Dag Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307. North-Holland Publishing Co., 1971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the rules for case by the rules case(inl(P ) M;N) MP; case(inr(P ) M;N) NP; case(5A B (P ) M;N) 5C (P ) A fundamental result about natural deduction is the fact that every proof (term) reduces to a normal form, which is unique up to renaming. This result was rst proved by Prawitz [24] for i . speci ed in Definition 3.3) is con uent. Equivalently, conversion in is Church Rosser. parallel reduction (see also Barendregt [2] Hindley and Seldin [15] or Stenlund [27] 13 (as in Definition 3.3) is strongly normalizing. 1971) 11] 1972) see also Gallier [7] ....

....P ) M) M )P; or casex inx( P ) of inx(t: x: A) N N [ t; P=x] casex(5 9tA (P ) M) 5C (P ) A fundamental result about natural deduction is the fact that every proof (term) reduces to a normal form, which is unique up to renaming. This result was rst proved by Prawitz [24] for i . speci ed in De nition 7.3) is con uent. Equivalently, conversion in is Church Rosser. parallel reduction (see also Barendregt [2] Hindley and Seldin [15] or Stenlund [27] strongly normalizing. 1971) 11] 1972) see also Gallier [7] If one looks carefully at the ....

D. Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proc. 2nd Scand. Log. Symp., pages 235-307. North-Holland, 1971.

.... have been established quite early by Turing (around 1941, published in [Gan80] and Sanchis [San67] This article employs a proof method that allows to show strong normalization for all typed terms without recourse to inclusive predicates such as strong computability [Tai67] or validity [Pra71] that are not formalizable in primitive recursive arithmetic. A simple induction on types veri es that (strong) normalizability is closed under application while closure under substitution has to be shown simultaneously. Separating concerns. With the prospect of computer checkable formalization ....

.... written Rec s t r) case analysis rR with R (x:s; y:t) commonly denoted by case r of inj 0 x ) s, of inj 1 y ) t) and generalized application rR with R (s; x:t) for r y written as t x fr; sg in [Sch99] The vector notation turns out to replace concepts like branches and endsegments [Pra71] which have been important elements of normalization arguments in systems with permutative conversions. Simply typed calculus. The basic system is dealt with in the rst four sections: In section 1 untyped calculus, function types, reduction and weak and strong normalizability are de ned. In ....

[Article contains additional citation context not shown here]

Dag Prawitz. Ideas and results in proof theory. In Jens E. Fenstad, editor, Proceedings of the Second Scandianvian Logic Symposium, pages 235-307. North{Holland, Amsterdam, 1971.

....a proof # from the relativized premises: # For then we get a proof of ZF # # , which is just ZF # #. So how we obtain # from # To be concrete, suppose we are working with a natural deduction formalization of first order logic. By the normal form theorem [17], since the conclusion of the proof is atomic, we can assume that # applies only elimination rules. We must modify # so that it accepts relativized versions of its premises and delivers a relativized version of its conclusion. The only hard cases involve quantifiers. Where # applies the ....

Dag Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Second Scandinavian Logic Symposium, pages 235--308. North-Holland, 1971.

....over P(WFF WFF . Thus the expression # A is a sequent where # (the antecedent) is a finite (possibly empty) set of formulas and A is a formula. As usual in Gentzen systems the meaning of operators and connectives is given by the rules for their introduction and elimination (cf. e.g. [17]) More precisely, this is true in the presence of the structural rules of exchange, duplication and contraction. Otherwise, the introduction and elimination rules have to be supplemented by rules for the structural meaning of the operators involved [8, 18] According to Definition 1 the usual ....

D. Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307, Amsterdam, 1971. North Holland.

.... x: x:fx ) f Similar problems arise if we try to enrich the calculus with extra rewrite rules which may be confluent by themselves, but which when taken in conjunction with j contraction fail to be confluent [16] Recently several researchers [2,15,20,19,22, 49] have adopted older proposals [41,62,68] that j conversion be interpreted as an expansion: t ) x:tx if t : A B and the resulting rewrite relation has been shown confluent. In these works infinite reduction sequences such as: f ) x:fx ) x: y:fy)x ) 1.2) are prohibited by imposing syntactic restrictions to limit the ....

....may be calculated by first contracting all fi redexes and then performing any remaining j expansions or, vice versa, by performing all j expansions and then contracting any remaining fi redexes. Historically the use of j expansions, as opposed to j contractions, can be traced back to Prawitz [68] and Huet [41] The formulation of the restrictions on expansion required to recover strong normalisation were originally proposed by Mints [62] although it is only recently that several researchers [2,15,19,22,49] using different proof strategies, have proved confluence and strong ....

[Article contains additional citation context not shown here]

D. Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proc. 2nd Scandinavian Logic Symposium, pages 235--307. North Holland, 1971.

.... have been established quite early by Turing (around 1941, published in [Gan80] and Sanchis [San67] This paper employs a proof method that allows to show strong normalization for all typed terms without recourse to inclusive predicates such as strong computability [Tai67] or validity [Pra71] that are not formalizable in primitive recursive arithmetic. A simple induction on types veri es that (strong) normalizability is closed under application while closure under substitution has to be shown simultaneously. Separating concerns. With the prospect of computer checkable formalization ....

