G. Gentzen. Untersuchungen uber das logische Schliessen. Mathematische Zeitschrift, 39:176--210,405--431, 1935.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of a set: to support quantifiers where the domain is the extension of a set. We assume some familiarity with axiomatic set theory (say, via Enderton [2] and also with the Gentzen formalism, which it is extended to include new rules for the local quantifiers (see Curry [2] Gallier [5] Gentzen [6], Kleene [7] 2. Syntax. We revise the syntax in [8] Chapter 9, which was intended for the particular system G, and introduce some changes in the standard formalism. In particular we make a distinction between free and bound variables (as in Gentzen [6] which we think is more convenient in ....

....(see Curry [2] Gallier [5] Gentzen [6] Kleene [7] 2. Syntax. We revise the syntax in [8] Chapter 9, which was intended for the particular system G, and introduce some changes in the standard formalism. In particular we make a distinction between free and bound variables (as in Gentzen [6]) which we think is more convenient in dealing with local quantifiers. This approach forces a distinction between formulas in general and normal formulas (see 2.3) 2.1. We consider a general first order language SET, where the primitive symbols are: set variables, X,Y, Z, operation symbols, ....

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Gentzen, G. (1934). Untersuchungen Uber das logische Schliessen. Mathematische Zeitschrift 39:176-210 and 405-431.

....check whether a formula is valid in . In the case of G# we obtain: F F 2 implies F G# where F is the number of (distinct) propositional variables occurring in F . 3 Sequent calculi of relations There are quite di#erent ways to interpret Gentzen s classical sequent calculus LK [17]. These lead to di#erent types of generalizations of Gentzen s calculus. One very useful interpretation of a sequent G 1 , G m is to understand it as expressing the assertion that either one of the F i (1 n) is false or one of the G j (1 m) is true. In this view a ....

....finite valued logic if we follow the above definitions. In fact, because of the presence of the standard structural rules, these calculi are just notational variants of the many placed sequent calculi or signed calculi as described, e.g. in [23, 10] The special case of classical logic LK [17] was already sketched at the beginning of the section. We can even get rid of the truth constants in the formulation of the calculi. The reason for this is that, obviously, any atomic formula of T that contains a constant can only be of form C i (c j ) and thus is either simply true or ....

Gentzen, G.: Untersuchungen uber das logische Schliessen I, II. Mathematische Zeitschrift, 39 (1934) and (1935) 176--210 and 405--431

....a finite set of types; iii) RG , the calculus of G, is a type logic. The non associative Lambek calculus NL serves as a base logic for the kinds of type logical grammars that the present paper applies to. The systems of inference rules shown below conform to the logical style of Gentzen s [4] sequent calculus. Definition 2.2 (Axiom) A #[A] #[#] Cut) #[ A B #) L) #[ # B A) L) # B) A B ( R) B #) B A ( R) Extensions of NL have been developed in the literature [11,13] which employ unary operators that are analogous to ....

Gentzen, G., Untersuchungen uber das logische Schliessen, Math. Zeitschrift 39 (1934), pp. 176--210, 405--431, english translation in [17]. 13

....preserving this fixed occurrence of L. i.e. we require that any subsequent application of ( oe) and ( preserves the fixed occurrence. Proposition1. Let P 1 ; Pn be a consistent set of formulas and G be a formula. The sequent P 1 ; Pn G is deducible in Gentzen s calculus LK [13] if and only if there exists a proof tree w.r.t. the initiale sequent P 1 ; Pn ; G G; h i in the calculus GD. Proof. A proof of this proposition may be obtained by extending a proof of the soundness and completeness of a calculus GD 2 [14] to the case of 1st order logic, taking ....

Gentzen, G. : Untersuchungen uber das Logische Schliessen. Math. Zeit. 39 (1934) 176--210.

....with such a performance rate, information capacity, and flexibility that programming complex intellectual processes became feasible. As a response to the emergence of computing machinery of that sort a series of papers appeared in which the issues of the implementation of Gentzen type calculi [9] and inference search methods relying on the results of Skolem [10] and Herbrand [11] were discussed. For more detail, the reader is referred to, for example, 12] 13] 14] 15] etc. It might be well to point out that in those first papers an answer to the principal question about a ....

Gentzen, G. Untersuchungen uber das Logische Schliessen. Math. Z. N 39 (1934) 176--210.

....: j are assigned by E. T E p : j iff P(p) T E p q iff P(p) P(q) In a slight abuse of notation we also write T E O iff O satisfies O For lists of judgements D we write T E D to mean that, for all j in D, T E j. Sequent Calculus In this chapter we introduce a Gentzen style [20] sequent calculus for proving properties of processes with respect to the modal calculus. Sequent calculi for this purpose have been considered in [4, 13, 14, 15, 19] The calculus introduced by Beattie [4] uses explicit induction and coinduction arguments to prove properties that are given by ....

