5.1 Axioms governing the complex terms

5.2 Principles governing encoding

We turn finally to the axioms governing our primitive predicate $A!$ in both exemplification and encoding predications.

6.1 The denotations of the terms of T

We can now say what the constants and $\lambda $ -expressions of T denote in our background theory. Thus far, we’ve assumed T includes primitive constants, primitive 1-place predicates, primitive 2-place predicates (including identity), and $\lambda $ -expressions. Where $\tau $ is any primitive individual constant of T, any primitive 1-place predicate constant of type $\langle i\rangle $ of T, any 2-place predicate constant of type $\langle i,i\rangle $ of T, or any 1-place $\lambda $ -expression of type $\langle t\rangle $ of T (for any type t), let $\tau _T$ be the T-indexed version of $\tau $ . By the Importation Principle, these indexed terms have the same type as their non-indexed counterparts. We now turn to the question of what these indexed terms denote.

We shall later argue (Section 7.2) for the view that the meaning of a mathematical term $\tau $ in theory T is the logical role it has within T. But we here assert an axiom that captures this view by stipulating that term $\tau $ of type t in theory T denotes the abstract object of type t that encodes exactly the properties (of type $\langle t\rangle $ ) that $\tau _T$ exemplifies in T. Formally, we assert the following Reduction Axiom Schema, which uses canonical descriptions to identify the denotations of the indexed mathematical terms imported into object theory. Where $\tau _T$ and $\alpha $ have type t, $A!$ and F have type $\langle t\rangle $ :

• Reduction Axiom Schema:

  

Note that the instances of this schema are not definitions, since the expressions on the left of the identity sign also appear on the right. But they are principles that are analytic, or so we will argue in the next section. Here are some simple examples of the above; these tell us exactly which abstract objects are denoted by $\emptyset _{\mathrm {ZF}}$ , $S_{\mathrm {ZF}}$ , $\in _{\mathrm {ZF}}$ (later we’ll discuss some more complex examples):

• Instances of the Reduction Axiom Schema:

  

  

  

These all have obvious readings.

We next focus just on the identification of the primitive constants and predicates of a particular theory, so that we can more easily see their consequences. The following Equivalence Theorem Schema is an immediate consequence of our Reduction Axiom Schema, by the Abstraction Principle for abstract objects and substitution of identicals:Footnote ²⁷

• Equivalence Theorem Schema:

  

This asserts that a term $\tau $ (individual or relation) of theory T encodes exactly the properties $\tau $ exemplifies in T. As somewhat more specific examples of this schema, we have:

In other words, for any first-level property G, the individual $\kappa _T$ encodes G iff $\kappa _T$ exemplifies G in T; for any second-level property of properties $\mathcal {G}$ , the property $\Pi ^1_T$ encodes $\mathcal {G}$ iff $\Pi ^1_T$ exemplifies $\mathcal {G}$ in T, and for any second-level property of relations $\mathcal {S}$ , the property $\Pi ^2_T$ encodes $\mathcal {S}$ iff $\Pi ^2_T$ exemplifies $\mathcal {S}$ in T.

Clearly, then, the following are instances of the Equivalence Theorem Schema:

That is, the empty set of ZF encodes exactly the properties G that it exemplifies in ZF; the ZF-property of being a set encodes exactly the second-level properties of properties that it exemplifies in ZF; and the membership relation of ZF encodes exactly the second-level properties of relations that it exemplifies in ZF.

Now the axioms introduced by the Importation Principle become salient. For the properties that can be abstracted from those claims can be instantiated into the above universal claims. In particular, the properties that are referenced in (H), (I), and (J) above may be instantiated, respectively, into the above claims to yield the following theorems:

Each primary term in the biconditionals displayed above has been given a formal identification in our theory. Moreover, since the right-hand side of each of the above equivalences is a theorem resulting from the Importation Principle, we have a proof of the following facts about , $S_{\mathrm {ZF}}$ , and $\in _{\mathrm {ZF}}$ :

In other words, it is provable in our framework that the empty set of ZF encodes the ZF-property of having no sets as members; the ZF-property of being a set encodes the (second-level) ZF-property of being a property such that nothing exemplifying it is a member of the empty set; and the membership relation of ZF encodes the (second-level) ZF-property of being a relation that no set bears to the empty set.

We conclude this section with two, somewhat more complex, examples. First, recall (H):

• (H)

We may now use the Reduction Axiom to identify the denotation of the $\lambda $ -expression as follows:

• $[\lambda x \: \neg \exists y(Sy \:\&\: y \in x)]_{\mathrm {ZF}} = $

  

This asserts that the ZF-property having no sets as members is identical to the abstract property that encodes all and only those second-level properties that are exemplified, in ZF, by the ZF-property of having no sets as members. As an example of such an encoded second-level property, consider being a property exemplified by the null set (). It is a fact about ZF that

and this gets imported as

That is, in ZF, the ZF-property having no sets as members exemplifies the ZF-property of properties being a property exemplified by the null set. So by the Equivalence Theorem Schema:

• $[\lambda x \: \neg \exists y(Sy \:\&\: y \in x)]_{\mathrm {ZF}}$ encodes

For the final example, recall (J):

• (J)

So, where R is a variable of type $\langle i,i\rangle $ , $\alpha $ is a variable of type $\langle \langle i,i\rangle \rangle $ , $A!$ has type $\langle \langle \langle i,i\rangle \rangle \rangle $ , and $\Gamma $ is a variable of type $\langle \langle \langle i,i\rangle \rangle \rangle $ , the Reduction Axiom Schema implies

We leave to the reader the formulation of a natural language gloss of this identification. And we leave it to the reader to find examples of properties that are exemplified by in ZF. By the Equivalence Theorem, these become encoded by .

6.2 Sentence reduction: True readings of mathematical theorems

Since we won’t be using relative interpretability as our standard of reduction, our methodology is to outline an alternative translation procedure that yields a true reading for every unprefixed theorem of each mathematical theory. We therefore show how to assign true object-theoretic readings to the theorems of mathematical theories when we consider those theorems in and of themselves, unprefixed by a theory operator. Platonist philosophers of mathematics believe that the unadorned claims of mathematics, such as ‘0 is a number’, ‘the empty set is an element of unit set of the empty set’, ‘two is less than $\pi $ ’, etc., are simply true, while fictionalist philosophers argue that they are false. Our view is that this disagreement is explained by the fact that these claims are ambiguous, for there is an exemplification reading on which they are false and an encoding reading on which they are true.

Take a simple atomic formula, e.g., the statement that ‘0 is a number’, when this is asserted as an axiom of PA. We’ve already seen that the prefixed claim “In PA, 0 is a number” is to be represented as

After importing the above into object theory, our analysis isFootnote ²⁸

Now we want to give a true reading of the unprefixed “0 is a number”. But we can infer such a reading from an instance of the Equivalence Theorem, since the encoding formula, $0_{{\mathrm {PA}}}N_{{\mathrm {PA}}}$ , is derivable. As a theorem, object theory regards $0_{{\mathrm {PA}}}N_{{\mathrm {PA}}}$ as a true reading of “0 is a number”. No such argument can be given for the exemplification reading, , of the unadorned claim “0 is a number”. In our framework, this exemplification claim is axiomatically false, by the contrapositive of AXIOM 5 and the fact that $N_{{\mathrm {PA}}}$ is an abstract property of individuals. This example shows that predications of the form ‘x is F’ in natural language are structurally ambiguous, and that in the case at hand, the encoding reading $xF$ is provably true while the exemplification reading $Fx$ is false.

Moreover, we take there to be a structural ambiguity in simple predications of natural language, for the unadorned claim “0 is a number” embodies not only a true atomic fact about a property that $0_{{\mathrm {PA}}}$ encodes but also a true atomic fact about a property that $N_{{\mathrm {PA}}}$ encodes, namely, . Once we import so as to yield the axiom , the Equivalence Theorem guarantees that the encoding formula is a theorem of our logical framework. Thus, we have another true reading of our unadorned mathematical claim. In a manner similar to the above, the exemplification reading, that $N_{{\mathrm {PA}}}$ exemplifies , is axiomatically false, by AXIOM 5 and the fact that is an abstract property of properties.

To see how to generalize the procedure for assigning true encoding readings for complex (unadorned) mathematical theorems, let us return to the statement, “No set is an element of the empty set”, said in the context of ZF. We’ve seen how the representation of the claim that this is a theorem of ZF gets imported as the following theorems of our logical framework (the first of which is an axiom by importation, the remainder of which are consequences):

• (G)

  (H)

  (I)

  (J)

But our question is, what reading should we assign to the unadorned ZF claim that “No set is an element of empty set”? Our answer is that we should assign the conjunction of the following three theorems:

These were all proved to be theorems at the end of Section 6.1. We suggest that the conjunction of all three theorems captures the facts embodied by the unadorned mathematical claim “No set is an element of the empty set”, for this latter claim is not only a fact about the empty set of ZF, but also a fact about the ZF-property of being a set and about the membership relation of ZF.

Indeed, we can, in general, abbreviate the conjunction of the above three theorems as a simple, intuitive formula:

We can intuitively think of this as a ternary encoding formula of the form $xyzH$ , where the type of H is that of a ternary relation among entities having the types of $x,y,z$ in that order. Of course, this isn’t a primitive formula of our logical framework, but it doesn’t need to be, for it simply serves as an abbreviation of a conjunction of well-formed formulas. In the particular case at hand, three places are needed because there are three primitive theoretical expressions in our target sentence (, , ). But it is straightforward to define a function that takes as input an unadorned theorem of ZF and yields as output a n-place encoding formula of the above form, where the output encoding formula is an abbreviation of n well-formed encoding formulas. We omit the details here, though interested readers are directed to [Reference Zalta61, pages 250, 251].

