Modal If-Thenist Paraphrase Strategy

Sharon Berry

doi:10.1017/9781108992756.012

12 - Modal If-Thenist Paraphrase Strategy

from Part III

Published online by Cambridge University Press: 10 February 2022

Sharon Berry

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Sharon Berry: Affiliation:
Ashoka University, India

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Summary

This chapter presents (and notes certain advantages of) a basic modal if-thenist strategy for nominalistically paraphrasing Platonism theories in response to indispensability arguments.

Keywords

Modal if-thenism Hilary Putnam Geoffrey Hellman nominalist mathematical nominalist paraphrase strategy nominalization Field

Information

Type: Chapter
Information: A Logical Foundation for Potentialist Set Theory , pp. 126 - 134

DOI: https://doi.org/10.1017/9781108992756.012 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2022

12 Modal If-Thenist Paraphrase Strategy

With this picture of indispensability worries in mind, let’s turn to the question of when the Nominalist can answer them.

In this chapter I will introduce a key tool in the arsenal of the Nominalist of Chapter 10: a general Nominalist paraphrase strategy for replacing claims about mathematical objects with claims about logical possibility.

This paraphrase strategy follows Reference HellmanHellman (1994) in putting a modal twist on familiar if-thenism, but is developed using the conditional logical possibility operator rather than Hellman’s machinery. Roughly speaking, the idea will be that our nominalistic translation $T (ϕ)$ of the Platonist’s sentence $ϕ$ says: it’s logically necessary, fixing the facts about all relevant non-mathematical structures, that if there were also mathematical structures then $ϕ$ .

I’ll show how this nominalization strategy can be applied to any Platonist sentence $ϕ$ satisfying a certain “definable supervenience” condition. Then I’ll note that, where defined, the Nominalist paraphrases provided by this strategy will let us answer classic Quinean Indispensability arguments in the sense specified in Section 11.2. That is, it will let us transform a Platonist theory into a nominalistic theory which – the Platonist must acknowledge – matches the intended (possible worlds) truth conditions for that Platonist theory.

One might worry that the if-thenist form of these paraphrases makes them objectionably instrumentalist and unexplanatory. But in Section 13.6, I’ll argue that this is not the case. In fact, in certain central cases, we’ll see that relevant Nominalist regimentations of scientific theories are plausibly explanatorily better (more general, powerful and illuminating) than corresponding versions of the same theories. The basic structure of this paraphrase strategy will also be useful to help explicate and develop a general neo-Carnapian philosophy of language and a more realist approach to mathematical objects outside set theory (as we will see in Chapter 15).

12.1 Modal If-Thenist Paraphrase Strategy

12.1.1 Motivating Example

To motivate and begin to concretely explain the modal if-thenist nominalization strategy, consider the following sentence:

CRITICS: Some critics only admire each other.

A Platonist who believes in sets of critics could Platonistically formulate CRITICS as follows:

CRITICS_P: There is a set-of-critics x such that, for all y and z, if $y \in x$ and y admires z then $z \in x$ .

Now our modal if-thenist paraphrase strategy lets us capture this claim as T(CRITICS_P).

T(CRITICS_P): $□_{critic, admires}$ [If there are (objects with the intended structure of) a single layer of sets-of-critics under elementhood, then (it’s true in this structure that) there’s a set-of-critics x such that, for all y and z, if $y \in x$ and y admires z then $z \in x$ .]Footnote ¹

This says (roughly) that necessarily if the actual structure of critics and admiration were supplemented with extra objects with the structure the Platonist takes the sets-of-critics to have, then the Platonist’s claim CRITICS_P would be true.

Intuitively (from a Platonist point of view) this claim has the same truth conditions as the original claim. The truth-value of CRITICS_P is completely determined by the structure of how critic, admiration, set-of-critics apply.

We can say there are objects with the intended structure of the sets-of-critics under elementhood (when considered under some otherwise unused relations S and E) by conjoining the following:

The claim that there are sets corresponding to “all possible ways of choosing” some critics: $□_{critic, S, E}$ (there’s a set which contains exactly the critics who are happy).Footnote ²
Intuitively this captures the appeal to all possible ways of choosing by saying that it’s logically necessary (fixing the structure of the critics, sets-of-critics and elementhood) that however “happy” applies there will be a set-of-critics which contained exactly the happy critics.
A collection of first-order conditions that are easy to formulate, e.g., claims that the sets of critics are extensional, and that sets-of-critics only have critics as elements.

