This paper has two main aims: first, to set forth an analysis of Timaeus 48E-52D and then to explore the significance of those pages for our understanding of Plato’s metaphysics. Students of the “Receptacle” in Plato’s Timaeus have given close attention to the many metaphors he offers in his explanation of its nature. Less attention has been given to the overall structure of the passage in which he presents it. In this paper, I attempt to show that Plato’s exposition there (...) exhibits a coherent and very careful design. But once that design is grasped, it can be seen that his extended account of the Receptacle must properly be understood to treat a rather different subject: the nature of an image, rather than the nature of the Receptacle, is the focal concern uniting the whole of 48E-52D. (shrink)

The Stanford Encyclopedia of Philosophy is an open access, dynamic reference work designed to organize professional philosophers so that they can write, edit, and maintain a reference work in philosophy that is responsive to new research. From its inception, the SEP was designed so that each entry is maintained and kept up to date by an expert or group of experts in the field. All entries and substantive updates are refereed by the members of a distinguished Editorial Board before they (...) are made public. (shrink)

Science provides us with representations of atoms, elementary particles, polymers, populations, genetic trees, economies, rational decisions, aeroplanes, earthquakes, forest fires, irrigation systems, and the world’s climate. It's through these representations that we learn about the world. This entry explores various different accounts of scientific representation, with a particular focus on how scientific models represent their target systems. As philosophers of science are increasingly acknowledging the importance, if not the primacy, of scientific models as representational units of science, it's important to (...) stress that how they represent plays a fundamental role in how we are to answer other questions in the philosophy of science (e.g. the scientific realism debate). This entry begins by disentangling ‘the’ problem of scientific representation, before critically evaluating the current options available in the literature. (shrink)

The so-called ergodic hierarchy (EH) is a central part of ergodic theory. It is a hierarchy of properties that dynamical systems can possess. Its five levels are egrodicity, weak mixing, strong mixing, Kolomogorov, and Bernoulli. Although EH is a mathematical theory, its concepts have been widely used in the foundations of statistical physics, accounts of randomness, and discussions about the nature of chaos. We introduce EH and discuss its applications in these fields.

This book tackles the issues that arise in connection with intensional logic -- a formal system for representing and explaining the apparent failures of certain important principles of inference such as the substitution of identicals and existential generalization -- and intentional states --mental states such as beliefs, hopes, and desires that are directed towards the world. The theory offers a unified explanation of the various kinds of inferential failures associated with intensional logic but also unifies the study of intensional contexts (...) and intentional states by grounding the explanation of both phenomena in a single theory. When an axiomatized realm of abstract entities is added to the metaphysical structure of the world, we can use them to identify and individuate the contents of directed mental states. The special abstract entities can be viewed as the objectified contents of mental files, and they play a crucial role in the analysis of truth conditions of the sentences involved in inferential failures. (shrink)

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. (shrink)

Some recently-proposed counterexamples to the traditional definition of essential property do not require a separate logic of essence. Instead, the examples can be analysed in terms of the logic and theory of abstract objects. This theory distinguishes between abstract and ordinary objects, and provides a general analysis of the essential properties of both kinds of object. The claim ‘x has F necessarily’ becomes ambiguous in the case of abstract objects, and in the case of ordinary objects there are various ways (...) to make the definition of ‘F is essential to x’ more fine-grained. Consequently, the traditional definition of essential property for abstract objects in terms of modal notions is not correct, and for ordinary objects the relationship between essential properties and modality, once properly understood, addresses the counterexample. (shrink)

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. (shrink)

The author describes an interpreted modal language and produces some clear examples of logical and analytic truths that are not necessary. These examples: (a) are far simpler than the ones cited in the literature, (b) show that a popular conception of logical truth in modal languages is incorrect, and (c) show that there are contingent truths knowable ``a priori'' that do not depend on fixing the reference of a term.