This, then, is a procedure for assigning a true (T-relative) reading to every theorem of an arbitrary mathematical theory T. It completes the reduction of mathematics to the above axiomatic system. In the next section we will show that each of the axioms of our system counts as logical or analytic.

We note here that the procedure and analysis described above offers a logical reconstruction of mathematical objects and relations as they are given antecedently by some specific mathematical theory. While this provides, for example, a complete reconstruction of ZF-sets, we are not claiming that the reconstruction is necessarily a complete theory of sets since ZF isn’t a complete theory of sets. Rather, we are offering the above as an analysis of any claim a mathematician might make in any context in which the mathematician is adopting all and only the assumptions of ZF. And this generalizes to other mathematical theories: in every context in which a mathematician assumes the principles of a mathematical theory, we can use the above methods to analyze their claims.

One final observation, about the completeness of mathematical objects and relations, is in order. Recall that our analysis imports theorems $\phi $ of T, i.e., formulas $\phi $ such that $\vdash _T \phi $ , as claims of the form $T \models \phi ^*$ . But for incomplete theories T, a property of the derivability relation $\vdash $ now becomes relevant, namely, that $\vdash _T (\phi \lor \psi )$ doesn’t imply $\vdash _T \phi $ or $\vdash _T \psi $ . Given our methodology above, this extends object theory with claims of the form $T\models (\phi \lor \psi )^*$ but not with claims of the form $T \models \phi ^*$ or claims of the form $T \models \psi ^*$ . For example, consider the Continuum Hypothesis (CH), where this is formulated as $2^{\aleph _0} = \aleph _1$ . Since $\vdash _{\mathrm {ZF}} (\textrm {CH} \lor \neg \textrm {CH})$ doesn’t imply $\vdash _{\mathrm {ZF}} \textrm {CH}$ or $\vdash _{\mathrm {ZF}} \neg \textrm {CH}$ , the object-theoretic claim $\textrm {ZF} \models (\textrm {CH} \lor \neg \textrm {CH})^*$ doesn’t imply $\textrm {ZF}\models \textrm {CH}^*$ or $\textrm {ZF}\models (\neg \textrm {CH})^*$ . Moreover, though it follows that ${\aleph _1}_{\mathrm {ZF}}$ encodes $[\lambda x\: (2^{\aleph _0} = x \lor \neg 2^{\aleph _0} = x)]_{\mathrm {ZF}}$ , it doesn’t follow either that ${\aleph _1}_{\mathrm {ZF}}$ encodes $[\lambda x\: 2^{\aleph _0} = x]_{\mathrm {ZF}}$ or that ${\aleph _1}_{\mathrm {ZF}}$ encodes $[\lambda x\: \neg (2^{\aleph _0} = x)]_{\mathrm {ZF}}$ .

Note that this example highlights an important difference between the standard of relative interpretability and our method of sentence reduction. The relative interpretation of a disjunction $\phi \lor \psi $ is the disjunction of the respective relative interpretations of $\phi $ and $\psi $ , while our reduction of a disjunctive mathematical theorem $\phi \lor \psi $ does not necessarily coincide with the disjunction of the respective reductions of $\phi $ and $\psi $ . Indeed, if neither $\phi $ itself nor $\psi $ itself is a theorem, then our reduction will not apply to $\phi $ and $\psi $ at all.

These facts will help us to argue that this is a form of logicism—the objects and relations are reified incomplete concepts. Even ‘complete’ mathematical theories (e.g., the first-order theory of real-closed fields) are, in some sense, about objects that have only mathematical properties and are thus incomplete with respect to what they encode. And even ‘(deductively) incomplete’ theories are complete in the sense that they completely describe the incomplete entities they are about.

7.1 Our framework axioms are logical or analytic

In this section, we shall not argue, but rather assume, that the principles of classical logic, the substitution of identicals, the axiom governing descriptions, and the principle of $\lambda $ -Conversion (i.e., the axioms discussed in Section 5 prior to Section 5.2) are logically true (as this notion is defined below) or analytic, where analyticity is defined in the usual way as truth in virtue of meaning (in this case, of the logical symbols). The principles of classical logic and the substitution of identicals have traditionally been regarded as logical. We add the law governing descriptions and $\lambda $ -Conversion (AXIOMS 1 and 2) to this list of logical truths on the grounds that they are true in virtue of the meaning of the expressions the (represented by the $\iota $ ) and being such that (represented by the $\lambda $ ).Footnote ³¹

Moreover, AXIOMS 4–7 (in Section 5.2) stipulate what is meant by the property of being abstract and, as such, are nothing more than meaning postulates. Once we take abstract objects to be those entities that are individuated by the properties they encode, these axioms articulate the conception of such objects in formal detail and, hence, are analytically true. Thus, we see the burden of the present paper as showing that the comprehension principle for abstract objects (AXIOM 3) is a logical truth, despite the fact that it baldly asserts the existence of abstract objects.

We begin with the observation that the classical understanding of the model-theoretic interpretation of the predicate calculus has overlooked one key feature of such interpretations. In particular, model-theoretic interpretations should include, in the domain of interpretation of the variables, everything that is required for the very possibility of predication, logically complex thought (including abstract mathematical thought), and logical consequences of those thoughts. That’s the point of (a) thinking that the predicate calculus is a fundamental system for expressing our thoughts and validating inferences, and (b) thinking that an interpretation of that system will give us an insight into what’s required for the possibility of having those thoughts and making those inferences.

To approach our thesis, let’s reconsider why $\lambda $ -Conversion is a logical truth. Consider one of its consequences, which is logically complex not only because it involves the $\lambda $ -expression but also because it involves the negation symbol:

• $[\lambda y \: \neg Gy]x \equiv \neg Gx.$

This holds for any property G: something exemplifies the negation of G iff it fails to exemplify G. There exists a logical exemplification pattern that underlies this fact, one that everything that fails to exemplify G has in common! After all, entities in the world do divide up into those that exemplify G and those that do not, and without the existence of the negation of G, we could not express that thought. How could two individuals a and b which both fail to be G not share the pattern of what is most-easily described as “exemplifying not-G”? If we treat this property of exemplifying not-G as reifying or representing this exemplification pattern, then it is required in any domain that contains the entities needed for truth of multiple predications of the form “x exemplifies not-G” ( $[\lambda y \: \neg Gy]x$ ). And $\lambda $ -Conversion also provides the logical justification as to why it is correct to infer one side of the biconditional from the other.

This same argument now applies to other instances of $\lambda $ -Conversion, e.g., those involving other complex formulas such as conjunctions, disjunctions, conditionals, etc. The instances of $\lambda $ -Conversion are true in any domain that contains the entities needed for the truth of predications involving complex predicates such as “x is G-and-H” ( $[\lambda y \; Gy \mathbin {\&} Hy]x$ ), etc., and provide the logical justification for inferences to and from such predicates, such as the inference from “x is G-and-H” to “x is G” (i.e., $[\lambda y \; Gy\mathbin {\&} Hy]x\vdash Gx$ ). The point also applies to relations and relational $\lambda $ -expressions. Complex reasoning about the converse of relation R ( $[\lambda xy\: Ryx]$ )Footnote ³² and relations like unrequited love ( $[\lambda xy\: Lxy \:\&\: \neg Lyx]$ ) assumes that the domain contains such relations. And, in general, the comprehension principle for relations, which we noted is derivable from $\lambda $ -Conversion (footnote 31), is logically true precisely because it postulates the entities that are required for such complex relational reasoning.

This leads us to a somewhat different philosophical conception of logical truth and logical consequence. Let $\mathcal {L}$ be any language that is an extension of the language of object theory. Then we say: a formula $\phi $ in $\mathcal {L}$ is logically true if and only if $\phi $ is true in every model of $\mathcal {L}$ that includes all the entities required for the possibility of thinking thoughts expressible in $\mathcal {L}$ . For any formula $\psi $ of $\mathcal {L}$ , the phrase “possibility of thinking” that $\psi $ refers to the activity of having the particular thought that $\psi $ , i.e., entertaining the particular propositional content that $\psi $ . So, by saying “required for the possibility of thinking” that $\psi $ , we also mean required for the existence of the propositional content that $\psi $ .Footnote ³³ Moreover, logic is committed to the existence of whatever entities are required for the possibility of drawing inferences when reasoning theoretically.Footnote ³⁴ Consequently, we also say that a formula $\phi $ in $\mathcal {L}$ is a logical consequence of a set $\Gamma \subseteq \mathcal {L}$ if and only if $\phi $ is true in every model of $\mathcal {L}$ that (i) includes all the entities required for the possibility of thinking thoughts expressible in $\mathcal {L}$ and (ii) makes every member of $\Gamma $ true.

The Tarskian and Fregean conceptions of logical truth dovetail in our conception: from Tarski we take the idea that logical truth is truth in all models of some given language $\mathcal {L}$ [Reference Tarski and Corcoran49]; from Frege we take the conception of logic as providing constitutive norms of thought and reasoning as such [Reference MacFarlane40], including constitutive norms for logically complex thought and reasoning.Footnote ³⁵ By combining the two, we end up with a definition of the logical truth of a formula $\phi $ in $\mathcal {L}$ as $\phi $ ’s truth in all models that include everything that is required for the possibility of having the (logically complex) thoughts expressible in $\mathcal {L}$ .Footnote ³⁶ Under this conception, not only is $\lambda $ -Conversion a logical truth, but as noted in the example discussed above, the inference from $\neg Gx$ to $[\lambda y\: \neg Gy]x$ is one of logical consequence.

