Types, Tableaus, and Gödels God

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The Limits of Understanding

An object x that has the God-like property is called God. He used a sort of modal plenitude principle to argue this from the logical consistency of Godlikeness. Note that this property is itself positive, since it is the conjunction of the infinitely many positive properties.


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We also say that x necessarily exists if for every essence P the following is true: in every possible world, there is an element y with P y. Since necessary existence is positive, it must follow from Godlikeness. Moreover, Godlikeness is an essence of God, since it entails all positive properties, and any nonpositive property is the negation of some positive property, so God cannot have any nonpositive properties.

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Since any Godlike object is necessarily existent, it follows that any Godlike object in one world is a Godlike object in all worlds, by the definition of necessary existence. Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required. From these hypotheses, it is also possible to prove that there is only one God in each world: by identity of indiscernibles , no two distinct objects can have precisely the same properties, and so there can only be one object in each world that possesses property G.

This was more to preserve the logical precision of the argument than due to a penchant for polytheism. This uniqueness proof will only work if one supposes that the positiveness of a property is independent of the object to which it is applied, a claim which some have considered to be suspect.

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And it shows if there exists an arbitrary large object of some type, then there must also be an infinite object of that type. It shows e. In our case also it means if there is a finite collection of positive properties then there must exist an infinite collection of positive properties. You need the set of positive properties to include existence. If you don't have that you don't have anything—literally. The only way to get to that is by the set itself having infinite positive properties. In Anderson's system, Axioms 1, 2, and 5 above are unchanged; however the other axioms are replaced with:.

These axioms leave open the possibility that a Godlike object will possess some non-positive properties, provided that these properties are contingent rather than necessary. Wang reads "Baptist Lutheran" where Wang has "baptized Lutheran".

North Holland, Springer: Lecture Notes in Artificial Intelligence, Sobel, — Cambridge University Press, Ursini and P. Agliano, — Dekker, Wahrheit und Beweisbarkeit , edited by B.

Types, Tableaus and Gödel's God in Isabelle/HOL

Buldt et al. Huffman, Brian, and Kuncar, Ondrej. Kirchner, Daniel. Proceedings , edited by Natasha Sharygina and Helmut Veith, — Lowe, Edward Jonathan. Moreland, K. Sweis, and C. Meister, 61— Oxford University Press, MacLane, Saunders. Nipkow, Tobias, Paulson, Lawrence C.

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Oppenheimer, Paul E. Pelletier, Francis J. Schulz, Stephan. Proceedings , edited by Kenneth L. Scott, Dana S. Springer: Lecture Notes in Mathematics, Sobel, Jordan H. MIT Press, Zalta, Edward N. Reidel, Export Citation. This work is licensed under the Creative Commons Attribution 4. Deontic Logic and Contrary-to-Dutiesl J.

Types, Tableaus, and Gödel's God by M. Fitting, Paperback | Barnes & Noble®

Carmo, A. Marti-Oliet, J. Logical Frameworks; D. Basin, S. Proof Theory and Meaning; G. Goal Driected Deductions; D. Gabbay, N. On Negation, Completeness and Consistency; A. Introduction to Mathematical Programming investigates the mathematical structures and principles underlying the design of efficient algorithms for optimization problems.

Recent advances in algorithmic theory have shown that the traditionally separate areas of discrete optimization, linear programming, and nonlinear optimization are closely linked. This book offers a comprehensive introduction to the whole subject and leads the reader to the frontiers of current research. The prerequisites to use the book are very elementary.

All the tools from numerical linear algebra and calculus are fully reviewed and developed. Rather than attempting to be encyclopedic, the book illustrates the important basic techniques with typical problems. The focus is on efficient algorithms with respect to practical usefulness. Algorithmic complexity theory is presented with the goal of helping the reader understand the concepts without having to become a theoretical specialist.

Further theory is outlined and supplemented with pointers to the relevant literature. The book is equally suited for self-study for a motivated beginner and for a comprehensive course on the principles of mathematical programming within an applied mathematics or computer science curriculum at advanced undergraduate or graduate level. The presentation of the material is such that smaller modules on discrete optimization, linear programming, and nonlinear optimization can easily be extracted separately and used for shorter specialized courses on these subjects.


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