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Is Complexity a Source of Incompleteness?
 IS COMPLEXITY A SOURCE OF INCOMPLETENESS
, 2004
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Proofs Without Syntax
 Annals of Mathematics
"... [M]athematicians care no more for logic than logicians for mathematics. Augustus de Morgan, 1868 Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatori ..."
Abstract

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[M]athematicians care no more for logic than logicians for mathematics. Augustus de Morgan, 1868 Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graphtheoretic), rather than syntactic. It defines a combinatorial proof of a proposition φ as a graph homomorphism h: C → G(φ), where G(φ) is a graph associated with φ and C is a coloured graph. The main theorem is soundness and completeness: φ is true if and only if there exists a combinatorial proof h: C → G(φ). 1.
What Is Logic?
"... It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form ..."
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It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form the theory of any part of logic seems to be a set of rules of inference. This answer already introduces some structure into a discussion of the nature of logic, for in an inference we can distinguish the input called a premise or premises from the output known as the conclusion. The transition from a premise or a number of premises to the conclusion is governed by a rule of inference. If the inference is in accordance with the appropriate rule, it is called valid. Rules of inference are often thought of as the alpha and omega of logic. Conceiving of logic as the study of inference is nevertheless only the first approximation to the title question, in that it prompts more questions than it answers. It is not clear what counts as an inference or what a theory of such inferences might look like. What are the rules of inference based on? Where do we find them? The ultimate end