Let’s then see how this conception can be used to understand why the lowest level instances of AXIOM 3 (i.e., the instances asserting the existence of abstract individuals) are logically true. The early Greek mathematicians adopted a method that has persisted until this day: they attempted to characterize objects whose only properties are the properties given by the defining principles.

Consider a simple example, such as discussions of The Equilateral Triangle in some language $\mathcal {L}_E$ of Euclidean geometry. Take any model of $\mathcal {L}_E$ that includes all the entities required for the possibility of thinking thoughts expressible in $\mathcal {L}_E$ . A mathematician might have thoughts expressible in $\mathcal {L}_E$ such as:

• The Equilateral Triangle has sides of equal length.

In thinking about this object in the abstract, the mathematician might logically infer that The Equilateral Triangle is not scalene.Footnote ³⁷ Here we have a logical conclusion in the form of a simple predication about The Equilateral Triangle, and the domain must have an object that exemplifies or encodes being an equilateral triangle for the thought (i.e., the propositional content) to exist and for the inference to be valid.

Thus, the mathematician has defined, objectified, and drawn inferences about a pattern of properties of individuals. The assertion that this pattern exists as an individual is true in the given model of $\mathcal {L}_E$ , since models include everything required for having complex thoughts and making inferences expressible in $\mathcal {L}_E$ and thus include the relevant pattern of properties of individuals. It is important to emphasize here that the existence of this pattern doesn’t commit us to saying that there is an object that exemplifies the properties defining the pattern. In fact, we have two options that avoid this commitment but which offer an object of thought: either treat the pattern as a property of properties in 3rd-order logic, or treat it as an abstract individual that encodes the properties in question. But the assertion of simple predications in $\mathcal {L}_E$ like the one displayed above suggests that the mathematician has conceived of The Equilateral Triangle as an abstract individual. AXIOM 3 is therefore a logical truth because it is true in every model that includes the entities required for having thoughts expressible in $\mathcal {L}_E$ .

More generally, we may reason about any other combinations of properties in $\mathcal {L}$ that might be of interest where these combinations could be considered as individuals. AXIOM 3 is a logical truth given that these individuals must be present in every model of $\mathcal {L}$ . There is an analogy with $\lambda $ -Conversion; if one is willing to accept $\lambda $ -Conversion as logical, on the grounds that, for any language $\mathcal {L}$ , $\lambda $ -Conversion is true in every model that includes the relations needed to express exemplification predications in $\mathcal {L}$ , then one should likewise be willing to accept AXIOM 3 as logical. In other words, if one recognizes that second-order comprehension is logical because it merely expresses the existence of entities presupposed for higher-order thinking and reasoning, then one should also recognize that comprehension over abstract individuals is logical because it (analogously) merely expresses the existence of entities presupposed for the possibility of such activities.

Similar conclusions now apply to higher-level $\lambda $ -Conversion and higher-level abstracta. For take the example:

• $[\lambda R \: \neg \forall x Rxx]S \equiv \neg \forall x Sxx.$

This asserts: relation S exemplifies being a non-reflexive relation iff S fails to be reflexive. There is a pattern of which every relation that fails to be reflexive is a part! Clearly, relations in the world do divide up into those that are reflexive and those that are not, and without the existence of the negation of the property of reflexivity $[\lambda R \: \neg \forall x Rxx]$ , how could two relations S and $S'$ which both fail to be reflexive not share the pattern of what is most-easily described as “being a non-reflexive relation”? If we treat this property of being a non-reflexive relation as reifying or representing this pattern, then it is required in any model that contains the second-level properties of relations needed for the truth of multiple predications of the form “S is non-reflexive” ( $[\lambda R \neg \forall xRxx]S$ ). And so on to other instances of higher-order $\lambda $ -Conversion. Thus, higher-order $\lambda $ -Conversion is logical because higher-order properties like being non-reflexive exist in every model that includes the entities needed to express such thoughts as “S is non-reflexive” and for reasoning to the conclusion that a non-reflexive relation is not an equivalence relation.

And this is likewise the case for higher-level comprehension over abstract entities. Thus, let’s consider why a particular instance of the comprehension principle for abstract relations, as applied to mathematical relations, is a logical truth. Consider the less-than relation ( $<_D$ ) as given by a language $\mathcal {L}_D$ and the theory of dense linear orderings without endpoints. This relation is given by the following theory $T_D$ , in which $<$ is not indexed:

• $\forall x,y,z (x <y \:\&\: y < z \to x < z).$ (Transitivity)

  $\forall x (x \not < x).$ (Irreflexivity)

  $\forall x,y (x \not = y \to (x < y \:\vee \: y < x)).$ (Connectedness)

  $\forall x,y \exists z (x <z < y).$ (Dense)

  $\forall x \exists y \exists z (z < x < y).$ (No Endpoints)

The theorems derivable from these axioms constitute the theory $T_D$ . What is more, we think of the $<_D$ relation itself as being constituted by that theory as well. The world itself doesn’t contribute any facts about $<_D$ and there is no guarantee that a relation exists that exemplifies the properties of the $<_D$ relation—all there is to $<_D$ are the properties it has been assigned in this theory. In other words, the truths that ground all the facts about $<_D$ are facts of the form “In the theory $T_D$ , $\mathcal {R}\!\! <_D$ ”, where $\mathcal {R}$ ranges over properties of relations. The theorems in the scope of the operator “In the theory $T_D$ , …” ascribe to relation $<_D$ various properties of relations, such as the properties of being transitive, irreflexive, connected, dense, and having no endpoints, and those that follow from them. There exists a pattern of predications, embedded within the theorems of $T_D$ governing $<_D$ , that we may articulate as a pattern of properties of relations, namely, the pattern: $T_D \models \mathcal {R}\!\! <_D$ . $<_D$ just is that pattern of properties of relations, but instead of representing this pattern as a property of type $\langle \langle \langle i,i\rangle \rangle \rangle $ (property of properties of relations), encoding predication turns that pattern into an abstract relation of type $\langle i,i\rangle $ that encodes the properties of relations $\mathcal {R}$ that satisfy the pattern $T_D \models \mathcal {R}\!\! <_D$ . (This is expressed in terms of our Reduction Axiom Schema, discussed in Section 6.1.)

Indeed, that relation must exist for us to have a mathematical thought, and draw inferences, about the relation $<_D$ . Hence, the notions of logical truth and consequence defined above have the following application: a sentence $\phi $ of $\mathcal {L}_D$ containing the term $<_D$ is logically true if and only if $\phi $ is true in all interpretations that include all the (abstract) objects required for the possibility of having thoughts expressible in $\mathcal {L}_D$ . $<_D$ is required for the possibility of having thoughts expressible in $\mathcal {L}_D$ . Thus, the following claim, which expresses the existence of $<_D$ , is logically true:

This asserts: there is an abstract relation that encodes all and only the properties of relations exemplified by $<_D$ in $T_D$ . And, generally, for any relation S of mathematical theory T, to have a thought about S, the following must be true:

Notice the theory $T_D$ is a simple case in which the only distinguished term of the mathematical theory is a relation term. More complex mathematical theories involve both distinguished relation terms and distinguished individual terms.

For example, PA has as primitives: the property being a number, the relation successor, and the individual Zero. In this case, the existence of the abstract property $\textit {being a number}_{\mathrm {PA}}$ , of the abstract relation property $\textit {successor}_{\mathrm {PA}}$ , and the abstract individual $\textit {Zero}_{\mathrm {PA}}$ are asserted by the relevant instances of comprehension AXIOM 3. Thus, when our analysis is applied to PA:

• • There are at least three kinds of exemplification patterns that exist in the sentences prefaced by the operator “In PA, …”, namely, patterns of properties of the property of being a number, patterns of properties of the successor relation, and patterns of properties of the individual Zero. (There are, additionally, patterns of properties of both the relations and properties definable in PA, but we’ll discuss those below.)

• • These particular patterns induce and ground three corresponding kinds of encoding patterns that exist in the data of the form “In PA, …”, namely, patterns of properties of the property of being a number in PA, patterns of properties of the successor relation in PA, and patterns of properties of the individual Zero in PA.

So, it follows that the instances of AXIOM 3 that assert the existence of the entities needed for the analysis given by the Reduction Axiom Schema are all logical truths.Footnote ³⁸

In this section, we have argued for the logicality of axioms that assert the existence of two general kinds of logical entities:

• • Those which exist as exemplification patterns among individuals, properties and relations and which, by comprehension, are logical objects within the domain of (higher-order) properties, i.e., those whose existence is asserted by AXIOM 2.

• • Those that exist as predication patterns (either exemplification or encoding patterns) among properties and relations, that, by comprehension for abstract individuals, comprehension for abstract properties, and comprehension for abstract relations, are logical objects within the respective domains, i.e., those whose existence is asserted by AXIOM 3.

Both kinds of entities are logical in so far as they are patterns of predications. The entities asserted to exist by AXIOMs 2 and 3 are abstracted from pure logical patterns formulable solely in terms of predications generally in our language. Given that the conditions under which they are asserted to exist correlate with pure logical patterns that exist in our language, what else could they be but logical objects? So in what follows, we’ll refer to the entities denoted by canonical descriptions as logical objects.

Note that if our axioms are logical, then any theorem we can derive is logical.Footnote ³⁹ Now if we can show that the claims of mathematics prefixed by the theory operator, which are imported when we apply object theory, are analytic, then it will follow that the unprefixed encoding claims of mathematics derived from the prefixed claims and object theory (at the end of Section 6.1) become logical or analytic. So we now turn to a defense of the idea that when we extend object theory in the application to mathematics, the new axioms are analytic.

