Intuitionism

Abstract

To extend this proof interpretation to the other connectives, it is convenient to have the following notation. (a, b) denotes the pairing of constructions, and (c) 0 , (c) 1 are the first and second projections of c.A proof of a conjunction A Ÿ B is a pair (a, b) of proofs such that a : A and b : B.Interpreting the connectives in terms of proofs means that, unlike classical logic, the disjunction has to be effective, one must specify for which of the disjuncts one has a proof. A proof of a disjunction A ⁄ B is a pair (p, q) such that p carries the information which disjunct is shown correct by this proof, and q is the proof of that disjunct. We stipulate that p OE {0, 1}. So if we have (p, q) : A ⁄ B then either p = 0 and q : A, or p = 1 and q : B.The most interesting propositional connective is the implication. Classically, A AE B is true if A is false or B is true, but this cannot be used now as it involves the classical disjunction. Moreover, it assumes that the truth values of A and B are known before one can settle the status of A AE B.Heyting showed that this is asking too much. Consider A = 'there occur twenty 7's in the decimal expansion of p,' and B = 'there occur nineteen 7's in the decimal expansion of p.' ÿA ⁄ B does not hold constructively, but in the proof interpretation, A AE B is obviously correct.It is, because, if we could show the correctness of A, then a simple construction would allow us to show the correctness of B as well. Implication, then, is interpreted in terms of possible proofs: p : A AE B if p transforms each possible proof q : A into a proof p(q) : B.

Error Correction for Discrete Tomography

“…In the proofs of the theorems we use the fact that there is redundancy in the information given by the line sums. Hajdu and Tijdeman [38] pursued an analysis of the redundacy by Van Dalen [39]. Their line sum relation lemma (Lemma 6.1) is the basis of the present paper.…”