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)
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)
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.
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)
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 develop results in modal metaphysics whose discovery was computer assisted, and 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)
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)
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)
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)
of my axiomatic theory of abstract objects.<sup>1</sup> 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.
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)
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.
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)
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. (shrink)
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. (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)
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 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)
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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 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 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 arguments of the dialetheists for the rejection of the traditional law of noncontradiction are not yet conclusive. The reason is that the arguments that they have developed against this law uniformly fail to consider the logic of encoding as an analytic method that can resolve apparent contradictions. In this paper, we use Priest  and  as sample texts to illustrate this claim. In , Priest examines certain crucial problems in the history of philosophy from the point of view (...) of someone without a prejudice in favor of classical logic. For each of these problems, the logic of encoding offers an alternative explanation of the phenomena---this alternative is not considered when Priest describes what options there are in classical logic for analyzing the problem at hand. (shrink)
This paper introduces a new typology and associated measure of social and environmental mission integration by conceptually framing a feature of hybrid organizations—the degree of integration of their revenue model and social–environmental mission. The SEMI measure is illustrated using a hand-collected sample of 256 North American Certified B Corporations. We explore the heterogeneity of SEMI scores by identifying external-facing correlates and demonstrate non-congruence with Certified B Corporation’s audit results. Overall, our findings advance existing knowledge of these hybrid organizations and how (...) they balance their social–environmental missions with their economic objectives. (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)
This entry explains Frege's Theorem by using the modern notation of the predicate calculus. Frege's Theorem is that the Dedekind-Peano axioms for number theory are derivable from Hume's Principle, given the axioms and rules of second-order logic. Frege's methodology for defining the natural numbers and for the derivation of the Dedekind-Peano axioms are sketched in some detail.
Forma logica argumenti ontologici reconsiderataHac in tractatione auctores veritatem praemissarum argumenti ontologici, quod in dissertatione sua anno 1991 publicata proposuerunt, examinant. Auctores praesertim de prima Anselmi praemissa, qua asseritur, dari cogitabile quid, quo maius cogitari nequit, dubitant. Primo scilicet argumentum, quod Anselmus pro hac assertione astruit, reiciunt; deinde ostendunt, aliam interpretationem formalem huius praemissae dari posse, secundum quam vera evenit. Haec interpretatione adhibita, argumentum Anselmi non solum validum, sed etiam efficax esse constat. Reconstructio praecisa argumenti in hoc sensu intellectinihilominus revelat, (...) conclusionem eius, scilicet „Deus existit“, sensum peculiarem acquirere, qui Anselmi intentioni originali haud satisfacit.Reflections on the Logic of the Ontological ArgumentThe authors evaluate the soundness of the ontological argument they developed in their 1991 paper. They focus on Anselm’s first premise, which asserts that there is a conceivable thing than which nothing greater can be conceived. After casting doubt on the argument Anselm uses in support of this premise, the authors show that there is a formal reading on which it is true. Such a reading can be used in a sound reconstruction of the argument. After this reconstruction is developed in precise detail, the authors show that the conclusion, a reading of the claim “God exists”, does not quite achieve the end Anselm desired. (shrink)
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. (shrink)
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)
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.
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. (shrink)
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  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)
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. (shrink)
In this paper, we develop an alternative strategy, Platonized Naturalism, for reconciling naturalism and Platonism and to account for our knowledge of mathematical objects and properties. A systematic (Principled) Platonism based on a comprehension principle that asserts the existence of a plenitude of abstract objects is not just consistent with, but required (on transcendental grounds) for naturalism. Such a comprehension principle is synthetic, and it is known a priori. Its synthetic a priori character is grounded in the fact that it (...) is an essential part of the logic in which any scientific theory will be formulated and so underlies (our understanding of) the meaningfulness of any such theory (this is why it is required for naturalism). Moreover, the comprehension principle satisfies naturalist standards of reference, knowledge, and ontological parsimony! As part of our argument, we identify mathematical objects as abstract individuals in the domain governed by the comprehension principle, and we show that our knowledge of mathematical truths is linked to our knowledge of that principle. (shrink)
In this paper, the authors show that there is a reading of St. Anselm's ontological argument in Proslogium II that is logically valid (the premises entail the conclusion). This reading takes Anselm's use of the definite description "that than which nothing greater can be conceived" seriously. Consider a first-order language and logic in which definite descriptions are genuine terms, and in which the quantified sentence "there is an x such that..." does not imply "x exists". Then, using an ordinary logic (...) of descriptions and a connected greater-than relation, God's existence logically follows from the claims: (a) there is a conceivable thing than which nothing greater is conceivable, and (b) if <em>x</em> doesn't exist, something greater than x can be conceived. To deny the conclusion, one must deny one of the premises. However, the argument involves no modal inferences and, interestingly, Descartes' ontological argument can be derived from it. (shrink)
This paper contains answers to the following Five questions, posed by the editors are answered: (1) Why were you initially drawn to the foundations of mathematics and/or the philosophy of mathematics? (2) What example(s) from your work (or the work of others) illustrates the use of mathematics for philosophy? (3) What is the proper role of philosophy of mathematics in relation to logic, foundations of mathematics, the traditional core areas of mathematics, and science? (4) What do you consider the most (...) neglected topics and/or contributions in late 20th century philosophy of mathematics? (5) What are the most important open problems in the philosophy of mathematics and what are the prospects for progress? (shrink)
The authors investigated the ontological argument computationally. The premises and conclusion of the argument are represented in the syntax understood by the automated reasoning engine PROVER9. Using the logic of definite descriptions, the authors developed a valid representation of the argument that required three non-logical premises. PROVER9, however, discovered a simpler valid argument for God's existence from a single non-logical premise. Reducing the argument to one non-logical premise brings the investigation of the soundness of the argument into better focus. Also, (...) the simpler representation of the argument brings out clearly how the ontological argument constitutes an early example of a ?diagonal argument? and, moreover, one used to establish a positive conclusion rather than a paradox. (shrink)
The formulation of propositional modal logic is revised by interposing a domain of structured propositions between the modal language and the models. Interpretations of the language (i.e., ways of mapping the language into the domain of propositions) are distinguished from models of the domain of propositions (i.e., ways of assigning truth values to propositions at each world), and this contrasts with the traditional formulation. Truth and logical consequence are defined, in the first instance, as properties of, and relations among, propositions. (...) -/- These definitions have certain interesting consequences. One is that they resolve a question that cannot be answered by the traditional analysis, in which Kripke models directly interpret modal language. The question is, is the reason a modal sentence has a different truth value at other possible worlds due to the fact that the sentence has a different meaning at that world or due to a change in the world? The philosophical conception elaborates the second answer. Another interesting consequence is that modal language, properly speaking, is not intensional, at least according to the classic tests for, and definitions of, intensionality. (shrink)
We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoretical foundations. Starting with an axiomatic theory of abstract objects, we identify a mathematical structure as an abstract object encoding the truths of a mathematical theory. From such foundations, we derive consequences that address the main questions and issues that (...) have arisen. Namely, elements of different structures are different. A structure and its elements ontologically depend on each other. There are no haecceities and each element of a structure must be discernible within the theory. These consequences are not developed piecemeal but rather follow from our definitions of basic structuralist concepts. (shrink)
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’. (shrink)