Call the above conjunction D (I’ll later call this the Definable Supervenience condition). Then we have the following translation:

T(CRITICS_P): $□_{critic,admires} [D \to CRITICS]$

12.1.2 Definitions

With this motivating example in mind, I will now explain the modal if-thenist translation strategy. This comes in two parts.

First, there’s a definable supervenience condition, which specifies the intended structure of all the “extra” objects and relations the Platonist believes in, in terms of their relationship to objects and relations the Platonist and Nominalist can agree on.Footnote ³ Second, there’s a modal if-thenist framework which we plug this definable supervenience description into.

To explain both elements above, let me start by introducing some definitions. One might try to define nominalistic vocabulary as vocabulary which, with metaphysical necessity, applies only to non-mathematical objects. However, even the predicate for “real numbers” would satisfy that definition if Nominalism is true (for in this case, it’s metaphysically necessary that nothing is a real number). So, instead, we use the following definition.

Definition 12.1 (Nominalistic Vocabulary). A predicate or relation $R$ counts as nominalistic vocabulary iff the Platonist accepts that it is metaphysically necessary that the extension of $R$ contains only objects that the Nominalist would admit exist. For example, “is a cat” is a nominalistic predicate and “is taller than” is a nominalistic relation.

Definition 12.2 (Platonistic Vocabulary). A predicate or relation is Platonistic iff it is not nominalistic.

Thus, Platonistic vocabulary includes not only pure mathematical vocabularyFootnote ⁴ but also applied mathematical vocabularyFootnote ⁵ and relations which (the Platonist thinks) relate mathematical objects to non-mathematical objects.Footnote ⁶

Now let’s turn to the definable supervenience condition. Intuitively, speaking, the definable supervenience condition says a description $D$ uniquely describes the mathematical structures the Platonist accepts, using only nominalistic facts. At each metaphysically possible world, $D$ uniquely “pins down” the pure and applied mathematical structures the Platonist believes in (given the facts about non-mathematical structures that Platonists and Nominalists agree on at that world).

We can specify what it takes for a sentence D to be a definable supervenience condition for a Platonist language formally, by generalizing the notion of categoricity to a concept of certain descriptions of a structure being “categorical over” the facts about a certain part of that structure.

The idea here is that (just as we can completely specify the structure the Platonist takes the natural numbers to have using logical vocabulary alone), we can completely specify the intended structure of the goats and sets of goats, using only facts about the goats and logical vocabulary. We believe certain things (expressible using the conditional logical possibility operator) about what the relationship between the goats and the sets of goats is supposed to be like, such that, for any way of fixing the goats structure there’s only one way that the overall goats-and-sets-of-goats structure could be (which would make our beliefs true).

So a description $D (N_{1}, \dots, N_{m}, P_{1}, \dots, P_{n})$ is categorical for the $P_{1}, \dots, P_{n}$ over $N_{1}, \dots, N_{m}$ if the facts about how $N_{1}, \dots, N_{m}$ apply completely determine how $P_{1}, \dots, P_{n}$ apply – and indeed the whole $N_{1}, \dots, N_{m}$ , $P_{1}, \dots, P_{n}$ structureFootnote ⁷ given that D is true.

We can define this notion using the conditional possibility operator, and our definition of isomorphism (Definition 7.4).

Definition 12.3 (Categorical Over). $D$ is a categorical description of the relations $P = P_{1}, \dots, P_{n}$ over $N = N_{1}, \dots, N_{m}$ (where $P \cap N = ϕ$ ) just if $(D [N_{1}, \dots, N_{m}, P_{1}, \dots, P_{n}] \land D [N_{1}, \dots, N_{m}, P_{1} / {P^{'}}_{1}, \dots, P_{n} / {P^{'}}_{n}] \to N \cup^{} P ≅ N \cup^{} P')$

Using this we can now define the definable supervenience condition, i.e., the condition we expect our Platonist paraphrase to satisfy.

