3.1. Operator principles

In the case of sentential operators, formalization is relatively straightforward. For example, in the case of conjunction, to say that the conjuncts can be recovered from any conjunction is to say that two conjunctive formulas $\varphi \wedge \varphi '$ and $\psi \wedge \psi '$ express the same proposition only if $\varphi $ and $\psi $ express the same proposition, and likewise for $\varphi '$ and $\psi '$ . In general, such a principle can be stated for any n-ary sentential operator $\circ $ :

• $(\mathrm {O}_\circ )$ $\circ (p_1,\dots,p_n)=\circ (q_1,\dots,q_n)\to p_1=q_1\wedge \dots \wedge p_n=q_n$ .

Such principles are implicit in many discussions of structured propositions. Explicit formulations can be found in formal developments of such views, such as [Reference Bealer1, p. 65, Axioms 10 and 11] and [Reference Hugly and Sayward10, p. 183, Principle 7].

The schematic principle $\mathrm {O}_\circ $ cannot be instantiated for $\prec $ , since this is not a sentential operator. But a straightforward analog can be stated as follows:

• $(\mathrm {O}_\prec )$ $(p\prec \textit{pp})=(q\prec qq)\to p=q\wedge \textit{pp}=qq$ .

The only remaining logical operators of L are quantifiers. But before considering the question how to formulate principles of logical structure for quantifiers, it is worth noting straight away that $\mathrm {O}_\prec $ is inconsistent. This is an immediate corollary of the following (variant of a) result by Russell [Reference Russell12, Appendix B] and Myhill [Reference Myhill1]:

Proof. By plural comprehension, there are $qq$ such that:

$$ \begin{align*}\forall q(q\prec qq\leftrightarrow\exists \textit{pp}(q\nprec \textit{pp}\wedge\varphi(\textit{pp})= q)).\end{align*} $$

Assume for contradiction that $\varphi (qq)\nprec qq$ . Then for all $\textit{pp}$ , $\varphi (\textit{pp})=\varphi (qq)$ only if $\varphi (qq)\prec \textit{pp}$ . Thus $\varphi (qq)=\varphi (qq)$ only if $\varphi (qq)\prec qq$ , whence $\varphi (qq)\prec qq$ . This contradicts the assumption to the contrary, so $\varphi (qq)\prec qq$ . So there are $\textit{pp}$ such that $\varphi (qq)\nprec \textit{pp}$ and $\varphi (\textit{pp})=\varphi (qq)$ . Since $\varphi (qq)\prec qq$ , it follows that $\textit{pp}\neq qq$ .

In the proof of Theorem 4, it is not obviously ruled out that there is no q such that $\exists \textit{pp}(q\nprec \textit{pp}\wedge \varphi (\textit{pp})=q)$ . To carry out the deduction in $\vdash '$ , an auxiliary argument has to be added for this special case: If there is no q such that $\exists \textit{pp}(q\nprec \textit{pp}\wedge \varphi (\textit{pp})=q)$ , then $\forall \textit{pp}(\varphi (\textit{pp})\prec \textit{pp})$ . Let $\textit{pp}$ be all the propositions, and $qq$ the propositions identical to $\varphi (\textit{pp})$ ; the existence of both follows from . Then $\varphi (qq)\prec qq$ , whence $\varphi (\textit{pp})=\varphi (qq)$ . But since there are at least two propositions (one true and one false), $\textit{pp}\neq qq$ .

3.2. Some inconsistent quantifier principles

Returning to the question of how to formulate logical structure principles for quantifiers, the most straightforward attempt treats a quantifier binding a particular variable as a unary sentential operator like negation. For example, the instances of such a principle for universal propositional quantifiers have the following form:

• $\mathrm {(A)}$ $\forall p\varphi =\forall p\psi \to \varphi =\psi $ .

A version of this principle is discussed by Church [Reference Church4, p. 514], who formalizes Russell’s theory of structured propositions in a simple type theory. (See axiom 11; axioms 8–10 are noteworthy as well in corresponding to $\mathrm {O}_\circ $ .) According to Church’s formulation, quantified propositions are identical only if corresponding instances are identical:

• $\mathrm {(C)}$ $\forall p\varphi =\forall p\psi \to \forall p(\varphi =\psi )$ .

These two principles are easily seen to be equivalent, in the sense that $\mathrm {A}\vdash \mathrm {C}$ and $\mathrm {C}\vdash \mathrm {A}$ , via $\mathrm {UG}$ and $\mathrm {UI}$ .

However, as Church [Reference Church4, p. 520] notes, these principles are also shown to be inconsistent by the Russell–Myhill theorem. They entail:

• $\forall p(p\prec \textit{pp})=\forall p(p\prec qq)\to \forall p((p\prec \textit{pp})=(p\prec qq)).$

Identical propositions are materially equivalent, so from this one obtains:

• $\forall p(p\prec \textit{pp})=\forall p(p\prec qq)\to \textit{pp}=qq$ .

And this is inconsistent by Theorem 4.

As is easily seen, this observation applies as well to existential quantifiers. A similar argument can also be given for plural propositional quantifiers. Consider the following instance of the analog of $\mathrm {C}$ for plural propositional quantifiers:

• $\forall rr(rr=\textit{pp})=\forall rr(rr=qq)\to \forall rr((rr=\textit{pp})=(rr=qq))$

Instantiating the quantification of the consequent using $\textit{pp}$ , one obtains with the material equivalence of identicals and $\textit{pp}=\textit{pp}$ :

• $\forall rr(rr=\textit{pp})=\forall rr(rr=qq)\to (\textit{pp}=qq)$ .

