3,082 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
Some Turing-Complete Extensions of First-Order Logic
We introduce a natural Turing-complete extension of first-order logic FO. The
extension adds two novel features to FO. The first one of these is the capacity
to add new points to models and new tuples to relations. The second one is the
possibility of recursive looping when a formula is evaluated using a semantic
game. We first define a game-theoretic semantics for the logic and then prove
that the expressive power of the logic corresponds in a canonical way to the
recognition capacity of Turing machines. Finally, we show how to incorporate
generalized quantifiers into the logic and argue for a highly natural
connection between oracles and generalized quantifiers.Comment: In Proceedings GandALF 2014, arXiv:1408.556
Turing jumps through provability
Fixing some computably enumerable theory , the
Friedman-Goldfarb-Harrington (FGH) theorem says that over elementary
arithmetic, each formula is equivalent to some formula of the form
provided that is consistent. In this paper we give various
generalizations of the FGH theorem. In particular, for we relate
formulas to provability statements which
are a formalization of "provable in together with all true
sentences". As a corollary we conclude that each is
-complete. This observation yields us to consider a recursively
defined hierarchy of provability predicates which look a lot
like except that where calls upon the
oracle of all true sentences, the recursively
calls upon the oracle of all true sentences of the form . As such we obtain a `syntax-light' characterization of
definability whence of Turing jumps which is readily extended
beyond the finite. Moreover, we observe that the corresponding provability
predicates are well behaved in that together they provide a
sound interpretation of the polymodal provability logic
The First-Order Theory of Sets with Cardinality Constraints is Decidable
We show that the decidability of the first-order theory of the language that
combines Boolean algebras of sets of uninterpreted elements with Presburger
arithmetic operations. We thereby disprove a recent conjecture that this theory
is undecidable. Our language allows relating the cardinalities of sets to the
values of integer variables, and can distinguish finite and infinite sets. We
use quantifier elimination to show the decidability and obtain an elementary
upper bound on the complexity.
Precise program analyses can use our decidability result to verify
representation invariants of data structures that use an integer field to
represent the number of stored elements.Comment: 18 page
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On the Logic of Belief and Propositional Quantification
We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is some-thing that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss
A Relational Formulation of the Theory of Types
This paper developes a relational---as opposed to a functional---theory of types. The theory is based on Hilbert and Bernays' eta operator plus the identity symbol, from which Church's lambda and the other usual operators are then defined. The logic is intended for use in the semantics of natural language
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