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This paper examines the philosophical significance of the consequence relation defined in the $\Omega$-logic for set-theoretic languages. I argue that, as with second-order logic, the hyperintensional profile of validity in $\Omega$-Logic enables the property to be epistemically tractable. Because of the duality between coalgebras and algebras, Boolean-valued models of set theory can be interpreted as coalgebras. In Section \textbf{2}, I demonstrate how the hyperintensional profile of $\Omega$-logical validity can be countenanced within a coalgebraic logic. Finally, in Section \textbf{3}, the philosophical (...) |
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This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) |
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Reinhardt 's strong mechanistic thesis, a formalization of “I know I am a Turing machine”, is shown to be consistent with Epistemic Arithmetic. |
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This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in the framework of first-order (...) |
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This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The categorical duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The hyperintensional profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal and hyperintensional profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely (...) |
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This piece continues the tradition of arguments by John Lucas, Roger Penrose and others to the effect that the human mind is not a machine. Kurt Gödel thought that the intensional paradoxes stand in the way of proving that the mind is not a machine. According to Gödel, a successful proof that the mind is not a machine would require a solution to the intensional paradoxes. We provide what might seem to be a partial vindication of Gödel and show that (...) |
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The psychological orientation treats semantics as a matter of idealized computation over symbolic structures, and semantic relations like denotation as relations between linguistic expressions and these structures. I argue that results similar to Gödel's incompleteness theorems and Tarski's theorem on truth create foundational difficulties for this view of semantics. |
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We propose that, for the purpose of studying theoretical properties of the knowledge of an agent with Artificial General Intelligence (that is, the knowledge of an AGI), a pragmatic way to define such an agent’s knowledge (restricted to the language of Epistemic Arithmetic, or EA) is as follows. We declare an AGI to know an EA-statement φ if and only if that AGI would include φ in the resulting enumeration if that AGI were commanded: “Enumerate all the EA-sentences which you (...) |
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New epistemic principles are formulated in the language of Shapiro's system of Epistemic Arithmetic. It is argued that some plausibility can be attributed to these principles. The relations between these principles and variants of controversial constructivistic principles are investigated. Special attention is given to variants of the intuitionistic version of Church's thesis and to variants of Markov's principle. |
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This paper outlines a framework for the abstract investigation of the concept of canonicity of names and of naming systems. Degrees of canonicity of names and of naming systems are distinguished. The structure of the degrees is investigated, and a notion of relative canonicity is defined. The notions of canonicity are formally expressed within a Carnapian system of second-order modal logic. |
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The Paradox of the Knower was originally presented by Kaplan and Montague [26] as a puzzle about the everyday notion of knowledge in the face of self-reference. The paradox shows that any theory extending Robinson arithmetic with a predicate K satisfying the factivity axiom K → A as well as a few other epistemically plausible principles is inconsistent. After surveying the background of the paradox, we will focus on a recent debate about the role of epistemic closure principles in the (...) |
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The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. (...) |
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The alleged proof of the non-mechanical, or non-computational, character of the human mind based on Gödel’s incompleteness theorem is revisited. Its history is reviewed. The proof, also known as the Lucas argument and the Penrose argument, is refuted. It is claimed, following Gödel himself and other leading logicians, that antimechanism is not implied by Gödel’s theorems alone. The present paper sets out this refutation in its strongest form, demonstrating general theorems implying the inconsistency of Lucas’s arithmetic and the semantic inadequacy (...) |
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We analyse Kreisel’s notion of human-effective computability. Like Kreisel, we relate this notion to a concept of informal provability, but we disagree with Kreisel about the precise way in which this is best done. The resulting two different ways of analysing human-effective computability give rise to two different variants of Church’s thesis. These are both investigated by relating them to transfinite progressions of formal theories in the sense of Feferman. |
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BAT is a logic built to capture the inferential behavior of informal provability. Ultimately, the logic is meant to be used in an arithmetical setting. To reach this stage it has to be extended to a first-order version. In this paper we provide such an extension. We do so by constructing non-deterministic three-valued models that interpret quantifiers as some sorts of infinite disjunctions and conjunctions. We also elaborate on the semantical properties of the first-order system and consider a couple of (...) |
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We systematically study several versions of the disjunction and the existence properties in modal arithmetic. First, we newly introduce three classes $\mathrm {B}$, $\Delta (\mathrm {B})$, and $\Sigma (\mathrm {B})$ of formulas of modal arithmetic and study basic properties of them. Then, we prove several implications between the properties. In particular, among other things, we prove that for any consistent recursively enumerable extension T of $\mathbf {PA}(\mathbf {K})$ with $T \nvdash \Box \bot $, the $\Sigma (\mathrm {B})$ -disjunction property, the (...) |
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This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) |
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Recently, in an ongoing debate about informal provability, non-deterministic logics of informal provability BAT and CABAT were developed to model the notion. CABAT logic is defined as an extension of BAT logics and itself does not have independent and decent semantics. The aim of the paper is to show that, semantically speaking, both logics are rather complex and they can be characterized by neither finitely many valued deterministic semantics nor possible word semantics including neighbourhood semantics. |
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This is a paper for a special issue of Semiotic Studies devoted to Stanislaw Krajewski’s paper. This paper gives some supplementary notes to Krajewski’s on the Anti-Mechanist Arguments based on Gödel’s incompleteness theorem. In Section 3, we give some additional explanations to Section 4–6 in Krajewski’s and classify some misunderstandings of Gödel’s incompleteness theorem related to AntiMechanist Arguments. In Section 4 and 5, we give a more detailed discussion of Gödel’s Disjunctive Thesis, Gödel’s Undemonstrability of Consistency Thesis and the definability (...) |
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Austrian-born Kurt Gödel is widely considered the greatest logician of modern times. It is above all his celebrated incompleteness theorems—rigorous mathematical results about the necessary limits... |
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