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Diamonds are a Philosopher's Best Friends. The Knowability Paradox and Modal Epistemic Relevance Logic (Extended Abstract)
 Journal of Philosophical Logic
, 2002
"... Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the ..."
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Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is derived that all truths are, in fact, known. Nevertheless, the solution o#ered is in the spirit of the constructivist attitude usually maintained by defenders of the antirealist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete.
LambdaCalculus and Functional Programming tions.
"... The lambdacalculus is a formalism for representing funcBy the second half of the nineteenth century, the concept of function as used in mathematics had reached the point at which the standard notation had become ambiguous. For example, consider the operator P defined on real functions as follows: ..."
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The lambdacalculus is a formalism for representing funcBy the second half of the nineteenth century, the concept of function as used in mathematics had reached the point at which the standard notation had become ambiguous. For example, consider the operator P defined on real functions as follows: ⎧f(x) – f(0) for x 0 P[f(x)] = ⎨ x ⎩f ′(0) for x = 0 What is P[f(x + 1)]? To see that this is ambiguous, let f(x) = x 2. Then if g(x) = f(x + 1), P[g(x)] = P[x 2 + 2x + 1] = x + 2. But if h(x) = P[f(x)] = x, then h(x + 1) = x + 1 P[g(x)]. This ambiguity has actually led to an error in the published literature; see the discussion in (Curry and Feys