Definition 12.4 (Definable Supervenience Condition). A sentence $D$ is a definable supervenience condition, specifying how the application of some Platonistic vocabulary $P$ definably supervenes on that of some nominalistic vocabulary $N$ if and only if the following conditions hold:

(from a Platonist POV) $D$ is metaphysically necessary;
$◊_{N}$ $D$ , i.e., the Platonist isn’t supposing the existence of incoherent objects, and indeed it’s logically necessary that the $N$ structure can be supplemented with Platonistic structure in the way that D requires;
D is content-restricted to $P, N$ ;
D is a categorical description of the $P, N$ structure over the $N$ structure.

Note that all the pure and applied mathematical structures commonly used in applied mathematics (reals, complex numbers, classes of physical objects, functions from physical objects to mathematical objects etc.) can be straightforwardly given such a definable supervenience condition.

When we have a suitable definable supervenience condition D, we can translate every sentence $ϕ$ which is content-restricted to the total list of relations in the Platonist’s language as follows:

T (ϕ) = □_{N} (D \to ϕ)

Intuitively, this says that it’s logically necessary, given the structure of objects satisfying the list of nominalistic relations $N$ , that if there were (objects with the intended structure of) relevant mathematical objects then $ϕ$ would be true. Note that the Platonist must believe it is always logically possible to supplement the actual objects with objects that behave like the platonic objects and satisfy $D$ , because they think such objects exist.

12.2 A More Detailed Example

To clarify how this strategy can be applied to more complex cases, consider a Platonist who believes in three types of mathematical objects: natural numbers, sets of goats and partial functions from goats to natural numbers.

Consider the following sentence:

GOATS: There are a prime number of goats.

The Platonist will formalize this statement with a sentence like the following:

GOATS: There’s a $1 - 1$ functionFootnote ⁸ $f$ , such that $f$ maps the goats onto an initial segment of the natural numbers, from 0 up to, but not including, some prime number $n$ .

Can we nominalize this sentence? Yes. Our first step is to note that GOATS is implicitly content-restricted to a certain list of relations: natural number, set of goats, etc. It doesn’t involve unrestricted quantification, and its truth-value must be the same in any logically possible scenarios which agree on this structure. Now, can we write down a definable supervenience sentence D (call it D[numbers, goats-to-numbers functions]) which categorically specifies how all the relations on this list apply in terms of how the Nominalist relations on the list apply? We can write such a D by conjoining the following:

A categorical description of the natural numbers ${PA}_{◊}$ (i.e., a sentence which uniquely pins down how the Platonist thinks $ℕ, S, +, \times$ apply, up to isomorphism).
A sentence which pins down the structure of “all possible” partial functions from goats to numbers,Footnote ⁹ given the structure of the goats (and numbers).
A collection of “Julius Caesar sentences,” i.e., sentences specifying how the mathematical objects are supposed to relate to the non-mathematical objects. For example, we might say that the numbers are supposed to be distinct from the sets of goats, functions from goats to numbers, etc.Footnote ¹⁰

Note that there are only two things that can’t be obviously formulated in first-order logic in my description of the supervenience description D[numbers, goats-to-numbers-functions] above: the categorical description of the natural numbers, and the description of the partial functions from goats to numbers.

Recall that we saw how to categorically describe the natural numbers with a sentence ${PA}_{◊}$ in Section 4.3.2.1. What about describing the structure of partial functions from the goats to the numbers? We can nominalistically formalize this in the same way. Assume the Platonist’s language has relations “function()” and “maps()” such that maps(f, x, y) iff f is a function that maps x to y, i.e., f(x) = y. We can informally pin down the structure we want by saying two things:

There are functions witnessing all possible ways of mapping some of the goats to some of the numbers.Footnote ¹¹
There are no more functions than needed to ensure this (i.e., every function maps only goats to numbersFootnote ¹² and the functions are extensional).

The second claim is easy to formalize in FOL. And we can write the first using second-order relation quantification as follows:Footnote ¹³

$\forall R$ [If R is functional and only relates goats to numbers then $(\exists x) (function (x) \land (\forall y) (\forall x) [maps (x, y, z) \leftrightarrow R (y, z)$ ).]

We can rewrite this in the language of logical possibility, using any two-place relation that doesn’t figure in the body of scientific theorizing we want to translate. For example, I will pick “eucratises”:Footnote ¹⁴

$□_{ℕ, function,maps,goat}$ [If eucratises applies functionally and only relates goats to numbers then $(\exists x)$ $(function (x) \land (\forall y) (\forall x) [maps (x, y, z) \leftrightarrow eucratises (y, z))$ .]Footnote ¹⁵

It’s logically necessary given the structure of the goats, numbers and functions from goats to numbers, that if eucrastises only relates goats to numbers and applies functionally there’s a function x that relates goats to numbers in the same way.

Given D[numbers, goats-to-numbers functions] the Nominalist can translate the Platonist’s formalization of the claim that there are a prime number of goats into a nominalistic version of this claim, $T (GOATS)$ , as follows:

$T (GOATS)$ : $□_{goat} (D [numbers, goats-to-numbers-functions] \to ϕ_{GOATS})$

Intuitively, this says that it’s logically necessary, given the structure of the goats, that if there were (objects with the intended structure of) the numbers, sets and functions from goats to numbers then $GOATS$ would be true.

Furthermore, we can show that the Platonist must agree that this translation is true at the correct set of metaphysically possible worlds (i.e., the worlds at which they take $ϕ$ to be true); they must think that it’s metaphysically necessary that $T (ϕ) \leftrightarrow ϕ$ .

At each possible world $w$ , the truth-value of $GOATS$ is completely determined by the structure of goats, functions and numbers at that world.Footnote ¹⁶ And (according to the Platonist) the latter structure is completely determined by the structure of the goats at $w$ together with our definable supervenience description D. D completely pins down what sets and functions (the Platonist thinks) there are at w, given the facts about nominalistic stuff at $w$ . There’s only one logically possible way (structurally speaking) to supplement the pattern of goats at $w$ with numbers and functions as required by the claim D, which the Platonist takes to be a metaphysically necessary truth. So $GOATS$ is true at w if and only if it’s logically necessary, given the facts about the goats at $w$ and $D$ , that $ϕ$ .

So our total translation will have the following form:Footnote ¹⁷

□_{goat} [ψ_{1} \land □_{ℕ, S} (ψ_{2}) \land □_{ℕ, function,maps,goat} (ψ_{3}) \to ϕ_{GOATS}]

As in the previous case, the Platonist must say that this statement is true at exactly the same metaphysically possible worlds where GOATS is true.

12.3 Clarifications and Advantages

12.3.1 Harmlessness of Platonist Science

We can show that the nominalistic paraphrase strategy produced by our translation strategy T preserves the desired inferential role of scientific sentences. It captures both inferences from applied mathematical sentences to other applied mathematical sentences, and inferences between applied mathematical sentences and observational sentences.Footnote ¹⁸

But we can also show (see Section D.1 in Appendix D) that where we know it’s metaphysically necessary that $◊_{ℕ} D$ (something my Platonist and Nominalist alike take themselves to knowFootnote ¹⁹) we have:

Theorem 12.1 Suppose that $Φ$ , $Ψ$ are content-restricted to $P \cup N$ and $⊢ Φ \to Ψ$ then $⊢ T (Φ) \to T (Ψ)$ . Furthermore if $⊢ T (Φ) \to T (Ψ)$ then $⊢ (D \land Φ) \to Ψ$ .

That is, Platonist scientific arguments from $(D \land Φ)$ to $Ψ$ (where the latter don’t involve unrestricted quantification) can be easily transformed into Nominalist scientific arguments, from $T (ϕ)$ to $T (ϕ)$ and vice versa. Note that, for any statement $ν$ that’s content-restricted to Nominalist vocabulary, $T (ν) \leftrightarrow ν$ is easily derivable.

Putting this together, we get that whenever a Platonist can use their Platonist assumption D conjoined with sentences $ρ_{1}, \dots, ρ_{n}$ content-restricted to nominalistic stuff to prove $ρ^{'}$ content-restricted to nominalistic stuff, there is a good argument from $ρ_{1}, \dots, ρ_{n}$ to $ρ^{'}$ .