And this is again inconsistent by Theorem 4.

The problem with principles $\mathrm {A}$ and $\mathrm {C}$ may appear to be the treatment of quantifiers as variable-binding operators. But the problem persists if one introduces a variable binding $\lambda $ -operator, and treats quantifiers as higher-order predicates, as suggested, e.g., by Stalnaker [Reference Stalnaker13]. For, consider an extension of L in which for every propositional variable p and formula $\varphi $ , there is a unary sentential operator $\lambda p.\varphi $ . Then $\forall p\varphi $ can be treated as an abbreviation for $\forall \lambda p.\varphi $ , which suggests a principle of logical structure for the quantifiers with the following instances:

• $(\forall \lambda p.\varphi )=(\forall \lambda p.\psi )\to (\lambda p.\varphi )=(\lambda p.\psi )$ .

But this is again inconsistent, for much the same reasons as before. Consider the following instance:

• $(\forall \lambda p.p\prec \textit{pp})=(\forall \lambda p.p\prec qq)\to (\lambda p.p\prec \textit{pp})=(\lambda p.p\prec qq)$ .

By $\mathrm {UG}$ , the consequent can be universally quantified, the result of which entails $\forall p((\lambda p.p\prec \textit{pp})p\leftrightarrow (\lambda p.p\prec qq)p)$ . With the elementary principle of extensional (or material) $\beta $ -conversion, according to which $(\lambda p.\varphi )\psi $ is materially equivalent to $\varphi [\psi /p]$ , this in turn entails $\forall p(p\prec \textit{pp}\leftrightarrow p\prec qq)$ . Thus, it follows:

• $(\forall \lambda p.p\prec \textit{pp})=(\forall \lambda p.p\prec qq)\to \textit{pp}=qq$ .

And this was noted to be inconsistent by Theorem 4.

3.3. Dispensing with order

Church’s formulation $\mathrm {C}$ of the logical structure principle for universal quantifiers is interesting since it corresponds in a natural way to $\mathrm {O}_\wedge $ . Consider the view on which universal and existential quantifiers serve to express long conjunctions and disjunctions, respectively. Roughly, this view holds that a universal quantification $\forall p\varphi (p)$ serves to express a conjunction of the form:

• $\bigwedge \varphi (p_0)\varphi (p_1)\cdots $ ,

where the propositions expressed by $p_0,p_1,\dots $ constitute a well-order of the propositions, and $\bigwedge $ is an infinitary sentential operator which takes as arguments a sequence of formulas of the relevant order type. (Set aside worries about there being more propositions than propositional variables. The relevant extension of L is only appealed to loosely to motivate principles which will themselves be stated in L.)

Assume now that $\forall p\varphi (p)=\forall p\psi (p)$ . On the view under consideration, this can equivalently be stated as:

• $\bigwedge \varphi (p_0)\varphi (p_1)=\bigwedge \psi (p_0)\psi (p_1)\cdots $

The relevant generalization of $\mathrm {O}_\wedge $ should allow us to obtain from this the following sequence of claims:

• $\varphi (p_0)=\psi (p_0)$ , $\varphi (p_1)=\psi (p_1)$ , $\dots $

But since $p_0,p_1,\dots $ are assumed to comprise all propositions, this is equivalent to the claim that $\forall p(\varphi (p)=\psi (p))$ , as Church’s principle states.

This correspondence between the two principles is interesting, since it suggests a natural weakening of Church’s principle $\mathrm {C}$ . Notice that $\mathrm {O}_\wedge $ is immediately inconsistent with the commutativity of conjunction, i.e., the following principle:

• $(p\wedge q)=(q\wedge p)$ .

The inconsistency follows from the fact that commutativity entails $(p\wedge \neg p)=(\neg p\wedge p)$ , which, with $\mathrm {O}_\wedge $ , leads to the absurd $p=\neg p$ .

But even someone who is attracted to propositions having some amount of logical structure might want to endorse the commutativity of conjunction (and analogously disjunction). As Dorr [Reference Dorr5, p. 124, en. 56] notes, one might adapt an idea of Williamson [Reference Williamson16, p. 259] and conceive of the formation of conjunctions as amounting to putting conjuncts into a conjunctive bag, from which conjuncts can be recovered, but not necessarily in order. This motivates the following weaker logical structure principle for binary sentential operators:

• $(\mathrm {O}^-_\circ )$ $(p\circ p')=( q\circ q')\to (p=q\wedge p'=q')\vee (p=q'\wedge p'=q)$ .

Returning to the case of quantifiers (illustrated using universal propositional quantifiers), one might correspondingly conceive of $\forall p\varphi (p)$ as serving to express a conjunction of the following form:

• $\bigwedge \{\varphi (p_0),\varphi (p_1),\dots \}$ ,

where $\bigwedge $ is now understood as a unary operator taking a set of formulas as an argument. The idea that conjuncts can be recovered from conjunctions, but not necessarily in order, thus suggests that $\forall p\varphi (p)$ is only identical to $\forall p\psi (p)$ if for every index i, $\varphi (p_i)$ is $\psi (p_j)$ , for some index j, and vice versa. In L, this can be stated more succinctly as follows:

• $\forall p\varphi (p)=\forall p\psi (p)\to \forall p\exists q(\varphi (p)=\psi (q))\wedge \forall p\exists q(\psi (p)=\varphi (q))$ .

