Books by G Aldo Antonelli
Grounded Consequence for Defeasible Logic
Papers by G Aldo Antonelli
Abstract Frege's logicist program requires that arithmetic be reduced to logic. Such a program ha... more Abstract Frege's logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the" neologicist" approach of Hale and Wright. Less attention has been given to Frege's extensionalist program, according to which arithmetic is to be reconstructed in terms of a theory of extensions of concepts. This paper deals just with such a theory.
In this paper, we explore Fregean metatheory, what Frege called the New Science. The New Science ... more In this paper, we explore Fregean metatheory, what Frege called the New Science. The New Science arises in the context of Frege's debate with Hilbert over independence proofs in geometry and we begin by considering their dispute. We propose that Frege's critique rests on his view that language is a set of propositions, each immutably equipped with a truth value (as determined by the thought it expresses), so to Frege it was inconceivable that axioms could even be considered to be other than true. Because of his adherence to this view, Frege was precluded from the sort of metatheoretical considerations that were available to Hilbert; but from this, we shall argue, it does not follow that Frege was blocked from metatheory in toto. Indeed, Frege suggests in Die Grundlagen der Geometrie a metatheoretical method for establishing independence proofs in the context of the New Science. Frege had reservations about the method, however, primarily because of the apparent need to stipulate the logical terms, those terms that must be held invariant to obtain such proofs. We argue that Frege's skepticism on this score is not warranted, by showing that within the New Science a characterization of logical truth and logical constant can be obtained by a suitable adaptation of the permutation argument Frege employs in indicating how to prove independence. This establishes a foundation for Frege's metatheoretical method of which he himself was unsure, and allows us to obtain a clearer understanding of Frege's conception of logic, especially in relation to contemporary conceptions.
Nonmonotonic Logic (Stanford Encyclopedia of Philosophy)
With the aid of a non-standard (but still first-order) cardinality quantifier and an extra-logica... more With the aid of a non-standard (but still first-order) cardinality quantifier and an extra-logical operator representing numerical abstraction, this paper presents a formalization of first-order arithmetic, in which numbers are abstracta of the equinumerosity relation, their properties derived from those of the cardinality quantifier and the abstraction operator.
While second-order quantifiers have long been known to admit non-standard, or "general" interpret... more While second-order quantifiers have long been known to admit non-standard, or "general" interpretations, first-order quantifiers (when properly viewed as predicates of predicates) also allow a kind of interpretation that does not presuppose the full power-set of that interpretation's first-order domain. This paper explores some of the consequences of such "general" interpretations for (unary) first-order quantifiers in a general setting, emphasizing the effects of imposing various further constraints that the interpretation is to satisfy.
Review: Dov M. Gabbay, C. J. Hogger, J. A. Robinson, D. Nute, Handbook of Logic in Artificial Intelligence and Logic Programming, Volume 3, Nonmonotonic Reasoning and Uncertain Reasoning
The Bulletin of Symbolic Logic, 2000
The Bulletin of Symbolic Logic, 2001
One of the most important developments over the last twenty years both in logic and in Artificial... more One of the most important developments over the last twenty years both in logic and in Artificial Intelligence is the emergence of so-called non-monotonic logics. These logics were initially developed by McCarthy [10], McDermott & Doyle [13], and Reiter [17]. Part of the original motivation was to provide a formal framework within which to model cognitive phenomena such as defeasible inference and defeasible knowledge representation, i.e., to provide a formal account of the fact that reasoners can reach conclusions tentatively, reserving the right to retract them in the light of further information.
Artificial Intelligence, Jan 1, 1999
This paper introduces a generalization of Reiter's notion of "extension" for default logic. The m... more This paper introduces a generalization of Reiter's notion of "extension" for default logic. The main difference from the original version mainly lies in the way conflicts among defaults are handled: in particular, this notion of "general extension" allows defaults not explicitly triggered to pre-empt other defaults. A consequence of the adoption of such a notion of extension is that the collection of all the general extensions of a default theory turns out to have a nontrivial algebraic structure. This fact has two major technical fall-outs: first, it turns out that every default theory has a general extension; second, general extensions allow one to define a well-behaved, skeptical relation of defeasible consequence for default theories, satisfying the principles of Reflexivity, Cut, and Cautious Monotonicity formulated by D. Gabbay.
Revision rules: An investigation into non-monotonic inductive definitions
unpublished notes, Jan 1, 1998
The purpose of this note is to acknowledge a gap in a previous paper -"The Complexity of Revision... more The purpose of this note is to acknowledge a gap in a previous paper -"The Complexity of Revision", see [1] -and provide a corrected version of argument. The gap was originally pointed out by Francesco Orilia (personal communication and [4]), and the fix was developed in correspondence with Vann McGee.
Journal of philosophical logic, Jan 1, 2000
This paper presents a bivalent extensional semantics for positive free logic without resorting to... more This paper presents a bivalent extensional semantics for positive free logic without resorting to the philosophically questionable device of using models endowed with a separate domain of "non-existing" objects. The models here introduced have only one (possibly empty) domain, and a partial reference function for the singular terms (that might be undefined at some arguments). Such an approach provides a solution to an open problem put forward by Lambert, and can be viewed as supplying a version of parametrized truth non unlike the notion of "truth at world" found in modal logic. A model theory is developed, establishing compactness, interpolation (implying a strong form of Beth definability), and completeness (with respect to a particular axiomatization).
Rivista di estetica, Jan 1, 2007
Notre Dame Journal of Formal Logic, Jan 1, 2010
This paper presents a formalization of first-order arithmetic characterizing the natural numbers ... more This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a non-standard (but still first-order) cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a non-reductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.
Journal of Symbolic Logic, Jan 1, 2002
A propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p, which... more A propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p, which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.
Virtuous Circles: From Fixed Points to Revision Rules
Chapuis and Gupta, Jan 1, 2000
Philosophia Mathematica, Jan 1, 2010
The logical status of abstraction principles, and especially Hume's Principle, has been long deba... more The logical status of abstraction principles, and especially Hume's Principle, has been long debated, but the best currently availeble tool for explicating a notion's logical characterpermutation invariance -has not received a lot of attention in this debate. This paper aims to fill this gap. After characterizing abstraction principles as particular mappings from the subsets of a domain into that domain and exploring some of their properties, the paper introduces several distinct notions of permutation invariance for such principles, assessing the philosophical significance of each.
The purpose of this note is to present a simplification of the system of arithmetical axioms give... more The purpose of this note is to present a simplification of the system of arithmetical axioms given in previous work; specifically, it is shown how the induction principle can in fact be obtained from the remaining axioms, without the need of explicit postulation. The argument might be of more general interest, beyond the specifics of the proposed axiomatization, as it highlights the interaction of the notion of Dedekind-finiteness and the induction principle.
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Books by G Aldo Antonelli
Papers by G Aldo Antonelli