66,645 research outputs found
Predicativity, the Russell-Myhill Paradox, and Church's Intensional Logic
This paper sets out a predicative response to the Russell-Myhill paradox of
propositions within the framework of Church's intensional logic. A predicative
response places restrictions on the full comprehension schema, which asserts
that every formula determines a higher-order entity. In addition to motivating
the restriction on the comprehension schema from intuitions about the stability
of reference, this paper contains a consistency proof for the predicative
response to the Russell-Myhill paradox. The models used to establish this
consistency also model other axioms of Church's intensional logic that have
been criticized by Parsons and Klement: this, it turns out, is due to resources
which also permit an interpretation of a fragment of Gallin's intensional
logic. Finally, the relation between the predicative response to the
Russell-Myhill paradox of propositions and the Russell paradox of sets is
discussed, and it is shown that the predicative conception of set induced by
this predicative intensional logic allows one to respond to the Wehmeier
problem of many non-extensions.Comment: Forthcoming in The Journal of Philosophical Logi
Fragments of Frege's Grundgesetze and G\"odel's Constructible Universe
Frege's Grundgesetze was one of the 19th century forerunners to contemporary
set theory which was plagued by the Russell paradox. In recent years, it has
been shown that subsystems of the Grundgesetze formed by restricting the
comprehension schema are consistent. One aim of this paper is to ascertain how
much set theory can be developed within these consistent fragments of the
Grundgesetze, and our main theorem shows that there is a model of a fragment of
the Grundgesetze which defines a model of all the axioms of Zermelo-Fraenkel
set theory with the exception of the power set axiom. The proof of this result
appeals to G\"odel's constructible universe of sets, which G\"odel famously
used to show the relative consistency of the continuum hypothesis. More
specifically, our proofs appeal to Kripke and Platek's idea of the projectum
within the constructible universe as well as to a weak version of
uniformization (which does not involve knowledge of Jensen's fine structure
theory). The axioms of the Grundgesetze are examples of abstraction principles,
and the other primary aim of this paper is to articulate a sufficient condition
for the consistency of abstraction principles with limited amounts of
comprehension. As an application, we resolve an analogue of the joint
consistency problem in the predicative setting.Comment: Forthcoming in The Journal of Symbolic Logi
Designing Normative Theories for Ethical and Legal Reasoning: LogiKEy Framework, Methodology, and Tool Support
A framework and methodology---termed LogiKEy---for the design and engineering
of ethical reasoners, normative theories and deontic logics is presented. The
overall motivation is the development of suitable means for the control and
governance of intelligent autonomous systems. LogiKEy's unifying formal
framework is based on semantical embeddings of deontic logics, logic
combinations and ethico-legal domain theories in expressive classic
higher-order logic (HOL). This meta-logical approach enables the provision of
powerful tool support in LogiKEy: off-the-shelf theorem provers and model
finders for HOL are assisting the LogiKEy designer of ethical intelligent
agents to flexibly experiment with underlying logics and their combinations,
with ethico-legal domain theories, and with concrete examples---all at the same
time. Continuous improvements of these off-the-shelf provers, without further
ado, leverage the reasoning performance in LogiKEy. Case studies, in which the
LogiKEy framework and methodology has been applied and tested, give evidence
that HOL's undecidability often does not hinder efficient experimentation.Comment: 50 pages; 10 figure
Cut-Simulation and Impredicativity
We investigate cut-elimination and cut-simulation in impredicative
(higher-order) logics. We illustrate that adding simple axioms such as Leibniz
equations to a calculus for an impredicative logic -- in our case a sequent
calculus for classical type theory -- is like adding cut. The phenomenon
equally applies to prominent axioms like Boolean- and functional
extensionality, induction, choice, and description. This calls for the
development of calculi where these principles are built-in instead of being
treated axiomatically.Comment: 21 page
The Strength of Abstraction with Predicative Comprehension
Frege's theorem says that second-order Peano arithmetic is interpretable in
Hume's Principle and full impredicative comprehension. Hume's Principle is one
example of an abstraction principle, while another paradigmatic example is
Basic Law V from Frege's Grundgesetze. In this paper we study the strength of
abstraction principles in the presence of predicative restrictions on the
comprehension schema, and in particular we study a predicative Fregean theory
which contains all the abstraction principles whose underlying equivalence
relations can be proven to be equivalence relations in a weak background
second-order logic. We show that this predicative Fregean theory interprets
second-order Peano arithmetic.Comment: Forthcoming in Bulletin of Symbolic Logic. Slight change in title
from previous version, at request of referee
Computer Science and Metaphysics: A Cross-Fertilization
Computational philosophy is the use of mechanized computational techniques to
unearth philosophical insights that are either difficult or impossible to find
using traditional philosophical methods. Computational metaphysics is
computational philosophy with a focus on metaphysics. In this paper, we (a)
develop results in modal metaphysics whose discovery was computer assisted, and
(b) conclude that these results work not only to the obvious benefit of
philosophy but also, less obviously, to the benefit of computer science, since
the new computational techniques that led to these results may be more broadly
applicable within computer science. The paper includes a description of our
background methodology and how it evolved, and a discussion of our new results.Comment: 39 pages, 3 figure
Dependent choice, properness, and generic absoluteness
We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to -preserving symmetric submodels of forcing extensions. Hence, not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in and. Our results confirm as a natural foundation for a significant portion of classical mathematics and provide support to the idea of this theory being also a natural foundation for a large part of set theory
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