1. Introduction
Since Benacerraf’s ‘What Numbers Could Not Be, ’ Benacerraf (Reference Benacerraf1965) there has been a growing interest in mathematical structuralism. One of the most influential forms of structuralism is the modal structuralism developed in Geoffrey Hellman’s Mathematics Without Numbers (Hellman Reference Hellman1994). Modal Structuralism is a nominalist philosophy of mathematics which maintains that mathematicians can systematically express truths even if there are no mathematical objects, by interpreting statements about mathematical objects as modal claims about what is logically possible. Specifically, Hellman uses claims about logical possibility and second order logic to provide intuitively correct truth conditions for mathematical utterances without quantifying over mathematical objects like numbers and sets.
I don’t ultimately find nominalism persuasive, and won’t defend it against standard objections. However, I think that Hellman’s modal structuralist paraphrases reveal a close relationship between logical possibility and pure mathematics which is of interest to realists and nominalists alike.Footnote 1 For they show us how to systematically pair ordinary (platonistic) mathematical sentences with modal sentences which have exactly the truth value a platonist would want to ascribe to the original, but make claims about logical possibility rather than quantifying over mathematical objects. So, for example, Hellman’s paraphrase of ‘there are infinitely many primes’ is a modal sentence which is (intuitively) true at all possible worlds and does not quantify over mathematical objects.
This is useful to, for example, deflationary realists who want to (somehow) ground mathematical existence facts in logical possibilityFootnote 2 as well as to nominalists who want to deny the existence of mathematical objects. Also, one part of Hellman’s story (his treatment of set theory) provides a natural way of developing an independently popular view about set theory called potentialism. Philosophers like Charles Parsons, who have no truck with blanket nominalism about mathematical objects, have been motivated by specific (i.e. specific-to-set-theory) apparent paradoxes concerning the height of the hierarchy of sets to understand higher set theory as an investigation of extendability (Parsons Reference Parsons2007). Thus, one might want to accept something like Hellman’s approach to set theory while being a straightforward realist about other mathematical objects and structures.
In this paper, I will show how to streamline Hellman’s modal structuralist paraphrases for mathematics by appealing to a single, intuitively motivated, notion of logical possibility given certain facts – thus avoiding the need for second order quantification.Footnote 3 In addition to its intrinsic interest, this simplification provides expository and philosophical benefits over Hellman’s approach.
First, existing potentialist and modal structuralist paraphrases for sentences of set theory (including Hellman’s) involve quantifying in to the 
of logical possibility. That is, they use sentences like 
, where the logical possibility operator is applied to a formula with free variables. There are significant controversies about the truth conditions, and indeed meaningfulness, of such statements. For example, there is disagreement about whether any two things that are actually distinct are necessarily distinct. There is also disagreement about what to say about statements which quantify into a world where an object doesn’t exist. For example, Kripke’s approach (which Hellman invokes) allows sentences like 
to be true, a consequence which Williamson and others have argued is extremely counterintuitive.Footnote 4 These controversies can raise doubts about whether our intuitions about quantifying in are reliable while, to my knowledge, no analogous paradoxes arise in the system I lay out.Footnote 5
There is also a Quinean strand of argument which claims that quantifying into modal contexts is meaningless.Footnote 6 Thus, it seems, at least, rhetorically desirable to demonstrate that Hellman’s program (as well as potentialist set theory) doesn’t require quantifying in or similarly controversial notions.
Second, Hellman himself (Hellman Reference Hellman1996) has raised worries about whether his use of second-order logic is nominalistically acceptable, and my modifications show that his programFootnote 7 can be accomplished without second order logic, using only concepts he relies on elsewhere in his program. This is not to say that my modifications definitely render Hellman’s approach nominalistically acceptable. Indeed, one might even take my demonstration that logical possibility can fill in for second order logic as an argument against the nominalistic acceptability of logical possibility itself. Rather, I show that Hellman can avoid any extra burden imposed specifically by his use of second order logic. Either my modifications render Hellman’s account nominalistically acceptable or the very notion of logical possibility employed by Hellman is inherently nominalistically unacceptable and his program fails regardless of the role of second order logic.
In later work Hellman considersFootnote 8 a modification to his core story which avoids second order quantification.Footnote 9 However, this story relies on an additional assumption (that ‘arbitrary [mereological] sums of any individuals independently recognized’ exist) which my story and Hellman’s original story avoid.Footnote 10 This later proposal also does not avoid the issues about quantifying in noted above.
2. Modal structuralism: the core picture
The key idea behind modal structuralism is to reformulate mathematical claims about abstract (non-set-theoretic) objects, like the natural numbers, as claims about how it is possible for objects to be related to one another. For example, something like the twin prime conjecture may be paraphrased as the claim that it would be possible for there to be objects with the structure of the natural numbers and that, necessarily, in any such structure there are infinitely many twin primes. Note that the notion of possibility here isn’t that of metaphysical possibility. For, as Charles Parsons points out, our willingness to talk in terms of large mathematical structures (e.g. the reals or the Hilbert space of square integrable functions) does not seem to be hostage to our conviction that it would be metaphysically possible for there to be that many non-mathematical objects (Parsons Reference Parsons2007). Thus, it seems like the notion of possibility which the modal structuralist is reaching for is something more like mathematical or logical possibility.
In articulating his modal structuralism, Hellman invokes a primitive notion of logical possibility which he does relatively little to describe. He does say that, ‘[when evaluating logical possibility] we are not automatically constrained to hold material or natural laws fixed.’ So it may be logically possible that 
, but physically impossible. And he adds that, “we are free to entertain the possibility of additional objects – even material objects – of a given type”, which allows us to say that it’s logically possible for there to be infinitely many objects even if there are only finitely many objects. Beyond this, however, he just suggests that his applications of logical possibility will make the notion he has in mind clear.
I will abbreviate the claim that it is logically possible that 
as 
, and the claim that it is logically necessary that 
(i.e. 
as 
). With this notion of logical possibility in place, the modal structuralist proposes to understand a mathematician’s claim that 
holds in some mathematical structure (such as the natural numbers), as really asserting a conjunction of two claims. First, it is logically possible for there to be some objects with the relevant structure (e.g. there could be an 
sequence of objects). And second, it is logically necessary that if there were such objects they would satisfy the description 
(e.g. if there were an 
sequence of objects, a version of 
would be true in it).
Hellman uses second-order quantification to give categorical descriptionsFootnote 11 of such structures, e.g. the 
sequence mentioned above. Employing these descriptions allows Hellman’s paraphrase strategy to ensure (assuming second order logic works in the usual way) that all well-formed claims about these structures are either true or false. For example, let 
be the standard second order categorical axiomatization of the natural numbers in terms of a successor relation S Footnote 12 (conjoined into a single sentence) and let 
be a sentence about the natural numbers. Using 
to denote the result of replacing every instance of 
in 
with the second order variable X and every instance of the successor relationFootnote 13S with the second order relation variable f Hellman’s paraphrase of the mathematical claim 
becomesFootnote 14:
The first half of this sentence says that it is logically possible for some objects to form an 
-sequence (with some relation f acting as the successor function). The second half says that it is logically necessary that if some objects (those in X) form an 
-sequence (under f) then 
(modified to use X and f instead of 
and S) is true of them.
This paraphrase strategy (assuming logical possibility and second order quantification operate as Hellman expectsFootnote 15) captures the intended truth conditions for most statements in pure mathematics. However, Hellman also wishes to provide paraphrases for statements of applied mathematics. Consider the claim that there are a prime number of rats. One cannot give correct truth conditions for this claim by only talking about what is logically possible simpliciter – for the truth of ‘there are a prime number of rats’ is not determined only by facts about what is logically possible. It also reflects contingent facts about the world.
Hellman addresses this problem by replacing appeals to logical possibility with appeals to logical possibility given the ‘material’Footnote 16 facts. So, for example, where the platonist takes ‘there are a prime number of rats’ to mean something like ‘there is a function which bijectively maps the rats to the natural numbers below some prime p’, Hellman will translate this claim approximately as follows. It is logically possible, given the material facts, that there are objects which behave like numbers (in the sense of satisfying 
). And it is logically necessary, given the material facts, that if there are objects which behave like numbers then there is a function which bijectively maps the rats to the natural numbers below p.
Hellman considers two approaches to understanding this crucial notion of logical possibility given the material facts. The first is to leave it as a primitive, “reject[ing] the demand” for further explanation of what it means to hold material facts fixed. The second is to cash out the notion of ‘holding the material facts fixed’ by using an actuality operator @, read as ‘it is actually the case that.’ In either case, we see that Hellman is already committed to something like a notion of logical possibility holding some facts fixed. The reader should bear this in mind when considering the particular notion of logical possibility I offer below.
3. Logical possibility sharpened and generalized
I will now introduce my preferred notion of logical possibility given certain facts. Let me begin by calling to mind some features of the standard notion of logical possibility which I take Hellman to be developing.
3.1. The conventional notion of logical possibility
It seems that we have an intuitive notion of logical possibility which applies to claims like 
and makes sentences like the following come out true.
-
It is logically possible that
-
It is not logically possible that
-
It is logically necessary that
-pagination
-paginationPhilosophers representing a range of different philosophies of mathematics have made use of this notionFootnote 17 and are comfortable applying it to non-first order sentences as well. If you are skeptical that there is such a notion, note that it is definable in terms of the even more common notion of validity (something is logically possible iff its negation is not logically necessary iff the inference from the empty premise to its negation is not valid).
To evaluate whether a claim 
requires something logically possible, we hold fixed the operation of logical vocabulary (like 
), but abstract away from any further constraints imposed by metaphysical necessity on the behavior of particular relations. Thus, we consider all possible ways for relations to apply whether or not these ways are describable in our language. For example, it is logically possible that 
, even if it would be metaphysically impossible for anything to be both a raven and a vegetable. During this evaluation we also abstract away from constraints on the size of the universe, Footnote 18 so that 
would be true even if the actual universe contained only a single object.
This notion of logical possibility is generally regarded as a fundamental notionFootnote 19 conceptually distinct from syntactic consistency, i.e. the impossibility of proving a contradiction. Instead, it corresponds to our intuitive sense that certain mathematical theories (like second-order Peano Arithmetic) require something coherent, while others (like Frege’s inconsistent theory of extensions) do not – a sense which is not restricted merely to first-order descriptions.
A core idea I will develop is that the above notion of logical possibility can be naturally generalized. A (pure) logical possibility operator doesn’t allow information to ‘leak out’, so merely adding such an operator to first order logic does little to increase its ‘power.’ This can make it appear somewhat surprising that, as we shall see, the tame-looking further step of considering logical possibility holding certain facts fixed (a concept Hellman already appeals to) is enough to let us relinquish our use of second order quantification. However, we observed above that the concept of logical possibility goes far beyond what is capturable in first order logic, so it’s not totally shocking that we can unlock that power by letting some information pass through (but not free variables).
3.2. Logical possibility generalized
Let us now develop the notion of logical possibility discussed in the previous section. Consider a sentence like, ‘Given what cats and baskets there are, it is logically impossible that each cat slept in a distinct basket.’ There’s an intuitive reading on which this sentence will be true if and only if there are more cats than baskets.Footnote 20 This reading employs a notion of logical possibility holding certain facts fixed (in this case, facts about what cats and baskets there are). Remember, Hellman’s use of logical possibility given the material facts commits him to the coherence of something very much like this notion.
Accordingly, I think we can intuitively understand a conditional logical possibility operator 
which takes a sentence 
and a finite (potentially empty) list of relation symbols 
and produces a sentence 
which says that it is logically possible for 
to be true, given how the relations 
apply. For ease of reading, I will sink the specification of relevant relations into the subscript as follows: 
Thus, for example, the claim, ‘given what cats and baskets there are, it is logically impossible that each cat slept in a distinct basket’ becomes:
Finally, note that by using this notion we can also make nested logical possibility claims, i.e. claims about the logical possibility of scenarios which are themselves described in terms of logical possibility. I have in mind sentences like the following:
This sentence says that it would be logically possible for there to be cats and baskets such that it would be logically necessary, given (the structural facts about) what cats and baskets there are in that scenario, that some cat lacked its own basket to sleep in. Note that in a nested claim with this form (
), the subscript freezes the facts about how the relation R applies in the scenario being considered, which may not be the state of affairs in the actual world. For example, 
CATS expresses a metaphysically necessary truth. For, whatever the actual world is like, it will always be logically possible for there to be, say, 3 cats and 2 baskets. This scenario is one in which it is logically necessary (holding fixed the structural facts about what cats and baskets there are) that: if each cat slept in a basket then multiple cats slept in the same basket. So it is metaphysically necessary that 
CATS even if the actual world contains more baskets than cats.
In what follows, I will often use mathematical-looking symbols or schematic-looking symbols (e.g. 
) for relations appearing in logical possibility statements rather than actual relations like ‘happy()’ and ‘loves()’. However, these symbols should be regarded merely as an abbreviation, so when I write 
it is shorthand for something like 
.
Note that the specific choice of relations does not mater, as when a relation occurs inside a 
or 
which does not subscript that relation, it contributes to the truth conditions for this sentence in exactly the same way that any other relation with the same arity would. For example, the sentence 
will hold if and only if 
does.
This reflects the fact that questions about logical possibility abstract away from all specific facts about the relations in question (other than their arity). Logical possibility involves considering all possibilities for the relations mentioned in the statement under consideration, whether we can describe them or not (this is the analog of requiring second order quantifiers to range over all possible collections). I emphasize this fact, because I will translate claims about mathematical objects using claims about how it would be logically possible for some arbitrarily chosen relations to apply (as Putnam does in Putnam Reference Putnam1967, 10–11) instead of using variables bound by second order quantifiers as Hellman does.
Some readers may still have questions about how holding relations fixed works. One could think about 
claims as holding fixed the particular objects in the extension of the relations 
– and then asking whether one could supplement them with other objects (and choose extensions for all other relations) so as to make 
true.Footnote 21 However, I take the intuitive notion of preserving the structural facts about how some relations apply (that is, the facts about what might be called the mathematical structure of the objects with respect to some relations as opposed to facts about any particular objects) to make sense without appeal to any notion of de re properties or object identity across logically possible scenarios.
In terms of the CATS example, preserving the structural facts about how cat and basket apply requires considering scenarios which agree with the actual world on the number of objects satisfying 
, the number of objects satisfying 
and the number of things in the extension of both 
and 
. This does not require preserving facts about identity. For example, if one cat died and an additional kitten was born, the structural facts about how cat and basket apply would remain unaltered.
Speaking in set theoretic terms, we might say that the ‘structural facts about 
’ are those facts which determine the isomorphism class of the objects falling underFootnote 22 some 
. However, I take conditional logical possibility to be a primitive notion which we can learn directly.
Note that this notion of relativized logical possibility is stronger than Hellman’s notion of unrelativized logical possibility supplemented by an actuality operator in one important way. In Appendix 4, I show that we can capture the same content Hellman expresses using his actuality operator by relativizing all our possibility operators to the relations whose extension in the actual world we wish to discuss.Footnote 23 In contrast, merely using Hellman’s actuality operator does not allow us to express claims about what is logically possible relative to scenarios which are themselves merely logically possible but not actual. This feature turns out to be very useful, as we will see.
4. Reformulating Hellman’s simple paraphrases
Now we turn to demonstrating that Hellman’s paraphrases of mathematical claims can be captured using only conditional logical possibility claims and first order vocabulary.
4.1. Strategy
My translations will have approximately the same logical form as Hellman’s. Given a description D of a mathematical structure and a statement 
about this structure, my translation for 
will still assert that it would be logically possible for the structure described by D to be realized, and that it is logically necessary that if some objects have this structure than (a suitably modified version of) 
will be true of them. However, we will need to replace all of Hellman’s use of second order logic in his translations of mathematical statements with logical possibility claims.
To illustrate this strategy, consider the case of mathematical statements about the natural numbers (I describe how to generalize this approach in Appendix 2). Recall that one can uniquely describe the intended structure of the natural numbers by combining the first four Peano Axioms (Weisstein Reference Weisstein2013) (which can be expressed using only first order logical vocabulary) with a second order Axiom of Induction, which can be expressed as followsFootnote 24:
Informally, this axiom says that if some property X applies to 0 and is closed under successor, Footnote 25 then it applies to all the numbers. We can express the same idea using 
(and a predicate we abbreviate as P Footnote 26) as follows.
This formula says that, given the facts about what is a number and a successor, (i.e. how 
and S apply), it would be logically impossible for 
to apply to 
Footnote 27 and be closed under the successor operation but not apply to all the numbers. Call the result of conjoining this sentence with the four first order axioms of Peano arithmetic 
.
Now we can slot this into Hellman’s paraphrase strategy, and so replace his translation of any first order sentence of number theory 
.Footnote 28 Thus, 
becomes:
where P is an arbitrary one place relation and R is an arbitrary two place relation. As noted above, logical possibility claims reflect facts about all possible ways that a predicate P could apply - whether describable or not. Thus, my translation of a sentence 
about the natural numbers intuitively has the same truth value as Hellman’s translation of that sentence (assuming second order quantification and logical possibility work as Hellman expects).Footnote 29 In the remainder of this paper, I will simply speak of the truth-values of Hellman’s translations or Hellman’s intended truth-values, but in both cases I mean the truth-values his translations would have if the above assumption were true. A similar story can be told for mathematical structures other than the natural numbers, as I show in Appendix 2.
Hellman argues for the bivalence of his translations by appealing to the categoricity of the second order descriptions of the mathematical structures under consideration. In other words, given any sentence 
in the appropriate language, either it or its negation will be necessitated by Hellman’s description D of the relevant structure. If you accept that my translations of mathematical sentences have the same truth-values as Hellman’s translations of these sentences, then my translations of sentences about these mathematical structures will also be bivalent. However, we need not go through Hellman to see that my translations yield bivalence in cases where it is intuitively desired (i.e. when we seem to have a suitably definite conception of the relevant mathematical structure).
To see how this plays out in the case of the natural numbers, note that Hellman’s translations are intuitively bivalent because he uses second order logic to express the idea that the numbers are as few as can be (and thereby rule out nonstandard models which add ‘points at infinity’), by saying that any second order X applying to 0 and closed under successor applies to all the natural numbers. My translations do that same work by asserting that it would be logically impossible for a predicate to apply to 0 and the successor of every number it applies to without applying to all numbers. Intuitively this has the same effect that Hellman intends his second order description to have, while not presuming anything about the behavior of second order quantifiers.
5. Hellman’s potentialist set theory
Now let us turn to Hellman’s translations for statements of (pure) set theory, which have a significantly different structure from his translations of claims about ordinary mathematical structures.
5.1. Motivations for potentialism
If we had a categorical description of the intended structure of the hierarchy of sets (in the language of second order logic), we could nominalistically paraphrase sentences in set theory using the strategy from the last section.
However, there are well-known reasons for doubting that we have any coherent and adequate conception of absolute infinity (the supposed height of the hierarchy of sets). The concern here is not simply that it might be impossible to cash the notion of absolute infinity out in other terms. After all, every theory will have to take some notions as primitive. Rather, the worry is that our intuitive notion isn’t even coherent – in the way that our naive conception of set is incoherent (as demonstrated by Russell’s paradox).
One might like to say that the hierarchy of sets goes all the way up – so no restrictive ideas of where it stops are needed to understand its behavior. However, if the sets really do go ‘all the way up’ in this sense, then it would seem that the ordinals should satisfy the following closure principle.
For any way some things could be well-ordered, there is an ordinal correspondingFootnote 30 to it.
But the ordinals themselves are well ordered, and there can be no ordinal corresponding to this well-ordering. If the sets are a definite totality, i.e. a logically possible collection of objects, this is a contradiction. Thus, this naive closure principle can’t be correct.
In response, we might try to find some other characterization of the sets as a definite structure (in particular, some other characterization of the intended height of the hierarchy of setsFootnote 31). However, it’s not clear that any intuitive conception of the intended height of the sets remains once the paradoxical well-ordering principle above is retracted. As Wright and Shapiro put it Shapiro and Wright (Reference Shapiro, Wright, Rayo and Uzquiano2006), all our reasons for thinking that sets exist in the first place appear to suggest that, for any given height which an actual mathematical structure could have, the sets should continue up past this height. Thus, taking set theory at face value can seem to force us to posit an unprincipled fact about where the sets stop.Footnote 32 This problem isn’t limited to realists, but applies to all philosophers (including modal structuralists) who take set theory to be the study of a single definite structure.
5.2. The potentialist approach to set theory
Potentialists, including Hellman, respond to this problem by taking a potentialist approach to set theory (along lines suggested by Putnam Putnam (Reference Putnam1967)). On this approach, mathematicians’ claims which appear to quantify over sets should really beFootnote 33 understood as claims about how it is (in some sense) possible to extend initial segments of the hierarchy of sets, i.e. collections of objects which satisfy our intuitive conception of the width of the hierarchy of sets but not the paradox-generating height requirement. Hellman, unsurprisingly, understands the relevant notion of possibility in terms of logical possibility (and I will follow him in so doing).Footnote 34
The potentialist takes set theorists’ singly-quantified existence claims, like 
, to really be saying that it would be possible for a collection of objects 
to satisfy (a version of) 
while containing a suitable object x (in this case, an x such that 
). The potentialist takes set theorists’ universal statements with a single quantifier like 
, to really say that it is necessary that any object x in a collection of objects satisfying 
would have the relevant property.
The potentialist handles nested quantification using claims about how collections of objects satisfying a version of 
could be extended. For example, Hellman would offer the following translation of 
: necessarily if 
satisfies 
and includes a set x, it is logically possible for there to be an extension, 
, Footnote 35 of 
, also satisfying 
and containing a set y such that 
(in the sense of 
relevant to 
).Footnote 36 Writing this out formally using Hellman’s notion of logical possibility gives us the following sentence (implicitly restricting 
and 
to range over collections of objects satisfying a version of 
and using 
to denote extension):
Note that by adopting this potentialist understanding of set theory, we avoid commitment to arbitrary limits on the intended height of the hierarchy of sets. We also avoid the assumption that there is (or could be) any single structure which contains ordinals witnessing all possible well-orderings, though every possible well-ordering is realized in some possible initial segment of the sets.
6. Formulating potentialist set theory
Now let us turn to the problem of articulating a suitable replacement for Hellman’s potentialist paraphrases which avoids second order quantification. I will explain my version of these potentialist paraphrases informally, but the interested reader should see Appendix 4 for more details. The appendix also reviews why bivalence holds for my translations of sentences in set theory.Footnote 37
To articulate potentialist paraphrases of set theory in terms of conditional logical possibility, we must first express the claim that some objects behave like a standard width initial segment of the hierarchy of sets. Hellman expresses this idea by using 
, a second order version of the ZFC axioms of set theory. One can show that 
suffices to pin down the intended width of the hierarchy of sets (though not their height). It’s not too hard to write a version of 
in terms of my notion of conditional logical possibility, by using a version of the trick for replacing second order quantification with claims about logically possible extendability demonstrated in Section 4 and generalized in Appendix 2. This approach lets us write out a sentence (as it were 
) using the logical possibility operator which says that the objects satisfying 
under the relation 
capture the behavior of an initial segment of the sets.
We now must duplicate the complex statements about extendability used to handle nested quantification in Hellman’s paraphrases. It is straightforward to define the claim that 
extends 
using only the logical possibility operator and first order vocabulary. This allows us to talk about possible extensions of initial segments of the sets. However, to fully represent potentialist paraphrases, we also need to mirror Hellman’s claims which fix an object x from among those which some relations 
apply to, and talk about how an element y in a potential extension 
relates to x. As stated, this claim involves quantifying in, but we must find another method.
The key idea behind my strategy is to require that each initial segment of objects satisfying 
be considered together with a relation 
which assigns each ‘variable’ from some countable collectionFootnote 38 to an object satisfying 
. Thus, 
behaves like an assignment function which associates each variable with some object within the initial segment 
. We can then preserve the behavior of this assignment function in relevant modal contexts by adding 
to the subscripts on relevant 
s and 
s and demanding that 
agree with 
everywhere except on the particular variable we want to select from 
. This allows us to preserve our choice of some sets x, y and z within 
while considering ways that one could choose an additional object w from within some logically possible 
extending 
. The overall effect will be to duplicate what Hellman achieves via quantifying in, through the use of the relations 
.
7. Conclusion
In this paper I have shown how to streamline Hellman’s modal structuralist approach to mathematics, by invoking a notion of logical possibility given certain facts. We saw that Hellman already accepts a notion of logical possibility holding the material facts fixed. Given this, it is only natural that he should also accept my notion of conditional logical possibility. However, once one does this there is no need to invoke second order quantification as an additional primitive.
The streamlining I propose also helps us evaluate the two apparent problems for modal structuralism mentioned in the introduction. We have seen that it is possible to completely eliminate the controversial practice of quantifying in from Hellman’s paraphrases.
I think the technical work in this paper demonstrates that there is no unavoidable special problem for modal structuralism caused by its reliance on second order logic. This is not to say that modal structuralism is ontologically innocent. Although logical possibility intuitively appears ontologically innocent, whether my simplification defends modal structuralism’s ontological innocence or reveals that (despite our intuitions) logical possibility is itself unsuitable for nominalist use depends on the right answer to certain controversial background questions. Specifically, it depends on whether we ought to take any other notion which does the same work as second order logic to be equally ontologically committal.
If similarity of mathematical behavior doesn’t require (or make a strong case for) similarity of ontological role, then my simplification allows modal structuralism to shake off the aspersions that have been cast on its nominalistic credentials. If it does, then we can respond by either giving up on the nominalistic acceptability of modal structuralism and admitting that the seemingly innocent notion of logical possibility (and possibly many other notions we don’t suspect) is actually ontologically committal or by reevaluating our reasons for thinking that second order logic is ontologically committal (since the results of this paper show that although second order logic is similar to set theory which looks ontologically committal, it is also similar to logical possibility which looks non-committal).Footnote 39
In conclusion, we’ve seen that by adopting a small generalization of Hellman’s notion of logical possibility (the meaningfulness of which he has already endorsed), we can significantly streamline modal structuralism – and perhaps solve some other problems as well.
Notes on contributors
Sharon Berryis a postdoctoral researcher at The Polonsky Academy for Advanced Study in the Humanities and Social Sciences. His research interests include philosophy of mathematics, epistemology, metaethics.
A more formal approach to conditional logical possibility
I take the notion of conditional logical possibility to be primitive and intuitive. However, one can provide approximately correct truth conditions for sentences involving nested applications of subscripted 
and 
operators, in terms of the more familiar language of set theory with ur-elements.Footnote 41
First let us define a formal language 
, which I will call the language of logical possibility (though this language may be not able to express all meaningful claims involving logical possibility). Fix some infinite collection of variables and a collection of relation symbols, and define 
to be the smallest language built from these variables using these relation symbols and equality closed under applications of the normal first order connectives, quantifiers, 
and 
(where the latter two operators can only be applied to sentences, so there is no quantifying in).
Specifically, if we ignore the possibility of sentences which demand something coherent but wouldn’t have a model in the sets, (such as sentences which require the existence of proper class many objects) and take all quantifiers appearing outside a logical possibility operator to be implicitly restricted to some set sized domain of non-mathematical objectsFootnote 42 we could say the followingFootnote 43:
Definition 1: A formula 
is true relative to a model 
and an assignment 
which takes the free variables in 
to elements in the domain of 
Footnote 44 just if the following conditions obtainFootnote 45 (note that only the last clause says something out of the ordinary):
-
and (as usual is the interpretation of by ).
-
and .
-
and is not true relative to .
-
and both and are true relative to .
-
and either or are true relative to .
-
and there is an assignment which extends by assigning a value to an additional variable not in and is true relative to
Footnote 46.
-
and there is another model and a bijection from to such that and is true relative to and the empty assignmentFootnote 47.
and 
(as usual 
is the interpretation of 
by 
).
and 
.
and 
is not true relative to 
.
and both 
and 
are true relative to 
.
and either 
or 
are true relative to 
.
and there is an assignment 
which extends 
by assigning a value to an additional variable 
not in 
and 
is true relative to 
and there is another model 
and a bijection 
from 
to 
such that 
and 
is true relative to 
and the empty assignment Note that in the last clause the models 
and 
need not share any elements. Rather the structure 
(those elements appearing in some tuple in the extension of some 
) must be isomorphic (under the relations 
) to (
).
Set Theoretic Approximation: A sentence in the language of logical possibility is true simpliciter iff it is true relative to a set theoretic model whose domain consists of the actual objects (which the quantifiers in our original non-mathematical language range over) and whose extensions for atomic relations reflects the actual extensions of these relations and the empty assignment function 
. Note that this definition gives statements lacking any necessity operators the same truth values as they have in the actual world.
Modal structuralist paraphrases for regular mathematics
In this appendix, I will give a general method for simplifying Hellman’s paraphrases of non-set theoretic mathematics.Footnote 48 I will follow Hellman in focusing on the case where the mathematical structure under consideration has a categorical second order description D, and provide a translation of Hellman’s paraphrases which we may assume is in the following formFootnote 49 (where all first order quantifiers in D and 
are restricted to M-pagination49Footnote 50 and no logical possibility operators appear in D or 
):
We may ignore the difference between quantification over classes and quantification over relations, by regarding class variables as unary relation variables. For visual clarity we will use capital letters for second order quantification over relations. We will also assume that no second order function quantifiers occur in D or 
, though the same mechanism can be easily extended to handle function quantifiers. Note that as all first order quantifiers are restricted to M, we only need concern ourselves with the behavior of relations and relation variables on elements of M.
We may now define my translation of Hellman’s paraphrase 
to be 
where t is defined via the following recursive definition (with 
).
We now argue that this translation preserves (intended) truth values. Except for the first two lines the translation is entirely homophonic, so as long as those equalities preserve (intended) truth values, the entire translation should do so. However, the first and second equalities simply express the fact that, understand as Hellman intends, second order relation variables on a domain M range over all logically possible relations on M and vice versa. Finally, the same consideration (on a given domain, 
ranges over exactly the collections it would be logically possible for a predicate to apply to) tells us that moving between 
and 
shouldn’t change the truth value (again assuming second order quantification operates in the usual fashion as Hellman expects).
Note about applied mathematics
Although the aim of this paper is to simplify Hellman’s story about pure mathematics, everything Hellman says about applied mathematicsFootnote 51 is also expressible using my notion of conditional logical possibility. I only mention this fact because it means that Hellman could adopt my simplifications without significant harm to his proposal.
As noted in Section 2, Hellman paraphrases sentences in applied mathematics, like ‘There are a prime number of rats’ with sentences of the form:
where 
is a sentence asserting that if there are objects behaving like the numbers, (or whatever mathematical objects are mentioned in the statement to be translated) then these objects are related to the material objects in some (second-order describable) fashion. For instance, 
might assert that if some things behave like the natural numbers, then there is a function which pairs up the rats in the actual world in a one-to-one fashion with those natural numbers up to some prime, thereby asserting that there are a prime number of rats.
It is possible to do equivalent work using my notion of conditional logical possibility. First we apply the technique outlined in Appendix 2 to replace second order quantification with conditional logical possibility. We then add all the non-mathematical relations mentioned in the sentence to be translated (in the example ‘there are a prime number of rats’ this would just be the predicate ‘rat()’) as subscripts to all the 
and 
operators in the sentence. The resulting sentence now simply holds fixed every material fact it actually makes use of, allowing it to be expressed in terms of conditional logical possibilityFootnote 52 (without appeal to a notion of holding all the material facts fixed).
Paraphrasing potentialist set theory
Potentialism about set theory replaces claims about a definite totality of sets with claims about how initial segments of the sets can extend each other. Hellman considers initial segments of the sets which satisfy 
and uses quantifying into formulate claims about how these segments can be extended. We reformulate Hellman’s potentialist understanding of first order set theory.Footnote 53 in the language of conditional logical possibility in two steps.
First, we replace the requirement that the initial segments satisfy 
with an equivalent characterization 
in terms of conditional logical possibility, using the technique described in Appendix 2.
Secondly, we can reformulate claims about how initial segments can be extended in a way that eschews quantifying in. Recall that potentialism translates sentences of set theory by replacing quantifiers over the sets with statements about how it would be possible to extend initial segments of the sets and choose elements from those initial segments, e.g. if 
is quantifier free then 
would translate to 
where this says that it would be logically possible for there to be an initial segment of the hierarchy of sets containing an object that satisfied 
.
To express potentialist truth conditions without quantifying in, I will require that each initial segment 
be paired with an associated assignment relation 
which (in effect) assigns each of the countably many variables 
, 
...in the first-order language of set theory to objects within 
. When we ask about the possibility of extending the current initial segment (
) we can place 
in the subscript of all further 
and 
expressions to pass along the information about variable assignments. We allow this choice of assignments for variables to be modified to allow variables to be assigned to objects in 
(an initial segment extending 
) by defining another assignment 
which must agree with 
everywhere except for on the (number representing) the variable allowed to range over 
.
I will use 
to abbreviate the claim that 
satisfy 
and 
behaves like (the relation corresponding to) an assignment function from the objects satisfying 
to those satisfying 
. More concretely this amounts to the conjunction of the following three claims:
-
, i.e. behaves like an initial segment of the hierarchy of sets.
-
satisfy .
-
behaves like a function from to
, i.e. 
behaves like an initial segment of the hierarchy of sets.
satisfy 
.
behaves like a function from 
to 
Remember that, as discussed on page 12, schematic relation symbols (like 
, 
and P) are used as a mnemonic device in place of suitable non-mathematical relations with the same arity.
Note that my only reason for using 
is that the natural numbers (under successor) contain infinitely many definable objects, which we can use to represent variables, for example 1 represents 
, 2 represents 
etc. In what follows, I will use 
, to abbreviate the formula where 
is replaced by a variable constrained to be the (unique) n-th successor of 0. Thus, for example, a claim of the form 
abbreviates 
. I will abbreviate the conditionalized logical possibility operators 
and 
by 
and 
respectively.
I will use 
to abbreviate the claim that the 
under 
extends the 
under 
and the assignment of variables 
agrees with 
everywhere except on i (where i is the code for 
). Put more concretely, this is to say that
We can now translate the set theoretic utterance 
into a claim about how it is logically possible for 
to be extended. First we rewrite this set theoretic statement in a regimented language with numbered variables as 
. Then we translate this sentence into:
That is, such 
sentences can be understood as making a claim with the following form. There could be a model of set theory 
and a relation 
assigning 1 (representing 
) to an element of 
so that it is necessary (holding fixed 
and the numbers) than any model of set theory 
extending 
and relation 
assigning 2 to an element of 
(while agreeing with 
about the assignment of 1) makes the interior of the above formula true when 
are replaced by the assignments of 1, 2 by 
and 
is replaced with 
.
The same strategy works more generally to produce paraphrases of arbitrary sentences in the language of pure set theory. We can use recursive applications of the following principles to translate every sentence in the first-order language of set theory into a claim about logically possible extendability.
In particular we define 
as follows:
-
is the claim that assigns i to an object the object it assigns to j i.e.
-
is the claim that assigns i to the same object it assigns j to i.e.
-
-
-
for
-
for
, -
-
is the claim that 
assigns i to an object 
the object it assigns to j i.e. 
is the claim that 
assigns i to the same object it assigns j to i.e. 
, 
The translation of a set theoretic sentence 
is 
. Note that the validity of the above translation relies on the fact that for any two structures satisfying 
one is isomorphic to an initial segment of the other. Hellman invokes a version of this claim in Chapter 2 Section 3 of Hellman (Reference Hellman1994) and I think an analogous argument can be made within my formal system, but reasons of space prevent me from demonstrating this here. Also note that in the above definition we can replace 
with 
without affecting the truth value of the translation. This allows us to translate sentences with arbitrarily many quantifier alternations using a fixed finite number of atomic relations.
Note that this translation honors the intuitive bivalence of the language of set theory. Consider an arbitrary set theoretic sentence 
. 
and 
. Thus either 
or 
will be true.
is the result of replacing the induction schema in Peano Arithmetic with a single induction axiom formulated in second order logic as described in Weisstein (
are often formulated using a successor function, it is easy enough to transform them into claims about a successor relation, by adding an axiom asserting that every member of 
has a unique successor in 
.
is understood to express the property of being a number and S the successor relation, so we cannot use them as variables by writing something like 
.
(for example because second order logic is ontologically committal and the necessary second order objects can’t exist) then the paraphrase Hellman provides for statements in arithmetic would fail.
iff it appears in some tuple in the extension of 
, i.e. 
.
.
.
don’t suffice to categorically determine the height of the set theoretic hierarchy. For example, if (as most mathematicians assume) the hierarchy of sets extends beyond the first inaccessible then the initial segment of the hierarchy below that inaccessible will satisfy ZFC/
.
is an element of 
and the second order relation quantifier Hellman uses to give 
its meaning on 
agrees with 
on 
(and, indeed, any element of 
which is 
an element of 
is also in 
).
is a sentence in the language of set theory, either my translation of 
or my translation of 
will express a truth.
from the collection of variables in the language of logical possibility to objects in 
, such that the domain of 
is finite and includes (at least) all free variables in 
.
and 
to be abbreviations for 
and 
respectively.
substitutes v for x everywhere where v occurs free in 
.
can’t have any free variables.
. But when D is categorical this is equivalent to the form above.
or 
.
