Logic, Truth and Description. Essays on Chateaubriand's Logical Forms Jairo J. da Silva (ed.) CDD: 160 CHATEAUBRIAND ON THE NATURE OF LOGIC FRANK THOMAS SAUTTER Department of Philosophy Federal University of Santa Maria Campus Universitário, km 9, Camobi 97105-900 SANTA MARIA, RS BRAZIL sautter@terra.com.br Abstract: In this paper Chateaubriand’s approach to solve some problems related to the nature of logic is confronted with the traditional approaches. It is shown that his hierarchy of logical types opens up new possibilities to characterize logical properties and logical truths and that it also sheds some new light on the foundations of mathematics. Key-words: Foundations of logic. Foundations of mathematics. Hierarchy of logical types. Chateaubriand. Frege. Tarski. The order in which a system is exhibited almost always does not correspond to the order in which it has been elaborated. The system proposed by Oswaldo Chateaubriand is no exception (see Chateaubriand 2001). Since it contains many original ideas and original interpretations of traditional ideas which were initially developed to refute the slingshotargument, especially the variant formulated by Gödel, the first chapter of the book with regard to the order of elaboration is the fourth chapter with regard to the order of exhibition. The diverse parts of the system form an organic whole making it difficult to apprehend them in isolation; this explains also the recurrence of some ideas. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 95-104, jan.-jun. 2004. 96 FRANK THOMAS SAUTTER Among the many philosophical questions which Chateaubriand’s work tries to answer there is one I am especially interested in: What is logic? This question can be answered in different ways corresponding to the possible senses the question might be given. In what follows, I shall identify these and examine Chateaubriand’s answers to the corresponding questions. A) MINIMAL CHARACTERIZATION OF LOGIC I use the expression “minimal” to refer to those characterizations in which characteristic marks are enumerated that are sufficient to distinguish what is so characterized from the rest; to put in Leibniz’ jargon, a minimal characterization of something is a nominal definition of that thing. In some passages Chateaubriand alludes to a traditional characterization of logic according to which logic is “universal in some sense” (Chateaubriand 2001, p. 26, p. 302). This characterization, which goes back at least to Kant, contains a positive thesis about logic. But, according to Kant (KrV, B3-4), (strict) universality is only one of those criteria by means of which one recognizes a pure a priori knowledge; consequently, if we respect Kant’s verdict we have to admit that it is insufficient to provide a minimal characterization of logic. Kant has also a negative thesis about logic according to which (general and pure) logic abstracts not only from particular contents, as the rationalists claimed, but also from all other content (KrV, B78), because it neither enlarges nor amplifies our knowledge (KrV, B86) and, consequently, concerns only the form of thought in general (KrV, B79). With regard to this negative thesis, Chateaubriand agrees with the common view that logic does not have, in a certain respect, ontological commitments, namely insofar as it does not imply the existence of non-logical entities, but he admits, in accordance with his realist conception, that logic treats of “specifically logical entities” (Chateaubriand 2001, p. 26). However, what Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 95-104, jan.-jun. 2004. CHATEAUBRIAND ON THE NATURE OF LOGIC 97 Chateaubriand considers here as “specifically logical entities” appears to correspond, in Kant’s terminology, more to the form of judgements than to their contents. According to another common characterization of logic, which is presumably a minimal one, logic is concerned with inferential patterns (schemes, forms) all of whose instances are correct. MacFarlane (2000, pp. 36-41) calls this the conception of schematic formality. Since, in the introduction of his book, Chateaubriand criticizes the grammatical conception of logic formulated by Quine which is closely connected with the conception of schematic formality, it appears that Chateaubriand does not accept this characterization of logic, too. The following two answers to the question “What is logic?” to be discussed presuppose some general knowledge of the structuring of reality by the hierarchy of logical types that is proposed by Chateaubriand in Chapter 9 (2001, pp. 297-339). He divides all entities into three categories: properties, objects and states of affairs. The differences between these categories are partly explained by referring back to Frege’s well-known metaphor that properties are insaturated (or incomplete) entities whereas objects and states of affairs are saturated (or complete) ones. On