In 1934, in Bernays preface to the 1st edn. of the 1st vol. of Hilbert-Bernays"Grundlagen der Mathematik", a nearly completed draft of the the finally two-volume monograph is mentioned, which had to be revoked because of the completely changed situation in the area of proof theory after Herbrand and Goedels revolutionary results. Nothing at all seems to be known about this draft and its whereabouts. A third of a century later, Bernays preface to the 2nd edn. (1968) of the 1st vol. of Hilbert-Bernays mentions joint work of Hasenjaeger and Bernays on the second edition. Bernays states there that it became obvious that the integration of the many new results in the area of proof theory would have required a complete reorganization of the book, i.e. that the inclusion of the intermediately found new results in the area of proof theory turned out to be unobtainable by a revision, but would have required a complete reorganization of the entire textbook. We document that - even after the need for a complete reorganization had become obvious - this joint work went on to a considerable extent. Moreover, we document when Hasenjaeger stayed in Zurich to assist Bernays in the completion of the 2nd edn. In May 2017, we identified an incorrectly filed text in Bernays scientific legacy at the archive of the ETH Zurich as a candidate for the beginning of the revoked draft for the 1st edn. or of a revoked draft for the 2nd edn. In a partial presentation and careful investigation of this text we gather only some minor evidence that this text is the beginning of the nearly completed draft of the 1st edn., but ample evidence that this text is part of the work of Hasenjaeger and Bernays on the 2nd edn. We provide some evidence that this work has covered a complete reorganization of the entire 1st vol., including a completely new version of its last chapter on the iota.

We investigate the elimination of quantifiers in first-order formulas via Hilbert's epsilon-operator (or -binder), following Bernays' explicit definitions of the existential and the universal quantifier symbol by means of epsilon-terms. This elimination has its first explicit occurrence in the proof of the first epsilon-theorem in Hilbert-Bernays in 1939. We think that there is a lacuna in this proof w.r.t. this elimination, related to the erroneous assumption that explicit definitions always terminate. Surprisingly, to the best of our knowledge, nobody ever proved confluence or termination for this elimination procedure. Even myths on non-confluence and the openness of the termination problem are circulating. We show confluence and termination of this elimination procedure by means of a direct, straightforward, and easily verifiable proof, based on a new theorem on how to obtain termination from weak normalization.

Free variables occur frequently in mathematics and computer science with ad hoc and altering semantics. We present the most recent version of our free-variable framework for two-valued logics with properly improved functionality, but only two kinds of free variables left (instead of three): implicitly universally and implicitly existentially quantified ones, now simply called "free atoms" and "free variables", respectively. The quantificational expressiveness and the problem-solving facilities of our framework exceed standard first-order and even higher-order modal logics, and directly support Fermat's descente infinie. With the improved version of our framework, we can now model also Henkin quantification, neither using quantifiers (binders) nor raising (Skolemization). We propose a new semantics for Hilbert's epsilon as a choice operator with the following features: We avoid overspecification (such as right-uniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the epsilon supports reductive proof search optimally.

In the middle of the 1980s, David Poole introduced a semantic, model-theoretic notion of specificity to the artificial-intelligence community. Since then it has found further applications in non-monotonic reasoning, in particular in defeasible reasoning. Poole tried to approximate the intuitive human concept of specificity, which seems to be essential for reasoning in everyday life with its partial and inconsistent information. His notion, however, turns out to be intricate and problematic, which - as we show - can be overcome to some extent by a closer approximation of the intuitive human concept of specificity. Besides the intuitive advantages of our novel specificity orderings over Poole's specificity relation in the classical examples of the literature, we also report some hard mathematical facts: Contrary to what was claimed before, we show that Poole's relation is not transitive in general. The first of our specificity orderings (CP1) captures Poole's original intuition as close as we could get after the correction of its technical flaws. The second one (CP2) is a variation of CP1 and presents a step toward similar notions that may eventually solve the intractability problem of Poole-style specificity relations. The present means toward deciding our novel specificity relations, however, show only slight improvements over the known ones for Poole's relation; therefore, we suggest a more efficient workaround for applications in practice.

Herbrand's Fundamental Theorem provides a constructive characterization of derivability in first-order predicate logic by means of sentential logic. Sometimes it is simply called "Herbrand's Theorem", but the longer name is preferable as there are other important "Herbrand theorems" and Herbrand himself called it "Th\'eor\`eme fondamental". It was ranked by Bernays [1957] as follows: "In its proof-theoretic form, Herbrand's Theorem can be seen as the central theorem of predicate logic. It expresses the relation of predicate logic to propositional logic in a concise and felicitous form." And by Heijenoort [1967]: "Let me say simply, in conclusion, that Begriffsschrift [Frege, 1879], L\"owenheim's paper [1915], and Chapter 5 of Herbrand's thesis [1930] are the three cornerstones of modern logic." Herbrand's Fundamental Theorem occurs in Chapter 5 of his PhD thesis [1930] --- entitled Recherches sur la th\'eorie de la d\'emonstration --- submitted by Jacques Herbrand (1908-1931) in 1929 at the University of Paris. Herbrand's Fundamental Theorem is, together with G\"odel's incompleteness theorems and Gentzen's Hauptsatz, one of the most influential theorems of modern logic. Because of its complexity, Herbrand's Fundamental Theorem is typically fouled up in textbooks beyond all recognition. As we are convinced that there is still much more to learn for the future from this theorem than many logicians know, we will focus on the true message and its practical impact. This requires a certain amount of streamlining of Herbrand's work, which will be compensated by some remarks on the actual historical facts.

