General-Elimination Stability

Abstract

Abstract.General-elimination harmony articulates Gentzen's idea that the eliminationrules are justified if they infer from an assertion no more than can already be inferred from the grounds for making it. Dummett described the rules as not only harmonious but stable if the E-rules allow one to infer no more and no less than the I-rules justify. Pfenning and Davies call the rules locally complete if the E-rules are strong enough to allow one to infer the original judgement. A method is given of generating harmonious general-elimination rules from a collection of I-rules. We show that the general-elimination rules satisfy Pfenning and Davies' test for local completeness, but question whether that is enough to show that they are stable. Alternative conditions for stability are considered, including equivalence between the introduction-and elimination-meanings of a connective, and recovery of the grounds for assertion, finally generalizing the notion of local completeness to capture Dummett's notion of stability satisfactorily. We show that the general-elimination rules meet the last of these conditions, and so are indeed not only harmonious but also stable.

On a Generality Condition in Proof‐Theoretic Semantics

“…From this perspective not only is qor a problematic connective, but the AAC is shown to be connected with the smooth composition of derivations, 18 See also Prawitz (1965) for a definition of maximal segment and von Plato (2001) for a discussion of the problem. 19 A similar but slightly more general diagnosis of qor can be found in Jacinto and Read (2017). which is to say with the unrestricted applicability of Cut. In this case we can avail ourselves of a very restricted form of Cut, i.e.,…”

Translations Between Linear and Tree Natural Deduction Systems for Relevant Logics

“…25 I have presented the (•E) rule that I did, rather than a general elimination-style rule to keep the correspondence with cascade systems. See Read (2010Read ( , 2014 or Jacinto & Read (2017) for more on general elimination-style rules. 26 See, for example, Prawitz (1965), Raggio (1965), Medeiros (2006), andFrancez (2014), among others.…”

“…For more on the trade offs involved in cascade and tree systems, seeRestall (2014) andHazen & Pelletier (2014).25 I have presented the (•E) rule that I did, rather than a general elimination-style rule to keep the correspondence with cascade systems. SeeRead (2010Read ( , 2014 orJacinto & Read (2017) for more on general elimination-style rules. 26 See, for example,Prawitz (1965),Raggio (1965),Medeiros (2006), andFrancez (2014), among others.27 SeeBrady (2006Brady ( , 2010 for detailed examples.…”

Stabilizing Quantum Disjunction

“…The need of generalizing the usual formulation of the no less aspect of harmony in order to properly capture stabilty has been recently pointed out by Jacinto and Read [7], who refer to the original formulation of the no less aspect of harmony as local completeness and propose to replace it in favour of what they call generalized local completeness.…”

“…Using Jacinto and Read's receipt, one can cook up the following derivation: (7,8) E E 4 (9,10) E However, also in this case there is no principled reason to claim that a derivation of the form:…”