Here, q is the first variable free for p in $\psi $ distinct from p. Note that since identity is symmetric, only one of the conjuncts in the consequent need be included in the formulation of this schematic principle. Thus, the following slightly simpler principle suffices:

• $(\mathrm {I}_{\forall p})$ $\forall p\varphi (p)=\forall p\psi (p)\to \forall p\exists q(\varphi (p)=\psi (q))$ .

The analog of this principle for existential propositional quantifiers is the following:

• $(\mathrm {I}_{\exists p})$ $\exists p\varphi (p)=\exists p\psi (p)\to \forall p\exists q(\varphi (p)=\psi (q))$ .

And, letting Q be either $\forall $ or $\exists $ , the analog of $\mathrm {I}_{Qp}$ for plural propositional quantifiers reads as follows:

• $(\mathrm {I}_{Qp{\kern-1pt}p})$ $Qp{\kern-2pt}p\varphi (\textit{pp})=Qp{\kern-2pt}p\psi (\textit{pp})\to \forall \textit{pp}\exists qq(\varphi (\textit{pp})=\psi (qq))$ .

Stepping back from the view of quantifiers as serving to express long conjunctions and disjunctions, the weaker principles of $\mathrm {O}^-_\circ $ and $\mathrm {I}_{Qv}$ (where $v$ may be a propositional or a plural propositional variable) encapsulate independently natural views of operators and quantifiers: operands and instances can always be recovered, but not necessarily in any particular order. And $\mathrm {I}_{Qv}$ does not appear to suffer from the same problem which plagued the inconsistent principles of logical structure for quantifiers considered in the previous section. For, consider the instance for the formulas which there led to inconsistency:

• $\forall p(p\prec \textit{pp})=\forall p(p\prec qq)\to \forall p\exists q((p\prec \textit{pp})=(q\prec qq))$ .

Assume for the sake of the argument that any formula of the form $p\prec \textit{pp}$ expresses one of two propositions t and f (the first being true and the second being false). With this, $\forall p\exists q((p\prec \textit{pp})=(q\prec qq))$ can be seen not to entail, in general, $\textit{pp}=qq$ : For example, consider distinct pluralities of propositions $\textit{pp}$ and $qq$ , which both have some proposition among them, and both some proposition not among them. Such pluralities must exist; for example, $\textit{pp}$ might be the truths and $qq$ the falsities. Then the instances of $\forall p(p\prec \textit{pp})$ will comprise t and f, and so will the instances of $\forall p(p\prec qq)$ .

3.4. Summary

The previous sections singled out the following principles, one for each logical connective, with the principles for binary sentential operators coming in an ordered and an unordered variant:

$$ \begin{align*}\mathrm{O}_\neg,\mathrm{O}^{(-)}_\wedge,\mathrm{O}^{(-)}_\vee,\mathrm{I}_{\forall p},\mathrm{I}_{\exists p},\mathrm{I}_{\forall \textit{pp}},\mathrm{I}_{\exists \textit{pp}},\mathrm{O}_\prec,\end{align*} $$

$\mathrm {O}_\prec $ was shown to be inconsistent since it allows one to recover a plural propositional parameter, which the Russell–Myhill result shows to be impossible. But none of the the other principles obviously shares this feature, which poses the question which of them are individually and jointly consistent. Sections 4 and 5 answer this question.

Section 4 shows that if $\circ $ is a binary operator and $Qp$ is a propositional quantifier, then the corresponding instances of $\mathrm {O}^-_\circ $ and $\mathrm {I}_{Qp}$ are jointly inconsistent; a fortiori, the corresponding instances of $\mathrm {O}_\circ $ and $\mathrm {I}_{Qp}$ are jointly inconsistent as well. Section 5 shows that these inconsistency results delineate precisely the consistent combinations of principles of logical structure. That is, it is there shown that following theories are consistent, and maximally so among sets of the principles considered here:

• $T_o:=\{\mathrm {O}_\neg,\mathrm {O}_\wedge,\mathrm {O}_\vee,\mathrm {I}_{\forall \textit{pp}},\mathrm {I}_{\exists \textit{pp}}\}$

• $T_i:=\{\mathrm {O}_\neg,\mathrm {I}_{\forall p},\mathrm {I}_{\exists p},\mathrm {I}_{\forall \textit{pp}},\mathrm {I}_{\exists \textit{pp}}\}$ .

5.1. Models

Models will be based on a set X of propositions. Each of these propositions will be associated with two items of information: first, its instances or operands (no distinction needs to be made between the two concepts), and second, its truth value, using $0$ and $1$ for falsity and truth, respectively. By Cantor’s theorem, there are more sets of propositions than propositions, so only some sets of propositions will serve as the set of instances of some proposition. These sets will be called well-behaved. The set W of well-behaved sets of propositions will be assumed to contain all finite sets. The correspondence between propositions and their instances and truth-values will be given by a bijection c mapping every pair $\langle I,t\rangle $ consisting of a well-behaved set I of instances and a truth-value t to a proposition. Finally, a model will contain a switch s, which is o or i depending on whether the model is intended to validate $T_o$ or $T_i$ .

Since c is a bijection, the instances and truth-value of a proposition x can be recovered as the first and second coordinate of $c^{-1}(x)$ , which will be notated $\pi _1c^{-1}(x)$ and $\pi _2c^{-1}(x)$ . For stating various definitions, it will be useful to extend c to a function $\tilde c$ which applies to $\langle I,t\rangle $ for all $I\subseteq X$ and $t<2$ . The particular choice of this extension is unimportant, but the simplest option is to let $\tilde c\langle I,t\rangle $ always be $c\langle \emptyset,t\rangle $ whenever $c\langle I,t\rangle $ is undefined.