7.2 The additional axioms for mathematics are analytic

Our goal in this section is to show that the axioms added to object theory in Section 6, namely, those introduced by the Importation Principle and the Reduction Axiom, are analytic. These are principles that underlie our analysis of mathematics.

We take it to be uncontroversial to claim that axioms introduced by the Importation Principle are analytic: we can put aside the controversial question of whether “ $\emptyset \in \{\emptyset \}$ ” is analytic, and yet still claim that “In ZF, $\emptyset _{\mathrm {ZF}} \in _{\mathrm {ZF}} \{\emptyset \}_{\mathrm {ZF}}$ ” is. The latter is true in virtue of the meaning of the terms ‘ZF’, ‘ $\emptyset _{\mathrm {ZF}}$ ’, ‘ $\in _{\mathrm {ZF}}$ ’, and ‘ $\{\emptyset \}_{\mathrm {ZF}}$ ’. Since ‘ZF’ denotes a theory, and a theory encodes its theorems, ‘In ZF, $\emptyset _{\mathrm {ZF}} \in _{\mathrm {ZF}} \{\emptyset \}_{\mathrm {ZF}}$ ’, when represented as $\textrm {ZF} \models \emptyset _{\mathrm {ZF}} \in _{\mathrm {ZF}} \{\emptyset \}_{\mathrm {ZF}}$ , is true in virtue of the meaning of all the terms used in the representation.

It remains to argue that axioms introduced by the Reduction Axiom are analytic. To do this, we argue that the meanings of the terms flanking the identity sign in instances of the Reduction Axiom are identical, i.e., that the meaning $m_{\tau }$ of a mathematical term $\tau $ is identical to the meaning of the canonical description used to identify the denotation of $\tau $ . As we shall see, this conclusion almost immediately implies LC.

So, to argue that the instances of the Reduction Axiom are analytic, consider any mathematical concepts and the mathematical theories in which they occur. As examples, we again consider the concepts and the axioms of ZF set theory. Recall, first of all, that we formulated the following instances of the Reduction Axiom Schema:

The right-hand side of these identity statements involve canonical definite descriptions (we call them canonical T-based descriptions below). These descriptions are formulated with the new indexed terms introduced when the mathematical theories are imported into object theory.

By referencing these descriptions, we may give the following argument for the claim that the instances of our Reduction Axiom Schema are analytic (and once we establish that, we give an argument for the thesis LC of Logicism about Concepts). Let $\tau $ be any unambiguous mathematical term used in a mathematical theory T, where $\tau $ is either an individual term or a predicate of T:

1. (P1) The meaning, $m_{\tau }$ , of a mathematical term $\tau $ is the inferential role of $\tau $ in the theory T.

2. (P2) The inferential role of $\tau $ in the theory T is the logical object denoted by the canonical T-based description for $\tau $ .Footnote ⁴⁰

3. (P3) The logical object denoted by the canonical T-based description is also the meaning of the canonical T-based description.

So by transitivity of identity, the meaning $m_{\tau }$ of a mathematical term $\tau $ is identical to the meaning of the canonical T-based description for $\tau $ . And by the uncontroversial principle that if the meanings of $\tau $ and $\tau '$ are identical, then $\tau = \tau '$ is analytic, it follows that:

1. (A) The instances of the Reduction Axiom Schema are analytic.

Here, then, is our support for the premises of the above argument.

Concerning (P1). We think it is reasonable to suppose that a mathematical concept $m_{\tau }$ is constituted by the systematic use of the mathematical term $\tau $ . In turn, the systematic use of $\tau $ can be grounded in a system of axioms and inferences in which that term appears. Thus, $m_{\tau }$ can be identified with the inferential role of $\tau $ in T. For example, $m_\emptyset $ , $m_\pi $ , $m_\in $ , $m_<$ , etc., are identical to the inferential roles of $\emptyset $ , $\pi $ , $\in $ , $<$ , etc., in their respective theories.

We take (P1) to be a principle that is consistent with the following historical antecedents. It is one way to spell out Wittgenstein’s meaning-as-use doctrine [Reference Wittgenstein, Anscombe and Rhees56], as well as the work of inferential role theorists such as Sellars [Reference Sellars and Sicha48] and Brandom [Reference Brandom8]. (P1) is also related to Schlick’s and Carnap’s view of theoretical terms in science, which in turn were influenced by Hilbert’s view of geometry in which the meanings of mathematical terms are determined completely by the theories in which they figure.Footnote ⁴¹ (But P1 should be even easier to accept than the corresponding view of theoretical terms in science since, unlike the latter, mathematical terms aren’t introduced with the idea of representing some empirical entity.) Finally, (P1) is consistent with Frege’s Context Principle, except that the meaning for a mathematical term is not given by any single reference-fixing sentence but rather by a whole theory.

Note also that by identifying the meaning of a mathematical term with its inferential role, (P1) doesn’t require us to invoke either the notion of an intension (in Carnap’s [Reference Carnap13] sense) or the notion of a concept (in Church’s [Reference Church and Henle14] sense). These notions are not needed in the semantics we give in the Appendix: meaning there is represented in terms of denotation. For any term $\tau $ , the meaning of that term is simply its denotation relative to (our interpretation and) an assignment to the variables, i.e., . Moreover, we are not assuming a modal framework, and so the notion of intensionality, i.e., functions from worlds to (sets of) entities, doesn’t apply. The rest of our argument will then be aimed at explicating the meaning, now identified as an inferential role, in terms of objects that will turn out to be logical.

Concerning (P2). We argue for P2 by considering examples such as the following: the inferential role of ‘ $\emptyset $ ’ in ZF is properly identified by the canonical description, . This holds because the abstract object denoted encodes all and only the properties of the null set derivable in ZF.Footnote ⁴² For instance, it is derivable in ZF both that $\emptyset \in \{ \emptyset \}$ and that . The latter gets imported into object theory as . As such, the property $[\lambda x\: x\in \{\emptyset \}]_{\mathrm {ZF}}$ is one of the properties that satisfies the formula in the matrix of the description . In object theory, the inferential role of the symbol $\emptyset $ in ZF is constituted by the object that encodes the totality of such properties. Its representation, $\emptyset _{\mathrm {ZF}}$ , as identified by our canonical ZF-based description, captures the inferential role.

Concerning (P3). Our argument for this premise begins with the inspection of the semantics of our formalism, which reveals that the terms of our formalism are assigned only one semantic value. We claim that this semantic value serves both as the denotation and meaning of the terms of our formalism. Our semantics assigns meanings by assigning denotations. Indeed, we take it that for our formalism, the distinction between the denotation and meaning of its terms just collapses.Footnote ⁴³ One doesn’t have to build a formal language with terms having both intensions and extensions in order to model the intensions and extensions of the terms of natural language. One simply needs to have (a) terms in the formal language that can represent the extensions of the terms of natural language as well as (b) terms in the formal language that can represent the intensions of the terms of natural language. That is what our system does.Footnote ⁴⁴

Given P1–P3, therefore, we’ve established (A), i.e., that the instances of our Reduction Axiom Schema are analytic. This completes our argument that the additional axioms added to object theory for the analysis of mathematics are analytic, and so we have established logicism in the form of LT. Since the theorems of mathematics are representable as theorems of extended object theory, and the axioms of extended object theory are all either logical truths or analytic truths, the theorems of mathematics are themselves either logical or analytic. This of course assumes that the rules of inference in classical logic preserve logicality and analyticity. We shall not argue for this claim.

Furthermore, premises P1 and P2 imply LC in the following form:

1. LC^′ The meaning of a mathematical term $\tau $ , $m_{\tau }$ , is identical to a logical object.

LC follows from LC $'$ by taking the meaning of a term to be its denotation.

8.1 Objections

One objection that might be raised is whether we have offered an analysis that does ‘too much’, in that it would give us a means of reducing theoretical terms in natural science to logic! The objection argues that our very same procedure, as outlined above, would give us denotations for theoretical terms like ‘electron’, namely, the abstract property that encodes exactly the properties of properties attributed to this property by our best available physics. But, here, we argue, there is a disanalogy that prevents one from properly applying the above analysis to theoretical terms of natural science. The disanalogy is that in natural science, the theoretical properties like being an electron are natural properties, whereas the theoretical properties of mathematics are not. Thus, in the case of natural science, one might distinguish the natural property from our various concepts of that property, as these concepts change from scientific theory to scientific theory. The property of being an electron, for example, is something there in the world, though our theories of the electron reflect our evolving concept of this property. The concept, but not the property, is tied to the inferential role.

Given this distinction, we would argue that our analysis above could not be applied to analyze the property of being an electron (though it might be applied to the concept electron as this might be embodied by some scientific theory). Thus, P1 fails in the case of the natural properties of physics: “the meaning of theoretical term $\tau $ in a physical theory” is a natural property, not a physical concept. Hence P1 is false. Whereas the physical concept might well be identical to an inferential role, as (the corresponding version of) P1 would have it, the physical property is not an inferential role at all. By contrast, in the case of mathematical properties, there is no distinction to be drawn between our concepts of a mathematical property and the property itself. Either the mathematical properties and our concepts of them collapse, since the former are not given by anything over and above the concepts, or there are no mathematical properties beyond our mathematical concepts. In the former case, we use the above analysis to identify both the property and the concept, collapsing the two; whereas in the latter case, we use the above analysis solely for understanding our mathematical concepts (in which case there is nothing else to understand).

