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  1. G. Aldo Antonelli, First-Order Quantifiers.
    In §21 of Grundgesetze der Arithmetik asks us to consider the forms: a a2 = 4 and a a > 0 and notices that they can be obtained from a φ(a) by replacing the function-name placeholder φ(ξ) by names for the functions ξ2 = 4 and ξ > 0 (and the placeholder cannot be replaced by names of objects or of functions of 2 arguments).
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  2. G. Aldo Antonelli (2013). On the General Interpretation of First-Order Quantifiers. Review of Symbolic Logic 6 (4):637-658.
    While second-order quantifiers have long been known to admit nonstandard, or 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 interpretationgeneral” interpretations for (unary) first-order quantifiers in a general setting, emphasizing the effects of imposing various further constraints that the interpretation is to satisfy.
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  3. G. Aldo Antonelli (2012). A Note on Induction, Abstraction, and Dedekind-Finiteness. Notre Dame Journal of Formal Logic 53 (2):187-192.
    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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  4. G. Aldo Antonelli (2012). Frege's Theorem. International Studies in the Philosophy of Science 26 (2):219-222.
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  5. G. Aldo Antonelli, Laurent Bienvenu, Lou van den Dries, Deirdre Haskell, Justin Moore, Christian Rosendal Uic, Neil Thapen & Simon Thomas (2012). University of California at Berkeley Berkeley, CA, USA March 24–27, 2011. Bulletin of Symbolic Logic 18 (2).
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  6. G. Aldo Antonelli (2010). Numerical Abstraction Via the Frege Quantifier. Notre Dame Journal of Formal Logic 51 (2):161-179.
    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 nonstandard cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist 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.
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  7. G. Aldo Antonelli (2010). The Nature and Purpose of Numbers. Journal of Philosophy 107 (4):191-212.
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  8. G. Aldo Antonelli, Non-Monotonic Logic. Stanford Encyclopedia of Philosophy.
    The term "non-monotonic logic" covers a family of formal frameworks devised to capture and represent defeasible inference , i.e., that kind of inference of everyday life in which reasoners draw conclusions tentatively, reserving the right to retract them in the light of further information. Such inferences are called "non-monotonic" because the set of conclusions warranted on the basis of a given knowledge base does not increase (in fact, it can shrink) with the size of the knowledge base itself. This is (...)
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  9. G. Aldo Antonelli (2007). Free Quantification and Logical Invariance. Rivista di Estetica 33 (1):61-73.
    Henry Leonard and Karel Lambert first introduced so-called presupposition-free (or just simply: free) logics in the 1950’s in order to provide a logical framework allowing for non-denoting singular terms (be they descriptions or constants) such as “the largest prime” or “Pegasus” (see Leonard [1956] and Lambert [1960]). Of course, ever since Russell’s paradigmatic treatment of definite descriptions (Russell [1905]), philosophers have had a way to deal with such terms. A sentence such as “the..
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  10. G. Aldo Antonelli (2005). Grounded Consequence for Defeasible Logic. Cambridge University Press.
    This is a title on the foundations of defeasible logic, which explores the formal properties of everyday reasoning patterns whereby people jump to conclusions, reserving the right to retract them in the light of further information. Although technical in nature the book contains sections that outline basic issues by means of intuitive and simple examples. This book is primarily targeted at philosophers interested in the foundations of defeasible logic, logicians, and specialists in artificial intelligence and theoretical computer science.
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  11. G. Aldo Antonelli & Robert C. May (2005). Frege's Other Program. Notre Dame Journal of Formal Logic 46 (1):1-17.
    Frege’s logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the “neo-logicist” approach of Hale & 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. We present a system of second-order logic augmented with a predicate representing the fact that an object x is the extension of (...)
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  12. G. Aldo Antonelli (2004). Logic. In Luciano Floridi (ed.), The Blackwell Guide to the Philosophy of Computing and Information. Blackwell.
    Logic is an ancient discipline that, ever since its inception some 2500 years ago, has been concerned with the analysis of patterns of valid reasoning. Aristotle first developed the theory of the syllogism (a valid argument form involving predicates and quantifiers), and later the Stoics singled out patterns of propositional argumentation (involving sentential connectives). The study of logic flourished in ancient times and during the middle ages, when logic was regarded, together with grammar and rhetoric (the other two disciplines of (...)
     