the other hand, properties and states of affairs are to be found at all levels of the hierarchy except level 0, whereas objects are to be found only at level 0. Properties correspond to fregean functions, whereas objects correspond to fregean objects. States of affairs, which are also called “objects of superior level”, are a novelty with regard to the fregean ontology. They are introduced, basically, because of the needs of the theory of truth proposed by Chateaubriand – the theory of “truth as identification” – according to which a sentence is true if and only if the state of affairs to which the sentence refers actually exists. The hierarchy proposed by Chateaubriand is distinguished from more traditional hierarchies such as the hierarchy tacitly employed by Frege and the hierarchy explicitly employed by Russell mainly by two Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 95-104, jan.-jun. 2004. 98 FRANK THOMAS SAUTTER features: the flexibility of degrees of properties and the accumulation of properties in the hierarchy. The flexibility of degrees of properties permits, among other envisaged theoretical ends, to represent, by means of the multi-degree property of Diversity, sets and extensions of properties in terms of states of affairs, and with that to do without them (Chateaubriand 2001, p. 311). As will be shown in more detail below, the accumulation of properties in the hierarchy serves to characterize logical properties, but it is introduced, it seems to me, to do justice to natural language. Thus, when we are speaking of the property of existence in a general way, we are not referring to one of the specific properties of existence at a given level, but are referring to the limit of the properties of existence that are to be found in the hierarchy. The following notation appears to be adequate for describing the logical type of an entity: The logical type τ of an object is 0. The logical type τ of a property of level λ and arity κ is: <<λ, κ>C(κ),τ0,τ1,...>C ’({<λi,κi>}) In the case that C(κ) is omitted, the property has the fixed arity κ, that is, it is a mono-degree property. The presence of C(κ), where κ is the parameter, indicates the conditions under which the arity of the property can vary its degree. In the case that C ’({<λi,κi>}) is omitted, the arguments of the property have a fixed type, that is, the property is non-cumulative. The presence of C ’({<λi,κi>}), where the level and the arity of the arguments are parameters, indicates the conditions under which a property is cumulative in the hierarchy. τ0,τ1, ... are logical types of the property’s arguments. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 95-104, jan.-jun. 2004. CHATEAUBRIAND ON THE NATURE OF LOGIC 99 · The logical type τ of a state of affairs of level λ is: <<<<λ, κ>C(κ),τ0,τ1, ...>C’({<λi,κi>}i∈κ),τ0,τ1, …>, τ0,τ1, …>, because a state of affairs of level λ is nothing more but the saturation of a property that is also of level λ. B) CHARACTERIZATION IN TERMS OF LOGICAL PROPERTIES Since a property can accumulate in the hierarchy, it is possible that some properties occur along final segments of the hierarchy. These properties are called “logical properties” and the limits of such properties “absolute properties”. The limit of a property does not belong to the hierarchy, and it is, therefore, not a property in the strict sense. In this way Chateaubriand conceives of the universality of a property as its omnipresence from a determined hierarchical level onward. The classical text on logical properties is a posthumous article by Tarski (1986). In the spirit of the Erlanger program of Klein, Tarski proposed as a criterion for the logical status of a notion (considered as a set theoretical entity) the invariance under permutations of the power set of the universe of discourse. Surprisingly, although Tarski’s proposal differs prima facie very strongly from the proposal made by Chateaubriand, they are similar with respect to the obtained results. For, both Tarski and Chateaubriand draw the conclusion that relations of cardinality (universality, vacuity, nonvacuity, uniqueness, etc.) are logical notions (properties). But, they also agree that the relations of the square of oppositions (total and partial inclusion, total and partial exclusion) have the status of logical relations. These similarities deserve a more careful examination. C) CHARACTERIZATION IN TERMS OF LOGICAL TRUTH In his article mentioned above, Tarski confines himself to the characterization of logical notions, omitting the discussion about logical Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 95-104, jan.-jun. 2004. 