## Γ ⊢ πM: A

### Abstract

Abstract. We make an argument that, for any study involving computational effects such as divergence or continuations, the traditional syntax of simply typed lambda-calculus cannot be regarded as canonical, because standard arguments for canonicity rely on isomorphisms that may not exist in an effectful setting. To remedy this, we define a “jumbo lambda-calculus ” that fuses the traditional connectives together into more general ones, so-called “jumbo connectives”. We provide two pieces of evidence for our thesis that the jumbo formulation is advantageous. Firstly, we show that the jumbo lambda-calculus provides a “complete” range of connectives, in the sense of including every possible connective that, within the beta-eta theory, possesses a reversible rule. Secondly, in the presence of effects, we see that there is no decomposition of jumbo connectives into non-jumbo ones that is valid in both call-byvalue and call-by-name. Finally, we apply the concept of jumbo connectives to systems with isorecursive types (Jumbo FPC) and multiple conclusions (Jumbo LK). At each stage, we see that various connectives proposed in the literature are special cases of the jumbo connectives. 1 Canonicity and Connectives According to many authors [GLT88,LS86,Pit00], the “canonical ” simply typed λ-calculus possesses the following types: A:: = 0 | A + A | 1 | A × A | A → A (1) There are two variants of this calculus. In some texts [GLT88,LS86] the × connective (type constructor) is a projection product, with elimination rules