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  13. G. Aldo Antonelli (2002). The Complexity of Revision, Revised. Notre Dame Journal of Formal Logic 43 (2):75-78.
    The purpose of this note is to acknowledge a gap in a previous paper, "The complexity of revision," and to provide a corrected version of the argument.
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  14. G. Aldo Antonelli & Richmond H. Thomason (2002). Representability in Second-Order Propositional Poly-Modal Logic. Journal of Symbolic Logic 67 (3):1039-1054.
    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.
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  15. G. Aldo Antonelli (2001). Introduction. Topoi 20 (1):1-3.
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  16. G. Aldo Antonelli (2001). Review: Solomon Feferman, In the Light of Logic. [REVIEW] Bulletin of Symbolic Logic 7 (2):270-277.
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  17. S. Feferman & G. Aldo Antonelli (2001). REVIEWS-Articles in In the Light of Logic. Bulletin of Symbolic Logic 7 (2):270-276.
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  18. G. Aldo Antonelli (2000). Book Review. [REVIEW] Grazer Philosophische Studien 60:217-28.
    Like Elvis, logical empiricism has been officially dead for decades. But just like Elvis, it stubbornly keeps resurfacing at one juncture or another in our philosophical landscape. In fact, the more the main characters of logical empiricism recede in the distance, the more frequently they reappear, to the point that it’s fair to say that we are witnessing a veritable renaissance in studies leading to the historical appraisal of the import and influence of the logical empiricist movement.
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  19. G. Aldo Antonelli (2000). Book Review To Appear in the Bulletin of Symbolic Logic. [REVIEW] Bulletin of Symbolic Logic 6 (4):480-84.
    The emergence, over the last twenty years or so, of so-called “non-monotonic” logics represents one of the most significant developments both in logic and artificial intelligence. These logics were devised in order to represent defeasible reasoning, i.e., that kind of inference in which reasoners draw conclusions tentatively, reserving the right to retract them in the light of further evidence.
     
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  20. G. Aldo Antonelli (2000). Handbook of Logic in Artificial Intelligence and Logic Programming, Volume 3, Nonmonotonic Reasoning and Uncertain Reasoning, Edited by Gabbay Dov M., Hogger CJ, and Robinson JA, with Nute D., Handbooks of Logic in Computer Science and Artificial Intelligence and Logic Programming, Clarendon Press, Oxford University Press, Oxford, New York, Etc., 1994, Xix+ 529 Pp.–. [REVIEW] Bulletin of Symbolic Logic 6 (4):480-484.
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  21. G. Aldo Antonelli (2000). Origins of Logical Empiricism. Grazer Philosophische Studien 60:217-228.
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  22. G. Aldo Antonelli (2000). Proto-Semantics for Positive Free Logic. Journal of Philosophical Logic 29 (3):277-294.
    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 (...)
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  23. G. Aldo Antonelli (2000). 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. [REVIEW] Bulletin of Symbolic Logic 6 (4):480-484.
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  24. G. Aldo Antonelli (2000). Virtuous Circles: From Fixed Points to Revision Rules. In Anil Gupta & Andre Chapuis (eds.), Circularity, Definition, and Truth. Indian Council of Philosophical Research. 1--27.
     
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  25. G. Aldo Antonelli & Robert C. May (2000). Frege's New Science. Notre Dame Journal of Formal Logic 41 (3):242-270.
    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 (...)
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  26. G. Aldo Antonelli (1999). A Directly Cautious Theory of Defeasible Consequence for Default Logic Via the Notion of General Extension. Artificial Intelligence 109 (1-2):71-109.
    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 (...)
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  27. G. Aldo Antonelli (1999). Conceptions and Paradoxes of Sets. Philosophia Mathematica 7 (2):136-163.
    This paper is concerned with the way different axiom systems for set theory can be justified by appeal to such intuitions as limitation of size, predicativity, stratification, etc. While none of the different conceptions historically resulting from the impetus to provide a solution to the paradoxes turns out to rest on an intuition providing an unshakeable foundation,'each supplies a picture of the set-theoretic universe that is both useful and internally well motivated. The same is true of more recently proposed axiom (...)
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  28. G. Aldo Antonelli (1999). Free Set Algebras Satisfying Systems of Equations. Journal of Symbolic Logic 64 (4):1656-1674.
    In this paper we introduce the notion of a set algebra S satisfying a system E of equations. After defining a notion of freeness for such algebras, we show that, for any system E of equations, set algebras that are free in the class of structures satisfying E exist and are unique up to a bisimulation. Along the way, analogues of classical set-theoretic and algebraic properties are investigated.
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  29. G. Aldo Antonelli (1996). Defeasible Reasoning as a Cognitive Model. In Krister Segerberg (ed.), The Parikh Project. Seven Papers in Honour of Rohit. Uppsala Prints & Preprints in Philosophy.
    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 (...)
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  30. G. Aldo Antonelli (1994). A Revision-Theoretic Analysis of the Arithmetical Hierarchy. Notre Dame Journal of Formal Logic 35 (2):204-218.
    In this paper we apply the idea of Revision Rules, originally developed within the framework of the theory of truth and later extended to a general mode of definition, to the analysis of the arithmetical hierarchy. This is also intended as an example of how ideas and tools from philosophical logic can provide a different perspective on mathematically more “respectable” entities. Revision Rules were first introduced by A. Gupta and N. Belnap as tools in the theory of truth, and they (...)
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  31. G. Aldo Antonelli (1992). Revision Rules: An Investigation Into Non-Monotonic Inductive Definitions. Dissertation, University of Pittsburgh
    Many different modes of definition have been proposed over time, but none of them allows for circular definitions, since, according to the prevalent view, the term defined would then be lacking a precise signification. I argue that although circular definitions may at times fail uniquely to pick out a concept or an object, sense still can be made of them by using a rule of revision in the style adopted by Anil Gupta and Nuel Belnap in the theory of truth.
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