Another objection might run as follows. Our analysis assumes that mathematical objects are identified in terms of actual theories, i.e., theories that someone has actually developed or asserted. Doesn’t this imply that the abstract realm of mathematical objects depends on the contingent actions of humans? To this, we may reply that by showing how all axiomatically developed mathematics consists of logical/analytic truths, we have shown a striking fact that achieves the goals of logicism. But a deeper response is also available, since the objection suggests that the theorems of mathematics are contingent claims.

In fact, they are not. To see why, note that we’ve analyzed the theorems of mathematics as encoding truths about the individuals and relations of mathematics. Though we didn’t develop the modal version of object theory here, one modal principle included in the theory is the claim that $\Diamond xF \to \Box xF$ , i.e., that if possibly an abstract object encodes a property, then it does so necessarily.Footnote ⁴⁵ So, though it may be a contingent fact as to what mathematicians have asserted by way of mathematical axioms, the theory-prefixed claims of the form “In theory T, p” are not contingent; they are analytic truths and given claims of that form, neither the theorems of T (as we’ve analyzed them) are contingent claims nor are the objects of the theory contingent objects.Footnote ⁴⁶

Thus, the realm of mathematical objects is not so closely tied to the contingent actions of human mathematicians. Though we haven’t developed modal object theory in this paper, it would be trivial to add a modal operator. If we adopt the axioms for S5 modal logic, then these possibility claims about possible authors are in fact necessary, and thus the realm of mathematics becomes defined on our view in terms of objects with no air of contingency about them.

Finally, a Platonist might object that when we extend a theory like ZF to ZFC, the mathematician is not talking about a different realm of sets, while our approach implies that they are. But in fact this is not the case: our approach is consistent with assigning “the” right denotations to set-theoretic terms and predicates and that perhaps these denotations are only incompletely described by both ZF and ZFC. Note that the Platonist claim presupposes both that sets exist independently of our theories of them and that when we move from ZF to ZFC, the new theory is simply characterizing the objects of ZF further. So, for the sake of argument, suppose that the sets do exist independently of our theories of them and that there is consequently a complete body of all set-theoretic truths. Introduce a proper name, say ‘ $\mathfrak {G}$ ’, for that body of truths and replace ‘ZF’ in our reduction axioms from above by ‘ $\mathfrak {G}$ ’. Then everything should go as before, and we should be able to reconstruct the mathematical terms of set theory as logical expressions. That is, we can plug $\mathfrak {G}$ into the machinery that we described above and the result, we claim, is a logicist reconstruction of the concept of set. Given this reconstruction, the theorems of ZF and ZFC will be true of the denotations that we have assigned to the terms and predicates of set theory $\mathfrak {G}$ since $\mathfrak {G}$ includes these theorems (assuming Choice is included in $\mathfrak {G}$ ). Of course, this is all completely hypothetical: we know that if the body of truths of set theory exists, it is not recursively axiomatizable, so we will never be “given” that body of truths in the form of a complete axiomatic system; nor does there seem to be any alternative manner in which we could be “given” that body of truths in a literal sense. Indeed, since Importation does not include the negative internalization principle (NI) as noted Section 6, our axiomatic theory does not include any axioms that would rule out that ZF, ZFC, and $\mathfrak {G}$ are pairwise identical. So none of this affects the principal logicist point that we want to make.

8.2 Comparison with other approaches

At the present time, we know of no other successful version of logicism, i.e., no successful attempt to establish LC and LT. While Frege’s version of logicism failed due to the inconsistency of his Basic Law V, Whitehead and Russell’s account of logicism was based on principles, such as the Axiom of Reducibility and the Axiom of Infinity, whose status as logical truths were unclear at best. Similarly, efforts by Hodes [Reference Hodes31] and Tennant [Reference Tennant51] both require an appeal to non-logical, or even non-analytic, axioms of infinity,Footnote ⁴⁷ and it is not clear how the methodology in Tennant [Reference Tennant50, Reference Tennant52] can be extended to the logicist analysis of an arbitrary mathematical theory T (i.e., it is not clear how, for an arbitrary theory T, to state introduction and elimination rules for the terms and predicates of T in a way that yields all and only the theorems of T).Footnote ⁴⁸

In recent years, neologicist theories have been developed that rely on abstraction principles.Footnote ⁴⁹ These neologicist theories add new abstraction principles for each new kind of mathematical object introduced and each of these new ‘double abstraction-identity principles’ (like Hume’s Principle, which introduces two abstractions, $\#F$ and $\#G$ , in the same principle) combines both a comprehension (or existence) claim and an identity claim for the new kind of mathematical object. Clearly, any reduction of mathematics to logic will have to use some definitions or principles for identifying the mathematical objects as logical objects. On our view, however, only a single comprehension principle for objects is needed and, moreover, a single identity principle for objects is specifiable independently of comprehension.

Our approach differs from previous approaches in the following ways. (a) We appeal only to principles that are arguably logical or analytic and, in particular, we don’t appeal to any non-logical axiom of infinity. Our unapplied and purely logical theory still has a finite model (described in the Appendix). When we apply the theory to mathematics, we sometimes import an infinite number of theorems into our own theory in the form of analytic truths. The infinity of mathematical entities that results from this extension is a presupposition of mathematical thought and thus counts as logical in our understanding of the term. (b) We don’t have to continually re-prove our system is consistent since we use a uniform method for analyzing every kind of mathematical object. The model we have proposed in the Appendix—even though it is merely a minimal model for our background logical theory—grounds our conjecture that no special steps need be taken to guarantee consistency each time a new part of mathematics is analyzed in the manner outlined in Section 6. Though early forms of neologicism faced the ‘bad company’ and ‘embarrassment of riches’ objections, recent forms have employed more general methods to guarantee consistency.Footnote ⁵⁰ (c) Our approach is not subject to the Julius Caesar problem.Footnote ⁵¹ (d) Our analysis is prepared even for not-yet-formulated mathematical theories and new kinds of mathematical objects. Finally, (e) our approach gives an account of the denotations of both the individual terms as well as the predicates of mathematical theories.

Finally, our logicist proposal also differs from Linnebo’s [Reference Linnebo37] recent theory of thin objects. Linnebo develops another closely related follow-up project to Frege in which a comprehensive class of abstraction principles is employed to introduce abstract objects that are “thin” in the sense that their existence does not make any substantial demand on the world. There are three main differences to Linnebo’s approach. (i) His abstraction principles are dynamic and predicative (“new” domains of objects are being introduced by iterated abstraction from “old” or given domains of objects), whereas the abstraction principles that are derivable from our theory are of the more traditional static and impredicative type.Footnote ⁵² (ii) We defend our basic principles to be logically or analytically true, while Linnebo does not and instead argues that his abstraction principles don’t make a substantial demand on the world. Indeed, Linnebo does not argue that his theory is a version of logicism. (iii) While Linnebo offers solutions for the standard problems for neo-logicism, we may not know how to show that some present or future mathematical theories are derivable from any of his abstraction principles. If this is right, our approach would seem preferable, as it is based on the actual presentations of mathematical theories in mathematical practice. We will have to leave a more substantial comparison with Linnebo’s theory of thin objects to future work.

The principles of object theory are general in the sense accepted by both Kant and Frege (as described in [Reference MacFarlane40]), namely, they are constitutive of, and provide a norm for, the possibility of having complex logical thoughts, including abstract mathematical thought. That is very different from the more standard conception of logic, since our conception allows that some existence claims can be logical truths. Indeed, we suggest that the argument (in the previous section) for the logicality of both of our comprehension principles (one for relations of every higher-order type and one for abstract objects of every type) justifies the early logicist view that logic may endorse existence claims, namely, those that assert the existence of the logical objects that Frege, Russell, and Whitehead used to reduce mathematics. The only existence claims logic is committed to are those required for the possibility of having complex logical thoughts.

So logic does have ontological commitments, but it commits one to nothing more than what is required for the possibility of formulating and interpreting complex predications. In particular, unapplied object theory has ontological commitments (see the model in the Appendix), but the unapplied theory is not committed to anything more that what is required for the possibility of reifying structural relations among relations, i.e., what is required to make sense of the abstract relations that emerge from patterns of exemplification predication that are available in first- and second-order logic.

As we’ve mentioned, our understanding of logic and logicality has consequences for certain controversies concerning existence claims. Logicians have faced the following issue: what should one say about the fact that standard first-order logic entails existence claims (such as $\exists x\: x\! =\! x$ ). This has traditionally been seen as an uncomfortable conclusion since it was thought, on the one hand, that logic should be free of existential commitment, whereas on the other hand, that first-order logic should be the background system for the assertion of any non-logical existence principles. However, we can comfortably accept the fact that logical axioms imply existence claims because a non-empty classical domain should come pre-stocked with a minimum group of abstract objects.

Moreover, some philosophers have argued that $\exists x\exists y (x\not = y)$ can’t be a logical truth, since it asserts the existence of more than a non-empty domain. Our arguments in Sections 7.1 and 7.2 show that this formula is a logical truth.Footnote ⁵³ We argued that some formulas are in fact logically true in the sense that they are true in all models that make it possible to have logically complex thoughts. Moreover, once we import mathematics into our system in the form of analytic truths, then we can derive the existence of new logical objects.

8.3 Epistemology redux

We claim, finally, that the epistemological benefits of logicism now accrue. By showing that mathematical statements are analytic, it follows that by knowing the meanings of (the terms in) these statements, we are equipped with all the tools we need to determine whether they are true. We can know mathematical theorems by deriving them solely from logically true statements and analytic statements. Thus, no special cognitive faculty for knowledge of mathematical truths is needed other than the faculty of understanding, which is a faculty we, like Benacerraf [Reference Benacerraf and French3], take to be explainable in naturalistic terms.