Assignment functions map propositional variables to members of X, and plural propositional variables to subsets of X. As usual, for any assignment function a, $a[\xi /v]$ is the function mapping $v$ to $\xi $ and every other variable $v'$ to $a(v')$ , and correspondingly for sequences of variables. Relative to an assignment function a, a model interprets any formula $\varphi $ using an element $[\![\varphi ]\!]^a$ . This can be specified by determining the instances (operands) and truth value separately. Consider, by way of illustration, the case of negation. To validate $\mathrm {O}_\neg $ , negation should preserve instances (operands), but flip the truth value. Thus, the set of instances and the truth value of the proposition expressed by $\neg \varphi $ should be $\iota ([\![\varphi ]\!]^a)$ and $1-\tau ([\![\varphi ]\!]^a)$ , respectively. Therefore, $[\![\neg \varphi ]\!]^a$ can be specified as $\tilde c\langle \iota ([\![\varphi ]\!]^a),1-\tau ([\![\varphi ]\!]^a)\rangle $ . Note that for such a specification to have the intended effect, it is important that the relevant set of instances be well-behaved. In the case of negation, this is easily seen to be guaranteed, but in other cases, verifying this will be an important aspect in showing that the models validate the intended theories.

The other connectives are treated similarly, depending on the theory which is to be validated. For example, if $s=o$ , then conjuncts should be recoverable from conjunctions, so the instances of the proposition expressed by a conjunction $\varphi \wedge \psi $ should be the propositions expressed by $\varphi $ and $\psi $ . If $s=i$ , then no particular constraint is imposed on conjunctions, so it may simply be stipulated that the set of instances is empty. Thus the set of instances of $[\![\varphi \wedge \psi ]\!]^a$ can be specified as $\{[\![\varphi ]\!]^a,[\![\psi ]\!]^a:s=o\}$ . Similarly, the set of instances of $[\![\forall p\varphi ]\!]^a$ can be specified as the set of propositions $[\![\varphi ]\!]^{a[x/p]}$ for $x\in X$ , assuming $s=i$ . Plural propositional quantifiers are treated similarly, independently of the choice of s. Finally, since neither $T_o$ nor $T_i$ impose any particular constraints on $\prec $ , the set of instances of a proposition expressed by $\varphi \prec \textit{pp}$ may always be assumed to be empty. The truth-value of this proposition is determined by whether $[\![\varphi ]\!]^a$ is a member of $a(\textit{pp})$ . Taking $0$ (falsity) to be $\emptyset $ and $1$ (truth) to be $\{\emptyset \}$ , this may be specified as $\{\emptyset :[\![\varphi ]\!]^a\in a(\textit{pp})\}$ .

Truth of a formula $\varphi $ relative to a model and assignment function a is determined by $\tau ([\![\varphi ]\!]^a)$ , and validity is defined as truth relative to every assignment function and model.

The clauses for negation and quantifiers can usefully be re-stated using the following definitions:

With this, the above interpretation clauses are equivalent to the following, assuming, in the case of the second, that $s=i$ (and similarly for existential quantifiers):

• $[\![\neg \varphi ]\!]^a=-[\![\varphi ]\!]^a$

• $[\![\forall p\varphi ]\!]^a=\tilde c\langle \mathrm {Inst}^a_p(\varphi ),\mathrm {min}\tau [\mathrm {Inst}^a_p(\varphi )]\rangle $

• $[\![\forall \textit{pp}\varphi ]\!]^a=\tilde c\langle \mathrm {Inst}^a_{\textit{pp}}(\varphi ),\mathrm {min}\tau [\mathrm {Inst}^a_{\textit{pp}}(\varphi )]\rangle $ .

To be able to use models in consistency proofs, $\vdash $ first needs to be shown to be sound with respect to validity. This relies on the following standard lemma:

Furthermore, soundness relies on connectives (primitive and defined) to obey the standard truth-conditions:

Proof. By the constraints on models, in general $\tau \tilde c\langle I,t\rangle =t$ . Thus, if $[\![\varphi ]\!]^a=\tilde c\langle I,t\rangle $ , then $[\![\varphi ]\!]^a_\tau =t$ . With this, the claims for the primitive connectives follow straightforwardly from the definition of $[\![\cdot ]\!]^\cdot $ . By way of example, consider the case of negation:

$$ \begin{align*}\mathfrak{M},a\vDash\neg\varphi\text{ iff }[\![\neg\varphi]\!]^a_\tau=1\text{ iff }1-[\![\varphi]\!]^a_\tau=1\text{ iff }[\![\varphi]\!]^a_\tau\neq 1\text{ iff }\mathfrak{M},a\nvDash\varphi.\end{align*} $$

The cases of the defined connectives follow from their definitions and the cases of the primitive connectives as usual.□

It remains to identify models which validate $T_o$ and $T_i$ . $\mathrm {O}_\neg $ can in fact be shown to be valid in all models, as $-$ is guaranteed to be an injection:

Proof. If $-x=-y$ , then

$$ \begin{align*}c\langle\pi_1c^{-1}(x),1-\pi_2c^{-1}(x)\rangle=c\langle\pi_1c^{-1}(y),1-\pi_2c^{-1}(y)\rangle.\end{align*} $$

Since c is bijective, it follows that $\pi _1c^{-1}(x)=\pi _1c^{-1}(y)$ and $1-\pi _2c^{-1}(x)=1-\pi _2c^{-1}(y)$ . Thus $c^{-1}(x)=c^{-1}(y)$ , whence $x=y$ .

