logicandlanguage.net

One thing that emerged in the comments here and here, it was that it is easy to get confused about whether Dummett's concept of harmony is identical with the satisfaction of local reduction. The idea of local reduction is reasonably straight forward: a logical constant has the local reduction property just in case, whenever its introduction rule is used to derive some formula A, and then its elimination rule is immediately used, taking A as the major premise, then the proof can be rewritten without that pair of steps. (See this post for more detail, with the caveat that the post ignores the complicated literature on the subject of local reduction with respect to negation.)

Harmony and Meaning
Harmony is slippery because it is something which Dummett has an intuitive grip on and is trying to make more precise. The idea is introduced through talk of meaning. Dummett thinks that verificationists and pragmatists have been investigating different parts of the meaning-elephant: verificationists identify the meaning of a statement with what is needed to establish its truth (verify it), pragmatists identify the meaning of a sentence with its consequences, but Dummett thinks neither sufficient:

Someone would not be said to understand the phrase 'valid argument', for instance, if he knew only how to establish (in a large range of cases) that an argument was valid but had no idea that, by accepting an argument as valid, he has committed himself to accepting the conclusion if he accepts the premises. The analogue holds good for a great many expressions...(213)

Rather, the meaning of an expression encompasses both principles governing how to verify it, and principles governing what follows from it. He wonders if these rules might somehow be in tension. Could they be inconsistent? Or might it be that one rule could be too lax or too restrictive given the other? Such a situation would be a failure of harmony. Since it is difficult to isolate the principles governing a particular expression in a natural language, Dumment takes the logical constants as a case study, since their verification and consequence principles are clearly established in their introduction and elimination rules.

Harmony and Logic
Dummett has other reasons for being interested in the logical constants as well. His interest in harmony stems from his interest in rejecting or accepting change in logic. He thinks that fear of lack of harmony makes us wary of changing logics. He gives two examples, first quantum logic's proposal to weaken the classical [VE] rule, so that this classically valid argument form becomes invalid:

And second the (fictional) proposal to strengthen counterfactual logics (John P. Burgess used to call these and modal logics "neo-classical logics") so that the following argument form is valid:

Wittgensteinian Conventionalisism
One possible view about such new proposals in logic - which Dummett attributes to Wittgenstein - is that any combination of introduction- and elimination- rules defines a legitimate logical constant. Some of these constants are not in use in any natural or formal language, but that is merely a matter of convention. In principle we could even adopt connectives like 'tonk' that allowed us to derive contradictions. (I suppose this might be the explanation of the Wittgenstein quotation at the beginning of Graham Priest's chapter on Paraconsistent Logic in the second edition of the Handbook of Philosophical Logic: "Indeed, even at this stage, I predict a time when there will be mathematical investigations of calculi containing contradictions, and people will actually be proud of having emancipated themselves from 'consistency.'")

Dummett is looking to resist the Wittgensteinian conventionalist view by finding a way to criticise some combinations of rules. The concept of harmony appears to promise a foundation for this criticism. It looks as if what is wrong with 'tonk' is that the introduction and elimination rules are not in harmony; we can derive far too much from a claim of the form 'A tonk B', given what was required to establish it.

Harmony and two notions of Conservative Extension
Dummett suggests that the notion of a conservative extension can help us to make the notion of harmony more precise. Conservative extension is normally defined over theories, and a theory T2 is a conservative extension of T1 if i) it can be obtained from T1 by adding new expressions, along with axioms (or, according to Dummett) rules of inference which govern those expressions and ii) "if we can prove in it no statement expressed in the original restricted vocabulary that we could not already prove in the original theory." (218)

(Just a thought, but it seems to me that the ideas of language, theory and proof system are all mixed up together here, a la Tarski and Carnap.)

Dummett then turns to natural languages, wondering whether there could be any disharmony in English, and says "if there is disharmony [between the rules governing an expression E,] it must manifest itself in consequences not themselves involving the expression E, but taken by us to follow from the acceptance of a statement S containing E."

