Nominalist Neologicism

Rafał Urbaniak

Nominalism is usually formulated as the thesis that only concrete entities exist or that no abstract entities exist. But where, as here, the interest is primarily in philosophy of mathematics, one can bypass the tangled question of how, exactly, the general abstract/concrete distinction is to be understood by taking nominalism simply as the thesis that there are no distinctively mathematical objects: no numbers, sets, functions, groups, and so on. As to the nature of such objects (if there are any), it can be said that it has come to be fairly widely agreed, under the influence of Frege and others, that they are very different both from paradigmatically physical objects (bricks, stones) and from paradigmatically mental ones (minds, ideas). Modern nominalism emerged in the 1930s as a response to the view of Frege and others that numbers, sets, functions, groups, and so on belong to a “third realm.”

Nominalism in formal ontology is still the thesis that the only acceptable domain of quantification is the first-order domain of particulars. Nominalists may assert that second-order well-formed formulas can be fully and completely interpreted within the first-order domain, thereby avoiding any ontological commitment to second-order entities, by means of an appropriate semantics called "substitutional". In this paper I argue that the success of this strategy depends on the ability of Nominalists to maintain that identity, and equivalence relations more in general, is first-order and invariant. Firstly, I explain why Nominalists are formally bound to this first-order concept of identity. Secondly, I show that the resources needed to justify identity, a certain conception of identity invariance, are out of the nominalist's reach.

We observe that Putnam’s model-theoretic argument against determinacy of the concept of second-order quantification or that of the set is harmless to the nominalist. It serves as a good motivation for the nominalist philosophy of mathematics. But in the end it can lead to a serious challenge to the nominalist account of mathematical objectivity if some minimal assumptions about the relation between mathematical objectivity and logical objectivity are made. We consider three strategies the nominalist might take to meet this challenge, and we argue that all these strategies are untenable.

Nominalist Neologicism Rafal Urbaniak∗ Chair of Logic, Methodology and Philosophy of Science, Gdańsk University, Poland Centre for Logic and Philosophy of Science, Ghent University, Belgium rfl.urbaniak@gmail.com entiaetnomina.blogspot.com December 22, 2009 1 Introduction Abstraction Principles (APs), initially used by Frege (1884, 1893, 1903) in his system of foundations of mathematics, are recently being reintroduced into foundational studies within a fairly new approach called neologicism (Wright 1983; Zalta 1983; Hale and Wright 2001). APs are expressions of the form: ‘f (σ) = f (τ ) ≡ σRτ ’, where R is an equivalence relation and f is a newly introduced operator (σ and τ can be first- or higher-order variables). Roughly speaking, on the neologicist’s view, an AP is ideally meant to fix the reference of abstract terms and explicate operation f which on the intended interpretation assigns abstract objects to things that σ and τ range over.1 For instance, Hume’s Principle says that the number of one concept is the same as the number of another concept if and only if those concepts are equinumerous (a notion defined independently of the notion of number). Thus by introducing this principle we’re supposed to: fix reference of expressions like ‘the number of F ’, determine an operation that assigns numbers to objects, and explicate our sortal concept of a number.2 Difficulties that neologicism runs into are fairly well-known (Fine 2002). The neologicist would like to claim that those APs which they found suitable are an∗ This paper has been written during my stay at Bristol as a Visiting Fellow of the British Academy and I owe gratitude to the British Academy for their support. I am also grateful to Dr Oystein Linnebo, who hosted my visit, and to Hannes Leitgeb and Leon Horsten for their support, comments and time. The issues discussed in this paper has been discussed during various meetings I was able to attend and I am also grateful to the audiences of all those events. 1 For instance, to individuals that already belong to the domain in case σ and τ are first-order variables, or to concepts reaching over the domain of individuals, if we take σ and τ to be second-order variables and interpret them Frege-style as ranging over concepts. 2 In fact, adding comprehension principle for concepts and Hume’s Principle to secondorder logic yields a consistent system which allows to derive second-order Peano Arithmetic (a consistency proof is given relative to the consistency of real analysis (Boolos 1987)). 1 alytically (or at least conceptually) true. But giving a rationale for such a claim is far from trivial. One can’t say that all APs are true, because (within a sensible logical framework) certain APs lead to straightforward contradictions. One can’t even require that all consistent APs are true, because there is no consistency test for APs. What’s worse, certain APs are separately consistent but mutually exclusive, and hence not all of them can be true.3 These and related problems give rise to the fairly open problem of finding sensible acceptability conditions of APs. All suggestions put forward so far are quite complicated, often require assumptions stronger than those provable in standard set theory, and there is no general agreement as to their plausibility and effectiveness. Other problems pertain to a rather technical question as to how the neologicist strategy can be generalized in order for more elaborate mathematical theories to become available. In particular, it is not clear how the full force of standard set theory (ZFC) is to be gained if the foundational role is to be played by abstraction principles.4 As it is well-known, the abstraction principle initially used by Frege to rule the behavior of extensions, called Basic Law V (BLV), leads to contradiction within the rather moderate framework of second-order logic with full comprehension.5 BLV says that the extensions of two concepts are identical if and only if exactly the same objects fall under those concepts: [BLV] Ext(F ) = Ext(G) ≡ (∀x)(F (x) ≡ G(x)) Roughly, it leads to contradiction because Cantor’s theorem together with comprehension support the requirement that there be more extensions over a domain than the objects, whereas the abstraction principle requires that extensions are to belong to the domain itself, thus claiming that there are at least as many objects as extensions.6 Some attempts has been made to reconstruct set theory by means of a restricted version of BLV which allows to apply BLV only to certain “good concepts” and identifies the extensions of all “bad concepts”: [RV] (∀P, Q)[Ext(P ) = Ext(Q) ≡ [(Bad(P ) ∧ Bad(Q)) ∨ (∀x)(P (x) ≡ Q(x))]] Even assuming that it is possible to provide an intuitively clear account of what it means for a concept to be bad, and to give a plausible explanation as to why extensionality should fail for bad concepts thus understood, many technical 3 Interestingly, there also is a class of APs such that no finite conjunction of them is inconsistent (within a fixed classical logical framework), and yet they all cannot be satisfied in a domain. 4 There is an interesting tension between Hume’s Principle (HP) and ZFC. On one hand, HP entails that the property of being self-identical has a cardinal number, the number of all objects whatsoever. But ZFC doesn’t support the claim that there exists the cardinal number of all the sets there are. (see Boolos 1997 and Wright 1999 for a more elaborate discussion). 5 Full comprehension in this case says that for any formula (with a single free individual variable) in the object language there is a concept such that an object falls under that concept if and only if it satisfies that formula. 6 Another way to think how a contradiction arises is to observe that BLV entails the existence of the extension of all those objects (extensions included) which are not elements of themselves, and this, for Russellian reasons, leads to contradiction once we ask whether this class itself is an element of itself or not. 2 difficulties pertaining to what (part of) set theory can be regained by means of RV remain. More philosophical issues have also been raised. It is doubtful that APs are capable of fixing reference of abstract terms. Given a domain of non-abstract objects, no AP determines unambiguously the set of abstract objects that have to be added to this domain in order for the principle to hold. Even if we restrict ourselves to a certain cardinality of such an extended model, certain principles will still have non-isomorphic models and all APs will be insensitive to permutations of abstract objects. Other qualms can be introduced by the following example. Say I introduce a new sort of objects, which I call ‘humwumumwas’. I insist that humwumumwas are correctly introduced by the following principle: x’s humwumumwa is the same as y’s humwumumwa iff x and y are both logicians and either the number of x’s hairs and the number of y’s hairs are both even, or they’re both odd. Now, prima facie, the left side of this principle (in a sensible context) entails the existence of humwumumwas, whereas the right side does no such a thing. How can I just postulate their equivalence and claim that it is conceptually true? Even if we accept the principle, how does it help in determining whether a given object (a chair, number two, or what have you) is a humwumuwa? Even if we were able to identify humwumumwas as a sort, how would we be able to assign them to appropriate objects? That is, given a humwumumwa, how do I find out if it’s mine or, say, Kit Fine’s?7 In general, one worry (stemming from (Boolos 1997)) is that it is unclear to what extent APs can be analytically or conceptually true, if what they do is they allow to derive new existential statements. For instance, HP allows to infer the existence of infinitely many numbers. It is also far from obvious how come that the right-hand side of HP which says something only about the existence of a 1-1 relation can be analytically or conceptually equivalent to a claim that states the existence of a number.8 The goal of this paper is to hint at an approach to abstraction principles that avoids most of those difficulties, not by restricting abstraction principles themselves, but by dropping certain assumptions about them and showing that they can still play an important foundational role. That is, instead of approaching abstraction principles from the predicativist perspective or instead of trying to find sensible acceptability criteria for a restricted class of abstraction prin7 This problem is widely known as the Caesar problem and was originally formulated in the following manner: Never [. . . ] can we decide by means of our definitions whether any concept is assigned the number Julius Caesar, or whether this conqueror [. . . ] is a number or not. (Frege 1884: §56) 8 Note that the first worry is a different worry from the qualm that one side of the equation has existential consequences that the other doesn’t. In the former case we’re talking about existential claims following from the whole principle. 