100 FRANK THOMAS SAUTTER truths. Bellotti (2003, p. 402) finds it astonishing that Tarski is able to characterize the logical notions independently of the logical truths about these notions. Chateaubriand’s approach to characterize logical notions goes exactly into the opposite direction: it is guided by the conviction that the characterization of logical truths depends on the characterization of logical properties. Since a state of affairs is simply the result of saturating a property, and since a statement (proposition) is true if and only if the corresponding state of affairs is to be found in the hierarchy, a logical truth is identified with the corresponding state of affairs to be found in the hierarchy that this state of affairs is the result of saturating a logical property by other logical properties. At this point two observations are instructive. First, although Chateaubriand makes extensive use of the metaphor of saturation, he employs it in a sense that is different from that proposed by Frege, for he allows that a property is saturated by another property and, with that, that an unsaturated entity is saturated by another unsaturated entity – this, however, is absolutely excluded by Frege. This dissonance with Frege cannot be resolved because Chateaubriand’s employment of the saturated/unsaturated dichotomy is fundamental to his characterization of the logical truths. Second, in contrast to Frege, Chateaubriand recognizes statements lacking a truth-value as legitimate within the realm of logic. Thus, although Chateaubriand agrees with the common view that ∀x (x = x) is a logical truth because the corresponding state of affairs (in symbols <Reflexivity, Identity>, where both Reflexivity and Identity are logical properties) is to be found in the hierarchy, he disagrees with the common view by maintaining that a = a is not a logical truth because an individual constant can fail to denote something. D) CHARACTERIZATION IN TERMS OF LOGICAL INFERENCE Within his discussion of the fundamental platonic forms that are cited in Plato’s Sophist – Being, Movement, Rest, Identity, Difference – Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 95-104, jan.-jun. 2004. CHATEAUBRIAND ON THE NATURE OF LOGIC 101 van Fraassen suggests that among these forms there are some nonextensional relations: “[…] in fact, every part of Difference is a part of Being, so Difference is a part of Being. Contrariwise, Being is part of Difference: whatever is, also is not in some sense. [...] So Being and Difference each are part of the other; any possible individual participates in both. The logician’s immediate impulse must be to say that they are the same, for the distinction between them thus corresponds to no conceivable distinction in fact (in the individuals). But Plato argues that they are distinct forms […]” (van Fraassen 1969, p. 490). Although he does not present any particular example, Chateaubriand seems to adopt the same opinion, insofar as he “hold[s] that there are non-extensional relations between properties” (Chateaubriand 2001, p. 72). This view clearly presupposes the legitimacy of modal notions and modal logic. However, modal logic aroused suspicion by some distinguished contemporary logicians, for instance, by Frege, because he considered modal logic as a part of psychology, and also by Gödel, despite the fact that he used it in several occasions, e.g., in his version of the ontological argument for the existence of God and in his argument for the irreality of time, because he was not convinced to have “any clear philosophy in the models for modal logic”. See Wang (1996, p. 82). In order to get a more complete picture of Chateaubriand’s system, it would be interesting to know what is his opinion about modal logics and possible world semantics would be like. And, with regard to practical reasoning, it would also be of interest to know whether he admits the application of the logical relations to the realm of ought or, at least, the application of such relations that are analogous to the logical relations within the realm of being. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 95-104, jan.-jun. 2004. 102 FRANK THOMAS SAUTTER E) CONNECTIONS WITH OTHER PHILOSOPHICAL DISCIPLINES A major part of the Introduction is dedicated to the question of what the connections between logic and other disciplines are. There, Chateaubriand gives a classification of the different conceptions of logic which refers to the relation between logic and other disciplines: the group of the linguistic conceptions of logic and the group of the ontological-epistemological conceptions of logic. Chateaubriand’s own conception belongs to this last group. According to him, the relation between logic and ontology is of the following kind: the fundamental notion of logic is the notion of truth and this notion refers to reality itself. The manner in which Chateaubriand formulates this point is inspired by Frege: logic is concerned with the laws of truth and these must be understood, in a certain respect, as laws of being. Another point showing the intimate relation between logic and ontology in Chateaubriand’s system is the peculiar manner