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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.

Introduction |
1 |

Itinerary I: Metatheoretical Properties of Axiomatic Systems |
3 |

1.1 Introduction | 3 |

1.2 Peano's school on the logical structure of theories | 4 |

1.3 Hilbert on axiomatization | 8 |

1.4 Completeness and categoricity in the work of Veblen and Huntington | 10 |

1.5 Truth in a structure | 12 |

Itinerary II: Bertrand Russell's Mathematical Logic |
15 |

2.1 From the Paris congress to the Principles of Mathematics 1900–1903 |
15 |

2.2 Russell and Poincare on predicativity | 19 |

2.3 On Denoting | 21 |

2.4 Russell's ramified type theory | 22 |

2.5 The logic of Principia |
25 |

2.6 Further developments | 26 |

Itinerary III: Zermelo's Axiomatization of Set Theory and Related Foundational Issues |
29 |

3.1 The debate on the axiom of choice | 29 |

3.2 Zermelo's axiomatization of set theory | 32 |

3.3 The discussion on the notion of "definit" | 35 |

3.4 Metatheoretical studies of Zermelo's axiomatization | 38 |

Itinerary IV: The Theory of Relatives and Löwenheim's Theorem |
41 |

4.1 Theory of relatives and model theory | 41 |

4.2 The logic of relatives | 44 |

4.3 Löwenheim's theorem | 46 |

4.4 Skolem's first versions of Löwenheim's theorem | 56 |

Itinerary V: Logic in the Hilbert School |
59 |

5.1 Early lectures on logic | 59 |

5.2 The completeness of propositional logic | 60 |

5.3 Consistency and completeness | 61 |

5.4 Axioms and inference rules | 66 |

5.5 Grundzüge der theoretischen Logik |
70 |

5.6 The decision problem | 71 |

5.6.1 The decision problem in the tradition of algebra of logic | 72 |

5.6.2 Work on the decision problem after 1920 | 73 |

5.7 Combinatory logic and ?-calculus | 74 |

5.8 Structural inference: Hertz and Gentzen | 76 |

Itinerary VI: Proof Theory and Arithmetic |
81 |

6.1 Hilbert's Program for consistency proofs | 81 |

6.2 Consistency proofs for weak fragments of arithmetic | 82 |

6.3 Ackermann and von Neumann on epsilon substitution | 87 |

6.4 Herbrand's Theorem | 92 |

6.5 Kurt Gödel and the incompleteness theorems | 94 |

Itinerary VII: Intuitionism and Many-valued Logics |
99 |

7.1 Intuitionistic logic | 99 |

7.1.1 Brouwer's philosophy of mathematics | 99 |

7.1.2 Brouwer on the excluded middle | 101 |

7.1.3 The logic of negation | 102 |

7.1.4 Kolmogorov | 103 |

7.1.5 The debate on intuitionist logic | 106 |

7.1.6 The formalization and interpretation of intuitionistic logic | 108 |

7.1.7 Gödel's contributions to the metatheory of intuitionistic logic | 110 |

7.2 Many-valued logics | 111 |

Itinerary VIII: Semantics and Model-theoretic Notions |
117 |

8.1 Background | 117 |

8.1.1 The algebra of logic tradition | 117 |

8.1.2 Terminological variations (systems of objects, models, and structures) | 118 |

8.1.3 Interpretations for propositional logic | 119 |

8.2 Consistency and independence for propositional logic | 120 |

8.3 Post's contributions to the metatheory of the propositional calculus | 123 |

8.4 Semantical completeness of first-order logic | 124 |

8.5 Models of first order logic | 129 |

8.6 Completeness and categoricity | 130 |

8.7 Tarski's definition of truth | 134 |

Notes | 141 |

Bibliography | 149 |

Index of Citations | 175 |