3 ciples, I will rather sketch a general framework where all abstraction principles are acceptable in an iterative hierarchy of abstraction principles. My approach will be inspired by certain ideas of Kotarbiński (1929), a Polish logician and Tarski’s teacher. On his view, there are no abstract objects and the only reason we introduce terms that seemingly refer to them is brevity and simplicity of our discourse. Thus, (on philosophical grounds) he divides singular terms into those that really refer to objects and those that only pretend to do that and behave like singular terms, but in fact don’t refer to anything. The latter he dubs onomatoids.9 My rather anti-Fregean diagnosis of the problems encountered by neologicism, is this: it’s not APs that cause the problem. It’s the belief that they establish functions into the domain, that is, that abstract terms introduced by means of APs really refer to objects. An alternative way to go about APs is to treat them APs as linguistic rules that tell us what onomatoids can be introduced and what differences between them we are to ignore to increase simplicity of our discourse, thus providing us with a systematic way to introduce fake names and to provide sentences in which they occur with truth conditions. Once this approach is taken, most of the neologicist’s problems disappear, and yet, we can still get our mathematical theories to behave the way we want them to behave. In the end, I will accommodate abstraction principles into a modal and iterative framework. Before we reach this stage, though, I intend to introduce step-by-step all its essential ingredients, starting with a modal approach to linguistic devices and higher-order quantification. I will begin with a rather straightforward example of a modal interpretation of (what amounts to) monadic second-order quantification. This will be done in section 2. Section 3 will introduce another ingredient – iteration. Section 4 will use the example of Hume’s Principle to explain how abstraction principles are to be built into the framework. In section 5 I will elaborate on how the same general strategy can be used to obtain set-theory ZC using unrestricted Basic Law V interpreted iteratively. Finally, section 6 will contain general remarks about the philosophical advantages of the advocated approach. 2 Quantified naming logic: a modal semantics In this section I will be concerned with a variant of the logic of plurals, Quantified Naming Logic (QNL). The logic in question will be the consequence operation that the language of QNL yields when given set-theoretic semantics.10 In QNL, name variables are the only admissible kind of variables and the only quantifiers are those that bind name variables. Well–formed formulas are constructed from name variables a1 , a2 , a3 , . . . ∈ V ar, Boolean connectives ¬, ∧, a sentential functor of two name arguments ε (read as ‘is one of’ or simply ‘is’), and the existential quantifier ∃ according to the following rules: 9 He didn’t say anything about abstraction principles, though. I will use ‘QNL’ and ‘the logic of plurals’ interchangeably, trusting this will not cause any confusion. 10 Sometimes, 4 • If α1 , α2 ∈ V ar, α1 ε α2 is a well-formed formula. • If φ1 , φ2 are wff‘s and α ∈ V ar, then ¬(φ1 ), (φ1 ) ∧ (φ2 ) and (∃α)(φ1 ) are wff’s. • Nothing else is a wff. Intuitively, name variables can behave like place-holders for countable name phrases, or ‘names’ for short, no matter whether those are empty, singular or refer to multiple objects, and ‘a ε b’ is true iff a names exactly one object and this object is also named by b (b may name other objects as well, but it does not have to). Some examples of QNL renderings of natural language sentences are as follows. ‘Socrates is a philosopher’ is translated as: Socrates ε philosophers ‘All cats are animals’ yields: (∀a)(a ε cats → a ε animals) ‘Some logicians admire only each other’ is interpreted as: (∃a)[(∀b)(b ε a → b ε logicians)∧(∀c)(∀d)(c ε a∧c admires d∧d ε d → ¬d ε c∧d ε a)] Since the main two alternative semantics for plural quantification are set– theoretic or substitutional, and one of the main argument for the claim that plural quantification commits one to sets relies on the claim that the substitutional interpretation is either theoretically or nominalistically unsatisfactory and that QNL with set-theoretic semantics carries commitment to sets, let’s review these approaches briefly. A set-theoretic QNL–model is a structure hD, Iset i where: • D is a non-empty domain of objects, • Iset maps V ar into the powerset of D, i.e. P (D). A substitutional QNL–model is a structure hN, Isub , V ali where: • N 6= ∅ is a set of name substituends, • Isub maps V ar into N , • V al maps {hx, yi | x, y ∈ N } into {1, 0}. The basic idea is that under the substitutional interpretation the truth of substitution instances of atomic formulas is taken to be primitive and V al is a function that assigns truth values to such instances. That is, the pure substitutional interpretation explicitly refuses to provide further analysis of truthconditions of substitution instances of atomic formulas, especially in terms of the reference of their constituents. No such things are mentioned: the most fundamental level provided by a model is just a function which determines which of 5 those are true, and that’s the end of a story: why the function is such-and-such is not even an appropriate question in this context. As for satisfaction, I skip the clauses for Boolean connectives and describe how it works for atomic and quantified formulas: • Atomic formulas: hD, Iset i |=set α ε β iff | Iset (α) |= 1, and Iset (α) ⊆ Iset (β). hN, Isub , V ali |=sub α ε β iff V al(Isub (α), Isub (β)) = 1. • Quantified formulas:11 hD, Iset i |=set α (∃α)φ iff hD, Iset i |=set φ, α for some Iset which differs from Iset at most at α. hN, Isub , V ali |=sub α (∃α)φ iff hN, Isub , V ali |=sub φ, α for some Isub which differs from Isub at most at α. What I call the received view (motivated by certain remarks by Quine, Gödel or Skolem) is the view that any logic that uses higher-order quantification comparable to at least monadic second-order logic (and QNL is such a logic) is committed to the existence of sets. One of the reasons why one might think that plural quantification bears commitment to abstract objects is the belief that plural quantifiers have to range over sets, possibly because another alternative, the substitutional reading of plural quantifiers is not satisfactory. One could imagine someone arguing as follows:12 11 It may seem that additional requirements should be put on V al so that certain formulas come out valid. For instance, (∀a, b, c)(a ε b ∧ b ε c → a ε c) is set-theoretically valid, but not substitutionally valid. Take the interpretation where N = {1, 2, 3}, V al(1, 2) = V al(2, 3) = 1 but V al(1, 3) = 0, and I(a) = 1, I(b) = 2, I(c) = 3. Actually, the point of the substitutional interpretation was to allow for disagreements of this sort (Dunn and Belnap 1968). One of the uses of the substitutional interpretation is to provide a better formal framework for the debate about truth conditions for fiction and propositional attitudes. In such contexts, arguably, certain principles valid in set-theoretic semantics are bound to fail. For instance (depending on the details of one’s views) the substitutability of identicals may not preserve truth in belief contexts and reflexivity of identity may fail in fiction. For now, even if there is a suitable set of restrictions on V al that would make QNL with the substitutional semantics equivalent to QNL with set-theoretic semantics, I put the question of what those conditions would be aside. 12 I have not encountered an argument formulated exactly this way, but it seems that the tendency can be traced back at least to Quine. In (1947) he remarks that any usage of general terms in the context of quantification commits one to abstract objects (pp. 74–75), and he explicitly says about quantifiers binding predicate letters: If we bind the schematic predicate–letters of quantification theory, we achieve a reification of universals which no device analogous to Fitch’s is adequate to explaining away. These universals are entities whereof predicates may thence- 6 Whenever we present a logical system we have to provide it with a formal semantics for it that we find adequate. When we use a logical system with a certain semantics we are committed to whatever the quantifiers of the language of the system under considerations range over according to the semantics we provided. The logic of plurals can have two kinds of semantics: a set–theoretic semantics (discussed before) or a substitutional semantics. QNL with substitutional semantics is not theoretically satisfactory, because for the nominalist names are finite sequences over a finite alphabet, and there can be at most countably many of those, which is not enough to emulate quantification over subsets of an infinite domain.13 Therefore it has to be given a set–theoretic semantics. But on the set–theoretic reading of plural quantifiers, they range over subsets of a domain. Subsets of a domain are abstract objects and hence whenever we use the logic of plurals we are committed to abstract objects. To counter this argument, I will reject the assumption that the substitutional and set-theoretic semantics are the only sensible options available. To do that, I will introduce a modal substitutional semantics which provides us with a sufficient repertoire of possible names. In the relational semantics for QNL the quantifiers range over ways (certain) names could be. The relational models defined below will introduce the notion formally in a fairly well–known framework. This will allows