So we don’t have to posit a causal information pathway, like the causal theory of reference, to explain how we come to understand the terms of mathematical statements. Our comprehension principles already constitute the paths by which we apprehend abstract objects: they just are the means by which we cognitively grasp the objects denoted by the terms of mathematical theory T.

Note that the claim that mathematical truths are logical or analytic truths does not entail that for each mathematical claim it would be easy to determine that it is a mathematical truth. Moreover, what we have not done in the present paper is to say anything about the a priori justification of the logic underlying object theory. To be clear, though, the epistemological situation is very different from that surrounding the foundational system of Principia Mathematica. Whitehead and Russell couldn’t very well argue that the axioms of reducibility and infinity are logical, even if they had tried to use the grounds we provided above: it is hard to see how such axioms are required for the possibility of (abstract) thought. Our logical framework, by contrast, requires no such axioms and the axioms it does assert are required for such thought. But in the present case, even if one accepts that our axioms are logical, there is still the question of whether they are justified. We have not addressed that latter question. We could argue that a logic such as ours is justified because it is presupposed somehow, or because the logic, through a process of reflective equilibrium, offers a rational reconstruction of the data (i.e., logical consequences we accept pretheoretically) that is better than other logical systems. But we have to leave this argument for another paper.

A.1 A bounded language

Our analysis of mathematics does not require the full unbounded language defined in the paper in Section 4. So, as we develop a model of typed object theory, we restrict our attention to the fragment we need. We shall therefore define the bounded language $\mathcal {L}_{n,m}$ , where n and m are bounds that set, respectively, the width and height of the types for the terms of the language. We begin by defining the functions h and w for the height and width, respectively, of a given type.

The width of type t, written $w(t)$ , is defined as:

• • $w(i) = 1,$

• • $w(\langle \: \rangle ) = 1,$

• • $w(\langle t_1,\ldots ,t_k \rangle ) = \sum _1^k w(t_k).$

The height of type t, written $h(t)$ , is defined as:

• • $h(i) =0,$

• • $h(\langle \: \rangle ) =1,$

• • $h(\langle t_1,\ldots ,t_k \rangle ) = 1 + \max \{h(t_1), \ldots , h(t_k)\}.$

Then we define $\mathcal {L}_{n,m}$ as the language that includes any well-formed expression of $\mathcal {L}$ that can be formulated only with terms $\tau $ of type t such that $w(t) \leq n$ and $h(t) \leq m$ .

Before we define a model for the bounded language $\mathcal {L}_{n,m}$ , a few observations are in order. Intuitively, we want to choose bounds that will yield the smallest language and model needed for our analysis. The following two considerations play a role in setting the bounds on $\mathcal {L}_{n,m}$ :

• • In Section 6.1, we analyze the property of being a ZF-set as an abstract property having type $\langle i\rangle $ . Abstract properties of this type encode properties of type $\langle \langle i\rangle \rangle $ . And properties of this latter type can be both ordinary and abstract, though the mathematical properties will encode only abstract properties of type $\langle \langle i\rangle \rangle $ . But these abstract properties must, in turn, encode properties of type $\langle \langle \langle i \rangle \rangle \rangle $ . Note this requires the bound on the height of the types in the language to be at least 3 and the bound on the width to be at least 1.

• • In Section 6.1, we analyze the membership relation of ZF as an abstract relation having type $\langle i,i\rangle $ . Abstract relations of this type encode properties of type $\langle \langle i,i\rangle \rangle $ . And properties of this latter type can be both ordinary and abstract, though the membership relation will encode only abstract properties of type $\langle \langle i,i\rangle \rangle $ . But these abstract properties must, in turn, encode properties of type $\langle \langle \langle i,i \rangle \rangle \rangle $ . Note this requires the bound on the height of the types in the language to be at least 3 and the bound on the width to be at least 2.

Given these facts, it should be clear that the minimal fragment our analysis of mathematics requires is the bounded language $\mathcal {L}_{2,3}$ . In line with what we said above, $\mathcal {L}_{2,3}$ includes any well-formed expression of $\mathcal {L}$ that can be formulated only with terms $\tau $ of type t such that $w(t) \leq 2$ and $h(t) \leq 3$ . So in specifying a general model for $\mathcal {L}_{n,m}$ , we shall occasionally focus on the model for the language $\mathcal {L}_{2,3}$ .

A.2 The smallest, extensional model for $\mathcal {L}_{n,m}$

We construct our model in two basic stages: first we construct the structural domains of the model, and second, we specify the domains of quantification and a proxy function (that assigns to each element in a domain of quantification to an element of a structural domain). The construction of the structural domain occurs in two stages: (1) the kernel of each type and (2) the abstract objects of each type.

A.2.2 Structural domains: Abstract objects

(2) Given m as the maximum height, we recursively define the domain $\mathbf {A}_t$ as follows:

$$\begin{align*}\mathbf{A}_t = \begin{cases} \wp(\mathbf{O}_{\langle t \rangle} \cup \mathbf{A}_{\langle t \rangle}), & \text{if } h(t) < m, \\ \emptyset, & \text{otherwise.} \\ \end{cases} \end{align*}$$

The identity conditions for elements of $\mathbf {A}_t$ depend on the higher-type elements of $\mathbf {O}_{\langle t \rangle }$ . Given these stipulations, we have the following consequences for $\mathbf {A}_{\langle i\rangle }$ and $\mathbf {A}_i$ when $m\!=\!1$ :

• $ \mathbf{A}_{\langle i\rangle} = \emptyset, $

• $ \mathbf{A}_i = \wp(\mathbf{O}_{\langle i\rangle} \cup \mathbf{A}_{\langle i\rangle}) $

• $ \kern15pt = \wp(\mathbf{O}_{\langle i\rangle} \cup \emptyset)) $

• 

• 

When $m=2$ :

• $ \mathbf{A}_{\langle\langle i\rangle\rangle} = \emptyset, $

• $ \mathbf{A}_{\langle i\rangle} = \wp(\mathbf{O}_{\langle\langle i\rangle\rangle} \cup \mathbf{A}_{\langle\langle i\rangle\rangle}) $

• $ \kern22pt = \wp( \wp(\mathbf{K}_{\langle i\rangle}) \cup \emptyset ) $

• $ \kern22pt = \wp( \{\ldots \text{all } 8 \text{ subsets of } \mathbf{K}_{\langle i\rangle}\ldots\} ) $

• $ \kern22pt = \{\ldots \text{all } 256 \text{ subsets of } \wp(\mathbf{K}_{\langle i\rangle}) \ldots\}, $

• $ \mathbf{A}_i = \wp(\mathbf{O}_{\langle i\rangle} \cup \mathbf{A}_{\langle i\rangle}) $

• $ \kern15pt = \wp(\mathbf{O}_{\langle i\rangle} \cup \{\ldots \text{all } 256 \text{ elements of } \mathbf{A}_{\langle i\rangle} \ldots\}) $

• 

• 

• $ \kern15pt = \wp ( \{ \ldots \text{all } 257 \text{ elements of } \mathbf{O}_{\langle i\rangle} \cup \mathbf{A}_{\langle i\rangle}\ldots \}) $

• $ \kern15pt = \{\ldots \text{all } 2^{257} \text{ subsets of } \mathbf{O}_{\langle i\rangle} \cup \mathbf{A}_{\langle i\rangle}\ldots \}. $

But as noted earlier, for the purposes of building a specific model for our analysis of mathematics, we will set the bound of m to 3.

A.2.3 Domains of quantification and the proxy function

The domains over which the variables of our language range are now defined simply as follows. Where $\mathbf {D}_t$ is the domain of quantification for type t:

$$\begin{align*}\mathbf{D}_t = \mathbf{A}_t \cup \mathbf{O}_t. \end{align*}$$

We next define an extended proxy function $\|\cdot \|$ in two steps. In the first step, we define a proxy function $|\cdot |$ so that it maps abstract individuals, abstract properties, and abstract relations to special individuals, special properties, and special relations, respectively. In the second step, we extend this function to the extended proxy function $\|\cdot \|$ which is defined on all the domains of quantification: it preserves what $|\cdot |$ assigns to the abstract entities but also assigns each ordinary element in each domain of quantification to itself as proxy.

The function $|\cdot |$ is defined generally as, for each type t:

$$\begin{align*}|\cdot | : \mathbf{A}_t \rightarrow \mathbf{S}_t. \end{align*}$$

In the minimal model, there is only one special object in each domain $\mathbf {S}_t$ , and so all the abstract objects in $\mathbf {A}_t$ get mapped to the same proxy.Footnote ⁵⁵ For example, in the minimal model:

• • where $\mathbf {a}_i$ is an abstract individual in $\mathbf {A}_i$ (i.e., where $\mathbf {a}_i$ is a set of first-level properties), then ,

• • where $\mathbf {a}_{\langle i\rangle }$ is a first-level abstract property in $\mathbf {A}_{\langle i\rangle }$ (i.e., where $\mathbf {a}_{\langle i\rangle }$ is a set of second-level properties of properties), then let , and

• • where $\mathbf {a}_{\langle i,i\rangle }$ is a first-level abstract relation in $\mathbf {A}_{\langle i,i\rangle }$ (i.e., where $\mathbf {a}_{\langle i,i\rangle }$ is a set of second-level properties of relations), then let .

We then extend $|\cdot |$ to the extended proxy function $\|\cdot \|$ as follows. Where t is any type and $\mathbf {D}_t$ is the domain of type t (as defined below), and is a variable ranging over the entities of domain $\mathbf {D}_t$ , then for all :

A.3 The model

We now introduce a bounded domain, $\mathbf {D}^{n,m}$ , as follows:

• $\mathbf {D}^{n,m} =_{\mathit {df}} \{ \mathbf {D}_t \: |\: w(t)\leq n \:\&\: h(t)\leq m \}.$

In other words, a bounded domain collects all the domains of the types t within the width and height bounds n and m.

