Logical harmony

The principle of harmony (English principle of harmony ) is a metalogical principle, which calls for calculi of natural deduction that the introduction and elimination rules should be located for logical operators in "harmony", that is, roughly speaking, that the elimination rule given for a operator does not allow the transition to logically stronger statements than is covered by the introduction rule. The term was coined by Michael Dummett; the idea that there must be a principle of harmony, however, was already anticipated by Gerhard Gentzen.

Background: Prior challenge for the proof theoretic semantics

Gentzen in 1934 suggested that the logical operators to characterize semantically by a pair of introduction and elimination rule is specified for each operator. The introduction rule specifies the conditions under which a statement may be inferred that contains the operator concerned, as the main operator; the elimination rule specifies what can be inferred from such a statement. For example, see the rules for the conjunction in a calculus of natural deduction as follows:

Gentzen and, following him, the representatives of the proof-theoretic semantics assume that the meaning of the logical operators is given by these rules. Arthur Prior has pointed out in his essay "The runabout inference ticket" to a potential problem of this assumption: What prevents us from a connective " tonk " introduce its introduction rule corresponds to the Adjunktionseinführung ( from A follows A tonk B ), whose elimination rule but the Konjunktionsbeseitigung corresponds ( from A tonk B follows A)? This would allow the derivation of any statement of any assumptions and lead to a trivialization of logic:

At first glance this is a very strong argument against the idea that the rules of use determine the meaning of an expression. JT Stevenson has drawn in his contribution to the triggered by Prior debate the doctrine, the meaning of the logical operators must be fixed from before inference rules are given, and the inference rules would have on this in advance (eg, by truth tables ) defined meaning to orient.

The harmony principle as a solution

Gentzen himself had already seen the need for harmony between introduction and elimination rule:

The introductions represent, so to speak, the "Definitions" of the character in question is, and the fixes are ultimately only consequences thereof, which can be expressed like this: When removing a mark shall the formula in question to their extreme character this is, only "as the be used, what it means due to the introduction of this character ." [ ... ] By specifying these thoughts, it may be possible that B [ eseitigungs ] conclusions demonstrate, on the basis of certain requirements as unique functions of the corresponding E [ inführungs ] conclusions.

The indicated here by Gentzen approach boils down to consider the introduction of rules to be semantically primary and derive the elimination rules from them; a way to do this is the inversion principle (see below ) to the hand. Belnap and Dummett hit it as a clarification of harmony claim before the definition, theoretical demand for conservatism or non- creativity.

Not creativity

Nuel Belnap argued in his reply to Prior, that notwithstanding the fact that rules determine the meaning of an expression, not any (pairs of ) rules are adapted. Introduction and elimination rule must be harmonious, and the Belnap explicated as follows: The rules for a new operator may not allow the derivation of statements that do not contain the newly introduced term and were not previously found. The operator tonk violated this requirement in more obvious ways.

The idea that a harmony between the Truth and Behauptbarkeitsbedingungen a statement and its consequences must exist in this sense, was extended by Michael Dummett on non- logical vocabulary:

A simple case would be a pejorative term did of, eg ' Boche '. The condition for applying the term to someone is did he is of German nationality; The Consequences of its application are did he is barbarous and more prone to cruelty than other Europeans. [ ... ] The addition of the term ' Boche ' to a language Which did not added anonymously containment it would produce a non- conservative extension, ie, one in Which Certain other statements Which did not contain, the term were inferable from other statements not containing Which it were not inferable added anonymously.

The inference sequence " Fritz is German; So Fritz is a Boche; So Fritz is cruel "shows that the introduction of the term ' Boche ' is a non-conservative language extension: The statement" Fritz is cruel " would be complete without the intermediate step of " Fritz is a Boche " not from the statement," Fritz is German " derivable. However, it is questionable whether Nichtkonservativität is a problem in nichtlogischem vocabulary. The demand for non- creativity for every newly introduced term would amount to a ban on real language extensions, which is not plausible for natural languages.

Inversion principle

Dag Prawitz has proposed following works of Paul Lorenzen to explicate harmony in the recourse to the so-called inversion principle: There must exist a symmetry between introduction and elimination rules in such a way that the content of a conclusion that attained by application of an elimination rule to a premise P is not allowed to go beyond what was already contained in the premises from which P was inferred by means of the corresponding introduction rule. If, for example, A ∧ BA is concluded, so that's why admissible because A ∧ B (typically ) is obtained by applying the Konjunktionseinführung on the premises A and B. The evidence for a winning means eliminating rule conclusion is therefore already in the proof of their premises included when the main premise was concluded by the corresponding introduction rule. The term inversion principle, to clarify that an application of an elimination rule makes certain extent only undo what has been done by the corresponding introduction rule.

Harmony in terms of logical strength

Neil Tennant has proposed the following criterion harmony in Natural Logic:

Introduction and elimination rules for a logical operator λ must be Formulated as did a sentence with λ dominant Express train the strongest proposition Which can be inferred from the stated premises When the conditions for λ -introduction are satisfied; while it express train the weakest proposition possible under the conditions Described by λ - elimination.

How to behave at the various criteria for harmony with each other ( ie whether they are for example, successive recyclable ) is unknown as yet.