5.2. Operand models

This section shows $T_o$ to be consistent. First, it will be shown that any model in which the switch s is o validates $\mathrm {I}_{Qpp}$ , for universal and existential plural propositional quantifiers, and $\mathrm {O}^-_\circ $ for conjunction and disjunction. Using a syntactic mapping, this is then extended to $\mathrm {O}_\circ $ .

Starting with $\mathrm {I}_{Qpp}$ , the crucial step is to show that $\mathrm {Inst}^a_{\textit{pp}}(\varphi )$ is always well-behaved. This is best shown by proving that for every formula $\varphi $ and assignment function a, the set of propositions $[\![\varphi ]\!]^b$ is finite, where b may be any assignment function differing from a in the interpretation of plural propositional variables.

Proof. By induction on the complexity of $\varphi $ . Exemplarily, consider two cases:

Assume $\varphi $ is a conjunction $\psi \wedge \chi $ . $\mathrm {Inst}^a_\pi (\psi \wedge \chi )$ is a subset of:

$$ \begin{align*}\left\{\tilde c\left\langle\left\{[\![\psi]\!]^{a[\bar{Y}/\bar{\textit{pp}}]},[\![\chi]\!]^{a[\bar{Y}/\bar{\textit{pp}}]}\right\},t\right\rangle:\bar{Y}\subseteq X,t<2\right\}.\end{align*} $$

This, in turn, is a subset of $\{\tilde c\langle \{x,y\},t\rangle :x,y\in \mathrm {Inst}^a_\pi (\psi )\cup \mathrm {Inst}^a_\pi (\chi ),t<2\}$ , which by IH is finite.

Assume $\varphi $ is a universal plural propositional quantification $\forall qq\psi $ . $\mathrm {Inst}^a_\pi (\forall qq\psi )$ is a subset of:

$$ \begin{align*}\left\{\tilde c\left\langle\left\{[\![\psi]\!]^{a[\bar{Y}/\bar{\textit{pp}}][Z/qq]}:Z\subseteq X\right\},t\right\rangle:\bar{Y}\subseteq X,t<2\right\}.\end{align*} $$

This, in turn, is a subset of $\{\tilde c\langle I,t\rangle :I\subseteq \mathrm {Inst}^a_\pi (\psi ),t<2\}$ , which by IH is finite.

With this, the bijectivity of c ensures the validity of $\mathrm {I}_{Qpp}$ , for plural propositional quantifiers:

Proof. Consider the universal case; the existential case is analogous.

If $\mathfrak {M},a\vDash \forall \textit{pp}\varphi (\textit{pp})=\forall \textit{pp}\psi (\textit{pp})$ , then for some $t_1,t_2<2$ :

$$ \begin{align*}\tilde c\langle\mathrm{Inst}^a_{\textit{pp}}(\varphi(\textit{pp})),t_1\rangle=\tilde c\langle\mathrm{Inst}^a_{\textit{pp}}(\psi(\textit{pp})),t_2\rangle.\end{align*} $$

By Lemma 20, $\mathrm {Inst}^a_{\textit{pp}}(\varphi (\textit{pp}))$ and $\mathrm {Inst}^a_{\textit{pp}}(\psi (\textit{pp}))$ are finite, and so members of W. Thus by the injectivity of c, $\mathrm {Inst}^a_{\textit{pp}}(\varphi (\textit{pp}))=\mathrm {Inst}^a_{\textit{pp}}(\psi (\textit{pp}))$ . So for every $Y\subseteq X$ there is a $Z\subseteq X$ such that $[\![\varphi (\textit{pp})]\!]^{a[Y/\textit{pp}]}=[\![\psi (\textit{pp})]\!]^{a[Z/\textit{pp}]}$ . By Lemma 12, the latter is $[\![\psi (qq)]\!]^{a[Z/qq]}$ . Thus $\mathfrak {M},a\vDash \forall \textit{pp}\exists qq(\varphi (\textit{pp})=\psi (qq))$ .

A similar argument shows the validity of $O^-_\wedge $ and $O^-_\vee $ , as $\{[\![\varphi ]\!]^a,[\![\psi ]\!]^a\}$ must be finite and so well-behaved:

Proof. Consider the conjunctive case; the disjunctive case is analogous.

Assume $\mathfrak {M},a\vDash p\wedge p'=q\wedge q'$ . Then for some $t_1,t_2<2$ :

$$ \begin{align*}\tilde c\langle\{a(p),a(p')\},t_1\rangle=\tilde c\langle\{a(q),a(q')\},t_2\rangle.\end{align*} $$

As $\{a(p),a(p')\}$ and $\{a(q),a(q')\}$ are finite, they are members of W. It therefore follows from the bijectivity of c that $\{a(p),a(p')\}=\{a(q),a(q')\}$ . By elementary set theory, it follows that either $a(p)=a(q)$ and $a(p')=a(q')$ , or $a(p)=a(q')$ and $a(p')=a(q)$ . Thus $\mathfrak {M},a\vDash (p=q\wedge p'=q')\vee (p=q'\wedge p'=q)$ .