with which he reinterprets the central role and the organic unity that classical propositional logic and classical elementary logic have in contemporary logic: while classical propositional logic is a theory of the predicates ‘is true’ and ‘is false’, i.e., studies the laws of truth, classical elementary logic is a general theory of objects and predicates, i.e., studies the laws of being. Just as logic is related to ontology via the notion of truth, it is related to epistemology via the notion of preservation of truth, which provides us with a precise notion of justification that is, however, probably neither necessary nor sufficient to account for the pre-theoretic notion. It is precisely because he considers the syntactic conception of proof and definition as epistemologically not significant that he rejects the linguistic conception of logic. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 95-104, jan.-jun. 2004. CHATEAUBRIAND ON THE NATURE OF LOGIC 103 F) LOGIC AND MATHEMATICS In his article mentioned above, Tarski does not take directly a position on the question: Is mathematics a part of logic? According to him, an adequate answer of this question depends on a decision of the status of the membership relation: when the universe of discourse is confined solely to individuals and the membership relation only induces a construction of classes and relations, the answer is affirmative, i.e., mathematics is a part of logic; on the other hand, when the universe of discourse contains, not only individuals, but also classes and relations, and the membership relation is a primitive notion, the answer is negative, i.e., mathematics is not a part of logic. Chateaubriand’s approach to answer this question refers to Frege’s strategy: it is necessary to show that the truths of mathematics can be reduced to the truths of logic by means of appropriate definitions. However, Chateaubriand rejects one of the Achilles’ heels of the fregean project: the postulation of logical and mathematical objects. His solution seems to mix Frege’s logicism with Dedekind’s structuralism; according to it, mathematical truths are really logical truths, but there are no mathematical and logical objects, but only structures. Finally, I would like to indicate the possibility of applying the instrument developed by Chateaubriand to the less well-known logicist program pursued by Gödel. In a recent article (Sautter 2003) I pointed out the main presuppositions of Gödel’s project. One of them is that logic is a theory of concepts, whereas mathematics is a theory of the extensions of concepts. But, since not all concepts have a consistent extension, mathematics is only a proper part of logic. Within the part belonging exclusively to logic several self-applied concepts are to be found, as, e.g., the concept of concept. Now, the hierarchy of logical types proposed by Chateaubriand could be adjusted to the requirements of the gödelian project in a rather simple way. There would be two possibilities: either one admits properties that have the same degree as Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 95-104, jan.-jun. 2004. 104 FRANK THOMAS SAUTTER one or more of its arguments (the only objection made by Chateaubriand against this possibility relies on the paradoxes), or one concedes that the limits of absolute “properties” are really properties in the strict sense. REFERENCES BELLOTTI, L. “Tarski on logical notions”. Synthese, 135, pp. 401-413, 2003. CHATEAUBRIAND, O. Logical Forms. Part I. Truth and Description. Campinas: Centro de Lógica, Epistemologia e História da Ciência/ UNICAMP, 2001. (Coleção CLE, 34) KANT, I. Kritik der reinen Vernunft (KrV). Second Edition. Riga: Johann Friedrich Hartknoch, 1787. Transl. By N. Kemp Smith, London: Macmillan, 1929. MACFARLANE, J. G. What does it mean to say that logic is formal? Doctoral thesis. University of Pittsburgh, 2000. SAUTTER, F. T. “O papel das classes próprias na fundamentação das ciências formais.” O que nos faz pensar?, 17, pp. 99-105, 2003. TARSKI, A. “What are logical notions?”. History and Philosophy of Logic, 7, pp. 143-154, 1986. VAN FRAASSEN, B. “Logical structure in Plato’s Sophist.” The Review of Metaphysics, 22, pp. 482-498, 1969. WANG, H. A Logical Journey: From Gödel to Philosophy. Cambridge/ London: The MIT Press, 1996. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 95-104, jan.