β and if S contains a rule “infer B from A1,..., An ” and A1,..., An are in β, then B is in β.

It is easy to show that every inference that is valid in the revised sense is classically valid. Suppose the inference from premise F to conclusion G is valid in the revised sense. Let T be a (classical) truth-assignment to the atomic sentences in F and G under which F comes out true; we must show that G also comes out true under T . Let S be the set of boundary rules containing “from no premise infer A” for every atomic sentence A to which T assigns truth, and “from A infer ⊥” for every other atomic sentence A. Obviously, there is only one set α that is closed under S and does not contain ⊥, namely, the set of atomic sentences assigned truth by T . But then f-S behaves classically on the connectives, so that α f-S F . Since the inference is valid in the revised sense, α f-S G. Hence G is true under T .

From this we can surmise that there will be no counterexamples of the alarming sort encountered above. However, validity in the revised sense still does not coincide with intuitionistic validity.

Counterexample 3Let A be an atomic sentence, and G and H any sentences. Then the inference from premise A → (G ∨ H ) to conclusion ( A → G) ∨ ( A → H ) is valid in the revised sense.

Proof Let S be a set of boundary rules, and suppose α is an S-closed set not containing ⊥ such that α f-S A → (G ∨ H ). Let β be the S-closure of α ∪ { A}. If ⊥ ∈ β, then α f-S A → F for every F , so α f-S ( A → G) ∨ ( A → H ); hence we may suppose⊥ ∈/ β. If β f-S G, then α f-S A → G, for if γ is any S-closed extension of α with γ f-S A then β ⊆ γ , so that γ f-S G; similarly if β f-S H then α f-S A → H ; in either case α f-S ( A → G) ∨ ( A → H ). But if neither, then β is an S-closed extension of α such that β f-S A while not β f-S G ∨ H , which contradicts the hypothesis that α f-S A → (G ∨ H ).

In the usual model-theory of intuitionistic logic, say via Kripke trees, one obtains a model of A → (G ∨ H ) that is not a model of ( A → G) ∨ ( A → H ) by having two nodes v1 and v2, one of which models A and G but not H , the other models A and H but not G. For this it is essential that there be no u with u ≤ v1 and u ≤ v2 that models A; for if the root of the tree is to model A → (G ∨ H ) any such u would have to model G ∨ H , and thus have to model G or model H , but every node above u would also model G or every node would also model H , thus defeating the example. The problem is that, using fand boundary rules, these strictures cannot be met. For example, suppose G and H are also atomic. Using boundary rules one can insure that there is a closed set containing A and G and a distinct one containing A and H , but then there will also be a closed set containing A that is a subset of each of those, and in order to insure that A → (G ∨ H ) holds, that subset will have to contain either G or H .

It may be helpful to translate the situation back into Dummett's proof-theoretic language. Again suppose G and H , as well as A, are atomic. The counterexample shows that any valid canonical argument for A → (G ∨ H ) can be transformed into one for ( A → G) ∨ ( A → H ). Suppose, then, there is a valid canonical argument from premises α to conclusion A → (G ∨ H ). This is just to say that the inference from α and A to G ∨ H is valid, which in turn means that every valid canonical argument for α and A can be transformed into one for G ∨ H . Since there is a valid canonical argument from α and A to α and A, there must be one from α and A to G ∨ H . The last step of this must be an application of ∨-introduction. Hence there is either a valid canonical argument from α and A to G or one from α and A to H , and so there is a valid canonical argument from α to ( A → G) ∨ ( A → H ). The idea is that there is only one way to demonstrate A, so to speak.

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