12.3.2 Conditional Logical Possibility and Field’s Conservativity

Let me end with two points of comparison between this strategy and that famously advocated by Hartry Field in Science Without Numbers (Reference FieldField 1980).

First, using the logical possibility operator and axioms I’ve proposed is helpful to those who would follow Field’s paraphrase strategy as well. In this way we can cash out an intuitively appealing “conservativity” argument made by Reference FieldField (1980) to account for the goodness of Platonist science from a Nominalist point of view, while avoiding worries about circularity which I will now explain.

Reference FieldField (1980) wants to explain why using mathematics in the sciences is harmless and indeed helpful, despite the fact that (as he wanted to say at the time) existence claims about mathematical objects are false. He wants to say this is true because mathematical axioms are conservative (in the sense below). Reasoning with these axioms just speeds up proofs; it doesn’t let us prove anything new about non-mathematical objects.

if $B$ is any sentence, $B^{*}$ is the result of restricting $B$ to non-mathematical entities, and $M_{1}, \dots, M_{n}$ are the axioms of a mathematical theory $M_{1}, \dots, M_{n}$ , the conservativeness of M can be expressed by the following schema:

(Field 1980)

Field argues, working in ZFC, that one can always take a model of just the non-mathematical entities recognized by a theory and produce a model which also recognizes a hierarchy of sets taking those objects as ur-elements. He is criticized in Reference BuenoBueno (2020) for circularly using set theory to justify the claim that assuming set-theoretic axioms won’t let you prove anything false about non-mathematical objects in this way.

But if we accept the notion of logical possibility and the axioms I’ve proposed for it, we can justify a version of Field’s desired conservativity result (for suitable axioms describing mathematical objects, like a hierarchy of sets $V_{α}$ up to some suitably definable height $V_{α}$ ) from modal principles that don’t assert the existence of mathematical objects.Footnote ²⁰

Second, my paraphrase strategy always produces finitely stateable theories where it applies, and in Chapter 14 I’ll argue that it can be applied to solve the physical magnitude problems (at least for purposes of Quinean indispensability, if not reference and grounding) which drove Field to appeal to infinitely many different sentences satisfying a schema, rather than producing a single sentence that formalizes the scientific theory at issue.Footnote ²¹

Footnotes

¹ Note that I use the terms set and ∈ for readability purposes only. Any sentence produced by uniformly substituting a predicate P and a two-place relation R (without collision) for “set-of-critics” and “ ∈ ” will work equally well.

² C.f. the tools we used to describe a full-width layer of sets in Section 11.2.

³ The latter terms whose extensions the Platonist and Nominalist can agree on might include relations like “dog,” “bites” and “is more massive than,” but not “number” “plus” or “the mass of … in grams is ….”

⁴ Such as the predicate “Is a number” or the relation +.

⁵ Such as the predicates “Is a set of goats,” or “is a function from the goats to numbers.”

⁶ “… has more than … fleas” and “is an element of” and “… is a function from cats to numbers which maps … to ….”

⁷ So, for example, if the sets of people, along with set membership, ( Speople,∈people ) is categorical over the people P it’s not just true that the number of sets of people is totally determined by what people exist but also facts such as whether or not any set of people is a person must also be determined.

⁸ Here I treat functions as just another kind of mathematical object.

⁹ I will treat these as free-standing mathematical objects.

¹⁰ This may include specifying that the numbers and sets of goats are distinct from all (the finitely) many types of non-mathematical objects relevant to the physical theory to be translated.

¹¹ Or in the limiting case of the partial function that’s not defined anywhere, pairing no goats with numbers.

¹² That is, (∀x)(∀y)f(x)=y→goat(x)∧number(y)]) .

¹³ For every relation R that only relates goats to numbers (in that (∀x)(∀y)Rxy→x is a goat and y is a number] which is functional (R is functional iff (∀x)(∀y)(∀z)[(Rxy∧Rxz)→y=z] ) corresponds to a function f .

¹⁴ This is the relation x and y stand in when x restores y to the correct balance of humors (eucrasia).

¹⁵ That is, □N,function,maps,goat [if nothing eucratises two distinct things and only goats eucratise and only numbers get eucratised then (∃x) (function(x) ∧(∀y)(∀x)[maps(x,y,z)↔eucratises(y,z) )].

