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## Boundary RulesTo 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 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 (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 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 The true nature of the difficulty should be apparent, by now. The intuitionist reading of → as being “impredicative”: 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 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 It is easy to show that every inference that is valid in the revised sense is classically valid. Suppose the inference from premise 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.
In the usual model-theory of intuitionistic logic, say via Kripke trees, one obtains a model of It may be helpful to translate the situation back into Dummett's proof-theoretic language. Again suppose |

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