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. (shrink)

of my axiomatic theory of abstract objects.¹ The theory asserts the ex- istence not only of ordinary properties, relations, and propositions, but also of abstract individuals and abstract properties and relations. The.

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. (shrink)

In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's *Grundgesetze*. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom that (...) (philosophical) logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege's conception of numbers and logical objects. (shrink)

After defining a standard modal language and semantics, we offer some clear examples of logical and analytic truths that are not necessary. These examples: (a) are far simpler than the ones cited in the literature, (b) show that a popular conception of logical truth in modal languages is incorrect, and (c) show that there are contingent truths knowable ``a priori'' that do not depend on fixing the reference of a term.

In the debate about the nature and identity of possible worlds, philosophers have neglected the parallel questions about the nature and identity of moments of time. These are not questions about the structure of time in general, but rather about the internal structure of each individual time. Times and worlds share the following structural similarities: both are maximal with respect to propositions (at every world and time, either p or p is true, for every p); both are consistent; both are (...) closed (every modal consequence of a proposition true at a world is also true at that world, and every tense-theoretic consequence of a proposition true at a time is also true at that time); just as there is a unique actual world, there is a unique present moment; and just as a proposition is necessarily true iff true at all worlds, a proposition is eternally true iff true at all times. In this paper, I show that a simple extension of my theory of worlds yields a theory of times in which the above structural similarities between the two are consequences. (shrink)

In its approach to fiction and fictional discourse, pretense theory focuses on the behaviors that we engage in once we pretend that something is true. These may include pretending to name, pretending to refer, pretending to admire, and various other kinds of make-believe. Ordinary discourse about fictions is analyzed as a kind of institutionalized manner of speaking. Pretense, make-believe, and manners of speaking are all accepted as complex patterns of behavior that prove to be systematic in various ways. In this (...) paper, I attempt to show: (1) that this systematicity is captured in the basic distinctions and representations that are central to the formal theory of abstract objects, and (2) that this formal theory need not be interpreted platonistically, but may instead have an interpretation on which the `objects' of the theory are things that pretense theorists already accept, namely, complex patterns of linguistic behavior. The surprising conclusion, then, is that a certain Wittgensteinian approach to meaning (e.g., the meaning of a term like `Holmes' is constituted by its pattern of use) bears an interesting relationship to a formal metaphysical theory and the semantic analyses of discourse constructed in terms of that theory---the former offers a naturalized interpretation of the latter, yet the latter makes the former more precise. (shrink)

The author engages a question raised about theories of nonexistent objects. The question concerns the way names of fictional characters, when analyzed as names which denote nonexistent objects, acquire their denotations. Since nonexistent objects cannot causally interact with existent objects, it is thought that we cannot appeal to a `dubbing' or a `baptism'. The question is, therefore, what is the starting point of the chain? The answer is that storytellings are to be thought of as extended baptisms, and the details (...) of this response receive attention in the paper. Once the storytelling is complete, and the characters have been baptized, a priori metaphysical principles linking the storytelling with the realm of nonexistent objects provide the referential, non-causal connection between the names used in the storytelling and the objects denoted by such names. [This is the original English version of an article that first appeared in German translation, translated into German by Arnold Günther and published in the Zeitschrift für Semiotik 9/1-2 (1987): 85-95. The version that appears here is, for the most part, unaltered.]. (shrink)

Many philosophers, including direct reference theorists, appeal to naively to 'modes of presentation' in the analysis of belief reports. I show that a variety of such appeals can be analyzed in terms of a precise theory of modes of presentation. The objects that serve as modes are identified intrinsically, in a noncircular way, and it is shown that they can function in the required way. It is a consequence of the intrinsic characterization that some objects are well-suited to serve as (...) modes that present individuals and while others are well-suited to serve as modes that present properties (though such modes do not `determine' the objects they present---there is no necessary connection between a mode m and the individual x or property F that it presents). Moreover, it is also a consequence that the modes for properties and individuals can be organized into complexes that are structurally identical to Russellian propositions having the represented properties and individuals as constituents. Not only is the relationship between modes and Fregean senses and modes of presentation explored in the paper, but also the idea that the theory of modes of presentation developed in the paper can serve as a theory of concepts. (shrink)