By soundness, operand models therefore witness the consistency of $\{\mathrm {O}_\neg,\mathrm {O}^-_\wedge,\mathrm {O}^-_\vee,\mathrm {I}_{\forall \textit{pp}},\mathrm {I}_{\exists \textit{pp}}\}$ in $\vdash $ . In order to extend this consistency result to $T_o$ , recall how any binary connective $\circ $ satisfying $\mathrm {O}^-_\circ $ can be used to define a binary connective $\hat \circ $ satisfying $\mathrm {O}_{\hat \circ }$ . $\circ $ and $\hat \circ $ are not truth-functionally equivalent, but given both $\mathrm {O}^-_\wedge $ and $\mathrm {O}^-_\vee $ , a variant definition $\bar \circ $ is available which is truth-functionally equivalent to $\circ $ , with $\mathrm {O}^-_\circ $ still entailing $\mathrm {O}_{\bar \circ }$ . This can be used to define a function mapping any formula $\varphi $ to a formula $\bar \varphi $ which replaces any occurrence of $\wedge $ and $\vee $ by $\bar \wedge $ and $\bar \vee $ , respectively. It can be shown that if $\varphi $ is entailed by $T_o$ , then $\bar \varphi $ is valid in any operand model, and this suffices for consistency.

$\bar \circ $ can be defined as follows:

• $\varphi \:\bar {\circ }\:\psi :=((\varphi \wedge \neg \varphi )\vee (\varphi \wedge \varphi ))\circ (\psi \wedge \psi )$ .

By , for $\circ \in \{{\wedge },{\vee }\}$ , $\vdash \varphi \:\bar {\circ }\:\psi \leftrightarrow \varphi \circ \psi $ . Define $\overline {\varphi }$ recursively, so that $\overline \cdot $ maps every atomic formulas to itself, commutes with all logical constants except $\wedge $ and $\vee $ , and satisfies the following two clauses:

• $\overline {\varphi \wedge \psi }:=\overline \varphi \:\bar {\wedge }\:\overline \psi $

• $\overline {\varphi \vee \psi }:=\overline \varphi \:\bar {\vee }\:\overline \psi $ .

To show that this has the intended effect, it suffices to show that if $T_o\vdash \varphi $ , then $\overline \varphi $ is valid in any operand model. With the results above, this follows from the following three lemmas:

Proof. Consider exemplarily the case of a quantifier $Qv$ ; the remaining case of negation is analogous.

Let $\varphi $ be an instance of $\mathrm {I}_{Qv}$ for complement clauses $\psi $ and $\chi $ . Then $\overline {\varphi }$ is provably equivalent to

$$ \begin{align*}Qv\overline{\psi}(v)=Qv\overline{\chi}(v)\to\forall v\exists v'(\overline{\psi}(v)=\overline{\chi}(v')),\end{align*} $$

which is an instance of $\mathrm {I}_{Qv}$ .

Proof. By induction on the length of proofs. For every instance $\varphi $ of an axiom schema of $\vdash $ , there is an instance $\varphi '$ which is equivalent to $\overline \varphi $ . For example,

$$ \begin{align*}\vdash\overline{(\forall v\psi\to\psi[\varepsilon/v])}\leftrightarrow(\forall v\overline\psi\to\overline\psi[\overline\varepsilon/v]).\end{align*} $$

Analogously for the two rules of $\vdash $ . For example, assume that $\vdash \psi $ using from $\vdash \varphi $ and $\vdash \varphi \to \psi $ . By IH, $\vdash \overline \varphi $ and $\vdash \overline {\varphi \to \psi }$ . By the latter, $\vdash \overline \varphi \to \overline \psi $ , so using , $\vdash \overline \psi $ .

5.3. Instance models

To show that $T_i$ is consistent, models will be used in which the switch s is set to i. This ensures that the interpretational clauses of quantified formulas behave as expected, with $[\![\forall p\varphi ]\!]^a$ being interpreted as a proposition determined by $\mathrm {Inst}^a_p(\varphi )$ and the minimum of the truth-values of these instances. However, for $\mathrm {I}_{Qv}$ to be valid, for all quantifiers, it must be shown that $\mathrm {Inst}^a_v(\varphi )$ is always well-behaved. And this requires W to contain some infinite sets.

To illustrate this, consider the formula $\forall p\ p$ . $\mathrm {Inst}^a_p(p)$ is simply X, the set of all propositions, so X must be well-behaved. Similarly, consider $\forall p\forall q\ p$ . For every p, the proposition expressed by $\forall q\ p$ is a distinct proposition with a single instance p, so $\mathrm {Inst}^a_p(\forall q\ p)$ is the infinite set of these propositions, which must be well-behaved as well. Similarly, $\mathrm {Inst}^a_p(\neg \forall q\ p)$ , the set of instances of the proposition expressed by $\forall p\neg \forall q\ p$ , is the set of negations of these propositions; this must also be well-behaved. Since quantifiers and negations can be nested, W must more generally be required to be closed under correspondingly iterated operations on sets of propositions, in addition to containing X. The next definition formulates these constraints in suitable generality. To state it, for any set A, $A^*$ is taken to be the set of finite strings of elements of A, i.e., $\bigcup _{n<\omega }A^n$ , and e the string of elements of length $0$ .