¹⁶ Note that GOATS can be written with quantifiers restricted to objects which at least one of the Platonistic or nominalistic relations just mentioned (e.g., “goat,” “set,” “… is an element of …” “number,” “function,” “… is a function that assigns … to …”) apply to.

¹⁷ Here ψ1 is the part of our descriptions of the numbers and functions from goats to numbers which is straightforwardly stateable in FOL.

¹⁸ At least, this claim holds on the plausible assumption that the former can be understood as content-restricted to some Platonist vocabulary and the latter can be understood as content-restricted to some Nominalist vocabulary.

I tentatively hypothesize that no non-negotiable scientific practice requires unrestricted quantification. To roughly motivate this idea, we might say that when concerned with physics or biology, scientists don’t (and needn’t) concern themselves with talking about what fictional characters, or marriage licenses, could be like. So, we shouldn’t need to use sentences whose quantifiers are restricted to range over literally all objects, including these irrelevant ones. It suffices to use quantifiers which range over, e.g., all (relevant applied mathematical objects and) physical particles and spatial points, rather than statements of universal quantification.

Also note that formulating all our physics room talk with restricted quantifiers doesn’t require us to have any illuminating conception of each of the types of physical objects we are quantifying over, like “quark” or “boson” (thanks to Vann McGee for pressing this point in conversation). We can take our quantifiers to range over objects satisfying some rather broad uninformative notion like “fundamental physical object” or physical object. Or, if a Leibnitizan regress appears to exist, that the physics relevant particles/particles at a certain level or lower are such and such. This is important because it means that we can ask questions like “How many different kinds of fundamental physical particles are there?” without having much of a sense of what these particles are like.

¹⁹ We will generally be able to derive this from the fact that □ND , i.e., that it’s logically necessary that however the nominalistic relations apply it’s possible, holding fixed those nominalistic relations, that D .

²⁰ Let M be supervenience description for a hierarchy of sets with ur-elements chosen from the physical objects with height ω . Let R1,…,Rn be some list of nominalistic vocabulary such that it’s metaphysically necessary that: only non-mathematical objects are related by these relations and every non-mathematical object has at least one of these relations apply to it.

Then for any list of nominalistic relations R1,…,Rn , we can use axioms like those proposed in Part II to show that □◊R1,…,RnM . That is, it’s necessary that whatever the nominalistic R1,…,Rn structure is like, this structure can be supplemented with some new objects playing the roles of sets in Vω with ur-elements.

And given □◊R1,…,RnD , we can prove the analog of Field’s conservativity statement. Consider an arbitrary nominalistic sentence β , corresponding to Field’s B* (a description of what’s supposed to be happening with regard to the physical objects), hence content-restricted to our list of Nominalist relations R1,…,Rn and satisfying ◇β .

We can prove that if □◊R1,…,RnM and ◇β , then ◇(β∧M1∧…∧Mn) , as follows. Assume that □◊R1,…,RnM and ◇β . Enter this ◊ context. Then we know β . We can import our assumption that □◇R1,…,RnM , as it is content-restricted to the empty list of relations. Then we can infer ◊R1,…,RnM by □ Elimination (Lemma B.4 of Section B of the online appendix). Now because β is content-restricted to R1,…,Rn we can infer that ◊R1,…,Rn(M∧β) by Axiom 8.6. So leaving the diamond context we have ◇◊R1,…,Rn(M∧β) , hence ◇R1,…,Rn(M∧β) by Axiom 8.2 (Diamond Elimination) and ◇(M∧β) by Axiom 8.3 (Diamond Ignoring).

Thus we can use modal nominalistic reasoning about logical possibility, rather than set theory (as Field does) to show that assuming the existence of a hierarchy of sets with ur-elements over the physical objects you are currently talking in terms of up to Vω is harmless, and doesn’t let you prove anything false or unjustified about these non-mathematical objects.

²¹ However, certain disadvantages may also be admitted. Most obviously, accepting the conditional logical possibility operator is controversial (though, recall, Field himself advocates accepting a primitive logical possibility operator and uses it in his argument for conservatism). Also, the kind of paraphrases of physical magnitude statements provided will not be as attractively “intrinsic” in the way Field wants.

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