And then: "A conservative extension in the logicians' sense is conservative with respect to formal provability. In adapting the concept to natural language, we must take conservatism or non-conservatism as relative to whatever means exist in the language for justifying an assertion or an action consequent upon the acceptance of an assertion. The concept thus adapted offers at least a provisional method of saying more precisely what we understand by 'harmony': namely that there is harmony between the two aspects of the use of any given expression if the language as a whole is, in this adapted sense, a conservative extension of what remains of the language when that expression is subtracted from it."(218-9)

I think this is wrong. It seems to me that it confuses lack of harmony (as he characterised it before: a lack of fit between introduction and elimination rules) with a particular KIND of lack of harmony. If the elimination rule of an expression licences too much, given what was needed to introduce the expression, then yes, we will be able to prove sentences not containing that expression which we were not able to prove before, and we will have a non-conservative extension. But if the elimination rule is too restrictive given the introduction rule, then it doesn't seem that we will be able to prove any more than before. Suppose, for an example, we start out with a classical propositional logic containing only '→' and '¬' and we add 'V', governed by the usual introduction rule, but taking as an elimination rule the weak, quantum version given in the picture above. Surely the result is a conservative extension of the original, on Dummett's definition. And if we had added the usual classical 'V' it would still be a conservative extension - all three systems are just classical proof systems, some of which are easier to use than others. Yet how could the introduction rule for 'V' be harmonious with two non-equivalent elimination rules?

Local Reducation
In chapter 11 of the same book, entitled "Proof Theoretic Justifications of the Logical Laws," we get another provisional definition of harmony, this time just for the logical constants, and it is here that local reducation/normalisation comes in:

"The analogue, within the restricted domain of logic, for an arbitrary logical constant c, is that it should not be possible, by first applying one of the introduction rules for c, and then immediately drawing a consequence from the conclusion of that introduction rule by means of an elimination rule of which it is the major premiss, to derive from the premisses of the introduction rule a consequence that we could not otherwise have drawn."

(I couldn't find an argument for this identification.)

Dummett has just claimed that susceptibility to local reduction (aka normalisation) (within some system) is the formal analogue of being a conservative extension of a theory (once that idea is adapted for natural language.) But since the notion of a conservative extension was originally a formal notion anyway, it looks as if he thinks that a logical constant will be susceptible to local reduction with respect to some proof system just in case adding it to that proof system results in a conservative extension. And, as noted earlier, he thinks that the rules governing a connective are in harmony with respect to a proof system just in case adding that connective to a proof system results in a conservative extension. So it seems that, for logical constants anyway, Dummett identifies all three of the notions we have touched on. (I'm thinking he is a lumper, rather than a splitter.)

Two Questions
There are two questions that I would be very interested in the answers to, if anyone out there knows them:

1) I have argued that Dummett is mistaken in identifying idea that the rules for a logical constant are in harmony (with respect to some proof system), with the idea that adding the constant to a proof system will result in a conservative extension. If the elimination rule allows too little, rather than too much, to be derived (given the introduction rule) then the addition will still result in a conservative extension, even though the rules are not in harmony. Am I right, or I am hopelessly confused on this and being unfair to Dummett?

2) Dummett says that adding a constant to a proof system will result in a conservative extension of that system if, and only if, all the "local peaks", (or non-normal pairs) can be removed - that is, if, and only if, the new connective has the local reduction property with respect to that proof system. I could not find the argument for this in Dummett, but he might be right anyway. Is he? What IS the relationship between local reduction (a.k.a. normalisation) and conservative extension?

Two comments:

1. Please read my doctoral dissertation! At least the first two chapters, where I critique some issues common to the Dummett & Martin-Loef account of harmony.

2. I agree entirely with what you say about lack of kinds of harmony vis-a-vis conservative extension: more precisely conservative extension is only assocaited with one side of the dual pair of harmonies Belnap talked about, that of what I call synthetic harmony: when this fails but we have the more fundamental analytic harmony, then we may be talking about an extensible concept in the sense that natural numbers give rise to arithmetic extensibility.

I must thank you (and all the other logic bloggers) for all this discussion of local reduction recently! I've got my qualifying exam in a week and a half, and the topic I'm least prepared on is about Dummett and anti-realism, but in my meeting this afternoon, once the topics of truth values and deduction rules came up, I was on top of that!

I agree entirely with what you say about lack of kinds of harmony vis-a-vis conservative extension: more precisely conservative extension is only assocaited with one side of the dual pair of harmonies Belnap talked about, that of what I call synthetic harmony: when this fails but we have the more fundamental analytic harmony, then we may be talking about an extensible concept in the sense that natural numbers give rise to arithmetic extensibility.