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Absolute Generality Towards a Modal Approach

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Absolute Generality Towards a Modal Approach Corso di Dottorato di ricerca in Filosofia e Scienze della Formazione ciclo XXX° Tesi di Ricerca Absolute Generality Towards a Modal Approach SSD: M-FIL 02; M-FIL 05 Coordinatore del Dottorato ch. prof.ssa Emanuela Scribano Supervisore ch. prof. Paolo Pagani Dottorando Filippo Costantini Matricola 820866 2 Summary Introduction p. 4 Chapter 1: Basic Law V and the Origin of Paradox p. 12 Chapter 2: Introduction to Indefinite Extensibility p. 24 Chapter 3: The Inexpressibility Objection p. 48 Chapter 4: Against Schematism p. 65 Chapter 5: From First-order to Higher-order Logic p. 81 Chapter 6: The Domain Principle and Indefinite Extensibility p. 139 Chapter 7: A theory of Concepts p. 153 Appendix: Indefinite Extensibility without Intuitionism p. 194 Conclusion p. 209 References p. 214 3 INTRODUCTION This dissertation defends the idea that absolute general discourses are possible, i.e. sometimes the quantifiers in our sentences range over absolutely everything. However, it also defends the idea that there are indefinitely extensible concepts, i.e. concepts associated with an indefinitely extensible sequence of objects falling under them. Prima facie, the two ideas seem incompatible and, as a matter of fact, indefinite extensibility has been used to challenge absolute generality1. The reason of this apparent incompatibility is that the standard theory of quantification requires a domain for the quantifiers to range over: if the quantifier is absolutely general, then its domain2 of quantification must contain absolutely everything. But if concepts as those of set, ordinal, cardinal etc. are indefinitely extensible, then also the concept of object (or the concept of ‘being self-identical’) is indefinitely extensible, since sets, ordinals and cardinals are objects. But this implies that no all-inclusive domain exists: given a domain, it is possible to find new objects not present in it. My strategy to argue for absolute generality and indefinite extensibility consists in arguing that standard quantification is not the only form of generality. One of the main thesis of the whole work is the claim that there exists a form of generality which behaves differently from standard quantification. This generality, which is formally captured by means of a modal approach to quantification, is firstly introduced in chapter 2, and then developed in chapters 6 and 7. The general context in which this work has been developed is the debate on absolute generality. This is a highly interesting debate on quantifiers: is it possible to quantify over everything? Is there a totally unrestricted quantification over an all-inclusive domain? In the contemporary discussion, one of the first scholar to raise doubts about the possibility of meaningful sentences about everything was Russell. Speaking about the paradoxes (both set-theoretic and semantics) Russell writes: Thus it is necessary […] to construct our logic without mentioning3 such things as “all propositions” or “all properties”, and without even having to say that we are excluding such things. The exclusion must result naturally and inevitably from our positive doctrines, which must make it plain that “all propositions” and “all properties” are meaningless phrases. (Russell [1908], p. 226). 1 For instance, Glanzberg [2004], Hellman [2006] and Button [2011] uses ideas connected to indefinite extensibility to challenge the possibility of absolute generality. 2 As I shall make clear in the following chapters (especially chapter 6), when I say that no all-inclusive domain exists, the word ‘domain’ denotes either a set or a plurality (in the sense of plural logic). 3 It is well-known that Russell was very sloppy with the use-mention distinction. Here he says that we should not mention things as “all propositions”, but by saying “all propositions” he is just mentioning it! What he should have said is that, although we cannot use such expressions, we can certainly mention it. 4 If we cannot speak of all propositions or all properties, a fortiori, we cannot meaningfully speak of everything. Russell’s doubts did not meet many supporters. The standard view during the twentieth century was to regard standard first-order unrestricted quantification as unproblematic. Quine’s essay On what there is is a case in point: to the question “what is there?” Quine answered ‘Everything’, where “Everything” is an unrestricted first-order quantifier. At a naïve sight, it seems weird to question the possibility of absolute general discourses. How is it possible to fail to refer to everything, if I intend to refer to everything? Of course, everybody acknowledges that many occurrences of ‘everything’ in our sentences are restricted to some particular domain. If I say, “all bottles of beers are empty”, it is unlikely that ‘all bottles’ refers to absolutely all bottles on this earth; rather it is likely it refers to some restricted domain of bottles, i.e. the bottle in my house4. But certainly, we can speak of everything, if we