Carnap’s Early Metatheory: Scope and Limits

Georg Schiemer, Richard Zach, and Erich Reck. 2017. “Carnap’s Early Metatheory: Scope and Limits,” Synthese 194(1), 33–65

In his Untersuchungen zur allgemeinen Axiomatik (1928) and Abriss der Logistik (1929), Rudolf Carnap attempted to formulate the metatheory of axiomatic theories within a single, fully interpreted type-theoretic framework and to investigate a number of meta-logical notions in it, such as those of model, consequence, consistency, completeness, and decidability. These attempts were largely unsuccessful, also in his own considered judgment. A detailed assessment of Carnap’s attempt shows, nevertheless, that his approach is much less confused and hopeless than it has often been made out to be. By providing such a reassessment, the paper contributes to a reevaluation of Carnap’s contributions to the development of modern logic.

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Heinrich Behmann’s 1921 lecture on the decision problem and the algebra of logic

Paolo Mancosu and Richard Zach. “Heinrich Behmann’s 1921 lecture on the decision problem and the algebra of logic,” Bulletin of Symbolic Logic 21 (2015), 164–187

Heinrich Behmann (1891–1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved—independently of Löwenheim and Skolem’s earlier work—the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented important programmatic remarks on the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included.

Vagueness, Logic and Use: Four Experimental Studies on Vagueness

Mind and Language 26 (2011) 540–573
(with Phil Serchuk and Ian Hargreaves)

Although arguments for and against competing theories of vagueness often appeal to claims about the use of vague predicates by ordinary speakers, such claims are rarely tested. An exception is Bonini et al. (1999), who report empirical results on the use of vague predicates by Italian speakers, and take the results to count in favor of epistemicism. Yet several methodological difficulties mar their experiments; we outline these problems and devise revised experiments that do not show the same results. We then describe three additional empirical studies that investigate further claims in the literature on vagueness: the hypothesis that speakers confuse ‘P‘ with ‘definitely P‘, the relative persuasiveness of different formulations of the inductive premise of the Sorites, and the interaction of vague predicates with three different forms of negation.

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The Development of Mathematical Logic from Russell to Tarski: 1900-1935

Leila Haaparanta, ed., The History of Modern Logic. New York and Oxford: Oxford University Press, 2009, pp. 318-471 (with Paolo Mancosu and Calixto Badesa)

Reprinted in Paolo Mancosu, The Adventure of Reason. Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940. Oxford: Oxford University press, 2010

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight “itineraries” concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski’s work on truth.

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Effective Finite-Valued Approximations of General Propositional Logics

Avron, Arnon; Dershowitz, Nachum; Rabinovich, Alexander (Eds.). Pillars of Computer Science: Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday. Lecture Notes in Computer Science 4800. Berlin: Springer, 2008. 107-129
(with Matthias Baaz)

Propositional logics in general, considered as a set of sentences, can be undecidable even if they have “nice” representations, e.g., are given by a calculus. Even decidable propositional logics can be computationally complex (e.g., already intuitionistic logic is PSPACE-complete). On the other hand, finite-valued logics are computationally relatively simple—at worst NP. Moreover, finite-valued semantics are simple, and general methods for theorem proving exist. This raises the question to what extent and under what circumstances propositional logics represented in various ways can be approximated by finite-valued logics. It is shown that the minimal m-valued logic for  which a given calculus is strongly sound can be calculated. It is also investigated under which conditions propositional logics can be characterized as the intersection of (effectively given) sequences of finite-valued logics.

First-order Gödel logics

Annals of Pure and Applied Logic 147 (2007) 23-47 (with Matthias Baaz and Norbert Preining)

First-order Gödel logics are a family of finite- or infinite-valued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that GV is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. Complete axiomatizations for each of these cases are given. The r.e. prenex, negation-free, and existential fragments of all first-order Gödel logics are also characterized.

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Kurt Gödel and computability theory

Logical Approaches to Computational Barriers Second Conference on Computability in Europe, CiE 2006, Swansea. Proceedings. LNCS 3988 (Springer, Berlin, 2006) 575-583

Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel’s 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church, Kleene, and Rosser. Church and his students studied Gödel 1931, and Gödel taught a seminar at Princeton in 1934. Seen in the historical context, Gödel was an important catalyst for the emergence of computability theory in the mid 1930s.

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Hilbert’s program then and now

Dale Jacquette, ed., Philosophy of Logic. Handbook of the Philosophy of Science, vol. 5. (Elsevier, Amsterdam, 2006), 411-447.

Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic systems. Although Gödel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and metatheory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.

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The epsilon calculus and Herbrand complexity

Studia Logica 82 (2006) 133-155
(with Georg Moser)

Hilbert’s $$\varepsilon$$-calculus is based on an extension of the language of predicate logic by a term-forming operator $$\varepsilon x$$. Two fundamental results about the $$\varepsilon$$-calculus, the first and second epsilon theorem, play a role similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand’s Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure.

Review: Mathematical Reviews 2205042 (2006k:03127) (Mitsuru Yasuhara)

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