A (Simplified) Supreme Being Necessarily Exists, says the Computer: Computationally Explored Variants of Gödel's Ontological Argument (original) (raw)
Related papers
A collection of papers from Paul Hertz to Dov Gabbay - through Tarski, Gödel, Kripke - giving a general perspective about logical systems. These papers discuss questions such as the relativity and nature of logic, present tools such as consequence operators and combinations of logics, prove theorems such as translations between logics, investigate the domain of validity and application of fundamental results such as compactness and completeness. Each of these papers is presented by a specialist explaining its context, import and influence.
A fundamental non-classical logic
We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations.
Expanding the Universe of Universal Logic (Forthcoming, Theoria, Fall 2014)
In [5], Béziau provides a means by which Gentzen’s sequent calculus can be combined with the general semantic theory of bivaluations. In doing so, according to Béziau, it is possible to construe the abstract “core” of logics in general, where logical syntax and semantics are “two sides of the same coin”. The central suggestion there is that, by way of a modification of the notion of maximal consistency, it is possible to prove the soundness and completeness for any normal logic (without invoking the role of classical negation in the completeness proof). However, the reduction to bivaluation may be a side effect of the architecture of ordinary sequents, which is both overly restrictive, and entails certain expressive restrictions over the language. This paper provides an expansion of Béziau’s completeness results for logics, by showing that there is a natural extension of that line of thinking to n-sided sequent constructions. Through analogical techniques to Béziau’s construction, it is possible, in this setting, to construct abstract soundness and completeness results for n-valued logics.
2008
I. Formal Theory of the Subject \ II. Great Logic 1: The Transcendental \ III. Great Logic 2: The Object \ IV. Great Logic 3: The Relation \ V. The Four Forms of Change \ VI. Theories of Points \ VII. What is a Body? \ Conclusion \ Appendices \ Bibliography
Handbook of the 5th World Congress and School on Universal Logic
Handbook of the 5th World Congress and School on Universal Logic, 2015
Handbook of the 5th World Congress and School on Universal Logic, Ed: Jean-Yves Beziau, Şafak Ural, Arthur Buchsbaum, İskender Taşdelen, Vedat Kamer, Mantık Derneği Yayınları, İstanbul, 2015, ss. 409. ISBN: 978-605-66311-0-8 PDF: http://bit.ly/hunilog2015
On the Semantics of Higher-order Logic
The question of the semantic interpretation of higher-order logics has long been a matter of contention. Even though second-order quantification is quite natural, entangled interpretations have famously caused philosophers of logic such as Quine to reject second-order logic completely. In this paper I take a liberal attitude, open to maximizing the scope of logic, but careful to avoid conflation with other disciplines – and to avoid epistemological confusion. Higher-order logic (HOL) is perfectly acceptable, but one should be careful as to which semantics deserves to be called " standard ".
A guide to quantified propositional Gödel logic
2001
Gödel logic is a non-classical logic which naturally turns up in a number of different areas within logic and computer science. By choosing subsets of the unit interval [0, 1] as the underlying set of truthvalues many different Gödel logics have been defined. Unlike in classical logic, adding propositional quantifiers to Gödel logics in many cases increases the expressive power of the logic, and motivates thorough investigation. In a series of recent papers , we have started a research program to investigate quantified Gödel logics in a systematic manner. In this paper, we survey the results obtained so far. In the conclusion, we outline the future directions of this research program.