us to get a better handle on the semantics and perhaps provide us with a better understanding of this notion. Suppose we start with a world which for our purposes is devoid of linguistic objects (especially names). The content of this world is our domain of (already existing) bare individuals, call it the bare world. It is also possible to introduce names which would either be empty or refer to one or more already existing objects. In this setting we can take a possible world to consist of two sorts of objects: bare individuals (those which do not name anything) and names that either do not name anything or name (one or more) pure individuals. The situation in a possible world determines what reference the names that exist in it have (if there are any). The bare world can be also interpreted as such a possible world, only at least one of the sorts would be empty. Starting with the bare world we can subsequently extend its repertoire of names by introducing (countably many) new names that refer to bare individuals. Let us idealize forward be regarded as names; they may be construed as attributes or as classes . . . The predicate letters, when thus admitted to quantifiers, acquire the status of variables taking classes as values. (pp. 77-78) He is even more explicit when he discusses Leśniewski’s Ontology (Quine 1952: 141). There, he criticizes Ajdukiewicz and Leśniewski for not attaching sufficient significance to the fact that “the variables which have been said to stand in places appropriate to general terms are subjected in Leśniewski’s theory to quantification.” This theory, according to Quine, “. . . surely commits Leśniewski to a realm of values of his variables of quantification; and all his would–be general terms must be viewed as naming these values singly.” 13 This criticism of the substitutional reading comes from (Küng and Canty 1970). 7 here: we are not putting any restrictions on which individuals can be named by a name, we assume that any bare individual (or any bare individuals) is (are) nameable. This gives raise to the so–called naming structure: Definition 1. A naming structure is a tuple hI, W i where I is a set (of bare individuals) and W is a set of possible worlds. A possible world is a tuple hN, δi where I ∩ N = ∅ and δ ⊆ N × I. A bare world is the possible world with N = ∅. The following conditions all have to be satisfied: • B = h∅, ∅i ∈ W (i.e. the naming structure contains a bare world). • for any B 6= w = hN, δi: – N is countable, and – N 6= ∅. • The accessibility relation on possible worlds is defined by the following condition. Let w = hN, δi, w′ = hN ′ , δ ′ i. Rww′ if and only if: – N ⊂ N ′, – {hx, yi | x ∈ N ∧ hx, yi ∈ δ ′ } = δ (i.e. δ ′ ∩ N × I = δ). If M is a naming structure and I is the set of its bare individuals, I say that I underlies M or that M is based on I. If M = hI, W i and w ∈ W I will sometimes write w ∈ M. A few words of explanation here. First, we start “constructing” a naming structure with a bare world. This bare world together with I represents the situation where we have a set of objects that can be named but we have not introduced any names yet. Since in principle (unless one has very specific religious beliefs) there is no reason to believe in unnameable individuals, the I in the bare worlds is simply the domain of objects. For now, we are considering ways individuals which are not names could be named. This means that introducing new names does not change the domain of individuals (that is why I is a set in the naming structure and does not vary with possible worlds). Also, this indicates that the only way we get from one possible world to another accessible world is by extending the repertoire of available names (so N has to be a proper subset of N ′ ). On the nominalist reading, quite plausibly, names are finite sequences of symbols (or phonemes) from a finite alphabet. This means that in any possible world there can be at most countably many names (hence the requirement that N be countable).14 Moreover, the basic idea is that by “going” to another possible world we are extending what we already have; not changing the 14 I conjecture that as far as assessment of the language of QNL goes, nothing would change if the N was finite, but the nominalist doesn’t have to make this assumption. If there are finitely many names, it is for less a priori reasons than the mere countability of the set of finite sequences over finite alphabets. 8 ways that names that we already have are. That is, we can add a new name and take it to refer to such–and–such objects, but the reference of the already existing names cannot be changed. In a sense, names are thought of as given with reference. Hence the denotation relation in an accessible world has to agree with the initial denotation relation on all names that already existed before we extended the set of names. This sort of structure will not ensure yet that every intuitively possible way of meaning will have a representation in a model. To do this, we have to require that for any subset of the domain of individuals in a possible world there be an accessible possible world in which there is a name which denotes all and only its elements. Definition 2. Let hN, δi = w ∈ W . A naming structure M = hI, W i is w–complete if and only if: (∀A ⊆ I)(∃w′ = hN ′ , δ ′ i)(Rww′ ∧ (∃x ∈ N ′ )(∀y ∈ I)(δ ′ (x, y) ≡ y ∈ A)) A naming structure M = hI, W i is complete iff for any w ∈ W M is w– complete. The basic idea is that a naming structure is w–complete iff for any set of individuals existing in this world, there is a world accessible from w where a name which names elements of this set exists. In other words, it is w–complete if it models the full range of ways names could be in w.15 I can now explain how quantification in QNL is supposed to range over ways names could be (or refer). I will be evaluating sentences in a name structure by evaluating them in its bare world. First, we have to define satisfaction of a formula at a world in a naming structure. Definition 3. An M–interpretation is a triple hM, w, vi, where M is a naming structure, w = hN, δi is a possible world in M and v either assigns to every variable in QNL an element of N , if N 6= ∅, or is the empty function on the set of variables of QNL otherwise. If M is a complete naming structure, then we say that this M–interpretation is complete. Satisfaction of formulas under M–interpretations is defined as follows. Definition 4. Let hM, w, vi be an M–interpretation, w = hN, δi. Also, let a and b be QNL–variables and φ and ψ be QNL–formulas. The relevant satisfaction clauses are:16 • Atomic formulas: hM, w, vi |=⋄ a ε b iff v(a) and v(b) are defined, ∃!x∈I hv(a), xi ∈ δ, and (∃y ∈ I)(hv(a), yi ∈ δ ∧ hv(b), yi ∈ δ). 15 The reason why B–completeness is not sufficient is that the class of B–complete naming structures would agree with classical set–theoretic semantics only on sentences with only one quantifier. In a naming structure, the iteration of quantifiers carries us deeper and deeper into the structure. 16 I skip the clause for conjunction because it is fairly standard, but I give the clause for negation because it is slightly different from the usual formulation. 9 • Complex formulas: hM, w, vi hM, w, vi |=⋄ |=⋄ ¬φ iff v is not the empty function and hM, w, vi 6|= φ. (∃a)φ iff for some w′ ∈ M, Rww′ and hM, w′ , v ′ i |= φ where v ′ differs from v at most in what it assigns to a. The basic idea is that we treat the class of names that exist in a certain world as the class of substituends when we evaluate a formula in this world. Since the bare world does not contain names the notion of satisfaction in the bare world as applied to open formulas is not very fascinating. No formula containing free variables will be satisfied at the bare world. The situation changes when it comes to evaluating sentences in bare worlds. First, let us say that a sentence is true in a naming structure M if and only if it is satisfied in its bare world under any valuation. A sentence is M–valid if and only if it is true in any naming structure. A sentence is M–complete valid if it is true in any complete naming structure. The following should come as no surprise:17 Fact 1. (a) For any set–theoretic QNL model there is a complete M-model which agrees with it on all formulas. (b) For any complete M-model there is a set theoretic QNL model which agrees with it on all formulas. The following observations are in order. First, modulo this relational semantics, QNL has an intuitive translation into a two-sorted first-order quantified modal logic with one additional primitive binary operator representing the naming relation. Second, the size limitation objection does not apply here. Quantifiers are interpreted as ranging over possible names, but not over possible names from one particular possible world but rather over names that belong to the union of all sets of names from all accessible possible worlds. The initial plausibility of the objection results from the ambiguity between: It is possible that for every subset of the domain there is a name which names all and only those objects which are elements of this subset. and For any subset of the domain it is possible that there is a name which names all and only all its elements. For instance, suppose the domain consists of all real numbers. It is false that there is a possible world in which all elements of the domain are named by 17 For more details pertaining to QNL and its modal interpretation and a proof of the following claim see a paper wholly devoted to the system [REFERENCE REMOVED TO PRESERVE ANONYMITY]. 