WVOQ #65 Willard Van Orman Quine - cleancut

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http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Quine.html - Mechanisation of these results could provide benefits to the community, allowing proposed improvements in algorithms to be formally assessed in terms of completeness preservation. Page 15 15 We alluded to the use of the...

WVOQ #62 Willard Van Orman Quine - cleancut

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http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Quine.html - A more abstract approach would be to apply the techniques of LCF, a logic which explicitly deals with termination and recursive definitions. Currently, there is a "leap of faith" [Har95] required to bridge the gap between...

WVOQ #61 Willard Van Orman Quine - cleancut

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http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Quine.html - mechanisation in this area is the work of Shankar on mechanising Godels incompleteness theorem [Sha94]. Page 14 14 5 Conclusion and Future Work We have presented a deterministic system of first order logic, proved soundness...

WVOQ #54 Willard Van Orman Quine - cleancut

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http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Quine.html - OCaML definition. lemma prove : prove s = (if s = [] then True else prove (step s)) At this point, we believe that if the OCaML function terminates with true, then the derivation is finite. Conversely, we believe that if the...

WVOQ #52 Willard Van Orman Quine - cleancut

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http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Quine.html - lemma finite-deriv, if the derivation is finite, then there is some n such that the nth iteration is empty, and the algorithm will terminate. However, if the derivation is not finite, then our algorithm will fail to...

WVOQ #50 Willard Van Orman Quine - cleancut

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http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Quine.html - all sequents at depth n, to obtain the list of sequents at depth n + 1. If there are no sequents at a given depth, then all branches have been closed and we have found our finite derivation. We define a global version of the...

WVOQ #49 Willard Van Orman Quine - cleancut

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http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Quine.html - s get false, it too must get false in the model. We have thus found our falsifying model, and s could never have been proved using any sound system of rules. The completeness lemma is as follows. lemma completeness: infinite...

WVOQ #29 Willard Van Orman Quine - cleancut

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http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Quine.html - (at least one of) P or Q gets false in the model, so P ∧ Q gets false in the model. Having shown soundness and completeness for our system, we can derive an algorithm for first order validity: the algorithm simply takes the...

WVOQ #25 Willard Van Orman Quine - cleancut

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http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Quine.html - We say that a formula A is tagged with n, and write A n , when talking about the numbered formula (A, n). Sequents ⊣ Γ are lists Γ of numbered formulae. Initially every formula in a sequent is tagged with 0. Any...

WVOQ #22 Willard Van Orman Quine - cleancut

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http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Quine.html - much weaker than HOL. This work is also interesting from an aesthetic standpoint, in that it nicely combines the areas of mathematics, metamathematics, logic, and algorithms. In the following sections, we give all the...

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