Cardinality considerations again suggest that these constraints are satisfiable. Starting from a countably infinite set X, the set of finite subsets of X is countable. And as $\{q,n\}^*$ is countable, so is the set $\{c^\sigma [X]:\sigma \in \{q,n\}^*\}$ , given any choice of c. Thus, W can be chosen to be countable, so that there exists a bijection c from $W\times \{0,1\}$ to X. The only difficulty is that in this line of reasoning, the choice of W is dependent on a choice of c, which itself depends on W. The difficulty can be overcome by dividing X into four countably infinite subsets: the set $X^1_f$ of true propositions with finitely many instances, the set $X^1_\infty $ of true propositions with infinitely many instances, and correspondingly sets $X^0_f$ and $X^0_\infty $ of false propositions. Fixing the interpretation of negation by a suitable function $\sim $ on X, the choice of c and W can be determined in three steps: First, the behavior of c can be fixed for pairs $\langle I,t\rangle $ with I finite, by choosing a suitable bijection with codomain $X^1_f\cup X^0_f$ . Second, W is determined by this, since $c^\sigma $ is determined by the behavior of c on such pairs and the behavior of negation. Finally, c can be completed by choosing a suitable bijection from pairs $\langle I,t\rangle $ with I an infinite member of W to $X^1_\infty \cup X^0_\infty $ . The following proof makes this line of argument precise.

Having defined instance models and demonstrated their existence, the next step is to show that they behave as intended, i.e., that $\mathrm {Inst}^a_v(\varphi )$ is always well-behaved. This follows from the following lemma:

6.1. Weakening logic

The axiomatic principles of $\vdash $ comprise, apart from elementary principles governing Boolean connectives and quantifiers, two principles which are specific to plural quantification: and . Weakening the elementary principles governing Boolean connectives and quantifiers won’t be considered here; many will consider them far more plausible than any principle of logical structure. The following therefore considers dropping or weakening one of and .

Inspecting the deduction of an inconsistency sketched in Section 4, it is easy to see that is nowhere appealed to. There is thus nothing to be gained by questioning . Furthermore, this shows that this proof could as well have been carried out in a more standard higher-order language in which plural propositional quantification is replaced with quantifiers binding variables taking the syntactic position of sentential operators. However, the models constructed in Section 5 do little to assure us of the viability of $T_o$ and $T_i$ in such a context: They may show that these theories are consistent in a proof system corresponding to $\vdash $ , and so a fortiori consistent in a proof system omitting the axiom corresponding to . But these models essentially validate , which is at least controversial in the case of quantification into operator position: roughly, such quantifiers can be read as ranging over properties of propositions, which are plausibly individuated non-extensionally. This raises the general question of which sets of logical structure principles are consistent in higher-order systems with non-extensional higher-order quantification, including more comprehensive type theories in which any finite sequence of types gives rise to a type of relational terms.

Consider now the option of restricting . The option of restricting to was already noted not to restore consistency. Another option is to restrict to predicative instances, in which $\varphi $ may not contain any plural propositional quantifiers or parameters. Walsh [Reference Walsh15] shows, in a similar type-theoretic setting, that the Russell–Myhill theorem essentially relies on impredicative instances of comprehension. This suggests that such a weakening of may be enough to render consistent all of the principles of logical structure discussed in Section 3. Assessing this is beyond the scope of this paper, but it is worth noting that even on such a restriction of , there are natural variants of $\mathrm {O}^-_\circ $ and $\mathrm {I}_{Qp}$ in an expanded language which are inconsistent.

To motivate this expansion of the language, note that principles $\mathrm {O}^-_\circ $ and $\mathrm {I}_{Qv}$ allow one to recover operands and instances, in the sense that propositions expressed by applications of $\circ $ are only identical if the operands are the same, and propositions expressed by applications of $Qv$ are only identical if the instances are the same. Those who find such principles attractive may naturally also want to be able to talk of the operands and the instances of any given proposition. Formally, they may thus want to add to the language binary sentential operators O and I satisfying the following principles:

• ( $\mathrm {O}^O_\circ $ ) $O(r,p\circ q)\leftrightarrow r= p\vee r= q$

• ( $\mathrm {I}^I_{Qv}$ ) $I(q,Qv\varphi (v))\leftrightarrow \exists v(q=\varphi (v))$ .

In the case of binary operations $\circ $ , there is little difference between $\mathrm {O}^-_\circ $ and $\mathrm {O}^O_\circ $ . After all, $\mathrm {O}_\circ ^O$ entails $\mathrm {O}^-_\circ $ , as is easily seen. And conversely, given $\mathrm {O}^-_\circ $ , one can define an operation $O'$ satisfying $\mathrm {O}^{O'}_\circ $ , as follows:

• $O'(p,q):=\exists r((p\wedge r)= q\vee (r\wedge p)= q)$ .

The case of $\mathrm {I}^I_{Qv}$ is different. While $\mathrm {I}^I_{Qv}$ is still easily seen to entail $\mathrm {I}_{Qv}$ , it is not clear how one would even define, on the assumption of $\mathrm {I}_{Qv}$ , what it takes for p to be an instance of q.

This additional strength of $\mathrm {I}^I_{Qv}$ can be harnessed to show an inconsistency result which requires only predicative instances of . Instead of the Russell–Myhill theorem, the relevant derivation makes use of the following result, adapted from [Reference Uzquiano14]:

Like the Russell–Myhill theorem, this relies on a version of plural comprehension which entails the existence of an empty plurality. But again, this is not essential: if there is no q such that $\neg \psi (q,q)$ , then it follows with $\psi (p,\varphi (\textit{pp}))\leftrightarrow p\prec \textit{pp}$ that $\varphi (\textit{pp})\nprec \textit{pp}$ for all $\textit{pp}$ . But this is inconsistent with the existence of the (non-empty) plurality of all propositions, which follows by an instance of predicative comprehension.