10 individual terms, because there are not enough names in any particular possible world. However, this does not mean that there are unnameable real numbers. Quite the contrary, every real number can be named in some accessible possible world. The abundance of numbers arises from the abundance of accessible possible worlds — the ways names could be.18 3 Cumulative naming logic Now, let’s try to generalize this approach to a cumulative hierarchy of possible names and see how close to set theory we can get. This will also allow us to take a first look at the idea of iteration as developed within the modal approach. The main idea here is just an extension of the account of naming developed in the previous section. Recall that there I provided a relational semantics for QNL by formalizing the idea that only those names can be introduced which can name only those object which already existed before the introduction of those names.19 In other words, before we can successfully introduce a name for an object, the object has to exist. The repertoire of moves that we could make was a little bit restricted, though. We started with a fixed domain of nameable individuals and we could introduce names which either did not refer to anything or referred to objects from this fixed domain. There is a viable way in which we can weaken the condition, still abiding by the restriction that only those objects which already exist can be named. Consider the following scenario. We start with a fixed domain of individuals which are not names themselves. At the next step, we introduce certain names which either do not name anything, or name individuals from that fixed set. Those will be (possible) names of level 1. What we get is another possible world, where among existing objects we have not only the individuals which are not names, but also those names that have been introduced. Now we can introduce new names which (i) either do not name anything or name objects from the domain of pure individuals and thus we extend the set of names of level 1, or (ii) have at least one name of level 1 among its referents but do not have names of higher level among its referents, and thus we extend the set of names of level 2, and so on. In general we can introduce names to refer to whatever objects we want insofar as they are among objects already available in the world.20 18 For a more elaborate paper on the modal interpretation of plural quantifiers see (Author year) [reference removed to preserve anonymity]. The paper also contains a historical account of earlier attempts to provide a nominalistically acceptable semantics for plural quantification and a philosophical discussion of the issues that the modal semantics raises. 19 This restriction will be soon weakened. For now, let’s see how far we can get without dropping it. 20 An objection might be raised that this view prohibits one from introducing names that refer to future objects even in cases which intuitively are not very problematic. For instance, a mother can decide what she will name her child if she has any even before the pregnancy. Even on the meta–linguistic level there are certain quite innocent uses of this practice. For instance, I can write the following pair of sentences. The next sentence will be written in 11 The basic idea behind the cumulative view is fairly simple. Any extra– linguistic objects that already exist can be named by a new token and any tokens thus introduced can be named by yet another new token without running into a contradiction. We can introduce the notion of a cumulative naming structure, which is a generalization of our notion of a naming structure (a more informal gloss follows the definition). Definition 5. A cumulative naming structure is a tuple hI, W i, where I is a set of bare individuals and W is a set of cumulative possible worlds. A cumulative possible world (c.p.w., for short) is a tuple hδ, (Ni )i∈N+ i, where (Ni )i∈N+ is a denumerable family of Ssets of names indexed with positive natural S numbers,21 δ ⊆ ( i∈N+ Ni × I ∪ i∈N+ Ni ), and the following conditions are satisfied: [ Nk }] (1) (∀i)[{y | (∃x ∈ Ni )hx, yi ∈ δ} ⊆ I ∪ k<i (∀i > 1)(∀x)[x ∈ Ni → (∃y ∈ Ni−1 )hx, yi ∈ δ] (2) B = hδ, (Ni )i∈N+ i ∈ W , where (∀i)Ni = ∅ For any i ∈ N+, Ni is countable. (3) (4) (∀w)[w = hδ, (Ni )i∈N+ i → (∃i)(∀k > i)Nk = ∅] (5) If w = hδ, (Ni )i∈N+ i and x ∈ Ni , we say that x is a name of level i in w. If (∀k > i) Nk = ∅ and i is the least such index, we say that w is inhabited up to i, or that w is of level i. A few words of explanation are due. Just like a naming structure, a cumulative naming structure is based on bare individuals I and contains possible worlds in W . Possible worlds, however, are slightly more complicated. First of all, instead of just one (possibly empty) level of names, a possible world contains a denumerable number of (possibly empty) levels of names. The intuition here is that N1 is the set of names which either do not denote anything or denote English. The previous sentence is true. These, and similar, equally innocent uses would be prohibited on the cumulative view of naming, one might argue. The key observation here is that the purpose of the cumulative view of naming is not to provide a complete list of ways name can be introduced in natural language, or to account for the introduction of any name which intuitively does not lead to troubles in natural language. Rather, the aim is to provide a fairly unified theory of which names for sure can be introduced without any serious doubts whether new paradoxes can arise. In a way, what I want to be true is that whenever a name can be introduced according to the cumulative view of naming, its introduction does not lead to a paradox. The opposite claim, that if a name cannot be introduced according to the cumulative view naming, its introduction is not innocent and potentially dangerous, is neither needed nor suggested here. Perhaps in natural language there are other names whose introduction violates the restrictions put on naming by the cumulative view, and yet their use is quite innocent. I am only concerned with formalizing a certain strategy of introducing names and seeing how far it gets us. Further on, I will weaken the restrictions on available ways of naming things, to accommodate some other moves that one can make when introducing names. 21 That is, N = N \ {0}. For the sake of simplicity I take N to be I. + 0 12 bare individuals only, and in general Ni is the set of those names which denote bare individuals or names up to level i − 1 only. Hence condition (1): δ is the naming relation and names can name only bare individuals or names of lower levels that exist in the same possible world. Condition (2) fixes the hierarchy: any empty name and all names whose referents are all bare individuals belong to level 1, and in order to belong to level i > 1 a name has to name at least one name of level i − 1. (3) says that the bare world has to be included in W . Requirement (4) has the same justification as before: names are taken to be finite sequences over a finite alphabet. The motivation for restriction (5) is this: in order to introduce a name of level i > 1 we have to already have at least one name of level i − 1 and thus getting to a world with all Ni non–empty would requires an infinite number of steps through the accessibility relation in a complete cumulative naming structure (notions to be defined below), but one cannot complete a non–terminating infinite sequence of steps.22 Later on, I will concede that it is possible to refer to infinitely many possible names on infinitely many possible levels in certain cases — those will be cases where an infinite sequence of “actions” has to be complete in order for reference to be established. Definition 6. Let hI, W i be a cumulative naming structure, w = hδ, (Ni )i∈N+ i, and let w′ = hδ ′ , (Ni′ )i∈N+ i belong to W . Then, Rww′ if and only if: (∀i)Ni ⊆ Ni′ (∀i)(∀x, y)[{hx, yi | x ∈ Ni ∧ hx, yi ∈ δ ′ } = {hx, yi | x ∈ Ni ∧ hx, yi ∈ δ}] (6) (7) This definition just extends the notion of accessibility used for QNL naming structures. Requirement (6) says that we introduce new names by extending their repertoire, and (7) says that when we introduce new names, the reference of previously existing names remains the same. Now we introduce the notion of completeness of a structure. Again, this is just an extension of the notion that has been already introduced. Definition 7. Let M = hI, W i be a cumulative naming structure and let w = hδ, (Ni )i∈N+ i ∈ W . For the sake of convenience we fix the notation as follows. Ni ’s with i ≥ 1 are sets of names, and N0 is just another name for I. By (5), there is the least k ∈ N such that (∀i > k)Ni = ∅.23 M is said to be w–cumulatively complete (w–complete, for short) if and only if for any S A ⊆ n∈N,n≤k Nn there is a possible world hδ, (Ni′ )i∈N i = w′ ∈ W such that w′ is at most of level k + 1 (that is, (∀i > k + 1)Ni′ = ∅), Rww′ and [ ′ (∃x ∈ Nk+1 )(∀y)(y ∈ Nn → (hx, yi ∈ δ ′ ≡ x ∈ A)) n∈N,n≤k 22 For more details on the impossibility of completing a non–terminating infinite sequence of steps see (Chihara 1965). A question arises as to whether we can allow that the number of names existing in a possible world be countable even if we deny that an infinite number of steps can be performed. I will return to this issue later. 23 Remember that N = N ∪ {0}. + 13 M is said to be cumulatively complete (complete) if and only if for any w ∈ W , M is w–cumulatively complete. Informally speaking, a naming structure is cumulatively complete with respect to a certain c.p.w. w if for any choice of objects already existing in this world (including bare individuals and names introduced so far) there is a c.p.w. accessible from w at most one level higher than w, where a name which names exactly those objects exists.24 Let’s try to define a language more similar to the language of set theory and interpret it in cumulative modal models. The language of cumulative naming logic (CNL) contains the standard logical symbols: (∃, )¬, ∧, =, (, ) variables that (under an interpretation) will take pure individuals as values: x 1 , x2 , x3 , . . . variables that (under an interpretation) will take either names or pure individuals as values: a 1 , a2 , a3 , . . . The quantifier can bind variables of both sorts (in a way, there are two quantifiers, for the semantics of quantification will depend on the sort of the bound variable). Besides, the language contains one primitive symbol ‘D’ which is a two-place predicate (which can take variables of both sorts as arguments in arbitrary combinations) that in the intended reading means ‘denotes’. A CNL term is either an individual variable, or an ai variable.25 Formation rules are standard. Definition 8. Complete cumulative name structures are intended models of the language of CNL. A CNL interpretation is a tuple hM, w, vi such that M is a complete cumulative naming structure, w is a c.p.w. which belongs to it, and v (i) maps individual variables into I if I 6= ∅ and v is undefined on individual variables otherwise, (ii) maps the variables ai into DO (w) if DO (w) 6= ∅ and v is undefined on ai variables otherwise. Let φ and ψ be CNL formulas and let α1 and α2 be CNL terms. The satisfaction under an interpretation is defined 24 If one feels uncomfortable with set theory being used in meta-language, the same definitions can be given in a slightly more long-winded manner using plurals. 25 I will use the standard simplifications regarding dropping indices and writing a, b, c, d instead of a1 , a2 , a3 , a4 and u, v, x, y, z instead of x1 , x2 , x3 , x4 , x5 . 