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

A. Plantinga develops a challenging critique of Castañeda's guise theory, by identifying fundamental intuitions that guise theory gives up and by developing several objections to the guise-theoretic world view as a whole. In this paper, I examine whether Plantinga's criticisms apply to the theory of abstract objects. The theory of abstract objects and guise theory can be fruitfully compared because they share a common intellectual heritage---both follow Ernst Mally [1912] in postulating a special realm of objects distinguished by their "internal" or "encoded" properties. Despite this common heritage, however, the theories organize, develop, and apply these special objects in distinctive ways. The two metaphysical systems, therefore, differ significantly, and these differences become important when one considers Plantinga's critique of guise theory. In this essay, the author shows that the theory of abstract objects anticipates and addresses most of Plantinga's concerns about guise theory, by preserving intuitions guise theory has abandoned.

The author examines the differences between the general intensional logic defined in his recent book and Montague's intensional logic. Whereas Montague assigned extensions and intensions to expressions (and employed set theory to construct these values as certain sets), the author assigns denotations to terms and relies upon an axiomatic theory of intensional entities that covers properties, relations, propositions, worlds, and other abstract objects. It is then shown that the puzzles for Montague's analyses of modality and descriptions, propositional attitudes, and directedness towards nonexistents can be solved using the author's logic.

In this paper, the author compares passages from two philosophically important texts and concludes that they have fundamental ideas in common. What makes this comparison and conclusion interesting is that the texts come from two different traditions in philosophy, the analytic and the phenomenological. In 1912, Ernst Mally published *Gegenstandstheoretische Grundlagen der Logik und Logistik*, an analytic work containing a combination of formal logic and metaphysics. In 1913, Edmund Husserl published *Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie*, a seminal work in phenomenology in which noemata are defined and given a crucial role in directing our mental states. In the passages from these two texts reproduced below, the author shows that the abstract `determinates' postulated by Mally in 1912 are assigned much the same role that Husserl assigned to noemata in 1913. Though Mally's determinates are not as highly structured as Husserl's noemata, they have a feature that explains how they manage to play the role assigned to them. The corresponding feature is missing, or at least, not emphasized in Husserl's account of noemata. Therefore, insights from both philosophers, and thus from both the analytic and phenomenological traditions, are needed to give a more complete account of directed mental states.

The authors provide an object-theoretic analysis of two paradoxes in the theory of possible worlds and propositions stemming from Russell and Kaplan. After laying out the paradoxes, the authors provide a brief overview of object theory and point out how syntactic restrictions that prevent object-theoretic versions of the classical paradoxes are justified philosophically. The authors then trace the origins of the Russell paradox to a problematic application of set theory in the definition of worlds. Next the authors show that an object-theoretic analysis of the Kaplan paradox reveals that there is no genuine paradox at all, as the central premise of the paradox is simply a logical falsehood and hence can be rejected on the strongest possible grounds—not only in object theory but for the very framework of propositional modal logic in which Kaplan frames his argument. The authors close by fending off a possible objection that object theory avoids the Russell paradox only by refusing to incorporate set theory and, hence, that the object-theoretic solution is only a consequence of the theory’s weakness.

In this paper, the author develops a theory of concepts and shows that it captures many of the ideas about concepts that Leibniz expressed in his work. Concepts are first analyzed in terms of a precise background theory of abstract objects, and once concept summation and concept containment are defined, the axioms and theorems of Leibniz's calculus of concepts (in his logical papers) are derived. This analysis of concepts is then seamlessly connected with Leibniz's modal metaphysics of complete individual concepts. The fundamental theorem of Leibniz's modal metaphysics of concepts is proved, namely, whenever an object x has F contingently, then (i) the individual concept of x contains the concept F and (ii) there is a (counterpart) complete individual concept y which doesn't contain the concept F and which `appears' at some other possible world. Finally, the author shows how the concept containment theory of truth can be made precise and made consistent with a modern conception of truth

The foregoing set of theorems forms an effective foundation for the theory of situations and worlds. All twenty-five theorems seem to be basic, reasonable principles that structure the domains of properties, relations, states of affairs, situations, and worlds in true and philosophically interesting ways. They resolve 15 of the 19 choice points defined in Barwise (1989) (see Notes 22, 27, 31, 32, 35, 36, 39, 43, and 45). Moreover, important axioms and principles stipulated by situation theorists are derived (see Notes 33, 37, and 38). This is convincing evidence that the foregoing constitutes a theory of situations. Note that worlds are just a special kind of situation, and that the basic theorems of world theory, which were derived in previous work, can still be derived in this situation-theoretic setting. So there seems to be no fundamental incompatibility between situations and worlds — they may peacably coexist in the foundations of metaphysics. The theory may therefore reconcile two research programs that appeared to be heading off in different directions. And we must remind the reader that the general metaphysical principles underlying our theory were not designed with the application to situation theory in mind. This suggests that the general theory and the underlying distinction have explanatory power, for they seem to relate and systematize apparently unrelated phenomena