NOL Seminar with Hannes Leitgeb
published: 2025-05-18
event date:
2025-05-26
The Nordic Online Logic Seminar
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Date Monday, 26 May 2025 at 16:00 CEST (UTC+2) on Zoom
Speaker Hannes Leitgeb (Chair and Co-Director of the Munich Center for Mathematical Philosophy, LMU München)
Title 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. Arthur Prior cast doubt on this by introducing rules
for his tonk operator that would allow for the derivation of any sentence
whatsoever from any sentence whatsoever. The obvious inferentialist reply was to
require some constraints on the defining rules, such as conservativeness
(Belnap) or harmony (Dummett). In this talk, I will defend and investigate a
different constraint: rules define a classical logical operator just in case
they translate into an explicit definition in pure classical second-order logic.
The right-hand side of this criterion will be found (i) to be philosophically
principled in taking the idea of rules as definitions perfectly seriously, (ii)
to explain how the semantic meaning of the operators can be determined from
their rules, (iii) to be local in a similar sense as harmony, (iv) to validate
the intuitionistic natural deduction rules and the intuitionistic/classical
sequent calculus rules as defining the classical propositional operators while
ruling out Prior’s rules for tonk, (v) to make clear why already the
intuitionistic natural deduction rules define the classical meaning of logical
operators so long as metavariables are interpreted as expressing classical
propositions, (vi) to validate the classical natural deduction rules as
analytic, (vii) to entail consistency, and, in the case of propositional
operators (not quantifiers), (viii) to be decidable and (ix) to determine
precisely those operators to be definable by rules that correspond to
truth-functions.