14 by: hM, w, vi |= D(α1 , α2 ) iff hv(α1 ), v(α2 )i ∈ δ hM, w, vi hM, w, vi |= |= α1 = α2 iff v(α1 ) = v(α2 ) φ ∧ ψ iff hM, w, vi |= φ and hM, w, vi |= ψ (9) (10) hM, w, vi |= ¬φ iff hM, w, vi 6|= φ (11) hM, w, vi |= (∃xi )φ iff hM, w, v ′ i |= φ for some v ′ which differs from v at most at xi (12) hM, w, vi |= ′ (8) ′ (∃ai )φ iff hM, w , v i |= φ for some w′ such that Rww′ and for some v ′ which differs from v at most at ai (13) In our metalanguage we can introduce certain useful abbreviations. I will use ‘⇔’ to mark an introduction of a definition. For instance, we can introduce a constant U (Urelement) which in any possible world under any possible valuation will refer to all and only elements of I: U (a)⇔(∃x)x = a (14) That is, U (a) is satisfied under an interpretation iff this interpretation assigns a bare individual to a.26 I will also make another notational convention. Instead of ‘D(α1 , α2 )’ I will just write ‘α1 ∈ α2 ’.27 Now, certain (translations of) principles that hold for sets in set theory ZC (with urelements but without replacement and infinity) hold also for possible names. I will list some of them.28 First, just like in Z urelements are atomic (i.e. do not have elements), bare individuals do not name anything: (∀a)(U (a) → ¬(∃b)b ∈ a) (15) This holds in cumulative models because no elements of I are first arguments of δ. In Z we can separate subsets of sets that we already have by defining them using formulas of the formal language of set theory. Something similar can be said about possible names. If we have a name, it is possible to introduce a name which names all and only those objects named by that name which satisfy a certain condition. If φ(c) is a CNL formula with c as its only variable, the following holds in any cumulative naming structure: (∀a)[¬U (a) → (∃b)(¬U (b) ∧ (∀c)(c ∈ b ≡ c ∈ a ∧ φ(c)))] 26 This (16) was not a very complex move: we just “predicatized” one sort of variables. here has a different meaning than it has in set theory, but whenever I will speak of the language of set theory I will use ‘∈’ to express elementhood as well, assuming that the context will decide what meaning should be given to this symbol. 28 The axiomatization of set theory that I have in mind to some extent follows the one from (Tourlakis 2003). 27 ‘∈’ 15 This holds in complete cumulative models since once we have a possible name a, every thing it denotes has to lie on a lower level, and since any assembly of things of lower levels than the one that a belongs to has a possible name in an accessible world, any set of such objects determined by a certain formula a fortiori has to have a possible name in an accessible world. Also, just like in Z the set of urelements exists, in a cumulative naming structure it is always possible to introduce a name which names all and only bare individuals: (∃a)(¬U (a) ∧ (∀b)(b ∈ a ≡ U (b))) (17) This, again, results from the completeness of cumulative structures.29 The existence of the empty set follows in set theory from the claim that the set of urelements exists and the axiom of separation. Similarly, the possible existence of an empty name follows by (16). It is enough to take c ∈ c for φ(c) (since the theory does not allow not well–founded names and no name refers to itself): (∃a)(¬U (a) ∧ ¬(∃b)b ∈ a) (18) In set theory the axiom of pair says that for any two objects (urelements or sets) there exists a set which contains them as its elements. Correspondingly, for any two elements of a domain of objects of a given c.p.w. it is possible to introduce a name which has those two among its referents. (∀a, b)(∃c)(a ∈ c ∧ b ∈ c) The union operator can be defined by: [ a∈ b⇔(∃c)(c ∈ b ∧ a ∈ c) (19) (20) That is, a is in the union of the name b if there is a name that b names which names a.30 The CNL rendering of the axiom of union is the following: (∀a)(∃b)(∀c, d)(c ∈ d ∧ d ∈ a → c ∈ b) (21) It says that for any possible name a it is possible to introduce a name b such that if a names a name d and d names an object c, b names c. The property of being an empty name can be defined by: ∅(a)⇔¬U (a) ∧ ¬(∃b)b ∈ a (22) The name–theoretic translation of the axiom of foundation is: (∀a)[¬U (a) ∧ ¬∅(a) → (∃b)(b ∈ a ∧ (∀c)(c ∈ b → c 6∈ a)] (23) 29 Strictly speaking, Z doesn’t have to be constructed with urelements. Since, however, the nominalist wants to be able to make sense of sets grounded in extra-mathematical objects admitting urelements seems like the way to go. 30 Note: the quantification here can be read ‘there is’ instead of ‘it is possible to introduce’ since no possible extension of the naming structure can change the reference of b, once it is given, so the only repertoire of names that b can refer to is those names which it already names. 16 It also holds in any complete cumulative naming structure. Since this is not as obvious as previous claims, let us take a while to understand why exactly this is the case. Names are divided into levels. Suppose then that an is of level n and assume (for reductio) that it violates the principle of foundation, that is, an is a non–empty name and for every object b that it names, there is an object c named by b such that an names c. First, n 6= 1, because a name of level one is either an empty name (but we assumed that ¬∅(a)) or names an individual (but individuals vacuously are witnesses for the existential quantifier in the consequent of (23)). Suppose n > 1, then (the case where n = 0 is excluded by ¬U (a) in the antecedent of (23)). Then (i) an names an object of level n−1, and (ii) for any k < n if an names an object of level k, say bk , there has to be an object of level k − 1, say ck−1 such that both an and bk name ck−1 . But this cannot be the case because n is a natural number only a finite number steps afar from 0 and there are no objects of level lower than 0. The axiom of choice, in the form: (∀a)[¬U (a) ∧ (∀b)(b ∈ a → (∃c)c ∈ b) → (∃d)(∀b ∈ a)(∃c ∈ b)c ∈ d] also holds. Now it says that for any possible name a which names only names b that name something, it is possible to introduce a name d which picks an object from each of the names b and names them together. This holds because if a is present at level k, then all the b-s have to be present at level k − 1, and all the objects named by b-s have to be present at k − 2. But the completeness requirement tells us that at level k − 1 exists a possible name that names all the “chosen” objects. Let us pause for a second and ask: is CNL good enough to mimic set theory? Alas, the answer is negative. There are three concerns here. First of all, we did not pose any restrictions against there being two distinct coextensive names.31 For this reason, the CNL rendering of the axiom of extensionality: (∀a, b)(¬U (a) ∧ ¬U (b) → ((∀c)(c ∈ a ≡ c ∈ b) → a = b)) (24) fails. In a way, one might think, we can find a way around this problem just by postulating that identity w.r.t. names means something different and define it as having coextensiveness as its necessary and sufficient condition. Even putting philosophical qualms about this strategy aside, this does not provide us with a theory of desired strength. Recall the size limitation objection against the substitutional interpretation of plural quantifiers and the story about inscriptions that it employed. The worry was that names are finite inscriptions over a finite alphabet and therefore in any possible world there can be at most countably many names. This is going to cause some problems with the axiom of powerset. Its name–theoretic translation is: (∀a)(∃b)(∀c)((∀d)(d ∈ c → d ∈ a) → c ∈ b) (25) It says that for any possible name a it is possible to introduce a name b which names all possible names c such that c is either empty or names some of the 31 Chihara (1990) makes this assumption with respect to open formulas. 17 objects that a names. The concern here is two–fold. First, CNL names cannot in general name names outside of the world in which they exist, so they cannot be said to name all possible names which satisfy certain condition, if this condition does not exclude the possibility of their existence outside of the c.p.w. under consideration. Another problem is that if a name names denumerably many objects (a possibility we have no good reasons to exclude), its “power name” would have to name uncountably many names. But uncountably many names cannot exist in a c.p.w.! The third worry is that even though we can derive the existence of infinitely many objects, we are still unable to prove the existence of an infinite set (well, the possibility of introducing a name that names infinitely many objects). For this reason, the CNL rendering of the axiom of infinity fails. To be able to regain the axiom of infinity we could call all the levels described in the hierarchy so far finite successor levels and postulate that one more level is available: the ω-level where names naming things that exist at finite successor levels are available. A variant of this move will be made once abstraction principles are introduced, but as things stand at this point, this wouldn’t help us deal with the powerset and extensionality axioms. I think that a better approach to the problem makes use of and is formulated in terms of abstraction principles. Accommodating abstraction principles has also another advantage of providing a way to sort out certain philosophical and technical problems surrounding neologicism. Let’s agree then that CNL, even though it constitutes a step towards a nominalistically more plausible interpretation of set theory, falls short of providing us with a sufficiently strong theory. Keeping these issues in mind we move on to a slightly different story intended to explain how these issues might be fixed. The story will involve so-called abstraction principles (APs), so far employed within a foundational project called neologicism. Although the way this project employs them leads to certain important difficulties, it doesn’t mean that abstraction principles should be abandoned in general. I will advocate a different approach to them, which, I will argue, avoids some of the most important problems and fits rather nicely with the modal framework developed so far. 4 Hume’s principle: a modal interpretation In this section I explain how Hume’s principle can be interpreted from the modalnominalist perspective on the assumption that abstraction principles don’t fix reference to abstract objects but rather systematically introduce onomatoids, expressions which syntactically behave like singular terms but don’t really refer to anything – their role is exhausted when a way of associating them with real names is given and truth-conditions of sentences containing them are provided. First, since we don’t believe in concepts, instead of using Fregean comprehension, we rather speak of possible names, along the lines of previous sections. That is, ‘(∃F ) . . .’ is read as: ‘it is possible to introduce a name (empty, referring to exactly one object or referring to more than one objects) F such that . . . ’ or 18 ‘there are no logical obstacles to introducing a name F such that . . . ’. Definition 9. hD, P, ∆i is an infinite naming structure (INS), if D denumerably infinite, ∆ ⊆ P × D, and for any A ⊆ D, there is a σ ∈ P such that (∀x ∈ D)[x ∈ A ≡ ∆(σ, x)] (A valuation function maps QNL variables into P , satisfaction conditions are mutatis mutandis as before.) For now, we are putting the issue of actual infinity aside and don’t worry about the assumption that the domain of objects is infinite. Now, P is the intended assembly of possible names over D satisfying comprehension. When we look at this from the perspective of the modal semantics developed before, it is worth noting that elements of P don’t really have to belong to one possible world; P can be rather thought of as the union of possible names of level 1 above an infinite domain. By the same token, there doesn’t have to be a single possible world where all elements of P exist: it is enough that each such element can exist in a possible world accessible from the bare world. For the sake of simplicity, we sweep the possible world talk under the carpet and talk about whole unions of levels, assuming that this manner of speaking can be recast in the setting of the modal semantics developed before. There will be a point where ignoring differences between names existing in different possible worlds might conceal an important philosophical move, but at that point I will come clean about it and argue for the plausibility of this move. In such a framework, Hume’s Principle tells us that for each possible name σ ∈ P we can introduce an onomatoid ‘N (σ)’ (‘the number of σ’). Moreover, it tells us that we are supposed to ignore all differences between onomatoids N (σ) and N (τ ) if σ and τ are equinumerous. This idea is captured in the following definition: Definition 10. A minimal HP-model is an INS hD, P, ∆i extended with a set NM = {N (σ) | σ ∈ P } and an equivalence relation ≈ on NM such that (∀σ, τ ∈ P )[N (σ) ≈ N (τ ) ≡ σ ∼ τ ] where ∼ is the equinumerosity relation between possible names. The equivalence relation is the “identification” relation between fake terms, and this requirement can be easily seen as embracing Hume’s Principle. We say that N (τ ) is the successor of N (σ) (i.e. S(σ, τ )) iff τ is equinumerous to a possible name that refers to all those objects which σ refers to, and one more object. So far, minimal HP-models allow us to introduce numerical terms of a very specific form: ‘the number of σ’. However, nothing prohibits us from introducing other numerical terms; they don’t have to be all of the form ‘N (σ)’. We can introduce, pretty much, any expression whatsoever, to play a role of a numerical term, as long as we know how to “identify” it with standard numerical terms in NM , and as long as the assembly of the newly introduced possible numerical terms can be handled systematically (we don’t want a notational system for 19 arithmetic which doesn’t give us a way of delivering a symbol for each “natural number” in an effective way). So, for instance, we don’t have to restrict ourselves to the expression “the number of things that are not identical with themselves”. Equally well, we can just say “zero” (well, we could have introduced “blabla” to play the same role; it just happened so that we didn’t). Extended HP-models are meant to handle this sort of moves. Definition 11. An extended HP-model is a minimal HP-model extended with an arbitrary set of possible numerical terms NA , such that ≈ is an equivalence relation on NM ∪ NA and (∀σ ∈ NA )(∃τ ∈ NM )σ ≈ τ . We can quite naturally extend our definition of successor relation to NM ∪NA (we just add that two numerical terms are in that relation iff they’re associated with two standard numerical terms that are). Here’s one way to get P A1 out of this. (i) If σ ∈ P , ¬(∃x ∈ D)∆(σ, x), then N (σ) is a natural numerical term (NNT) and all elements of NM ∪NA ≈-related to it are NNT’s. (ii) If σ is an NNT and S(σ, τ ), τ is an NNT. (iii) Nothing else is an NNT. Say NA is just: 0, S(0), S(S(0)), . . ., so that 0 is ≈-related to the minimal NNT’s in NM , and the behavior of S mirrors the successor relation on NNT’s. Then, quantification in the language of PA1 might be taken to range over possible singular terms from NA (or NA ∪ NM ). Addition and multiplication are introduced in a fairly standard manner. Boolean connectives have standard satisfaction conditions. Identity in the model is just taken to be ≈, and quantification ‘(∃x) . . .’ is taken to express ‘it is possible to introduce a natural numerical term x such that. . .’. Constants are attached to arbitrary representatives of their numerical terms.32 Thus we obtain a non-trivial modal semantics for PA1 in terms of possible onomatoids whose introduction is governed by Hume’s Principle. The situation we get so far can be represented as follows: ··· S(S(0)) S(0) S(S(S(0))) 0 O O O O O O O O O O O O O O O O O O O O O O O O N (F ) O N (G) O N (H) O N (I) O ··· F G H I ··· y •s •w • ··· There are a few things one can play around in this framework. For instance, we can make a comprehension-HP-comprehension sandwich to provide a model for second-order Peano Arithmetic. 32 Another way would be to take constants to bear multiple reference relation to all its numerical terms in the model, but we don’t need to get into these things. 20 Definition 12. A second-order minimal HP-model is a minimal HP-model hD, P, ∆, NM , ≈i extended with a set P ′ and a relation ∆′ ⊆ P ′ × NM such that for any A ⊆ NM closed under ≈ there is a σ ∈ P ′ such that: (∀x ∈ NM )∆′ (σ, x) ≡ x ∈ A and for no x, y ∈ NM it is the case that: x ≈ y ∧ (∃σ ∈ P ′ )(∆′ (σ, x) ∧ ¬∆′ (σ, y)) The underlying idea here is that for any assembly of possible numerical terms it is possible to introduce a name that refers to them, but obeys the restriction posed on the identification relation. That is, newly introduced names cannot “see” the difference between those numerical terms that have been identified. To interpret the language of PA2 it is now enough to: assign constants to arbitrary representatives of appropriate equivalence classes of numerical terms, take individual variables to range over natural numerical terms in NM , interpret identity between numbers to be the identification relation, and to understand second-order quantifiers as ranging over possible names from P ′ . One worry here is that infinity was built in from the beginning, and from the nominalistic point of view this doesn’t seem like a kosher move. In fact, this assumption was made only for the sake of simplicity and there are at least two strategies to deal with this issue. The first strategy is to go modal about existence of objects. Thus, instead of saying that the domain of objects is infinite, we might say that no matter how many objects exist, it is still possible that there exist more objects. This, on one hand, forces us to retract the assumption that D is infinite in a particular naming structure, and to consider many naming structures differing in size on the other. Then, quantifiers in arithmetic would not only contain a modal factor of the sort ‘given the way the extra-linguistic world is, it is possible to introduce a natural numerical term such that. . . ’ but also a certain modal claim about the way the extra-linguistic world is: ‘given the way the extra-linguistic world, for all I know, there could be so many objects that one could introduce a natural numerical term such that. . . ’. Definition 13. A D-modal naming structure (DNS) is a structure hW, Π, ∆i, where W is a non-empty family of sets such that at least one finite D is in W , and for any finite Di ∈ W there is a Dj ∈ W with Di ⊂ Dj . Each Di in W is associated with a Pi in the family of sets Π so that for any Di , for any A ⊆ Di , there is a σ ∈ Pi such that ∆(σ, x) iff x ∈ A, for any x. ∆ is an “internal” naming relation, which means that it behaves like ∆ in INS in referring elements of Pi to elements of Di only, for each i. Also, if Di ⊂ Dj then Pi ⊂ Pj and ∆ in Pj restricted to Pi is just the way it was over Di .33 33 Observe that as far as arithmetic is concerned, we can also assume that each Di is finite. 21 If we now appropriately proceed with adding HP on top of each ‘mininaming-structure’ (that is, for each Di together with Pi and ∆),34 we get a model for arithmetic pretty much the same way we did before, with the slight modification that constants can refer to representatives of numerical terms in other “mini-naming-structures’, and quantifiers can carry a formula across to different elements of W (so now, the arithmetical (∃x)φ as assessed at Di says: ‘at some Dj with Di ⊆ Dj , it is possible to introduce a natural numerical term such that. . .’). This strategy works, but is quite dissimilar from the way neologicists think they can pull numbers out of their hats. The (quasi-)Fregean strategy was to start with a single empty domain, and show that even such a domain can be inflated by means of HP to provide all the numbers that arithmetic needs. In fact, this approach can be emulated within the current framework as well. Recall that the reasoning in the initial setting was roughly as follows. Even if there are no objects, there is an empty concept. But this means that there is the number of this empty concept, number zero. But now we also have the concept of number zero. HP tells us that its number exists. Call it ‘1’. Take the concept ‘is identical to 0 or is identical to 1’, take its number, and you have three objects. But hey, we can keep on doing this! How do we mimic these moves? Well, instead of assuming that the domain is infinite, and instead of considering multiple possible finite domains, we can iterate our use of HP. That is, instead of making a three-layered comprehensionHP-comprehension sandwich, we now pile things up infinitely. This means, we start with a domain of objects D (without saying anything about its size), add comprehension over D, that is, a level of possible names P1 with a naming relation ∆1 such that every subset of D has a corresponding name in P1 . Then add a level HP1 of fake numerical terms governed by HP. Then, continue by adding ≈-sensitive comprehension over HP1 , that is a level of possible names P2 with ∆2 such that every ≈-closed subset of HP1 has a corresponding name in P2 , and no name in P2 allows to distinguish between ≈identified elements of HP1 . Next, add a level of fake numerical terms HP2 , this time not only identifying numerical terms of equinumerous names within HP2 , but rather within HP1 ∪ HP2 . The next level of ≈-sensitive comprehension, P3 , is over HP1 ∪ HP2 . Then, we continue adding numerical terms for any S HP Pn , building the identification relation appropriately to n , and adding i≤n S ≈-sensitive comprehension over i≤n HPn for Pn+1 . Supposing that we start with an empty domain, a part of such construction might look like this: 34 By the way, in this setting HP can be consistently put on top a finite domain without inflating it. This is the case because HP has nothing to do with reference to objects. 