Bill Craig at the blackboard, 1988

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# A conference in honor of William Craig

Craig's interpolation theorem is part of the standard logic curriculum. This and other results of Craig's have had a profound significance in logic, philosophy of science, philosophy of logic, and computer science. Six internationally distinguished speakers will reflect on the importance and impact of Craig's work: Solomon Feferman (Stanford), Michael Friedman (Stanford), Cesare Tinelli (University of Iowa), Dana Scott (Carnegie Mellon), Jouko Väänänen (University of Amsterdam and University of Helsinki), Johan van Benthem (University of Amsterdam and Stanford University). The organizers are Branden Fitelson, John MacFarlane, Paolo Mancosu, and Sherri Roush (Berkeley, Philosophy).

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# Where and when

The conference will take place in Howison Library, in Moses Hall, at the University of California, Berkeley, on May 13, 2007.

# Schedule

Time | Speaker/event |
---|---|

9.00-9.10 | Welcome |

9.10-9.55 | Solomon Feferman (Stanford), "Harmonious Logic: Craig's Interpolation Theorem and Its Descendants" |

9.55-10.05 | Q&A |

10.05-10.50 | Michael Friedman (Stanford), "Wissenschaftslogik: The Role of Logic in Philosophy of Science" |

10.50-11.00 | Q&A |

11.00-11.20 | Coffee Break |

11.20-12.05 | Jouko Väänänen (Amsterdam, Helsinki), "The Interpolation Theorem in Abstract Model Theory" |

12.05-12.15 | Q&A |

12.15-2.30 | Lunch Break |

2.30-3.15 | Dana Scott (Carnegie Mellon), "The Algebraic Interpretation of Classical and Intuitionistic Quantifiers" |

3.15-3.25 | Q&A |

3.25-4.10 | Cesare Tinelli (Iowa), "The Impact of Craig's Interpolation Theorem in Computer Science" |

4.10-4.20 | Q&A |

4.20-4.40 | Coffee Break |

4.40-5.25 | Johan van Benthem (Stanford, Amsterdam), "Interpolation, Annotated Proofs, and Inference Across Models" |

5.25-5.35 | Q&A |

5.35-5.45 | Bill Craig will say a few words |

5.45-6.45 | Reception in 301 Moses Hall |

7.00 | Dinner for Bill, speakers, and co-organizers |

# Abstracts of talks

**Harmonious Logic: Craig's Interpolation Theorem and Its Descendants**

Solomon Feferman (Stanford)

Though deceptively simple and plausible on the face of it, Craig's interpolation theorem (published 50 years ago) has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. I shall trace its early history and some of its subsequent generalizations and their applications, especially of many-sorted interpolation theorems. The history is also of interest because of the rare interaction between proof theory and model theory that took place surrounding these results. It has been suggested that any reasonable logic (with two-valued semantics) ought to have the interpolation and compactness properties among others. The talk will conclude with an introduction to the quest for reasonable proper extensions of first order logic.

**Wissenschaftslogik: The Role of Logic in Philosophy of Science**

Michael Friedman (Stanford)

Hempel's well-known paper, "The Theoretician's Dilemma," famously introduced Craig's theorem on recursive axiomatizability (1953) into philosophy of science, and applied this theorem (as Craig himself had briefly suggested in 1956) to the problem of theoretical terms. Hempel sets this discussion in a broadly Carnapian framework, and, in particular, treats Carnap's use of the Ramsey sentence as an alternative approach to dispensing with theoretical terms. This raises the more general question of the relevance of technical results in logic to the philosophy of the empirical sciences, and I discuss Carnap's conception of Wissenschaftslogik against this background.

**The Interpolation Theorem in Abstract Model Theory**

Jouko Väänänen (Amsterdam and Helsinki)

We discuss the importance of the Interpolation Theorem in abstract model theory, mainly in higher order logic, infinitary logic and logics with generalized quantifiers. We pay special attention to Ehrenfeucht-Fraisse games and factors that lead interpolation to fail or succeed.

**The Algebraic Interpretation of Classical and Intuitionistic Quantifiers**

Dana Scott (Carnegie Mellon and Berkeley)

In their 1963 book, "The Mathematics of Metamathematics", Rasiowa and Sikorski present an approach to completeness theorems of various logics using algebraic methods. This idea can of course be traced back to Boole, but it was revived and generalized by Stone and Tarski in the 1930s; however, the most direct influence on their work came from their well-known colleague, Andrzej Mostowski, after WW II. Mostowski's interpretation of quantification can as well be given for intutionistic as classical logic. The talk will briefly review the history and content of these ideas and raise the question of why there was at that time no generalization made to higher-order logic, or algebraic logic, or set theory.

**The Impact of Craig's Interpolation Theorem in Computer Science**

Cesare Tinelli (Iowa)

Logic has a central role in Computer Science by providing the mathematical foundations of many of its areas. Craig's interpolation theorem has had a strong and lasting impact in several of these areas, both at the theoretical and the practical level. Craig's result has proven particularly useful in the development of formal methods and tools for the design or the analysis of computer systems. This talk will discuss the theorem's impact by presenting a couple of very popular formal methods directly based on the computation of Craig interpolants. Implementations of these methods are currently used in industry for hardware and software verification.

**Interpolation, Annotated Proofs, and Inference Across Models**

Johan van Benthem (Stanford and Amsterdam)

An interpolation theorem provides additional information about how an inference from premises to a conclusion actually takes place. I will discuss this 'surplus' in three settings: (a) Bolzano's view of inference as needing a decision what is fixed and what is variable vocabulary, (b) Calculi of proof which make the role of changing vocabulary explicit, and (c) Generalized semantic entailment along any relation between models, merging interpolation and preservation theorems, and sometimes suggesting new 'model-crossing' logical operators.

References: (1) J. Barwise & J. van Benthem, 1999, 'Interpolation, Preservation, and Pebble Games',

Journal of Symbolic Logic64:2, 881-903. (2) J. van Benthem, 2003, 'Is there still Logic in Bolzano's Key?', in E. Morscher, ed.,Bernard Bolzanos Leistungen in Logik, Mathematik und Physik, Academia Verlag, Sankt Augustin, 11-34.