6,155 research outputs found
The Relevant Logic E and Some Close Neighbours: A Reinterpretation
This paper has two aims. First, it sets out an interpretation of the relevant logic E of relevant entailment based on the theory of situated inference. Second, it uses this interpretation, together with Anderson and Belnap’s natural deduc- tion system for E, to generalise E to a range of other systems of strict relevant implication. Routley–Meyer ternary relation semantics for these systems are produced and completeness theorems are proven
Weakly complete axiomatization of exogenous quantum propositional logic
A weakly complete finitary axiomatization for EQPL (exogenous quantum
propositional logic) is presented. The proof is carried out using a non trivial
extension of the Fagin-Halpern-Megiddo technique together with three Henkin
style completions.Comment: 28 page
Topic-Sensitive Epistemic 2D Truthmaker ZFC and Absolute Decidability
This paper aims to contribute to the analysis of the nature of mathematical modality, and to the applications of the latter to unrestricted quantification and absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of two-dimensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the priority and relation between epistemic mathematical modality and metaphysical mathematical modality. The discrepancy between the modal systems governing the parameters in the two-dimensional intensional setting provides an explanation of the difference between the metaphysical possibility of absolute decidability and our knowledge thereof. I also advance an epistemic two-dimensional truthmaker semantics, if hyperintenisonal approaches are to be preferred to possible worlds semantics. I examine the relation between epistemic truthmakers and epistemic set theory
Doing and Showing
The persisting gap between the formal and the informal mathematics is due to
an inadequate notion of mathematical theory behind the current formalization
techniques. I mean the (informal) notion of axiomatic theory according to which
a mathematical theory consists of a set of axioms and further theorems deduced
from these axioms according to certain rules of logical inference. Thus the
usual notion of axiomatic method is inadequate and needs a replacement.Comment: 54 pages, 2 figure
Indeterminateness and `The' Universe of Sets: Multiversism, Potentialism, and Pluralism
In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
Relational Quantum Mechanics
I suggest that the common unease with taking quantum mechanics as a
fundamental description of nature (the "measurement problem") could derive from
the use of an incorrect notion, as the unease with the Lorentz transformations
before Einstein derived from the notion of observer-independent time. I suggest
that this incorrect notion is the notion of observer-independent state of a
system (or observer-independent values of physical quantities). I reformulate
the problem of the "interpretation of quantum mechanics" as the problem of
deriving the formalism from a few simple physical postulates. I consider a
reformulation of quantum mechanics in terms of information theory. All systems
are assumed to be equivalent, there is no observer-observed distinction, and
the theory describes only the information that systems have about each other;
nevertheless, the theory is complete.Comment: Substantially revised version. LaTeX fil
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