Polarizing Double Negation Translations

Double-negation translations are used to encode and decode classical proofs in intuitionistic logic. We show that, in the cut-free fragment, we can simplify the translations and introduce fewer negations. To achieve this, we consider the polarization of the formul{\ae}{} and adapt those translation to the different connectives and quantifiers. We show that the embedding results still hold, using a customized version of the focused classical sequent calculus. We also prove the latter equivalent to more usual versions of the sequent calculus. This polarization process allows lighter embeddings, and sheds some light on the relationship between intuitionistic and classical connectives

Normalisation Control in Deep Inference via Atomic Flows

We introduce `atomic flows': they are graphs obtained from derivations by tracing atom occurrences and forgetting the logical structure. We study simple manipulations of atomic flows that correspond to complex reductions on derivations. This allows us to prove, for propositional logic, a new and very general normalisation theorem, which contains cut elimination as a special case. We operate in deep inference, which is more general than other syntactic paradigms, and where normalisation is more difficult to control. We argue that atomic flows are a significant technical advance for normalisation theory, because 1) the technique they support is largely independent of syntax; 2) indeed, it is largely independent of logical inference rules; 3) they constitute a powerful geometric formalism, which is more intuitive than syntax

Inter-species variation in colour perception

Inter-species variation in colour perception poses a serious problem for the view that colours are mind-independent properties. Given that colour perception varies so drastically across species, which species perceives colours as they really are? In this paper, I argue that all do. Specifically, I argue that members of different species perceive properties that are determinates of different, mutually compatible, determinables. This is an instance of a general selectionist strategy for dealing with cases of perceptual variation. According to selectionist views, objects simultaneously instantiate a plurality of colours, all of them genuinely mind-independent, and subjects select from amongst this plurality which colours they perceive. I contrast selectionist views with relationalist views that deny the mind-independence of colour, and consider some general objections to this strategy

Algebraic and logistic investigations on free lattices

Lorenzen's "Algebraische und logistische Untersuchungen \"uber freie Verb\"ande" appeared in 1951 in The journal of symbolic logic. These "Investigations" have immediately been recognised as a landmark in the history of infinitary proof theory, but their approach and method of proof have not been incorporated into the corpus of proof theory. More precisely, Lorenzen proves the admissibility of cut by double induction, on the cut formula and on the complexity of the derivations, without using any ordinal assignment, contrary to the presentation of cut elimination in most standard texts on proof theory. This translation has the intent of giving a new impetus to their reception. The "Investigations" are best known for providing a constructive proof of consistency for ramified type theory without axiom of reducibility. They do so by showing that it is a part of a trivially consistent "inductive calculus" that describes our knowledge of arithmetic without detour. The proof resorts only to the inductive definition of formulas and theorems. They propose furthermore a definition of a semilattice, of a distributive lattice, of a pseudocomplemented semilattice, and of a countably complete boolean lattice as deductive calculuses, and show how to present them for constructing the respective free object over a given preordered set. This translation is published with the kind permission of Lorenzen's daughter, Jutta Reinhardt.Comment: Translation of "Algebraische und logistische Untersuchungen \"uber freie Verb\"ande", J. Symb. Log., 16(2), 81--106, 1951, http://www.jstor.org/stable/226668

Disadvantage, Autononomy, and the Continuity Test

The Continuity Test is the principle that a proposed distribution of resources is wrong if it treats someone as disadvantaged when they don’t see it that way themselves, for example by offering compensation for features that they do not themselves regard as handicaps. This principle – which is most prominently developed in Ronald Dworkin’s defence of his theory of distributive justice – is an attractive one for a liberal to endorse as part of her theory of distributive justice and disadvantage. In this paper, I play out some of its implications, and show that in its basic form the Continuity Test is inconsistent. It relies on a tacit commitment to the protection of autonomy, understood to consist in an agent deciding for herself what is valuable and living her life in accordance with that decision. A contradiction arises when we consider factors which are putatively disadvantaging by dint of threatening individual autonomy construed in this way. I argue that the problem can be resolved by embracing a more explicit commitment to the protection (and perhaps promotion) of individual autonomy. This implies a constrained version of the Continuity Test, thereby salvaging most of the intuitions which lead people to endorse the Test. It also gives us the wherewithal to sketch an interesting and novel theory of distributive justice, with individual autonomy at its core

Intuitionistic fixed point theories over Heyting arithmetic

In this paper we show that an intuitionistic theory for fixed points is conservative over the Heyting arithmetic with respect to a certain class of formulas. This extends partly the result of mine. The proof is inspired by the quick cut-elimination due to G. Mints

Petitio principii: the case for non-fallaciousness

This paper presents a case for the non-fallaciousness of petitio principii in the context where the only evidence which can confirm the conclusion of an argument has a content which is identical to the content of the conclusion. The more usual rhetorical and dialectical frameworks for the analysis off allacies are challenged for what I describe as their proscriptive stance. As an alternative to proscription, I recommend an analysis of the context in which petitio arguments occur. Such an analysis, I argue, suggests the relaxation of a priority condition described by Waiton (1985) and the relevance to the present case of Sorensen's (1991) analysis ofthe non-circularity of certain 'P, therefore, P' arguments

Semantic A-translation and Super-consistency entail Classical Cut Elimination

We show that if a theory R defined by a rewrite system is super-consistent, the classical sequent calculus modulo R enjoys the cut elimination property, which was an open question. For such theories it was already known that proofs strongly normalize in natural deduction modulo R, and that cut elimination holds in the intuitionistic sequent calculus modulo R. We first define a syntactic and a semantic version of Friedman's A-translation, showing that it preserves the structure of pseudo-Heyting algebra, our semantic framework. Then we relate the interpretation of a theory in the A-translated algebra and its A-translation in the original algebra. This allows to show the stability of the super-consistency criterion and the cut elimination theorem

Fregean Description Theory in Proof-Theoretical Setting

We present a proof-theoretical analysis of the theory of definite descriptions which emerges from Frege’s approach and was formally developed by Kalish and Montague. This theory of definite descriptions is based on the assumption that all descriptions are treated as genuine terms. In particular, a special object is chosen as a designatum for all descriptions which fail to designate a unique object. Kalish and Montague provided a semantical treatment of such theory as well as complete axiomatic and natural deduction formalization. In the paper we provide a sequent calculus formalization of this logic and prove cut elimination theorem in the constructive manner