Higher-level cognition includes logical reasoning and the ability of question answering with common sense. The RatioLog project addresses the problem of rational reasoning in deep question answering by methods from automated deduction and cognitive computing. In a first phase, we combine techniques from information retrieval and machine learning to find appropriate answer candidates from the huge amount of text in the German version of the free encyclopedia "Wikipedia". In a second phase, an automated theorem prover tries to verify the answer candidates on the basis of their logical representations. In a third phase—because the knowledge may be incomplete and inconsistent—we consider extensions of logical reasoning to improve the results. In this context, we work toward the application of techniques from human reasoning: We employ defeasible reasoning to compare the answers w.r.t. specificity, deontic logic, normative reasoning, and model construction. Moreover, we use integrated case-based reasoning and machine learning techniques on the basis of the semantic structure of the questions and answer candidates to learn giving the right answers.

Using Heijenoort's unpublished generalized rules of quantification, we discuss the proof of Herbrand's Fundamental Theorem in the form of Heijenoort's correction of Herbrand's "False Lemma" and present a didactic example. Although we are mainly concerned with the inner structure of Herbrand's Fundamental Theorem and the questions of its quality and its depth, we also discuss the outer questions of its historical context and why Bernays called it "the central theorem of predicate logic" and considered the form of its expression to be "concise and felicitous".

We investigate models of first-order logic designed to give semantics to reductive proof-search systems, with special attention to the so-called γ- and δ-rules controlling quantifiers. The key innovation is the use of syntax and semantics with (finitely supported) name-symmetry, in the style of nominal techniques.

In the middle of the 1980s, David Poole introduced a semantical, model-theoretic notion of specificity to the artificial-intelligence community. Since then it has found further applications in non-monotonic reasoning, in particular in defeasible reasoning. Poole tried to approximate the intuitive human concept of specificity, which seems to be essential for reasoning in everyday life with its partial and inconsistent information. His notion, however, turns out to be intricate and problematic, which --- as we show --- can be overcome to some extent by a closer approximation of the intuitive human concept of specificity. Besides the intuitive advantages of our novel specificity ordering over Poole's specificity relation in the classical examples of the literature, we also report some hard mathematical facts: Contrary to what was claimed before, we show that Poole's relation is not transitive. The present means to decide our novel specificity relation, however, show only a slight improvement over the known ones for Poole's relation, and further work is needed in this aspect.

In this position paper, we briefly review the development of automated inductive theorem proving and computer-assisted mathematical induction. We think that the current low expectations on progress in this field result from a faulty projection. On an abstract but hopefully sufficiently descriptive level, we explain why we believe that future progress in the field is to result from human-orientedness and descente infinie.

Using Heijenoort's unpublished generalized rules of quantification, we discuss the proof of Herbrand's Fundamental Theorem in the form of Heijenoort's correction of Herbrand's "False Lemma" and present a didactic example. Although we are mainly concerned with the inner structure of Herbrand's Fundamental Theorem and the questions of its quality and its depth, we also discuss the outer questions of its historical context and why Bernays called it "the central theorem of predicate logic" and considered the form of its expression to be "concise and felicitous".

Using a human-oriented formal example proof of the lim +-theorem (that the sum of limits is the limit of the sum), we exhibit a non-permutability of beta-steps and delta(+)-steps (according to SMULLYAN'S classification), which is not visible with non-liberalized delta-rules and dissolves into a problem of mere inefficiency with further liberalized delta-rules, such as the delta(++)-rules. Beside a careful presentation of the human-oriented search for a formal proof of (lim +), our main intention is to show where sequent and tableau calculi are in conflict with human-oriented proof construction. (C) 2011 Elsevier Ltd. All rights reserved.

Reductive proof-search tries to reduce a goal to tautologies. In the presence of quantifiers this becomes a complex design problem by which proof-theory shades into ‘proof-engineering’. There is no single right answer here, but there are a lot of practical problems with strong roots in theory. In this work we consider a nominal semantics for this design space. The reduction in complexity is striking, and we get an elementary account of reductive proof-search with quantifiers.

Using a human-oriented formal example proof of the lim+-theorem (that the sum of limits is the limit of the sum), we exhibit a non-permutability of β-steps and δ+-steps (according to Smullyan’s classification), which is not visible with non-liberalized δ-rules and dissolves into a problem of mere inefficiency with further liberalized δ-rules, such as the δ++-rules. Beside a careful presentation of the human-oriented search for a formal proof of (lim+), our main intention is to show where sequent and tableau calculi are in conflict with human-oriented proof construction.

The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year 1925, when he was 17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding. Inevitably, when all introductory words are said in the annual postgraduate course on logic and automated deduction, the professor will feel the urge to point out to the young students that there are things beyond the latest developments of computer technology or the fabric of the Internet: eternal truths valid on planet Earth but in all those far away galaxies just as well. And as there are still ten minutes to go till the end of the lecture, the students listen in surprise to the strange tale about the unknown flying objects from the far away, now visiting planet Earth and being welcomed by a party of human dignitaries from all strata of society. Not knowing what to make of all this, the little green visitors will ponder the state of evolution on this strange but beautiful planet: obviously life is there — but can it think?

We present a convenient notation for positive/negative-conditional equations. The idea is to merge rules specifying the same function by using case-, if-, match-, and let-expressions. Based on the presented macro-rule-construct, positive/negative-conditional equational specifications can be written on a higher level. A rewrite system translates the macro-rule-constructs into positive/negative-conditional equations.