-jun. 2004. Logic, Truth and Description. Essays on Chateaubriand's Logical Forms Jairo J. da Silva (ed.) CDD: 160 LOGIC AND MODALITY: REPLY TO FRANK SAUTTER OSWALDO CHATEAUBRIAND Department of Philosophy Pontifical Catholic University of Rio de Janeiro Rua Marquês de São Vicente, 225, Gávea 22453-900 RIO DE JANEIRO, RJ BRAZIL oswaldo@fil.puc-rio.br Abstract: In §1 I examine the connections between my account of logical properties and Tarski’s account of logical notions. In §2 I briefly present some of my views on modality and the basis for my claim that there are intensional as well as extensional relations between properties. In §3 I compare my views on the nature of logic and of mathematics with Gödel’s views. Key-words: Tarski. Logical notion. Logical property. Modality. Gödel. Frank gives an excellent summary of my views on the nature of logic and raises three main questions. (1) What is the relation between my view of logical properties and Tarski’s view of logical notions? (2) What are my views on modality and possible world semantics and how do I justify my claim that there are non-extensional relations between properties? (3) What is the relation between my views and Gödel’s logicist program? I will discuss these in turn 1 . 1 There are many complex issues involved in this discussion and although I cannot deal with them in detail in the context of this reply, there will be a more systematic discussion of some of the issues in Chapter 19. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 105-114, jan.-jun. 2004. 106 OSWALDO CHATEAUBRIAND 1. TARSKI ON LOGICAL NOTIONS Tarski’s view of logical notions presented in his posthumously published paper “What are Logical Notions?” 2 is that the logical notions are those that are “invariant under all possible one-one transformations of the world onto itself” (Tarski, 1986, p. 149). This is a very interesting idea and there has been a fair amount of discussion of it in the literature. At the end of the lecture Tarski distinguishes two methods of constructing set theory: the type-theoretic method of Principia Mathematica and the first-order method of the formulations of set theory by Zermelo, von Neumann, and others. About the former he says: Using the method of Principia mathematica, set theory is simply a part of logic. The method can be roughly described in the following way: we have a fundamental universe of discourse, the universe of individuals, and then we construct out of this universe of individuals certain notions, classes, relations, classes of classes, classes of relations, and so on. However, only the basic universe, the universe of individuals, is fundamental. A transformation is defined on the universe of individuals, and this transformation induces transformations on classes of individuals, relations between individuals, and so on. ... When we speak of transformations of the ‘world’ onto itself we mean only transformations of the basic universe of discourse, or the universe of individuals ... Using this method it is clear that the membership relation is certainly a logical relation. It occurs in several types, for individuals are elements of classes of individuals, classes of individuals are elements of classes of classes of individuals, and so on. And by the very definition of an induced transformation it is invariant under every transformation of the world onto itself. (Tarski 1986, p. 152) Let me make some remarks about this and compare it with what I do in Chapter 9. One basic difference with the ontology that I adopt is that my ontology is an ontology of properties rather than an ontology of sets – 2 I was fortunate to be a participant at the Conference on the Nature of Logic held in honor of Tarski at the State University of New York at Buffalo in 1973 where he gave this lecture on logical notions. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 105-114, jan.-jun. 2004. REPLY TO FRANK SAUTTER 107 which is how Tarski is interpreting the ontology of Principia Mathematica. This is not essential though, because Tarski’s idea could be applied just as well to my ontology of properties and states of affairs. And, in fact, using my criterion of universality and omnipresence throughout the hierarchy we seem to get the same classification of properties into logical and non-logical 3 . Tarski’s criterion in terms of one-one transformations is sharper than mine, and it is also mathematically more interesting because of the connection he draws with Klein’s Erlangen Program, but in my mind it raises a problem concerning the universe of individuals. I hold that the question of what properties are logical properties should be completely independent of which individuals happen to exist in the world and of what sort of things these individuals are 4 . Even if there were no individuals at all, there would still be logical properties, and they would be exactly the same properties as the logical properties in a universe that contains individuals. Here is a place where it makes a difference that my ontology is an intensional ontology of properties and Tarski’s is an extensional ontology of sets. In any case, it seems to me that the characterization of logical properties should be intrinsic and not depend on contingencies. Of course, we could use Tarski’s characterization in terms of possible universes of individuals 5 , or in terms of models, and this might be compatible with my characterization. 