.... written Rec s t r) case analysis rR with R (x:s; y:t) commonly denoted by case r of inj 0 x : s, of inj 1 y : t) and generalized application rR with R (s; x:t) for r y written as t x fr; sg in [Sch99] The vector notation turns out to replace concepts like branches and endsegments [Pra71] which have been important elements of normalization arguments in systems with permutative conversions. Simply typed calculus. The basic system is dealt with in the rst four sections: In section 1 untyped calculus, function types, reduction and weak and strong normalizability are de ned. In ....

[Article contains additional citation context not shown here]

Dag Prawitz. Ideas and results in proof theory. In Jens E. Fenstad, editor, Proceedings of the Second Scandianvian Logic Symposium, pages 235-307. North{Holland, Amsterdam, 1971.

....induction on the structure of the given derivations, using extensive case analysis. From right to left the theorem follows by case analysis on local or commuting reductions. 24 The calculus further satisfies a normalization theorem. This can be proven either directly via Tait s method as in [Pra71], by a detour via cut elimination as in [Pfe99] via CPS translation as in [dG99] or via an interpretation into a simply typed lambda calculus with disjunction [BBdP98] The latter is in many ways the simplest and easily extends to additional connectives. We map both 2A and 3A as A so that ....

Dag Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307. North-Holland Publishing Co., 1971.

....of that type A. This constructor T corresponds to the modality # just as the constructors in the ordinary typed lambda calculus correspond to in propositional formulas. They gave a natural deduction system for PLL and prove a strong normalization theorem by using the method in Prawitz [Pra97] (see also Tait [Tai67] and Troelstra [Tro73] Gol81] argued for an application of the logic in Grothendieck s topology. He extracted the principle (#) A is locally true at # i# A is true at all points close to # For instance, two functions f and g are said to be equivalent, or to have the same ....

D. Prawitz, Ideas and results in proof-theory, Proceedings of the second Scandinavian logic symposium, North-Holland, 1971, pp. 235-307.

....] n h is encoded without side conditions, in a similar way as v i [t] n (see Definition 3.3) Define d[d # ] h = d[d # ] 0 h . 13 PROOF REDUCTION 20 13 Proof reduction To illustrate how the defined machinery can be used to manipulate proof objects, we define Prawitz proof reduction rules [Pra71]. 15 The goal is to remove detours, as in the following tree. #; # # d [ # # # # (#, d) # # # # # e [ # # # # (# (#, d) e) # Instead of first assuming # to build a proof d of #, introduce the implication # # # and then eliminate it immediately by plugging ....

D. Prawitz. Ideas and results in proof theory. In Jens Erik Fenstad, editor, Proceedings of the Scandinavian Logic Symposium, pages 235307, Amsterdam, 1971, North-Holland.

....acts, in addition to the connectives of classical logic. As indicated above, the pragmatic connectives 1 The philosophical position adopted here seems to us in agreement with the point of view expressed by Professor Solomon Feferman in many occasions, in particular in his review of Prawitz [12] in the J.S.L. 5] A PRAGMATIC INTERPRETATION OF SUBSTRUCTURAL LOGICS 5 are given Heyting s semantics of proofs . A formula is called pragmatically valid (or p valid) if it represents relations between illocutionary acts which hold in all circumstances. Like the notion of provability in ....

D. Prawitz. Ideas and Results in Proof Theory. In Proceedings of the Second Scandinavian Logic Symposium, ed. Fenstad, North-Holland, 1971.

No context found.

Dag Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307. North-Holland Publishing Co., 1971.

No context found.

Dag Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307. North-Holland Publishing Co., 1971.

No context found.

D. Prawitz, Ideas and results in proof theory, in: J.E. Fenstad (ed) Proc. 2nd Scandinavian Logic Symp. (North-Holland, 1971) 235--307.

No context found.

Dag Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307. North-Holland Publishing Co., 1971.

No context found.

Dag Prawitz. Ideas and results in proof theory. In Proc. Second Scandinavian Logic Symposium

No context found.

D. Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, nd Scandinavian Logic Symposium, pages 235--307. North Holland, 1971.

No context found.

D. Prawitz, Ideas and results in proof theory, in Proceedings of the Second Scandinavian Logic Symposium, 1971, pp. 235--307.

No context found.

D. Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, volume 63 of Studies in Logic and The Foundations of Mathematics, pages 235-307. North-Holland, 1971.

No context found.

D. Prawitz. Ideas and Results of Proof Theory. In Proceedings of the 2nd Scandinavian Logic Symposium, volume 63 of Studies in Logic and the Foundations of Mathematics, pages 235--307. North-Holland, 1971.

No context found.

Dag Prawitz. Ideas and results in proof theory. In Jens Erik Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, volume 63 of Studies in Logic and the Foundations of Mathematics, pages 235--307, Amsterdam, 1971. North-Holland. Cited on pages 8, 25, 26, and 77.

No context found.

Dag Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proc. 2 Scandianvian Logic Symposium, pages 235-307. North{ Holland, 1971. 22

No context found.

Prawitz, D. Ideas and Results in Proof Theory. in: Proceedings of the Second Scandinavian Logic Symposium, University of Oslo, June 18--20,

No context found.

Prawitz, D., Ideas and Results in Proof Theory, Proc. 2nd Scand. Log. Symp., 1970.

No context found.

D. Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proceedings Second Scandinavian Logic Symposium, pages 235--307. 1971.

No context found.

D. Prawitz. Ideas and results of proof theory. In J.E. Fenstad, editor, The 2nd Scandinavian Logical Symposium, pages 235307. North-Holland, 1970.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC