Edward Zalta

The views of David Lewis and the Meinongians are both often met with an incredulous stare. This is not by accident. The stunned disbelief that usually accompanies the stare is a natural first reaction to a large ontology. Indeed, Lewis has been explicitly linked with Meinong, a charge that he has taken great pains to deny. However, the issue is not a simple one. "Meinongianism" is a complex set of distinctions and doctrines about existence and predication, in addition to the famously large ontology. While there are clearly non-Meinongian features of Lewis' views, it is our thesis that many of the characteristic elements of Meinongian metaphysics appear in Lewis' theory. Moreover, though Lewis rejects incomplete and inconsistent Meinongian objects, his ontology may exceed that of a Meinongian who doesn't accept his possibilia. Thus, Lewis explains the truth of "there might have been talking donkeys" by appealing to possibilia which are talking donkeys. But the Meinongian need not accept that there exist things which are talking donkeys. Indeed, we show that a Meinongian even need not accept that there are nonexistent things which are talking donkeys!

The simplest quantified modal logic combines classical quantification theory with the propositional modal logic K. The models of simple QML relativize predication to possible worlds and treat the quantifier as ranging over a single fixed domain of objects. But this simple QML has features that are objectionable to actualists. By contrast, Kripke-models, with their varying domains and restricted quantifiers, seem to eliminate these features. But in fact, Kripke-models also have features to which actualists object. Though these philosophers have introduced variations on Kripke-models to eliminate their objectionable features, the most well-known variations all have difficulties of their own. The present authors reexamine simple QML and discover that, in addition to having a possibilist interpretation, it has an actualist interpretation as well. By introducing a new sort of existing abstract entity, the contingently nonconcrete, they show that the seeming drawbacks of the simplest QML are not drawbacks at all. Thus, simple QML is independent of certain metaphysical questions

In "Actualism or Possibilism?" (Philosophical Studies, 84 (2-3), December 1996), James Tomberlin develops two challenges for actualism. The challenges are to account for the truth of certain sentences without appealing to merely possible objects. After canvassing the main actualist attempts to account for these phenomena, he then criticizes the new conception of actualism that we described in our paper "In Defense of the Simplest Quantified Modal Logic" (Philosophical Perspectives 8: Philosophy of Logic and Language, Atascadero, CA: Ridgeview, 1994). We respond to Tomberlin's criticism by showing that we wouldn't analyze the problematic claim (e.g., "Ponce de Leon searched for the fountain of youth") in the way he suggests.

In this paper, the authors describe their initial investigations in computational metaphysics. Our method is to implement axiomatic metaphysics in an automated reasoning system. In this paper, we describe what we have discovered when the theory of abstract objects is implemented in PROVER9 (a first-order automated reasoning system which is the successor to OTTER). After reviewing the second-order, axiomatic theory of abstract objects, we show (1) how to represent a fragment of that theory in PROVER9's first-order syntax, and (2) how PROVER9 then finds proofs of interesting theorems of metaphysics, such as that every possible world is maximal. We conclude the paper by discussing some issues for further research.

Mathematical pluralism can take one of three forms: (1) every consistent mathematical theory consists of truths about its own domain of individuals and relations; (2) every mathematical theory, consistent or inconsistent, consists of truths about its own (possibly uninteresting) domain of individuals and relations; and (3) the principal philosophies of mathematics are each based upon an insight or truth about the nature of mathematics that can be validated. (1) includes the multiverse approach to set theory. (2) helps us to understand the significance of the distinguished non‐logical individual and relation terms of even inconsistent theories. (3) is a metaphilosophical form of mathematical pluralism and hasn't been discussed in the literature. In what follows, I show how the analysis of theoretical mathematics in object theory exhibits all three forms of mathematical pluralism.

Formulations of Anselm’s ontological argument have been the subject of a number of recent studies. We examine these studies in light of Anselm’s text and (a) respond to criticisms that have surfaced in reaction to our earlier representations of the argument, (b) identify and defend a more refined representation of Anselm’s argument on the basis of new research, and (c) compare our representation of the argument, which analyzes that than which none greater can be conceived as a definite description, to a representation that analyzes it as an arbitrary name.

In a recent book, K. Lambert argues that philosophers should adopt Mally's Principle of Independence (the principle that an object can have properties even though it lacks being of any kind) by abandoning a constraint on true predications, namely, that all of the singular terms in a true predication denote objects which have being. The constraint may be abandoned either by supposing there is a true predication in which one of the terms denotes a beingless object (Meinong) or by supposing there is a true predication in which one of the terms denotes nothing at all (free logic). However, Lambert's conclusions can be undermined by showing that the data he produces in support of his position can be explained by either of two recent theories of abstract and nonexistent objects, both of which are couched in languages which conform to the traditional constraint.

Principia Logico-Metaphysica contains a foundational logical theory for metaphysics, mathematics, and the sciences. It includes a canonical development of Abstract Object Theory [AOT], a metaphysical theory that distinguishes between ordinary and abstract objects.This article reports on recent work in which AOT has been successfully represented and partly automated in the proof assistant system Isabelle/HOL. Initial experiments within this framework reveal a crucial but overlooked fact: a deeply-rooted and known paradox is reintroduced in AOT when the logic of complex terms is simply adjoined to AOT’s specially formulated comprehension principle for relations. This result constitutes a new and important paradox, given how much expressive and analytic power is contributed by having the two kinds of complex terms in the system. Its discovery is the highlight of our joint project and provides strong evidence for a new kind of scientific practice in philosophy, namely, computational metaphysics.Our results were made technically possible by a suitable adaptation of Benzmüller’s metalogical approach to universal reasoning by semantically embedding theories in classical higher-order logic. This approach enables one to reuse state-of-the-art higher-order proof assistants, such as Isabelle/HOL, for mechanizing and experimentally exploring challenging logics and theories such as AOT. Our results also provide a fresh perspective on the question of whether relational type theory or functional type theory better serves as a foundation for logic and metaphysics.

In this presentation, I first outline three different notions of concepts: one derives from Leibniz, while the other two derive from Frege. The Leibnizian notion is the subject of his “calculus of concepts” (which is really an algebra). One notion of concept from Frege is what we would call a “property”, so that when Frege says “x falls under the concept F”, we would say “x instantiates F” or “x exemplifies F”. The other notion of concept from Frege is that of the notion of sense, which played various roles within Frege's theory. This notion of concept can be generalized and, as such, accounts for our intuitive talk of “x's concept of …”, where the ellipsis can be filled in with a name for individual, a property, or a relation, etc. After outlining these three notions, I then discuss how (axiomatic) object theory offers a distinct, precise regimentation of each of the three notions.

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.

Principia Logico-Metaphysica contains a foundational logical theory for metaphysics, mathematics, and the sciences. It includes a canonical development of Abstract Object Theory [AOT], a metaphysical theory that distinguishes between ordinary and abstract objects. This article reports on recent work in which AOT has been successfully represented and partly automated in the proof assistant system Isabelle/HOL. Initial experiments within this framework reveal a crucial but overlooked fact: a deeply-rooted and known paradox is reintroduced in AOT when the logic of complex terms is simply adjoined to AOT’s specially formulated comprehension principle for relations. This result constitutes a new and important paradox, given how much expressive and analytic power is contributed by having the two kinds of complex terms in the system. Its discovery is the highlight of our joint project and provides strong evidence for a new kind of scientific practice in philosophy, namely, computational metaphysics. Our results were made technically possible by a suitable adaptation of Benzmüller’s metalogical approach to universal reasoning by semantically embedding theories in classical higher-order logic. This approach enables one to reuse state-of-the-art higher-order proof assistants, such as Isabelle/HOL, for mechanizing and experimentally exploring challenging logics and theories such as AOT. Our results also provide a fresh perspective on the question of whether relational type theory or functional type theory better serves as a foundation for logic and metaphysics.

Our computational metaphysics group describes its use of automated reasoning tools to study Leibniz’s theory of concepts. We start with a reconstruction of Leibniz’s theory within the theory of abstract objects (henceforth ‘object theory’). Leibniz’s theory of concepts, under this reconstruction, has a non-modal algebra of concepts, a concept-containment theory of truth, and a modal metaphysics of complete individual concepts. We show how the object-theoretic reconstruction of these components of Leibniz’s theory can be represented for investigation by means of automated theorem provers and finite model builders. The fundamental theorem of Leibniz’s theory is derived using these tools.

In this paper, the authors briefly summarize how object theory uses definite descriptions to identify the denotations of the individual terms of theoretical mathematics and then further develop their object-theoretic philosophy of mathematics by showing how it has the resources to address some objections recently raised against the theory. Certain ‘canonical’ descriptions of object theory, which are guaranteed to denote, correctly identify mathematical objects for each mathematical theory T, independently of how well someone understands the descriptive condition. And to have a false belief about some particular mathematical object is not to have a true belief about some different mathematical object.

Current versions of nominalism in the philosophy of mathematics have the benefit of avoiding commitment to the existence of mathematical objects. But this comes with the cost of not taking mathematical theories literally. Jody Azzouni's _Deflating Existential Consequence_ has recently challenged this conclusion by formulating a nominalist view that lacks this cost. In this paper, we argue that, as it stands, Azzouni's proposal does not yet succeed. It faces a dilemma to the effect that either the view is not nominalist or it fails to take mathematics literally. After presenting the dilemma, we suggest a possible solution for the nominalist.

An axiomatic theory of abstract objects is developed and used to construct models of Plato's Forms, Leibniz's Monads, Possible Worlds, Frege's Senses, stories, and fictional characters. The theory takes six primitive metaphysical notions: object ; n-place relations ,G,...); x,...x exemplify F x...x); x exists ; it is necessary that "); and x encodes F "). Properties and propositions are one place and zero place relations, respectively.objects are objects which necessarily fail to exist E!x"). The two most important proper axioms are that no possibly existing object encodes any properties E!x F)xF)), and for every expressible condition on properties, there is an abstract object which encodes just the properties satisfying the condition x) )), where has no free x's). Semantically, an abstract object encodes a property iff the property is an element of the set of properties correlated with the object. Two abstract objects will be identical just in case they encode the same properties. Abstract objects may exemplify properties as well--being non-red, not having spatial location, etc. ;The models of Forms, Monads, and Possible Worlds consist in: identifying these philosophical entities as different species of abstract objects through definitions of the object language, and either showing that there are consequences of the theory which are reasonable facsimiles of assertions made by important philosophers or showing that there are many interesting theorems which capture our own intuitions about these entities . The model for Frege's Senses consists in: developing a language in which the abstract objects generated by the theory serve as the senses of names and descriptions, and showing that the data may be consistently translated into the language preserving truth. Finally, the model for fictional characters consists in: assigning abstract objects to serve as the denotations of names of stories and characters, and showing that such an assignment allows us to understand how the intuitively true sentences about stories and characters have the truth value that they do. ;The material is structured as follows. In Chapters I, III, and V, the elementary, modal, and typed theories of abstract objects are developed, respectively. In Chapters II, IV, and VI, these theories are applied, respectively. Each of the theoretical chapters is subdivided as follows: 1--The Language; 2--The Semantics; 3--The Logic; and 4--The Proper Axioms. Thus, the proper axioms of the theory are couched in a precisely defined and interpreted language, and the logic allow us to prove the consequences of the theory

In this book, Zalta attempts to lay the axiomatic foundations of metaphysics by developing and applying a (formal) theory of abstract objects. The cornerstones include a principle which presents precise conditions under which there are abstract objects and a principle which says when apparently distinct such objects are in fact identical. The principles are constructed out of a basic set of primitive notions, which are identified at the end of the Introduction, just before the theorizing begins. The main reason for producing a theory which defines a logical space of abstract objects is that it may have a great deal of explanatory power. It is hoped that the data explained by means of the theory will be of interest to pure and applied metaphysicians, logicians and linguists, and pure and applied epistemologists.

In (1991), Meinwald initiated a major change of direction in the study of Plato’s Parmenides and the Third Man Argument. On her conception of the Parmenides , Plato’s language systematically distinguishes two types or kinds of predication, namely, predications of the kind ‘x is F pros ta alla’ and ‘x is F pros heauto’. Intuitively speaking, the former is the common, everyday variety of predication, which holds when x is any object (perceptible object or Form) and F is a property which x exemplifies or instantiates in the traditional sense. The latter is a special mode of predication which holds when x is a Form and F is a property which is, in some sense, part of the nature of that Form. Meinwald (1991, p. 75, footnote 18) traces the discovery of this distinction in Plato’s work to Frede (1967), who marks the distinction between pros allo and kath’ hauto predications by placing subscripts on the copula ‘is’.

In this book, Dennett determines just how far we can push the idea that mental states are distinguished by intentionality, that is, by the fact that they have content in virtue of being about, or directed towards, the world at large. Intentionality is characteristic of such states as belief and desire, since all belief is belief of something or that something be the case. In contrast to the physical stance and the design stance, the intentional stance is the predictive attitude or strategy philosophers or cognitive scientists adopt when, in order to explain the behavior of a sufficiently complex system, they treat it as if it had beliefs and desires and acted rationally on its beliefs to satisfy its desires. Dennett insists that this stance is essentially a heuristic that informs the formation of scientific theories which are void of mental state terms. For Dennett, the interesting question, whether we are considering mechanical systems, computer systems, enzymes, microorganisms, vervet monkeys, humans, or martians, is just how far should we go in employing the intentional strategy? At what point must we abandon a strict realism about beliefs, desires, and the other concepts of folk psychology?

This paper introduces a new method of interpreting complex relation terms in a second-order quantified modal language. We develop a completely general second-order modal language with two kinds of complex terms: one kind for denoting individuals and one kind for denoting n-place relations. Several issues arise in connection with previous, algebraic methods for interpreting the relation terms. The new method of interpreting these terms described here addresses those issues while establishing an interesting connection between λ and ε calculi. The resulting semantics provides a precise understanding of the theory of relations.

In this paper, the author responds to D. Jacquette's paper, "Mally's Heresy and the Logic of Meinong's Object Theory'' (History and Philosophy of Logic, 10, 1989, 1-14), in which it is claimed that Ernst Mally's distinction between two modes of predication, as it is employed in the theory of abstract objects, is reducible to, and analyzable in terms of, a single mode of predication plus the distinction between nuclear and extranuclear properties. The argument against Jacquette's claims consists of counterexamples to his reductions and analyses. Reasons are offered for thinking that no such reduction/analysis of the kind Jacquette proposes could be successful.

In this paper, the author shows how one can independently prove, within the theory of abstract objects, some of the most significant claims, hypotheses, and background assumptions found in Kripke's logical and philosophical work. Moreover, many of the semantic features of theory of abstract objects are consistent with Kripke's views — the successful representation, in the system, of the truth conditions and entailments of philosophically puzzling sentences of natural language validates certain Kripkean semantic claims about natural language.

The appeal to possible worlds in the semantics of modal logic and the philosophical defense of possible worlds as an essential element of ontology have led philosophers and logicians to introduce other kinds of `worlds' in order to study various philosophical and logical phenomena. The literature contains discussions of `non-normal worlds', `non-classical worlds', `non-standard worlds', and `impossible worlds'. These atypical worlds have been used in the following ways: (1) to interpret unusual modal logics, (2) to distinguish logically equivalent propositions, (3) to solve the problems associated with propositional attitude contexts, intentional contexts, and counterfactuals with impossible antecedents, and (4) to interpret systems of relevant and paraconsistent logic. However, those who have attempted to develop a genuine metaphysical theory of such atypical worlds tend to move too quickly from philosophical characterizations to formal semantics.

Mark Balaguer’s project in this book is extremely ambitious; he sets out to defend both platonism and fictionalism about mathematical entities. Moreover, Balaguer argues that at the end of the day, platonism and fictionalism are on an equal footing. Not content to leave the matter there, however, he advances the anti-metaphysical conclusion that there is no fact of the matter about the existence of mathematical objects.1 Despite the ambitious nature of this project, for the most part Balaguer does not shortchange the reader on rigor; all the main theses advanced are argued for at length and with remarkable clarity and cogency. There are, of course, gaps in the account but these should not be allowed to overshadow the sig-

In an author-meets-critics session at the March 1992 Pacific APA meetings, the critics (Christopher Menzel, Harry Deutsch, and C. Anthony Anderson) commented on the author's book *Intensional Logic and the Metaphysics of Intentionality* (Cambridge, MA: MIT/Bradford, 1988). The critical commentaries are published in this issue together with these replies by the author. The author responds to questions concerning the system he proposes, and in particular, to questions concerning the treatment of modality, the semantics of belief reports, and the general efficacy of the metaphysical foundations as compared to that of set theory.