# Proof-theoretic Harmony

## Intensional aspects from a second-order perspective

### Luca Tranchini

The talk will focus on harmony, the widely discussed condition that two collections of introduction and of elimination rules for a certain logical constant should satisfy in order to perfectly match each other. Attention will be restricted to rules for propositional connectives in single-conclusion natural deduction format.

I will first argue that a decent criterion of harmony should be (hyper-)intensional, in the sense that taken two equivalent collections of elimination rules (i.e. such that each rule of the first collection is derivable from the rules in the second collection, and vice versa), it may well be that only one of the two collections is in harmony with a given collection of introduction rules.

To precisely formulate an intensional criterion for harmony we propose to modify the (extensional) criterion for harmony recently proposed by Schroeder-Heister (for short SH-harmony) and based on the idea that to any collection of introduction and of elimination rules one can associate a formula of intuitionistic propositional logic with propositional quantification (IPC2). Two collections of introduction and elimination rules for a certain connective are in SH-harmony if the two formulas associated to them are interderivable in IPC2.

An intensional criterion is achieved by imposing on the two formulas a condition stricter than intederivability: isomorphim. We will present the notion of isomorphism (well-known from category theory and computational study of proof systems), stress its significance for proof-theoretic semantics, and finally apply it to clarify the relationship between different ways of generating collections of elimination rules from a given collection of introduction rules known in the literature.

# A Modal Theory of Free Choice Sequences

### Ethan Brauer

Free choice sequences (also called ‘infinitely  proceeding sequences’) are a concept from intuitionistic mathematics that are central to the development of the intuitionistic theory of the continuum. Free choice sequences have, however, been largely rejected or ignored both by classical and other constructive mathematicians. In this talk I argue that the objections to free choice sequences can be overcome by grounding the concept in our temporal intuition and formalizing the theory in modal logic. I will present a theory of free choice sequences as a modal extension of classical second-order arithmetic. The resulting theory is able to prove modal versions of the intuitionist’s axioms for so-called lawless sequences, suffices for the development of a theory of real number generators, and captures many results distinctive of intuitionistic analysis including the non-existence of functions on real numbers with definable discontinuities.

# Indexicals, Time, and Compositionality

### Peter Pagin

Kaplan’s official argument in “Demonstratives” for Temporalism, the view that some English sentences express propositions that can vary in truth value across time, is the so-called Operator Argument: temporal operators, such as “sometimes”, would be vacuous without such propositions.

Equally important is the argument from compositionality. Without temporal propositions, the sentences

(I) It is raining where John is
(2) It is raining where John is now

would express the same proposition. But they embed differently:

(3) Sometimes, it is raining where John is
(4) Sometimes it is raining where John is now

(3) and (4) express distinct propositions, so if they both are of the form “Sometimes, p”, and if (1) and (2) express the same proposition, we have a violation of compositionality.

In this talk it is shown that with Switcher Semantics, which allows for a generalized form of compositionality, we can have the result that (1) and (2) agree in content when unembedded (assertoric content) but differ in content when embedded under temporal operators (ingredient sense).

We can also show that Switcher Semantics, over Kaplan’s models, preserves the validities in the Logic of Demonstratives. All in all, the arguments for Temporalism are substantially undermined.