Encoding many-valued logic in {\lambda}-calculus

We extend the well-known Church encoding of two-valued Boolean Logic in $\lambda$-calculus to encodings of $n$-valued propositional logic (for $3\leq n\leq 5$) in well-chosen infinitary extensions in $\lambda$-calculus. In case of three-valued logic we use the infinitary extension of the finite $\lambda$-calculus in which all terms have their B\"ohm tree as their unique normal form. We refine this construction for $n\in\{4,5\}$. These $n$-valued logics are all variants of McCarthy's left-sequential, three-valued propositional calculus. The four- and five-valued logic have been given complete axiomatisations by Bergstra and Van de Pol. The encodings of these $n$-valued logics are of interest because they can be used to calculate the truth values of infinitary propositions. With a novel application of McCarthy's three-valued logic we can now resolve Russell's paradox. Since B\"ohm trees are always finite in Church's original $\lambda{\mathbf I}$-calculus, we believe their construction to be within the technical means of Church. Arguably he could have found this encoding of three-valued logic and used it to resolve Russell's paradox.Comment: 15 page

Simulation of annual leaf carbon fluxes and analysis of stand structure of poplars and balck locusts in an alley-cropping system, Brandenburg, Germany

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The Evolution of an Industrial Sector with a Varying Degree of Roundaboutness of Production

The evolutionary model presented in this paper depicts an industrial sector with a varying degree of economic roundaboutness, i.e. vertical division of labour between producers and users of different types of intermediate products that are ultimately used for the production of a single final product. To include this vertical aspect of industrial dynamics, the model adds the concept of production trees to the evolutionary models of Schumpeterian competition. The specification of this concept suggests the use of the notions of graph theory and the related algorithms of computer science in the treatment of industrial novelty, including structural innovations. Although the model is developed within the Nelson and Winter tradition, the introduction of the 'Austrian' issue of roundaboutness implies a major extension of the research agenda, including production- structure innovations, the emergence and functioning of markets for intermediate products, ways of coping with the instability of upstream markets, the spread of the effects of an upstream innovation, and the measurement of the degree of roundaboutness and of overall productivity. The model reflects a Schumpeterian version of the Böhm-Bawerkian vision of the emergence of increased long-term roundaboutness of production. The Schumpeterian approach implies an innovation- and entrepreneur-driven process of vertical disintegration and reintegration.Roundaboutness, production graphs, evolutionary economic modelling, Nelson and Winter

Comparing long-term crop yields of a short rotation alley cropping agroforestry system and of a standard agricultural field in northern Germany

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Glueability of Resource Proof-Structures: Inverting the Taylor Expansion

A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be expanded into a set of resource proof-structures: its Taylor expansion. We introduce a new criterion characterizing those sets of resource proof-structures that are part of the Taylor expansion of some MELL proof-structure, through a rewriting system acting both on resource and MELL proof-structures