intend to. Or at least this seems to be the case. Williamson [2003] fully articulates what he dubs as naïve absolutism, i.e. the intuitive view according to which there are no problem in quantifying over everything. Naïve absolutism is supported by a number of different examples where it seems clear that we manage to achieve absolute generality. For instance, ontological and metaphysical discourses seem to presuppose absolute general claims5. Quine’s answer above presupposes that ‘everything’ is to be taken as totally unrestricted. But also, logic requires unrestricted quantification. The laws of logic are valid not just in some restricted domain, but in all domains. Therefore, it seems that we need absolute generality to state them. In any case, there are also examples from ordinary talk. If you deny that there exists something, let’s say Pegasus, then you are committed to the sentence ‘Everything is such that it is not Pegasus’. Here the quantifier must be taken as absolutely unrestricted, if that sentence has its intended meaning; otherwise, if the quantifier has only a restricted domain, then the truth of that sentence is compatible with the existence of (an object equal to) Pegasus in a broader domain6. In addition, standard logic seems to suggest the possibility that unrestricted quantification is present also when the quantifiers are restricted. Suppose you say that ‘all dogs bark’ or that ‘there exist a black swan’. These are examples of restricted quantification: the first quantifier is restricted to the domain of dogs, while the second one is restricted to the domain of swans. These sentences are usually formalized as follows: ∀�(�� → ��) and ∃�(�� ∧ ���), where � =being a dog; � =barking; � =being a swan; �� =being black. In both cases, the quantifier lies outside the parenthesis, and binds the whole formula inside the parenthesis (it binds, respectively, the formulas �� → �� and �� ∧ ���), not only the formulas that respectively say that we are speaking 4 Stanley and Szabo [2000] provides an insightful analysis of how quantifier domain restriction works in natural language. I shall not bother with this topic in the present work. 5 For a different opinion, see Glanzberg [2004]. 6 This is a form of the inexpressibility objection, which we have deeply analyzed in chapter 3. More reasons to support the plausibility of unrestricted quantification can be found in Williamson [2003]. 5 of dogs (��) and swans (��). This standard way of formalization leads quite naturally to think that the range of the quantifier is always everything, and the predicate � and � select a sub-domain of everything. If this is the case, then every restricted quantification should be analyzed as composed by an unrestricted quantifier, and a predicate that restricts its domain7. What we have just seen suggests that naïve absolutism enjoys prima facie plausibility. However, it has been challenged in many ways. In the literature, there are four big objections that have been raised. They are the objection from paradox, the objection from the indeterminacy of reference, the objection from ontological relativity and the objection from sortal restriction. We shall now briefly explain what these objections amount to. The objection from paradox. This is the most important objection, and the one more discussed in this dissertation. So, we just outline here his general aspect. The set theoretic paradoxes show that no universal set exists. Because, if it existed, then we could consider the set R of all non-self-membered sets. But if R belongs to itself, it does not belong to itself; if it does not belong to itself, it belongs to itself. We thus have a contradiction. The universal set thus cannot exist. But since standard model theoretic semantics is based on set theory, unrestricted quantification over everything needs the universal set as its own domain of quantification. The non-existence of this set implies that no unrestricted quantification is possible. A slightly different version of the argument exploits the paradoxes to show that any domain can be expanded. Suppose you consider a domain D, which purports to be the domain of everything (the universal set). Then you can derive Russell’s paradox. But at this point you can exploit the paradox to argue that the set R is not one of the objects included in D. Therefore, you can expand D by adding R to it. In this way, you find a more comprehensive domain �v = � ∪ �. But D was arbitrary. Any domain that presents itself as the all-inclusive domain can be expanded. This is the argument at the root of indefinite extensibility. More on this in chapters 1 and 2. The objection from indeterminacy of reference. This objection is based on the Löwenheim-Skolem theorem for first-order logic. Since the theorem claims that a first- order theory with an infinite model has models of any other infinite cardinality, and in particular it has a countable model, the objection claims that if we have a first-order theory whose quantifiers are supposed to be absolutely unrestricted, then it is undetermined if they range over everything or over a countable subset of everything.
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