Gerhard Gentzen. Untersuchungen uber das logische Schliessen. Mathematische Zeitschrift, 39:176--210 and 405--431, 1935.

....Z # P (X) # (#Y # Z)Y # X This is, of course, the well known power set operation, which was denounced as being impredicative in [8] although a last ditch e#or was made in Chapter 8 to provide a predicative foundation. At this stage we do not support anymore this e#ort (see also note 8. 5 in [8] This is only a partial rejection. We have no problem with P (#) but we think P (P (#) is problematic (see Appendix A in [8] In an operational system, set existence must be expressed via explicit operations which are controlled by operational axioms. In [8] we gave a number of ....

....is derivable in the classical calculus, where # is the universal closure of the rule axiom: #, # # # (#Y # f(X) p(X, Y ) proof. Informally, we note that # # asserts that p(X, Z) holds for some Z = h(X, Y 1 , Yn ) From the axiom it follows that Z # f(X) hence (#Y # f(X) p(X, Y ) 2 5 The Reduction Theorem In this section we fix the (k 1) ary p and denote by # the normal formula (#Y )p(X, Y ) where as usual X is the input list X 1 , X k . 16 The theorem we prove below is a revised version of Theorem 9.6 in [8] In fact, it is a stronger version, because here we ....

Gentzen, G. (1934) Untersuchungen Uber das logische Schliessen. Mathematische Zeitschrift 39:176-210 and 405-431.

....the formula depends. If I introduce an hypothesis A in a proof, I add a new label, a singleton of a new index standing for tion [153] modelled after Lemmon s textbook, used in the U. K. for many years. Logicians on continental Europe are much more likely to use Prawitz [214] or Gentzen style [111, 112] natural deduction systems. This geographic distribution of pedagogical techniques (and its resulting influence on the way research is directed, as well as teaching) is remarkably resilient across the decades. The recent publication of Barwise and Etchemendy s popular textbook introducing logic ....

....23, 2001 http: www.phil.mq.edu.au staff grestall 32 FACT 12 (CUT IS ADMISSIBLE IN THE LAMBEK CALCULUS) If X # A is provable in the Lambek calculus with the aid of the cut rule, it can also be proved without it. PROOF Lambek s proof of the cut admissibility theorem parallels Gentzen s own [111, 112]. You take a proof featuring a cut and you push that cut upwards to the top of the tree, where it evaporates. So, given an instance of the cut rule, if the formula featured in the cut is not introduced in the rules above the cut, you permute the cut with the other rules. You show that you could ....

GERHARD GENTZEN. "Untersuchungen uber das logische Schliessen". Math. Zeitschrift, 39:176--210 and 405--431, 1934. Translated in The Collected Papers of Gerhard Gentzen [112].

....and compare the use of tableaux, automata and games for modal and temporal logics, giving a brief overview of how they cope with eventualities in solving the problems mentioned above. 2 Tableaux Tableau systems were first developed as syntactical devices to obtain Gentzen systems for modal logics [Gen35]. Later, it was discovered that they basically coincide with semantical tableau systems [Bet53, Kri59] From then on, tableau systems have been widely used in the study of modal logics in particular. A tableau is a finite tree whose nodes are labelled with (sets of) formulas. The annotation of ....

G. Gentzen. Untersuchungen uber das Logische Schliessen. Math. Zeitschrift, 39:176--210,405--431, 1935.

....to investigate the design of logic programming languages via goal directed provability for a multiple conclusioned system for intuitionistic logic, and to compare the results with the single conclusioned case. 2 Preliminaries 2. 1 Sequent Calculi Sequent calculi are due originally to Gentzen [7] and are often used in the analysis of proof systems. This is because sequent calculus rules are local (and hence conceptually straightforward to implement) and there is a natural distinction between programs and goals. A sequent may be thought of as stating that if all the formulae in are ....

....to be at least two distinct such generalisations. F F Axiom F; F Cut F1 ; F2 ; F1 F2 ; R F; F L F1 ; F2 ; F1 F2 L ; F1 F2 ; F1 F2 ; R ; F1 ; F2 ; F1 F2 L LK has the cut elimination property [7], i.e. that any proof containing occurrences of the Cut rule can be replaced with a (potentially much larger) proof in which there are no occurrences of the Cut rule. Both of the other sequent calculus systems used in this paper (LJ and LM) also have the cut elimination property. 2.2 ....