In order to preserve the information about whether objects in the domain are ordinary or abstract, we define two indicator functions and defined as follows. Where is again a variable ranging over the entities of domain $\mathbf {D}_t$ , then for all :

We next define an interpretation V that assigns to each constant $\kappa $ of the language an element of an appropriate domain of quantification, i.e.,

• If $\kappa $ is a constant of type t, $\mathbf {V}(\kappa )\in \mathbf {D}_t.$

$\mathbf {V}$ also assigns a special entity to each predicate $A!$ of the language:

• $\mathbf {V}(A!^{\langle t\rangle }) = \mathbf {S}_t.$

That is, the interpretation of the predicate constant $A!^{\langle t\rangle }$ is the set $\mathbf {S}_t$ of proxy elements of type t. Using the definitions above, we then define a model $\mathbf {M}$ as a structure of the form:

where $\mathbf {D}^{n,m}$ is a bounded domain, $\|\cdot \|$ is an extended proxy function, and are the indicator functions that identify which elements of the bounded domain are abstract and ordinary, respectively, and $\mathbf {V}$ is an interpretation function. These elements have all been defined as above.

As noted earlier, for analysis of mathematics developed in this paper, we need models with bounded domain $\mathbf {D}^{2,3}$ . We leave the complete list of types included within this bound to a footnote.Footnote ⁵⁶ Note that many of these types don’t play a role in our analysis of mathematics and aren’t represented in Figure A.1.

Figure A.1

A fragment of the minimal model with unrestricted typed comprehension for abstracta. The domains of ordinary objects, from the bottom up, are: the kernel of propositions $\mathbf {K}_{\langle \:\rangle }$ (= $\mathbf {O}_{\langle \,\rangle } \cup \mathbf {S}_{\langle \,\rangle }$ ); the kernel of individuals $\mathbf {K}_i$ (= $\mathbf {O}_i \cup \mathbf {S}_i$ ); the kernel of properties of individuals $\mathbf {K}_{\langle i\rangle }$ (= $\mathbf {O}_{\langle i\rangle } \cup \mathbf {S}_{\langle i\rangle }$ ); the kernel of binary relations among individuals $\mathbf {K}_{\langle i,i\rangle }$ (= $\mathbf {O}_{\langle i,i\rangle } \cup \mathbf {S}_{\langle i,i\rangle }$ ); the kernel of properties of properties of individuals $\mathbf {K}_{\langle \langle i\rangle \rangle }$ ; the kernel of properties of relations among individuals $\mathbf {K}_{\langle \langle i,i\rangle \rangle }$ ; and so on. The domains of abstract objects, from the top down, are: the abstract individuals $\mathbf {A}_i$ (= the power set of $\mathbf {O}_{\langle i\rangle }\cup \mathbf {A}_{\langle i\rangle }$ ); the abstract properties of individuals $\mathbf {A}_{\langle i\rangle }$ (= the power set of $\mathbf {O}_{\langle \langle i\rangle \rangle }\cup \mathbf {A}_{\langle \langle i\rangle \rangle }$ ); and the abstract relations among individuals $\mathbf {A}_{\langle i,i\rangle }$ (= the power set of $\mathbf {O}_{\langle \langle i,i\rangle \rangle }\cup \mathbf {A}_{\langle \langle i,i\rangle \rangle }$ ); and so on.

A.4 Simultaneous definition of denotation and truth

A.5 Axioms

Since we’ve assumed classical logic in the description of our model, it remains to show that the axiom groups of Sections 5.1 and 5.2 are true in the above model. It is easy to see that the following lemma holds:

• Substitution Lemma. If is defined and $\phi $ is any formula, then satisfies $\phi ^{\tau }_{\alpha }$ if and only if satisfies $\phi $ .

In other words, satisfies $\phi ^{\tau }_{\alpha }$ whenever the assignment just like except that it assigns to $\alpha $ satisfies $\phi $ (assuming is defined). This lemma holds because, according to our semantics, the truth value of an atomic formula is calculated in terms of the denotations of its terms, assuming they all have such, and this feature is inherited by all the molecular and quantified formulas built out of such formulas.

In what follows, we omit the proofs that the axioms for classical logic hold in our model. We also omit proofs for the [Reference Hintikka29] axiom for descriptions (AXIOM 1) and AXIOMS 4–7. These are obviously true in the model (since the model was constructed in part to make these axioms true).Footnote ⁵⁸ It remains to show only that the three distinctive principles, the AXIOM for the substitution of identicals, and AXIOMS 2 and 3, are true in the model.

A.6 Axiom: Substitution of identicals

The proof that substitution of identicals is true in the model is by cases. The cases are:

• $x=y,$

  $F=G,$

  $R=S,$

  $p=q.$

Consider the first case:

• Assume some assignment, say , satisfies $x=y$ . We want to show satisfies $\phi $ if and only if satisfies $\phi '$ , where $\phi '$ is the result of replacing 0 or more free occurrences of x by y in $\phi $ . Now our assumption implies, by definition of $x=y$ , that satisfies

  • $(O{!}x \:\&\: O{!}y\:\&\: \forall F(Fx \equiv Fy)) \lor (A{!}x \:\&\: A{!}y\:\&\: \forall F(xF \equiv yF)).$

  But since there are no ordinary individuals in the model (the domain $\mathbf {O}_i$ is empty), it follows that satisfies

  • $A{!}x \:\&\: A{!}y\:\&\: \forall F(xF \equiv yF).$

  So by S4, satisfies $\forall F(xF \equiv yF)$ . Now by S3, we know

  • satisfies ‘ $xF$ ’ iff (a) is defined, (b) , and (c) .

  And we know something analogous for satisfies $yF$ . Now suppose for reductio that . Since both objects are in $\mathbf {A}_i$ , they are both sets of type $\langle i\rangle $ properties, so there must be a property that is an element of one, say, , and not the other, that is, . Let $\kappa $ be such a property, so that we know and not . Now consider the variable assignment . Since , it follows by S3 that satisfies $xF$ . And since , it follows by S3 that doesn’t satisfy $yF$ . So by the biconditional clause of S4, doesn’t satisfy $xF \equiv yF$ . Then, by the universal quantifier clause of S4, doesn’t satisfy $\forall F(xF \equiv yF)$ , This contradicts the assumed fact that does satisfy $\forall F(xF \equiv yF)$ . Hence, . So by reasoning from the Substitution Lemma, we can argue as follows: satisfies $\phi $ if and only if satisfies $\phi $ iff satisfies $\phi $ iff satisfies $\phi '$ .

Given the definitions for identity in Section 4, the proof of the remaining cases, i.e., $F=G$ , $R=S$ , and $p=q$ , are similar to the above.Footnote ⁵⁹

A.7 AXIOM 2: $\lambda $ -Conversion

To see AXIOM 2 holds, note that the axiom is trivially true in the case where $[\lambda \alpha _1\ldots \alpha _n \: \phi ]$ is an elementary $\lambda $ -expression. For in that case, the $\lambda $ -expression has the form

• $[\lambda \alpha _1\ldots \alpha _n \: \Pi \alpha _1\ldots \alpha _n],$

where $\alpha _1,\ldots ,\alpha _n$ have types $\langle t_1,\ldots ,t_n\rangle $ , respectively, and $\Pi $ is any simple n-place relation term (i.e., constant or variable) of type $\langle t_1,\ldots ,t_n\rangle $ . So, by the consequence noted at the end of D3, the following holds:

• $[\lambda \alpha _1\ldots \alpha _n \: \Pi \alpha _1\ldots \alpha _n]\beta _1\ldots \beta _n \equiv \Pi \beta _1\ldots \beta _n.$

For there are two cases: if $\Pi $ denotes an ordinary relation, then the denotation of the $\lambda $ -expression is just the denotation of $\Pi $ itself, and if $\Pi $ denotes an abstract relation, then $\Pi $ denotes the empty set and both sides of the biconditional $[\lambda \alpha _1\ldots \alpha _n \: \Pi \alpha _1\ldots \alpha _n]\beta _1\ldots \beta _n \equiv \Pi \beta _1\ldots \beta _n$ are false (given that the contrapositive of AXIOM 5 tells us that abstract relations aren’t exemplified).

When $[\lambda \alpha _1\ldots \alpha _n\: \phi ]$ is non-elementary, then $\phi $ is any propositional formula that is free of descriptions (by the clause for Complex terms in the definition of our language). Our semantic definition of truth requires us to show that every assignment function satisfies

• $[\lambda \alpha _1\ldots \alpha _n \: \phi ]\beta _1\ldots \beta _n \equiv \phi ^{\beta _1\ldots \beta _n}_{\alpha _1,\ldots ,\alpha _n}$ , provided $\beta _i$ is substitutable for $\alpha _i$ in $\phi $ ( $1\leq i\leq n$ ).

For simplicity and ease of readability, we prove this only for the 1-place case, so that where $\alpha , \beta $ are variables of some type t, our task is to show that every assignment function satisfies

• $[\lambda \alpha \: \phi ]\beta \equiv \phi ^{\beta }_{\alpha }$ , provided $\beta $ is substitutable for $\alpha $ in $\phi .$

Moreover, if we use $x, y, z, \ldots $ as arbitrarily chosen variables of type t, we simply have to show

• $[\lambda x\: \phi ]y \equiv \phi ^y_x$ , provided y is substitutable for x in $\phi .$

So suppose is an arbitrary assignment function. Then, by the clauses in S4, we have to show satisfies $[\lambda x\: \phi ]y$ if and only if satisfies $\phi ^y_x$ .