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. (shrink)

In this paper, we describe "metaphysical reductions", in which the well-defined terms and predicates of arbitrary mathematical theories are uniquely interpreted within an axiomatic, metaphysical theory of abstract objects. Once certain (constitutive) facts about a mathematical theory T have been added to the metaphysical theory of objects, theorems of the metaphysical theory yield both an analysis of the reference of the terms and predicates of T and an analysis of the truth of the sentences of T. The well-defined terms and (...) predicates of T are analyzed as denoting abstract objects and abstract relations, respectively, in the background metaphysics, and the sentences of T have a reading on which they are true. After the technical details are sketched, the paper concludes with some observations about the approach. One important observation concerns the fact that the proper axioms of the background theory abstract objects can be reformulated in a way that makes them sound more like logical axioms. Some philosophers have argued that we should accept (something like) them as being logical. (shrink)

Three different notions of concepts are outlined: 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. (shrink)

This entry introduces the reader to the main ideas in Frege's philosophy of logic, mathematics, and language.

The author resolves a conflict between Frege's view that the cognitive significance of coreferential names may be distinct and Kaplan's view that since coreferential names have the same "character", they have the same cognitive significance. A distinction is drawn between an expression's "character" and its "cognitive character". The former yields the denotation of an expression relative to a context (and individual); the latter yields the abstract sense of an expression relative to a context (and individual). Though coreferential names have the (...) same character, they may have distinct cognitive characters. Propositions involving these abstract senses play an important role in explaining de dicto belief contexts. (shrink)

The principal goal of this entry is to present Frege's Theorem (i.e., the proof that the Dedekind-Peano axioms for number theory can be derived in second-order logic supplemented only by Hume's Principle) in the most logically perspicuous manner. We strive to present Frege's Theorem by representing the ideas and claims involved in the proof in clear and well-established modern logical notation. This prepares one to better prepared to understand Frege's own notation and derivations, and read Frege's original work (whether in (...) German or in translation). Moreover, this should prepare the reader to understand a number of scholarly books and articles in the secondary literature on Frege's work. (shrink)

In this paper, the author replies to a question raised about theories of nonexistent objects. The question concerns the way names of fictional characters, when analyzed as names which denote nonexistent objects, acquire their denotations. Since nonexistent objects cannot causally interact with existent objects, it is thought that we cannot appeal to a‘dubbing’or a‘baptism’. The question is, therefore, what is the starting point of the chain? The answer is that storytellings are to be thought of as extended baptisms, and the (...) details of this response receive attention in the paper. Once the storytelling is complete, and the characters have been baptized, a priori metaphysical principles linking the storytelling with the realm of nonexistent objects provide the referential, non‐causal connection between the names used in the storytelling and the objects denoted by such names. (shrink)

The modal object calculus is the system of logic which houses the (proper) axiomatic theory of abstract objects. The calculus has some rather interesting features in and of itself, independent of the proper theory. The most sophisticated, type-theoretic incarnation of the calculus can be used to analyze the intensional contexts of natural language and so constitutes an intensional logic. However, the simpler second-order version of the calculus couches a theory of fine-grained properties, relations and propositions and serves as a framework (...) for defining situations, possible worlds, stories, and fictional characters, among other things. In the present paper, we focus on the second-order calculus. The second-order modal object calculus is so-called to distinguish it from the second-order modal predicate calculus. Though the differences are slight, the extra expressive power of the object calculus significantly enhances its ability to resolve logical and philosophical concepts and problems. (shrink)

In this paper, I respond 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. (shrink)

In this entry, Frege's logic is introduced and described in some detail. It is shown how the Dedekind-Peano axioms for number theory can be derived from a consistent fragment of Frege's logic, with Hume's Principle replacing Basic Law V.

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. (shrink)