22 P3 HP2 P2 HP1 P1 u2 u1 u0 y % N (sv1 ) /o /o o/ N (sv2 ) O O " N (v3 ) O |< <| < | |< |< < | |< sv1 sv2 v3 |< |< < | |< |< |< yu %) N (v1 ) /o /o o/ N (v2 ) O O v1 v2 D Here’s a short story about the picture. We start with an empty domain. In P1 it is possible to introduce two different empty names, v1 and v2 . For each of those names, it is possible to introduce a standard numerical term in HP1 : N (v1 ) and N (v2 ). We ≈-identify those, because v1 and v2 are equinumerous (in general, we measure equinumerosity up to ≈ now). Next, in P2 we introduce two different names both naming exactly one (up to ≈) numerical term (that is, both referring to identified numerical terms only). For those names we introduce corresponding numerical terms in HP2 , and identify them (so N (sv1 ) is “our” number 1). Since earlier in P2 we also added v3 , an empty name over HP1 , we also introduce its corresponding numerical term in HP2 , N (v3 ). Since it is just another copy of our “number 0”, we identify it across levels with numerical terms in HP1 . At level P3 we introduce a possible name u2 that names “number 0” and “number 1”, a possible name u1 that names “number 1”, and a possible name u0 that names our “number 0”. Of course, this can proceed further, but I hope by now the reader has a rather clear picture of what this looks like. To interpret the first-order language of arithmetic in such a structure we just take individual variables to range over the union of all HP -levels, and constants to name arbitrary representatives of appropriate ≈-equivalence classes A minor glitch is represented by the fact that now at level P2 we have a name that’s not closed under identification relation. Since for now we were concerned with the language of P A1 which doesn’t have variables ranging over P2 , this doesn’t make much of a difference. If, however, we want to move to second-order arithmetic, the problem should be fixed by allowing names of numerical terms to extend their reference appropriately as the stages proceed. Thus, for instance, sv2 should not only refer to N (v2 ) and N (v1 ), but also to N (v3 ), and any other numerical term identified with N (v1 ). This might seem a little odd and prima facie involves some sort of self-reference. For instance, suppose a name σ names exactly one object from D. We introduce a numerical term N (σ). Then we 23 introduce a possible name τ which refers only to N (σ) and its ≈-comrades. But we also can introduce N (τ ), and since N (τ ) will be equinumerous to N (σ), τ will also have to refer to N (τ ). Come to think of it, however, this is quite innocent, and there is nothing unusual about the fact that the number of ‘number 1’ is number 1. On this approach, the Caesar problem doesn’t arise. Of course, we can consider a language with variables ranging over individuals and over numerical terms together (the language can also include constants); we can even allow for mixed identity statements to be well-formed and meaningful. The truth conditions for identity statements would take identity in the language to be real identity, if one of the arguments refers to (or is assigned) an individual, and identification relation otherwise. Any mixed identity statements, however, will come out false: none of the real individuals is identical to a number because there are no numbers. We don’t need a way of identifying the sort of numbers, because there is no such a sort of objects. Numerical terms, on this view, don’t play the role of real names, they are just the result of a hypostasis, and so we no longer have to answer questions which would be valid only if they were real names. Troubles with fixing the reference don’t bother us anymore. Abstract terms don’t refer and the story ends when we specify how to introduce them and how to assign truth-values to sentences that contain them. There are no objects to be assigned to them and hence reference questions are not hard but rather misguided. This doesn’t mean that mixed identities are ill-formed. We can, in English, ask whether Julius Caesar is number zero, but the answer is quite straightforward. ‘Julius Caesar = 0’ is false, because ‘Julius Caesar’ is not an English numerical term identified with ‘0’. Also, we don’t encounter the problem of existential import of HPs. The left side of an HP only says that it is possible to introduce certain expressions (and indeed, Hume’s Principle on the modal interpretation is in this sense satisfiable even on finite domains: we just can’t introduce numerical terms for cardinalities greater than that of the domain). This interpretation also deals with the worry that the two sides of HP cannot be conceptually equivalent because the lefthand side makes commitments that the right-hand side doesn’t make. The left-hand side doesn’t make any commitment beyond the commitment that the right-hand side makes: certain expressions can be introduced and identified just in case certain possible names are equinumerous (equinumerosity can be also explicated in the modal framework: the possibility of introduction of a binary predicate, such that. . . ). I promised to come clean about moves that might cause difficulties when we try to recast the above account in terms of possible world talk as it was developed in previous sections. At least two such moves need to be mentioned. First, observe that in the modal semantics for second-order Peano arithmetic a possible name in the level above the onomatoids introduced by HP has to refer to all possible numerical expressions identified by the fake identity relation. But equinumerous possible names can exist in different possible worlds, and their corresponding numerical expressions introduced by HP can belong to different possible worlds too. This means that a possible name above the level of ono24 matois might have to refer outside of the possible world in which it exists, even though the “chain of reference” is still downward-looking. This move, I think, is nominalistically acceptable, even without postulating that those possible names in some sense have to exist for another names to refer to them. In the case at hand there is nothing more to “establishing reference” to such possibilia than producing a rigified description of a certain sort. When I say: “any possible name (of level 1) equinumerous with ‘the number of flying elephants in this room’ ”, I’m in this sense referring to many names that don’t exist without any serious commitment. Second, when we iterate HP to obtain arithmetic in a more Fregean fashion and decide to identify numerical expressions of equinumerous names introduced at different levels, we also need to allow possible names to refer to possible “future” expressions. This, again, is not really problematic at least as long as no circularity or ill-foundedness: there is no serious commitment behind using terms like “any possible numerical token of a name equinumerous with name σ”, even if it is clearly possible to introduce names equinumerous with σ in a level higher than the HP level right above the level on which σ occurs. 5 Going modal about BLV The strategy from the previous section can be extended to other abstraction principles. For instance, we might apply the moves we employed before (modalized iteration of APs, reference “outside” of a possible world, and, in a way, reference “ahead” of the hierarchy) to regain set theory using Basic Law V. Now, we start with (a possible empty) domain of urelements, and iterate BLV and (≈-sensitive) comprehension. At level 0 we have just the domain D. At level 1 we have two layers: a layer of possible names over D, P1 for which full comprehension is granted, and a layer of possible set-theoretic terms introduced by means of BLV, E1 . At any level S n + 1 (n > 1) there are two layers: a layer of possible names Pn+1 above D ∪ i≤n Ei (with the restriction that each name in Pn+1 names at least one possible name at En ), whereas the comprehension requires that each ≈-closed subset has a corresponding possible name and that no possible name allows to make a distinction between ≈-identified objects, and a layer of possible set-theoretic terms generated above Pn+1 by the n + 1-th iterated use of BLV. The identification relation goes between those set-theoretic terms that are BLV-assigned to coextensive names (due to the above-mentioned restriction, no set-theoretic terms at different levels are ≈-related). Finally, we allow an ω level which starts with possible names Pω with (≈-sensitive) comprehension above ∪i∈N Ei , and we allow successor levels above Pω in the same manner we did for levels below ω. The iterated BLV on this reading tells us that for any possible name σ that can be reached in the hierarchy it is possible to introduce a corresponding set-theoretic term ‘Ext(σ)’, so that set-theoretic terms assigned to coextensive possible names are identified. When we take the language of set theory, variables are taken to range over 25 set-theoretic terms that can be introduced this way.35 All constants in the language can be assigned arbitrary representatives from among the set-theoretic terms in the structure. For instance, ‘∅’ can be taken to refer to any set-theoretic term associated with an empty name. All the standard abbreviations can be introduced in a rather straightforward manner, mimicking the development of set theory. The existential quantifier in the language now is interpreted as saying ‘it is possible to introduce a set-theoretic term, such that. . . ’. Elementhood of Ext(σ) is identified with falling under σ and identity is real identity in D, and fake identity on any level of set-theoretic terms. All the axioms that held in CNL still hold, pretty much for the same reasons for which they did in CNL. As for extensionality, it now holds by fiat, because our identification relation on purpose identifies set-theoretic terms of coextensive possible names. As for the powerset axiom, we now allow names referring outside of possible worlds they’re in, as long as the relation S goes downwards. That is, suppose we have a certain ≈-closed A ⊆ D ∪ i≤n Ei (only for such subsets there are set-theoretic terms at Ei+1 ). Now, comprehension for level n tells us that at Pn+1 there is a possible name that names all elements of B, for any ≈-closed B ⊆ A. This means that at En+1 there is a term Ext(B) for any such B. But then, also the set of all such Ext(B) is ≈-closed and so at level Pn+2 there is a possible name that names all and only such terms. Take its corresponding set-theoretic term at Pn+2 and this will be the powerset of Ext(A). In other words, the powerset axiom on the modal reading says that for any extension term Ext(A) it is possible to introduce an extension term Ext(σ) such that anything that is an element of Ext(σ) (that is, falls under σ) is an extension term of a possible name naming (in an ≈-sensitive manner) some things falling under A. Notice that the powerset axiom requires only two successor steps in the hierarchy, and so doesn’t rely on the limit ω-jump in the iteration. The limit jump at ω, on the other hand, is needed to ensure the axiom of infinity. The successor steps starting from D won’t necessarily lead to a point where a name that names infinitely many ≈-different possible set-theoretic terms (not to mention, infinitely many ≈-different possible set-theoretic terms that mimic the contents of the ω set). Once level ω is reached, however, a name that names all terms ≈-identified with, say: Ext(∅), Ext(∅ ∪ Ext(∅)), . . . can be introduced, and hence its corresponding extension term occurs at level ω. Nothing forces us to go with the hierarchy further than to successors of ω - infinity requires only ω, and powerset can be reached by means of successor steps. This doesn’t mean that the model cannot be extended to allow for greater 35 If we want to, we can build arbitrary set-theoretic terms into the model if we want, in the same fashion we did it for HP. Another option is not to have arbitrary constants in the model, but to have them outside of it, in the interpreted language. For the sake of simplicity, this is the way I will go. 26 limit jumps – but since we don’t have to do that to obtain ZC, I won’t bother doing that. As for replacement, I won’t discuss its validity here, because its formulation would require a separate account of relations (which I don’t think should be constructed as certain sets). But my conjecture is that once all preliminaries are in place, replacement will come out valid (or valid in slightly modified models). The consistency of the theory thus constructed follows from the fact that models thus defined can be constructed within set theory.36 6 Wrapping up As it is the case with pretty much any philosophical account of anything, this approach also has its challenges. First of all, I have used modalities and possibleworld talk just granting that these are nominalistically acceptable. Of course, a more elaborate justification of this assumption is needed, but it lies beyond the scope of this already long paper. The goal of current considerations was only to see what sense can be made of set theory in a nominalistic fashion granted these assumptions. Another main worry is that in the explanations I have used set theory when giving the semantics. Here is a rather cheap way out. A distinction between set theory as a mathematical theory without any deep philosophical interpretation, and between set theory interpreted in some sense realistically should be made. Now, the enterprize was to use set theory as an uninterpreted mathematical theory, together with a certain philosophical (hopefully nominalistically acceptable) story to explain that set theory was ontologically innocent to start with. In fact, to clarify the question of how strong the metatheory really has to be it might be desirable to have an axiomatization of this account independent of set theory, and an interpretability proof for set theory within an axiomatic system thus formulated. In such case, a defence of nominalistic acceptability of those axioms would be needed. This, however, is another project for another day. Let’s take a look at the worries that we had about neologicism now: 1. Not all APs can be accepted because some are inconsistent within a certain sensible logical background theory. 2. There are no settled acceptability criteria for APs. 3. BLV cannot be used to obtain set theory and the successes of various restrictions of BLV are limited. 36 Technically speaking one might be tempted to say that BLV and set theory are sufficient insofar as we are interested in the foundational issues and no other APs are needed. There are, however, philosophical reasons to avoid “identifying” prima facie non set-theoretic mathematical objects with sets. For a neat display of this worry with respect to numbers see the classic (?). In general, once a general framework for abstraction principles is developed, HP can live side-by-side with set theory, even if numerical terms are taken to be onomatoids sui generis. This of course doesn’t mean that we cannot carefully draw identification lines across different types of abstract terms introduced by means of different APs, if we have a reason to do that. 27 4. APs are unable to fix the reference of abstract terms. 5. APs are unable to identify their associated sorts of objects. 6. APs are unable to guide our assessment of mixed identity statements. 7. Given the existential import of (the left-hand side of) APs, it’s unclear how they are to be analytically or conceptually true. Granting that modalities that I employed are nominalistically acceptable, these qualms are dealt with in the following manner: ad 1 and 2. Since on this setting no AP puts any requirements on the size of the domain above which it’s laid out, any AP is acceptable, as long as we grant that no two different APs introduce the same onomatoids.37 Even the infamous Basic Law V can be used iteratively. ad 3. By iterative use of BLV set theory can be regained. ad 4, 5 and 6. No reference of abstract terms has to be fixed. What’s needed is a systematic way of introducing fake names of a certain sort and of assessing the sentences containing them. Realism about truth-value of such sentences doesn’t have to be realism about the reference of terms occurring in them. Similarly, there is no problem with identifying sorts of objects because there are no abstract objects to start with. For a similar reason, the Caesar problem doesn’t arise. Because no fake singular term a refers, it’s simply false for any really singular term s that s = a. Similarly, there is no question of the matter as to abstract objects introduced by different abstraction principles are the same: if for some reason we find it convenient and consistent, we may make such identification sometimes, but the relation is conventional fake identification between possible name tokens, not really identity. ad 7. Since the left-hand side of an AP is non-committing, the balance between the commitment of both sides is preserved. The right-hand side carries no commitment, and the left-hand side is just an abbreviation of what’s being talked about on the right-hand side and as such carries no commitment either. Since the rules pertaining to the introduction of fake singular terms and their identification are conventional, any AP can be thought of as giving such a set of instructions and as such, as an analytic claim. The foundational role of modally interpreted APs is not the only role they can play. Most likely, other applications can be developed in linguistics and philosophy of language (for instance, to account for the type-token distinction or to replace commitment to propositions understood as abstract objects) or in metaphysics (to provide a parsimonious account of our talk about abstract objects in general). These issues, however, lie beyond the scope of the present paper. 37 This requirement can be weakened by allowing “equivalent” (modulo notational changes and logical equivalence) AP’s to introduce the same onomatoids. 28 References Boolos, G. (1987). The consistency of Frege’s foundations of arithmetic. In On Being and Saying: Essays for Richard Cartwright. MIT Press. repr. Logic, Logic, and Logic, Harvard University Press, 1998. Boolos, G. (1997). Is Hume’s Principle analytic? In Language, thought, and logic, pages 245–261. Oxford University Press. Chihara, C. S. (1965). On the possibility of completing an infinite process. The Philosophical Review, 74:74–87. Chihara, C. S. (1990). Constructibility and Mathematical Existence. Oxford University Press, Oxford. Dunn, J. M. and Belnap, N. (1968). The substitution interpretation of the quantifiers. Nous, 2:177–185. Fine, K. (2002). The Limits of Abstraction. Clarendon Press. Frege, F. (1884). Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl. W. Koebner. Trans. by J. L. Austin as The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number, Oford: Blackwell, second revised edition, 1974. Frege, F. (1893). Grundgesetze der Arithmetik, volume (band 1). Verlag Hermann Pohle. Partial translation by M. Furth as The Basic Laws of Arithmetic, Berkeley: University of California Press, 1964. Frege, F. (1903). Grundgesetze der Arithmetik, volume (band 2). Jena: Verlag Hermann Pohle. Hale, B. and Wright, C., editors (2001). The Reason’s Proper Study. Essays Towards a Neo-Fregean Philosophy of Mathematics. Clarendon Press. Kotarbiński, T. (1929). Elementy Teorii Poznania, Logiki Formalnej i Metodologii Nauk. Ossolineum, Lwów. Küng, G. and Canty, J. T. (1970). Substitutional quantification and Leśniewskian quantifiers. Theoria, 36:165–182. Quine, W. (1952). Review: On the notion of existence. some remarks connected with the problem of idealism, by Kazimierz Ajdukiewicz. The Journal of Symbolic Logic, 17:141–142. Quine, W. V. (1947). On universals. The Journal of Symbolic Logic, 12:74–84. Tourlakis, G. (2003). Lectures in Logic and Set Theory. Volume 2: Set Theory. Cambridge University Press. Wright, C. (1983). Frege’s Conception of Numbers as Objects. Aberdeen University Press. 29 Wright, C. (1999). Is Hume’s Principle analytic? Notre Dame Journal fo Formal Logic, 40:6–30. Zalta, E. (1983). Abstract Objects: An Introduction to Axiomatic Metaphysics. D. Reidel. 30

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