Note that by clause S1, satisfies $[\lambda x\: \phi ]y$ if and only if:

1. (a) is defined.

2. (b) , i.e., , by definition of

3. (c)

But (a), (b), and (c) hold if and only if

• ,

  where this latter set is an element of

by clause D3.Footnote ⁶⁰ So by set-abstraction, the fact that $\phi $ is propositional and description-free, and the Lemma on Proxies (see below), the above holds if and only if

i.e., by the Substitution Lemma, if and only if

For this conclusion to hold, it remains only to show:

• Lemma on Proxies: Let x be a variable of type t. Then if $\phi $ is a description-free propositional formula and and and have the same proxy, then:

  1. L1 for any term $\tau $ in $\phi $ , , and

  2. L2 satisfies $\phi $ iff satisfies $\phi .$

Proof. The proof proceeds by induction on the $\lambda $ -rank of propositional formulas $\phi $ , i.e., how deeply nested is the deepest $\lambda $ expressions in $\phi $ . But in what follows, the notion of $\lambda $ -rank applies to any formula or term.Footnote ⁶¹ (We ignore the trivial case where x doesn’t occur free in $\phi $ .) For L2, without loss of generality, we need only prove the left to right direction, i.e., that if satisfies $\phi $ , then satisfies $\phi $ . So assume satisfies $\phi $ .

The base case is where $\phi $ is a formula of $\lambda $ -rank 0, i.e., without $\lambda $ -expressions. We first consider atomic formulas $\phi $ of the form $\Pi ^n\tau _1\ldots \tau _n$ ( $n\geq 0$ ) where none of $\Pi ^n$ , $\tau _1,\ldots ,\tau _n$ are $\lambda $ -expressions. Note that to prove L1, we must show that (1) and (2) for each $\tau _i$ , . We will prove (1) and (2) in the course of proving L2. There are two subcases for L2: $n\geq 1$ or $n=0$ . When $n\geq 1$ , then each of $\Pi ^n$ , $\tau _1,\ldots ,\tau _n$ is either a constant or a variable. And when $n=0$ , then $\phi $ has the form $\Pi $ where $\Pi $ is a constant or a variable of the empty type. We cover these two subcases in turn.

In the first subcase ( $n\geq 1$ ), $\phi $ is governed by S1. So $\phi $ has the form $\Pi ^n \tau _1\ldots \tau _n$ and contains no $\lambda $ -expressions. So we knowFootnote ⁶² (i) x is either $\Pi ^n$ or one of the $\tau _i$ , (ii) $\tau _1, \ldots ,\tau _n$ have any types and (iii) $\Pi $ is of an appropriate type to relate $\tau _1, \ldots ,\tau _n$ . Then, since satisfies $\phi $ , it follows that:

1. (a) are all defined.

2. (b)

3. (c) .

We’re trying to show satisfies $\phi $ , i.e., all of (d)–(f) have to hold:

1. (d) are all defined.

2. (e) .

3. (f) .

In the second subcase ( $n=0$ ), $\phi $ has the form $\Pi $ where $\Pi $ is of the empty type. So $\Pi $ can only be a constant or a variable. If $\phi $ is a constant or variable of type $\langle \:\rangle $ , S2 applies. Then either $\Pi $ is either a constant, or a variable other than x, or x itself. If $\Pi $ is a constant or a variable other than x, then satisfies $\Pi $ iff (by S2) iff (since $\Pi $ doesn’t contain a free occurrence of x) iff satisfies $\Pi $ (by S2). If $\Pi $ is x, then satisfies $\phi $ iff (by S2) iff iff (see below) iff iff satisfies $\phi $ . To see that iff , recall that (by hypothesis) and since $\mathbf {T} \not \in \mathbf {S}_{\langle \:\rangle }$ , the only object with $\mathbf {T}$ at its proxy is $\mathbf {T}$ itself. So iff . Note that this proves part (1) of L1 where $\phi $ falls under the second subcase. There is no part (2) of L1 for this subcase.

We now have that L1 and L2 hold for atomic formulas $\phi $ of $\lambda $ -rank 0. We conclude this base case by noting that L1 and L2 hold for complex formulas $\phi $ of the form $\neg \psi $ , $\psi \to \chi $ , and $\forall \alpha \psi $ . By S4 and D5, the truth of L1 and L2 for these complex formulas is grounded in the truth of L1 and L2 for the atomic formulas with no $\lambda $ -expressions given that complex formulas of rank 0 contain no $\lambda $ -expressions.

Inductive cases: IH: The lemma holds for $\psi $ with $\lambda $ -rank of n or less, i.e., for $\psi $ with $\lambda $ -rank of n or less we may assume:

• IH-L1: , for any term $\tau $ in $\psi $ , and

• IH-L2: satisfies $\psi $ if and only if satisfies $\psi $ .

We need to show that it holds for $\phi $ with $\lambda $ -rank of $n+1$ .

To show L1, we need to show that , for any term $\tau $ in $\phi $ . We first consider the case where $\phi $ is atomic. Fix an arbitrary such $\tau $ and consider its denotation:

• • D1 and D2 only apply if $\tau $ has a $\lambda $ -rank of 0, so the result follows immediately from IH-L1.

• • D3 applies when $\tau $ is of the form $[\lambda \alpha _1\ldots \alpha _n \:\psi ]$ . Now if $\tau $ has $\lambda $ -rank n or less, then the result follows by IH-L1. So we need only be concerned with the case when $\tau $ has $\lambda $ -rank $n+1$ where $\psi $ has $\lambda $ -rank n. We have to show . Note that D3 defines the denotation of the $\lambda $ -expression as the set of n-tuples (of proxies of objects) such that satisfies $\psi $ . But $\psi $ has $\lambda $ -rank n (one less than $\tau $ ), so by IH-L2, satisfies $\psi $ if and only if satisfies $\psi $ , so the set of proxy-tuples must be the same and L1 holds.

• • D4 would apply if $\tau $ were a description. But $\phi $ is description-free.

We now have the L1 holds for atomic formulas $\phi $ of $\lambda $ -rank up to $n+1$ . We can then conclude that L1 holds for complex formulas $\phi $ of the form $\neg \psi $ , $\psi \to \chi $ , and $\forall \alpha \psi $ . By D5, the truth of L1 for these complex formulas is grounded in the truth of L1 for the atomic formulas with $\lambda $ -rank up to $n+1$ given that a complex formula can not have a $\lambda $ -rank higher than its component atomic formulas.

To show L2, we have to show: satisfies $\phi $ if and only if satisfies $\phi $ , when $\phi $ has an $\lambda $ -rank of $n+1$ . We now make use of L1, which has been proved for all $\lambda $ -ranks. So we know that for any term $\tau $ in $\phi $ , . We first consider the case where $\phi $ is atomic. Moreover, we need only consider formulas governed by S1, for (a) if the $\lambda $ -rank of $\phi $ is some n other than 0, then $\phi $ is not a constant or variable, and so S2 doesn’t apply, (b) S3 doesn’t apply since that governs encoding formulas, which aren’t propositional. So we are consider $\phi $ of the form $\Pi \tau _1\ldots \tau _n$ where any of $\Pi $ , $\tau _1$ ,…, $\tau _n$ can have a $\lambda $ -rank of $n+1$ . Without loss of generality, we need only prove the left to right direction, i.e., that if satisfies $\phi $ , then satisfies $\phi $ . So assume satisfies $\phi $ . By assumption, it follows that:

1. (a) are all defined.

2. (b) .

3. (c) .

We’re trying to show satisfies $\phi $ , i.e., all of (d)–(f) have to hold:

1. (d) are all defined.

2. (e) .

3. (f) .

Proof of (f)

By assumption (b), we know that the denotation of $\Pi $ is ordinary. And since the denotation of $\Pi $ is ordinary, the proxy of the denotation is just equal to the denotation itself. So:

Moreover, we also know, by L1, that the proxies of the denotations for each $\tau $ are identical, that is for each $\tau _i$ . So the tuple in (f) is identical to the tuple in (c). Hence it follows that (f) must hold if (c) holds.

We now have that L2 holds for atomic formulas $\phi $ of $\lambda $ -rank $n+1$ . We conclude this inductive case by noting that L2 holds for complex formulas $\phi $ of the form $\neg \psi $ , $\psi \to \chi $ , and $\forall \alpha \psi $ . By S4, truth of L2 for these complex formulas is grounded in the truth of L2 for the atomic formulas with $\lambda $ -rank less than or equal to $n+1$ , given that complex formulas only have rank $n+1$ in virtue of the ranks of their atomic components. That is, S4 doesn’t increase the $\lambda $ -rank.

A.8 AXIOM 3: Comprehension principles for abstracta

At the beginning of the Appendix, we noted that our analysis of mathematics does not require the full unbounded language. In general we need only validate comprehension up to a given fixed type height h. So we fix n and m such that the bounded language $\mathcal {L}_{n,m}$ has $m> h$ . Now we will show AXIOM 3 holds in the model for any arbitrary t such that $h(t) < m$ . Recall that by definition (2) of $\mathbf {A}_t$ (in the Appendix subsection ‘Structural domains: Abstract objects’), when $h(t) < m$ , we have that $\mathbf {A}_t = \wp (\mathbf {O}_{\langle t \rangle } \cup \mathbf {A}_{\langle t \rangle })$ . We now show that $\exists \alpha (A!\alpha \:\&\: \forall F(\alpha F \equiv \phi ))$ holds by showing that $\{ F \, |\, \phi \}$ defines an abstract object of type t.