258 research outputs found
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
Lewis meets Brouwer: constructive strict implication
C. I. Lewis invented modern modal logic as a theory of "strict implication".
Over the classical propositional calculus one can as well work with the unary
box connective. Intuitionistically, however, the strict implication has greater
expressive power than the box and allows to make distinctions invisible in the
ordinary syntax. In particular, the logic determined by the most popular
semantics of intuitionistic K becomes a proper extension of the minimal normal
logic of the binary connective. Even an extension of this minimal logic with
the "strength" axiom, classically near-trivial, preserves the distinction
between the binary and the unary setting. In fact, this distinction and the
strong constructive strict implication itself has been also discovered by the
functional programming community in their study of "arrows" as contrasted with
"idioms". Our particular focus is on arithmetical interpretations of the
intuitionistic strict implication in terms of preservativity in extensions of
Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years
later
Hilbert's epsilon as an Operator of Indefinite Committed Choice
Paul Bernays and David Hilbert carefully avoided overspecification of
Hilbert's epsilon-operator and axiomatized only what was relevant for their
proof-theoretic investigations. Semantically, this left the epsilon-operator
underspecified. In the meanwhile, there have been several suggestions for
semantics of the epsilon as a choice operator. After reviewing the literature
on semantics of Hilbert's epsilon operator, we propose a new semantics with the
following features: We avoid overspecification (such as right-uniqueness), but
admit indefinite choice, committed choice, and classical logics. Moreover, our
semantics for the epsilon supports proof search optimally and is natural in the
sense that it does not only mirror some cases of referential interpretation of
indefinite articles in natural language, but may also contribute to philosophy
of language. Finally, we ask the question whether our epsilon within our
free-variable framework can serve as a paradigm useful in the specification and
computation of semantics of discourses in natural language.Comment: ii + 73 pages. arXiv admin note: substantial text overlap with
arXiv:1104.244
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