827 research outputs found
On Quine's Ontology: quantification, extensionality and naturalism (or from commitment to indifference)
Much of the ontology made in the analytic tradition of philosophy nowadays is founded on some of Quine’s proposals. His naturalism and the binding between existence and quantification are respectively two of his very influential metaphilosophical and methodological theses. Nevertheless, many of his specific claims are quite controversial and contemporaneously have few followers. Some of them are: (a) his rejection of higher-order logic; (b) his resistance in accepting the intensionality of ontological commitments; (c) his rejection of first-order modal logic; and (d) his rejection of the distinction between analytic and synthetic statements. I intend to argue that these controversial negative claims are just interconnected consequences of those much more accepted and apparently less harmful metaphilosophical and methodological theses, and that the glue linking all these consequences to its causes is the notion of extensionality
Kolmogorov Complexity in perspective. Part II: Classification, Information Processing and Duality
We survey diverse approaches to the notion of information: from Shannon
entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov
complexity are presented: randomness and classification. The survey is divided
in two parts published in a same volume. Part II is dedicated to the relation
between logic and information system, within the scope of Kolmogorov
algorithmic information theory. We present a recent application of Kolmogorov
complexity: classification using compression, an idea with provocative
implementation by authors such as Bennett, Vitanyi and Cilibrasi. This stresses
how Kolmogorov complexity, besides being a foundation to randomness, is also
related to classification. Another approach to classification is also
considered: the so-called "Google classification". It uses another original and
attractive idea which is connected to the classification using compression and
to Kolmogorov complexity from a conceptual point of view. We present and unify
these different approaches to classification in terms of Bottom-Up versus
Top-Down operational modes, of which we point the fundamental principles and
the underlying duality. We look at the way these two dual modes are used in
different approaches to information system, particularly the relational model
for database introduced by Codd in the 70's. This allows to point out diverse
forms of a fundamental duality. These operational modes are also reinterpreted
in the context of the comprehension schema of axiomatic set theory ZF. This
leads us to develop how Kolmogorov's complexity is linked to intensionality,
abstraction, classification and information system.Comment: 43 page
Intensional Models for the Theory of Types
In this paper we define intensional models for the classical theory of types,
thus arriving at an intensional type logic ITL. Intensional models generalize
Henkin's general models and have a natural definition. As a class they do not
validate the axiom of Extensionality. We give a cut-free sequent calculus for
type theory and show completeness of this calculus with respect to the class of
intensional models via a model existence theorem. After this we turn our
attention to applications. Firstly, it is argued that, since ITL is truly
intensional, it can be used to model ascriptions of propositional attitude
without predicting logical omniscience. In order to illustrate this a small
fragment of English is defined and provided with an ITL semantics. Secondly, it
is shown that ITL models contain certain objects that can be identified with
possible worlds. Essential elements of modal logic become available within
classical type theory once the axiom of Extensionality is given up.Comment: 25 page
An Intensional Concurrent Faithful Encoding of Turing Machines
The benchmark for computation is typically given as Turing computability; the
ability for a computation to be performed by a Turing Machine. Many languages
exploit (indirect) encodings of Turing Machines to demonstrate their ability to
support arbitrary computation. However, these encodings are usually by
simulating the entire Turing Machine within the language, or by encoding a
language that does an encoding or simulation itself. This second category is
typical for process calculi that show an encoding of lambda-calculus (often
with restrictions) that in turn simulates a Turing Machine. Such approaches
lead to indirect encodings of Turing Machines that are complex, unclear, and
only weakly equivalent after computation. This paper presents an approach to
encoding Turing Machines into intensional process calculi that is faithful,
reduction preserving, and structurally equivalent. The encoding is demonstrated
in a simple asymmetric concurrent pattern calculus before generalised to
simplify infinite terms, and to show encodings into Concurrent Pattern Calculus
and Psi Calculi.Comment: In Proceedings ICE 2014, arXiv:1410.701
Non‐Classical Knowledge
The Knower paradox purports to place surprising a priori limitations on what we can know. According to orthodoxy, it shows that we need to abandon one of three plausible and widely-held ideas: that knowledge is factive, that we can know that knowledge is factive, and that we can use logical/mathematical reasoning to extend our knowledge via very weak single-premise closure principles. I argue that classical logic, not any of these epistemic principles, is the culprit. I develop a consistent theory validating all these principles by combining Hartry Field's theory of truth with a modal enrichment developed for a different purpose by Michael Caie. The only casualty is classical logic: the theory avoids paradox by using a weaker-than-classical K3 logic.
I then assess the philosophical merits of this approach. I argue that, unlike the traditional semantic paradoxes involving extensional notions like truth, its plausibility depends on the way in which sentences are referred to--whether in natural languages via direct sentential reference, or in mathematical theories via indirect sentential reference by Gödel coding. In particular, I argue that from the perspective of natural language, my non-classical treatment of knowledge as a predicate is plausible, while from the perspective of mathematical theories, its plausibility depends on unresolved questions about the limits of our idealized deductive capacities
Invention, Intension and the Limits of Computation
This is a critical exploration of the relation between two common assumptions in anti-computationalist critiques of Artificial Intelligence: The first assumption is that at least some cognitive abilities are specifically human and non-computational in nature, whereas the second assumption is that there are principled limitations to what machine-based computation can accomplish with respect to simulating or replicating these abilities. Against the view that these putative differences between computation in humans and machines are closely related, this essay argues that the boundaries of the domains of human cognition and machine computation might be independently defined, distinct in extension and variable in relation. The argument rests on the conceptual distinction between intensional and extensional equivalence in the philosophy of computing and on an inquiry into the scope and nature of human invention in mathematics, and their respective bearing on theories of computation
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