A Model of Intuitionistic Affine Logic from Stable Domain Theory (Revised and Expanded Version)

Girard worked with the category of coherence spaces and continuous stable maps and observed that the functor that forgets the linearity of linear stable maps has a left adjoint. This fundamental observation gave rise to the discovery of Linear Logic. Since then, the category of coherence spaces and linear stable maps, with the comonad induced by the adjunction, has been considered a canonical model of Linear Logic. Now, the same phenomenon is present if we consider the category of pre dI domains and continuous stable maps, and the category of dI domains and linear stable maps; the functor that forgets the linearity has a left adjoint. This gives an alternative model of Intuitionistic Linear Logic. It turns out that this adjunction can be factored in two adjunctions yielding a model of Intuitionistic Affine Logic; the category of pre dI domains and affine stable functions. It is the goal of this paper to show that this category is actually a model of Intuitionistic Affine Logic, and to show that this category moreover has properties which make it possible to use it to model convergence/divergence behaviour and recursion

Incorrect Responses in First-Order False-Belief Tests:A Hybrid-Logical Formalization

In the paper (Braüner, 2014) we were concerned with logical formalizations of the reasoning involved in giving correct responses to the psychological tests called the Sally-Anne test and the Smarties test, which test children’s ability to ascribe false beliefs to others. A key feature of the formal proofs given in that paper is that they explicitly formalize the perspective shift to another person that is required for figuring out the correct answers – you have to put yourself in another person’s shoes, so to speak, to give the correct answer. We shall in the present paper be concerned with what happens when answers are given that are not correct. The typical incorrect answers indicate that children failing false-belief tests have problems shifting to a perspective different from their own, to be more precise, they simply reason from their own perspective. Based on this hypothesis, we in the present paper give logical formalizations that in a systematic way model the typical incorrect answers. The remarkable fact that the incorrect answers can be derived using logically correct rules indicates that the origin of the mistakes does not lie in the children’s logical reasoning, but rather in a wrong interpretation of the task

Cut-Elimination for Full Intuitionistic Linear Logic

We describe in full detail a solution to the problem of proving the cut elimination theorem for FILL, a variant of (multiplicative and exponential-free) Linear Logicintroduced by Hyland and de Paiva. Hyland and de Paiva's work used a term assignmentsystem to describe FILL and barely sketched the proof of cut elimination. In this paper, as well as correcting a small mistake in their paper and extending thesystem to deal with exponentials, we introduce a different formal system describing the intuitionistic character of FILL and we provide a full proof of the cut eliminationtheorem. The formal system is based on a notion of dependence between formulae within a given proof and seems of independent interest. The procedure forcut elimination applies to (classical) multiplicative Linear Logic, and we can (with care) restrict our attention to the subsystem FILL. The proof, as usual with cutelimination proofs, is a little involved and we have not seen it published anywhere