3 It is interesting to note in this connection that also Gödel seems to characterize logical properties in terms of universality. Wang says: “For Gödel, logic deals with formal – in the sense of universally applicable – concepts. From this perspective the concepts of number, set and concept are all formal concepts”. (Wang 1996, p. 267) 4 At the end of the passage that I quoted above Tarski says that the universe of individuals may be taken to be the universe of physical objects. 5 I.e., we could say that a property P (of a certain type) is a logical property if and only if in any possible universe of individuals P is invariant with respect to every one-one transformation of the universe onto itself. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 105-114, jan.-jun. 2004. 108 OSWALDO CHATEAUBRIAND From the point of view of universality, the level 1 unary logical properties are Existence and Nonexistence, which correspond to the universe of individuals and to the empty set in Tarski’s characterization. The binary logical relations are Identity, Diversity, the Universal relation and its complement the Self-Difference relation, just as for Tarski. Moreover, since I hold that there are relations of arbitrarily high (finite and infinite) arity, there will be infinitely many Identity and pairwise Diversity relations, as well as mixed Identity-Diversity relations. An example of the latter is the ternary relation that holds between three individuals if and only if the first and second are identical and are different from the third. These are the level 1 logical properties. At level 2 we get again Existence and Nonexistence (of level 2) and all the cardinality properties. We also get again the binary relations Identity, Diversity, Universal, Self-Difference as well as (like Tarski) the Aristotelian binary relations of Subordination, Nonsubordination, Exclusion and Nonexclusion, and all the complex properties that we can define by means of the usual logical notions. Among these we get binary relations between level 1 properties and individuals, and in particular the Application relation and its converse the Instantiation relation – where the latter corresponds to the membership relation for sets. And so on for all higher levels, where my classification generally coincides with Tarski’s. But differences will appear because the specific nature of the universe of individuals will introduce limitations for Tarski’s classification. For suppose that the universe of individuals is of cardinality κ (finite or infinite). If λ is larger than κ, then the pairwise settheoretic diversity relation of cardinality λ will be empty – because there does not exist any λ-sequence of pairwise different individuals 6 . Therefore the pairwise Diversity relations of cardinality greater than κ 6 I am talking here of a level 1 relation. If one goes up enough along the hierarchy of (set) levels, then one will find such a pairwise diversity relation for entities of level lower than it. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 105-114, jan.-jun. 2004. REPLY TO FRANK SAUTTER 109 will not have a set-theoretic counterpart. One way to avoid this is to postulate the Axiom of Units that I suggest in Chapter 9 (pp. 315-17), which ensures that one never runs out of individuals – i.e., that there are as many individuals as there are sets. 2. MODALITY AND POSSIBLE WORLD SEMANTICS Frank suggests that modal logic “aroused suspicion” in Gödel because he did not think that there is “any clear philosophy in the models for modal logic” (p. 101). I think that this may be a misinterpretation 7 . The problem is not with modal logic but with the attempted accounts of modal logic. I make some remarks along these lines on pp. 355-56 saying that while I agree that the technical work in possible world semantics has brought a fair amount of light to issues in modal logic, I do not think that this work gives a philosophical account of the basic notions of modal logic. Rather than explaining the notions of necessity and possibility, the notion of possible world presupposes them. If we have a good account of possibility, then we may have an account of the notion of possible world. But possible world semantics as such is not such an account8 . In those same pages I also complained that the usual accounts of properties in terms of possible worlds do not give us any clear insight into the nature of properties because these accounts are basically extensional – since a property is supposed to be a function that assigns an extension to each possible world. Although this idea actually meshes 7 The small paragraph where Gödel makes this remark goes as follows: “When I entered the field of logic, there were 50 percent philosophy and 50 percent mathematics. There are now 99 percent mathematics and only 1 percent philosophy; even the 1 percent is bad philosophy. I doubt whether there is really any clear philosophy in the models for modal logic.” (Wang 1996, p. 82) 8 Although we could take a specific system of possible world semantics to be something like an “axiomatic” or “implicit” account of modality. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 105-114, jan.-jun. 2004. OSWALDO CHATEAUBRIAND 110 quite well with the idea of properties as identity conditions that I suggest in p. 421 9 , it does not involve an intrinsic (or intensional) characterization of properties, but characterizes them in terms of the totality of their instances (in all possible worlds). What I attribute to Plato and to Kripke is an intrinsic characterization of a property in terms of the nature of its instances. I think that the notions of necessity and possibility may have to be taken as primitive notions that cannot be explained in more basic terms. We have various kinds of intuitions about these notions however, and they help to give some bite to our pronouncements about them. Some of these intuitions are based on our ability to conceive, or imagine, or describe certain situations that we then take to be possible. Other intuitions have a more formal character and derive from our understanding of the meaning of the notions in question, and this is what we try to codify into systems of modal logic (both syntactic and semantic). We are often very sloppy when we argue in terms of conceivability or imaginability. Thus people used to say that they can easily imagine a world in which Sherlock Holmes exists; or a world in which unicorns exist; and so on. One of the great merits of Kripke’s work in Naming and Necessity was to show that such claims are based on misconceptions 10 . But we might still say that we can imagine a spaceship that travels faster than the speed of light, or at the speed of light, or close to the speed of light. But can we really? What are we imagining when we imagine a spaceship traveling faster than the speed of light? Is imagining (or conceiving) a spaceship traveling faster than the speed of light different than imagining (or conceiving) a spaceship traveling at 20,000 kilometers an hour? In which way? I am not saying that we cannot conceive or imagine or describe various kinds of situations, but that this 9 See also §2 of my reply to Richard Vallée. My views on modality were deeply influenced by Kripke’s book – much more so than by possible world semantics as such. 10 Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 105-114, jan.-jun. 2004. REPLY TO FRANK SAUTTER 111 conceiving or imagining or describing should involve more than simply assuming that such a situation is the case. Which is also not to deny that within the framework of a given theory the possibility of assuming consistently that something is the case may be good enough as a mark of possibility. So even though I think that there are various ways in which we can back up our intuitions of possibility and of necessity, and that there is some insightful formal work on modal logic, including possible world semantics, I do not think that this amounts to an analysis of the fundamental modal notions. With respect to the claim in p. 72 (note 18) that there are nonextensional relations between properties, what I had in mind is something quite simple. When we say that all men are mortal, for instance, we normally mean the extensional subordination of the property of being human to the property of being mortal – i.e., that all humans die. It seems to me however, that there is a stronger connection between these properties in that it is in the very nature of humans to die. We don’t just happen to die, but it is an aspect of our biological nature that we must die. Whether this is right or wrong does not matter; the point is that if there is such a connection, then it is a necessary connection11 . 3. GÖDEL’S VIEWS ON LOGIC AND MATHEMATICS As I mention in the Preface, Gödel was a major influence in the development of my views. When I wrote my book I had read his published works and the account of his views in Wang From Mathematics to Philosophy. I am particularly sympathetic to his view that logic is a theory of concepts, which is essentially the view that I defend in my book in terms of properties. If a theory of concepts not involving type distinctions can be developed in a natural and consistent way along some of 11 I do not really know what to say about Frank’s question concerning the “realm of ought” since I have not thought about this subject in any detail. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 105-114, jan.-jun. 2004. 