By Dieter Probst, Peter Schuster

This e-book presents the reader with examine bobbing up from the Humboldt-Kolleg 'Proof' held in Bern in fall 2013, which accumulated prime specialists actively concerned with the concept that 'proof' in philosophy, arithmetic and computing device technological know-how. This quantity goals to do justice to the breadth and intensity of the topic and provides suitable present conceptions and technical advances that includes 'proof' in these fields.

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Extra resources for Concepts of Proof in Mathematics, Philosophy, and Computer Science

Sample text

Accordingly, the nonmonotonic semantics for such inference relations will also be based on worlds. The resulting nonmonotonic formalism will provide a complete characterization of the reasoning with causal theories of [McCain and Turner, 1997]. In addition, causal inference relations will be shown to correspond to biconsequence relations in a particular four-valued language. Finally, it will be shown that causal inference relations provide also a natural logical representation for logic programs under a broad range of declarative semantics for the latter.

In what follows, we will always assume that the sequents of a sequent theory do not contain repeated occurrences of propositions. , a h A, A, b is reducible to a \- A, b, any sequent theory can be safely assumed to be of this form. We will begin with the description of supported theories. For any locally . finite sequent theory A, we will define its completion, comp(A), as the set of all classical formulas of the form A^\J{/\(aU^b) | ah 6,A G A}, plus the set {-> /\a\ a\-£ A}. If we will treat propositions of the underlying language as propositional atoms, it should be clear that the classical models of the above classical propositional theory will stand in one-to-one correspondence with certain sets of propositions.

It also forms a mediating link between the preceding formalism of biconsequence relations, and systems of production and causal inference described in the next chapter. Chapter 8, Production and Causal Inference, introduces a first full- Introduction 17 fledged logical formalism of this study that is based on the classical language. Production inference relation is an inference system for causal rules of the form A => B that originates in input/output logics of [Makinson and van der Torre, 2000].