G. Gentzen. Untersuchungen uber das logische Schliessen. Math. Zeit., 39:176--210,405-- 431, 1934.

....discuss our conclusions and some prospects for further work. For the convenience of the general reader, we give the classical and linear sequent calculi in the appendices. 1 2 Logical Preliminaries An elegant and convenient proof system for classical logic is provided by the sequent calculus [2, 8, 3]. 3 Each derivation in the calculus yields a sequent Gamma Gamma Delta, which can be thought of as stating that if the conjunction of the formulae in (the antecedent) Gamma is true, then the disjunction of the formulae in (the succedent) Delta is true. For each connective, the calculus ....

G. Gentzen. Untersuchungen uber das logische Schliessen. Mathematische Zeitschrift 39, 176-210, 405-431, 1934.

....of our analysis of logic programming. We illustrate the ideas with an extended example. The logically sophisticated reader may wish to skip over this Section. We briefly consider how an analysis of goaldirected provability applies to linear logic [21] 2. 1 Linear logic The sequent calculus [7] is a presentation of the proofs of classical logic in which sequents of the form Gamma Delta are manipulated. Such sequents are to be read as the conjunction of the formulae from the set Gamma entails the disjunction of the formulae from the set Delta . In this setting, there are two ....

G. Gentzen. Untersuchungen uber das logische Schliessen. Math. Zeit. 39, 176-210, 405-431, 1934.

....characterized by calculations of solutions of sets of Boolean equations generated by searches. Our characterization encompasses lazy (or local) eager (or global) and intermediate (mixed local and global) strategies. 1 Introduction The formulation of linear logic [4] as a sequent calculus [3, 4, 13, 2] makes essential use of the multiplicative formulation of binary rules, in which the resources available to derive the principal formula in each of the premisses must be combined to form the resources available to derive the principal formula of the conclusion. For example, in each of the rules ....

G. Gentzen. Untersuchungen uber das logische Schliessen. Math. Zeit. 39, 176-210, 405-431, 1934.

....systems are incorporated into Partially supported by the Poland s Scientific Research Committee grant No. 8 T11 C018 11. 1 the resolution diagrams as a mechanism of decomposing formulae. While in clausal resolution the formulas are converted to the conjunctive normal form beforehand. Gentzen [2] proposed two distinct formalizations of deduction systems. One of them he called natural deduction. The second he called the logistic calculus. In this presentation we distinguish the second by calling it the sequent calculus. In a sense both formalizations are very close to each other. Rasiowa ....

G.Gentzen, Untersuchungen uber das Logische Schliessen, Mathem. Zeitschrift 39(1934), 176-210, 405-431

....space for the auxiliary B formula in a TRANS inference is not an option. For proof theoretic investigation of the categorial type logics one introduces a Gentzen presentation which is shown to be equivalent to the axiomatic presentation. The main result for the Gentzen calculus (the Hauptsatz of [Gentzen 34] then states that the counterpart of the TRANS rule, the Cut inference, can be eliminated from the logic without affecting the set of derivable theorems. An immediate corollary of this Cut Elimination Theorem is the subformula property which limits proof search to the subformulae of the theorem ....

Gentzen, G. (1934), `Untersuchungen uber das logische Schliessen'. Mathematische Zeitschrift 39, 176--210, 405--431.

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G. Gentzen. Untersuchungen uber das logische Schliessen. Mathematische Zeitschrift, 39:176--210,405--431, 1935.

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G. Gentzen. Untersuchungen uber das logische Schliessen. Mathematische Zeitschrift, 39:176--210,405--431, 1935.

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G. Gentzen. Investigations into logical deduction. First published as `Untersuchungen uber das logische schliessen' in Mathematische Zeitschrift, 39:176--210, 1935. Translation published in M. E. Szabo, editor, The Collected Papers of Gerhard Gentzen, North-Holland, 1969.

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G. Gentzen, Untersuchungen uber das logische Schliessen, Mathematische Zeitschrift, 39 (1934), pp. 176--210, 405--431. English translation in [2], pp. 68-131.

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G. Gentzen, Untersuchungen uber das logische Schliessen, Matematische Zeitscrift, 39, pp.176-210 and 405-431,

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G. Gentzen. Untersuchungen uber das logische schliessen I. Mathematische Zeitschrift, 39:176--210, 1934.

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G. Gentzen. Untersuchungen uber das logische Schliessen. Mathematische Zeitschrift 39, 176-210, 405-431, 1934.

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G. Gentzen. Untersuchungen uber das logische Schliessen. Mathematische Zeitschrift 39, pp. 176-210, 405-431, 1934.

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Gentzen, G. Untersuchungen uber das logische Schliessen. Math. Zeitschrift 39(1934), pp. 176-210, 405-431.

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G. Gentzen. Untersuchungen uber das logische Schliessen. Mathematische Zeitschrift 39, 176-210, 405-431, 1934.

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