112 OSWALDO CHATEAUBRIAND the directions that Gödel suggests, then I might choose it over a typed ontology. I just do not see at this point how it could be done. With respect to mathematics the situation may be a little different. I understand quite well what Gödel means by saying that mathematics is essentially a theory of extensions – or set theory. The problem is that I have a certain difficulty accepting that mathematical entities are objects, or that extensions (or sets) – in the sense in which they are used in mathematics – are objects. This is one of the reasons for my speculations in chapters 9 and 10 about extensions, sets and structures. As I discuss in my reply to Abel Casanave, I think that mathematics is also fundamentally a theory of properties (concepts), but it is primarily a theory of structural properties. Logic, on the other hand, is a general theory of properties as such, as well as a specific theory of logical properties–or of formal concepts, in Gödel’s terminology. This characterization of logic would seem to agree with Gödel’s, and maybe he would also agree with the characterization of mathematics as a theory of structural properties. Where Gödel would disagree is with my characterization of the hierarchy of properties. From the reports in Wang (1996), Gödel thought that a hierarchical approach to properties (or concepts) is a way of avoiding the fundamental problems. Wang quotes him as saying: Even though we do not have a developed theory of concepts, we know enough about concepts to know that we can have also something like a hierarchy of concepts (or also of classes) which resembles the hierarchy of sets and contains it as a segment. But such a hierarchy is derivative from and peripheral to the theory of concepts; it also occupies a quite different position; for example, it cannot satisfy the condition of including the concept of concept which applies to itself or the universe of all classes that belong to themselves. To take such a hierarchy as the theory of concepts is an example of trying to eliminate the intensional paradoxes in an arbitrary manner. 12 (Wang 1996, p. 278, 8.6.20) 12 Notice, by the way, that whereas in the quotations from Tarski ‘class’ is used synonymously with ‘set’, Gödel uses ‘class’ in the sense of ‘proper class’. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 105-114, jan.-jun. 2004. REPLY TO FRANK SAUTTER 113 As it is clear from the Russell paper and from Wang (1996), Gödel placed a lot of emphasis on the “intensional paradoxes, of which the most important is that of the concept of not applying to itself”, and for him a theory of concepts should give a solution to it and not merely escape it. In the Russell paper Gödel suggests that It might ... be possible to assume every concept to be significant everywhere except for certain “singular points” or “limiting points,” so that the [intensional] paradoxes would appear as something analogous to dividing by zero. (Gödel 1944, p. 150) But both in this paper and in the reports in Wang (1996, p. 268) Gödel maintains that no such theory is at hand. In this connection we could consider whether there really is a property property – or a concept concept – as well as a property non-selfapplicable. Is there a property object, for instance? In my discussion at the beginning of Chapter 9 (p. 322, note 1) I mentioned that Frege characterized objects as non-functions. In terms of identity conditions we may ask: What are the identity conditions for being an object? And what are the identity conditions for being a property? (Cf. my discussion in pp. 421-25.) At the end of Chapter 9 (pp. 319-20) I introduced the notion of ‘notion’ – which I actually took from Gödel (1944, pp. 137-38) – as a way of talking in this very general way. Also my idea of cumulativity for properties was a way of trying to make up for these lacks. I gather from various remarks in Wang (1996) that none of this would be satisfactory for Gödel. But Wang also quotes the following remark: The general concept of concept is an Idea [in the Kantian sense]. The intensional paradoxes are related to questions about Ideas. Ideas are more fundamental than concepts. The theory of types is only natural between the first and the second level; it is not natural at higher levels. Laying the foundations deep cannot be extensive. (Wang 1996, p. 268) This suggests a distinction between concepts and ideas (about which we find nothing in Gödel’s published work) that may perhaps have a similar Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 105-114, jan.-jun. 2004. 114 OSWALDO CHATEAUBRIAND role to the one that I am attributing to the notion of ‘notion’. Evidently there will be many problems in finding the correct interpretation of Gödel’s views, especially since he did not develop them systematically. REFERENCES GÖDEL, K. “Russell’s Mathematical Logic”. In Schilp, P. A. (ed.) The Philosophy of Bertrand Russell, pp. 125-53. New York, N.Y.: Tudor, 1944. KRIPKE, S. Naming and Necessity. Cambridge, Mass.: Harvard University Press, 1980. TARSKI, A. “What are Logical Notions?” History and Philosophy of Logic, 7, pp. 143-54, 1986. WANG, H. A Logical Journey. Cambridge, Mass.: The MIT Press, 1996. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 105-114, jan.-jun. 2004.