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Switch to: Citations
References in:
Causal interpretation of Gödel's ontological proof
In Kordula Świętorzecka (ed.), Gödel's Ontological Argument: History, Modifications, and Controversies. Semper. pp. 163.201 (2015)
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We want to show that Aristotle’s general conception of syllogism includes as its essential part the logical concept of necessity, which can be understood in a causal way. This logical conception of causality is more general then the conception of the causality in the Aristotelian theory of proof (“demonstrative syllogism”), which contains the causal account of knowledge and science outside formal logic. Aristotle’s syllogistic is described in a purely intensional way, without recourse to a set-theoretical formal semantics. It is shown (...) |
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Aristotelian causal theories incorporate some philosophically important features of the concept of cause, including necessity and essential character. The proposed formalization is restricted to one-place predicates and a finite domain of attributes (without individuals). Semantics is based on a labeled tree structure, with truth defined by means of tree paths. A relatively simple causal prefixing mechanism is defined, by means of which causes of propositions and reasoning with causes are made explicit. The distinction of causal and factual explanation are elaborated, (...) |
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This is a wide-ranging 2004 book about arguments for and against beliefs in God. The arguments for the belief are analysed in the first six chapters and include ontological arguments from Anselm to Gödel, the cosmological arguments of Aquinas and Leibniz, and arguments from evidence for design and miracles. The next two chapters consider arguments against belief. The last chapter examines Pascalian arguments for and against belief in God. There are discussions of Cantorian problems for omniscience, of challenges to divine (...) |
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In his text 'The modern development of the foundations of mathematics in the light of philosophy' from around 1961, Go¨del announces a turn to Husserl's phenomenology to find the foundations of mathematics. In Go¨del's archive there are two draft letters that shed some further light on the exact strategy that he formulated for himself in the early 1960s. Transcriptions of these letters are presented, together with some comments. |
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We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t: F that read t is a justification for F. Justification Logic absorbs basic principles originating from both mainstream epistemology and the mathematical theory of proofs. It contributes to the studies of the well-known Justified True Belief vs. Knowledge problem. We state a general Correspondence Theorem showing that behind each epistemic modal logic, there is a (...) |
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It is a widely known but little considered fact that Albert Einstein and Kurt Gödel were best friends for the last decade and a half of Einstein's life. The two walked home together from Princeton's Institute for Advanced Study every day; they shared ideas about physics, philosophy, politics, and the lost world of German science in which they had grown up. By 1949, Gödel had produced a remarkable proof: In any universe described by the Theory of Relativity, time cannot exist (...) |
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We describe a KB Gödelian ontological system, and some other weak systems, in a fully formal way using theory of types and natural deduction, and present a completeness proof in its main and specific parts. We technically and philosophically analyze and comment on the systems (mainly with respect to the relativism of values) and include a sketch of some connected aspects of Gödel's relation to Kant. |
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Gödel's philosophical views were to a significant extent influenced by the study not only of Leibniz or Husserl, but also of Kant. Both Gödel and Kant aimed at the secure foundation of philosophy, the certainty of knowledge and the solvability of all meaningful problems in philosophy. In this paper, parallelisms between the foundational crisis of metaphysics in Kant's view and the foundational crisis of mathematics in Gödel's view are elaborated, especially regarding the problem of finding the “secure path of a (...) |
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A new semantics is presented for the logic of proofs (LP), [1, 2], based on the intuition that it is a logic of explicit knowledge. This semantics is used to give new proofs of several basic results concerning LP. In particular, the realization of S4 into LP is established in a way that carefully examines and explicates the role of the + operator. Finally connections are made with the conventional approach, via soundness and completeness results. |
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A propositional logic of explicit proofs, LP, was introduced in [S. Artemov, Explicit provability and constructive semantics, The Bulletin for Symbolic Logic 7 1–36], completing a project begun long ago by Gödel, [K. Gödel, Vortrag bei Zilsel, translated as Lecture at Zilsel’s in: S. Feferman , Kurt Gödel Collected Works III, 1938, pp. 62–113]. In fact, LP can be looked at in a more general way, as a logic of explicit evidence, and there have been several papers along these lines. (...) |
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Causal slingshots are formal arguments advanced by proponents of an event ontology of token-level causation which, in the end, are intended to show two things: (i) The logical form of statements expressing causal dependencies on token level features a binary predicate ‘‘... causes ...’’ and (ii) that predicate takes events as arguments. Even though formalisms are only revealing with respect to the logical form of natural language statements, if the latter are shown to be adequately captured within a corresponding formalism, (...) |
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The book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, (...) |
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