A family of Gödel hybrid logics

In this paper, we define a family of fuzzy hybrid logics that are based on Gödel logic. It is composed of two infinite-valued versions called ${\mathsf{GH}}_{\infty }$ and ${\mathsf{WGH}}_{\infty }$, and a sequence of finitary valued versions ${({\mathsf{GH}}_n)}_{0<n<\infty }$. We define decision procedures for both ${\mathsf{WGH}}_{\infty }$ and ${({\mathsf{GH}}_n)}_{0<n<\infty }$ that are based on particular sequents and on a set of proof rules dealing with such sequents. As these rules are strongly invertible the procedures naturally allow one to generate countermodels. Therefore we prove the decidability and the finite model property for these logics. Finally, from the decision procedure of ${\mathsf{WGH}}_{\infty }$, we design a sound and complete sequent calculus for this logic.     

Bridging Curry and Church's typing style

There are two versions of type assignment in the λ-calculus: Church-style, in which the type of each variable is fixed, and Curry-style (also called “domain free”), in which it is not. As an example, in Church-style typing, ${\lambda }_{x:A}.x$ is the identity function on type A, and it has type $A\to A$ but not $B\to B$ for a type B different from A. In Curry-style typing, ${\lambda }_x.x$ is a general identity function with type $C\to C$ for every type C. In this paper, we will show how to interpret in a Curry-style system every Pure Type System (PTS) in the Church-style without losing any typing information. We will also prove a kind of conservative extension result for this interpretation, a result which implies that for most consistent PTSs of the Church-style, the corresponding Curry-style system is consistent. We will then show how to interpret in a system of the Church-style (a modified PTS, stronger than a PTS) every PTS-like system in the Curry style.     

The psychology of inferring conditionals from disjunctions: A probabilistic study

There is a new probabilistic paradigm in the psychology of reasoning that is, in part, based on results showing that people judge the probability of the natural language conditional, if $A$then $B$, $P\left(ifAthenB\right)$, to be the conditional probability, $P(B\mid A)$. We apply this new approach to the study of a very common inference form in ordinary reasoning: inferring the conditional if not-$A$then $B$ from the disjunction $A$ or $B$. We show how this inference can be strong, with $P$(if not-$A$then $B$) “close to” $P\left(AorB\right)$, when $A$ or $B$ is non-constructively justified. When $A$ or $B$ is constructively justified, the inference can be very weak. We also define suitable measures of “closeness” and “constructivity”, by providing a probabilistic analysis of these notions.