# Workshop on Logic and Philosophy of Mathematics

22.04.2023

## Idea & Motivation

Following the tradition of the *Luxemburger Zirkel*, the talks at the workshop are going to deal with questions and problems in logic, the philosophy of logic, and the foundations and philosophy of mathematics.

## Confirmed Speakers

## Program

09:00 | Welcome |

09:15-10:15 | Volker Halbach: Logical Constants |

10:30-11:30 | Martin Fischer: Reflection in the Setting of Kripkean Truth |

11:45-12:45 | Philip Welch: Dark Classes |

13:00-14:00 | Lunch (at venue) |

14:00-15:00 | Leon Horsten: Truth and Finite Conjunction |

15:15-16:15 | Levin Hornischer: Iterating Both and Neither: With Applications to the Paradoxes |

16:30-17:30 | Hannes Leitgeb: When Rules Define Logical Operators: Rules as Second-Order Definitions |

19:00 | Dinner |

## Abstracts

**Volker Halbach: Logical Constants**

Abstract: Semantic definitions of logical consequence as truth preservation under all interpretations rely on a distinction between the logical and the non-logical vocabulary, as only non-logical expressions are open to (re-)interpretation. I follow the standard strategy of using permutation invariance as a criterion of logicality. However, I apply the criterion not to semantic values, but rather in a more direct way to expressions in the language. The criterion is not relativized to domains, unlike on most other accounts. Moreover, I discuss further conditions going beyond invariance that should be satisfied by logical expressions.

### Martin Fischer: Reflection in the setting of Kripkean truth

Abstract: Reflection principles have been employed in the setting of Kripkean truth for different purposes. Probably the most well-known application is Feferman’s notion of reflective closure in his 'Reflecting on incompleteness’ (1991). Another less well-known application is provided by Reinhardt in his 1986 paper 'Some remarks on extending and interpreting theories with a partial predicate for truth’. Whereas his suggestion to focus on the significant part of KF, also known as Reinhardt’s program, is well studied, his suggestion to employ reflection principles for the justification of the non-significant axioms of KF has not received much attention. Reinhardt’s reflection principle is closer to set-theoretic reflection principles rather than proof-theoretic ones. In the talk I want to present and discuss some interpretations of Reinhardt’s suggestion. The talk is based on joint work with Carlo Nicolai and Mario Piazza.

### Philip Welch: Dark Classes

Abstrakt: Set theorists countenance proper classes usually only in the form of the minimal von-Neumann-Gödel-Bernays models: those classes defined by formulae in ZFC. We consider reasons for thinking about more complex classes within the murk of Kelley-Morse class theory, in particular some which pertain to thinking about "width reflection" (so-called) of V. Epistemological and ontological problems arise.

### Leon Horsten: Truth and Finite Conjunction

(Joint work with Guanglong Luo and Sam Roberts.)

Abstract: This talk constitutes a critical response to Kentaro Fujimoto’s “new conservativeness argument” (Fujimoto, Mind, 2022), which centers around the notion of finite conjunction. Fujimoto’s arguments turn on a quite specific way of formalising the notions of finite conjunction and finite conjunction in first-order logic. We argue that by instead formalising these arguments in a natural way in second-order logic, Fujimoto’s new conservativeness argument can be resisted.

**Levin Hornischer: Iterating Both and Neither: With Applications to the Paradoxes**

Abstract: A common response to the paradoxes of vagueness and truth is to introduce the truth-values ‘neither true nor false’ or ‘both true and false’ (or both). However, this infamously runs into trouble with higher-order vagueness or the revenge paradox. This, and other considerations, suggest iterating ‘both’ and ‘neither’: as in ‘neither true nor neither true nor false’. We present a novel explication of iterating ‘both’ and ‘neither’. Unlike previous approaches, each iteration will change the logic, and the logic in the limit of iteration is an extension of paraconsistent quantum logic. Surprisingly, we obtain the same limit logic if we use (a) both and neither, (b) only neither, or (c) only neither applied to comparable truth-values. These results promise new and fruitful replies to the paradoxes of vagueness and truth.

**Hannes Leitgeb: When Rules Define Logical Operators: Rules as Second-Order Definitions**

Abstract: Logical inferentialists hold that the meaning of logical operators is given by their rules of inference. Prior cast doubt on this by introducing rules for his so-called tonk operator that seemed to allow for the derivation of any sentence whatsoever from any sentence whatsoever. The obvious inferentialist reply was to require constraints on the defining rules, such as conservativeness (Belnap) or harmony (Dummett). In my talk, I will propose a different criterion for when rules define logical operators that (i) is philosophically principled in taking the idea of rules as definitions perfectly seriously, (ii) explains how the semantic meaning of the operators can be determined from their rules, (iii) is local in the same sense as harmony is, (iv) validates the intuitionistic natural deduction rules and the intuitionistic/classical sequent calculus rules as defining the classical logical operators while ruling out Prior's rules for tonk, (v) makes clear why already the intuitionistic natural deduction rules define the classical meaning of logical operators so long as metavariables are interpreted as expressing arbitrary classical propositions, (vi) validates the classical natural deduction rules as analytic, and (vii) does not guarantee conservativeness in Belnap's sense but in a closely related one that still entails consistency. The basic idea will be: rules define a classical logical operator just in case they translate into an explicit definition in pure classical second-order logic.

## Organizer

## Registration

Although registration is not strictly required, it would be helpful to be able to estimate the number of attendees, given that the workshop will take place on a Saturday. If you wish to attend please send an email to Office.Leitgeb@lrz.uni-muenchen.de

## Venue

Prof.-Huber-Pl. 2, Lehrturm, W 401