Let $\vdash ^-$ be $\vdash $ , with restricted to predicative instances. In this system, an instance of the schema just shown to be inconsistent can be derived from $\mathrm {O}^-_\circ $ and $\mathrm {I}^I_{Qp}$ :

While $\mathrm {I}^I_{Qp}$ may be stronger than $\mathrm {I}_{Qp}$ , it isn’t inconsistent on its own, even in $\vdash $ , as can be shown by adapting the above model constructions. Interpreting O and I as follows, it is easily seen that operator models and instance models validate $\mathrm {O}^O_\circ $ and $\mathrm {I}^I_{Qp}$ respectively, for binary Boolean connectives $\circ $ and propositional quantifiers ${Qp}$ :

• $[\![ O(\varphi,\psi )]\!]^a=[\![ I(\varphi,\psi )]\!]^a=\tilde c\langle \emptyset,\{\emptyset :[\![\varphi ]\!]^a\in [\![\psi ]\!]^a_\iota \}\rangle $ .

6.2. Weakening logical structure

Consider now the option of weakening the logical structure principles, in order to obtain a consistent theory which nevertheless imposes a natural form of logical structure on propositions. In the case of $\prec $ , it is hard to see how any principle weaker than $\mathrm {O}_\prec $ would encode the idea that propositions inherit the logical structure of sentences of the form $\varphi \prec \textit{pp}$ . It seems therefore that the idea of logical structure has to be restricted to cases other than those arising from sentences of the form $\varphi \prec \textit{pp}$ . The matter is different in the case of the jointly inconsistent principles $\mathrm {O}^-_\circ $ and $\mathrm {I}_{Qp}$ , for a binary operator $\circ $ and a propositional quantifier $Qp$ . In both cases, there are some independent reasons one might have for thinking that these principles are too strong.

In the case of $\mathrm {I}_{Qp}$ , one might note that even if propositions reflect some of the logical structure of quantified sentences expressing them, it is plausible this does not include the order of quantifiers in any string of the same quantifiers, and it is plausible that the identity of propositions expressed is invariant under relabeling bound variables. Illustrating this using universal quantifiers, the following identifications are plausible:

• $\forall p\forall q\varphi =\forall q\forall p\varphi $

• $\forall q\forall p\varphi =\forall p\forall q(\varphi [p/q,q/p])$ .

Consider the instances of these principles for $\varphi $ being p. Together, they entail that $\forall p\forall q\ p=\forall p\forall q\ q$ . But with $I_{\forall p}$ , it follows that $\forall p\exists q(\forall q\ p=\forall q\ q)$ . And any instance of this universal claim for a truth p is false. Thus and are inconsistent with $\mathrm {I}_{\forall p}$ .

These considerations may motivate restricting $\mathrm {I}_{Qp}$ to cases in which the complement clause $\varphi $ is a complex formula which does not itself start with a quantifier. But only such a restricted instance is appealed to in the results of Section 4, so even this restricted version of $\mathrm {I}_{Qp}$ , for a propositional quantifier $Qp$ , is inconsistent with $\mathrm {O}^-_\circ $ , for any binary sentential operator $\circ $ .

In the case of binary sentential operators, recall how $\mathrm {O}_\circ $ was noted to be inconsistent with the commutativity of $\circ $ , which motivated considering the weaker principle $\mathrm {O}^-_\circ $ . As one might hold on to some version of the idea that propositions exhibit conjunctive and disjunctive structure while arguing that conjunction and disjunction are commutative, one might similarly hold on to this idea while arguing that conjunction and disjunction are associative. On this view, the following principle of associativity holds for both $\wedge $ and $\vee $ :

• $(p\circ (q\circ r))=((p\circ q)\circ r)$ .

Someone might want to endorse this even if they think that conjuncts may be recovered from conjunctions; on their view, there may simply be one conjunctive proposition with three conjuncts p, q and r. As a commutative view of conjunction and disjunction can be illustrated by thinking of conjoining and disjoining propositions as akin to putting the conjuncts or disjuncts in a (conjunctive or disjunctive) bag, one can illustrate an associative view of conjunction and disjunction by thinking of conjoining and disjoining propositions as akin to gluing the conjuncts or disjuncts together (using a conjunctive or disjunctive glue).

The associativity of conjunction is inconsistent with $\mathrm {O}^-_\wedge $ : If $(p\wedge (\neg p\wedge \neg p))=((p\wedge \neg p)\wedge \neg p)$ , then by $\mathrm {O}^-_\wedge $ , p must be $p\wedge \neg p$ or $\neg p$ , which cannot be the case if p is true. A similar argument shows that the associativity of disjunction is inconsistent with $\mathrm {O}^-_\vee $ . Thus, one might argue that there are independent reasons for thinking that $\mathrm {O}^-_\circ $ is too strong. On this view, the recovery of conjuncts and disjuncts has to be restricted to conjuncts and disjuncts which are not themselves conjunctions or disjunctions, respectively. That is, the relevant weakening of $\mathrm {O}^-_\circ $ would be:

• $(\mathrm {O}^{{-}\kern2pt{-}}_\circ )$ $\forall r(r=p\vee r=p'\vee r=q\vee r=q'\to \forall p\forall q (r\neq p\circ q))\to $ $\phantom{(\mathrm {O}^{{-}\kern2pt{-}}_\circ )}((p\circ p')= (q\circ q')\to (p=q\wedge p'=q')\vee (p=q'\wedge p'=q))$ .

But as it turns out, even such a weakened principle is inconsistent with any principle $\mathrm {I}_{Qp}$ , assuming a couple of natural auxiliary assumptions governing negation. The first is $\mathrm {O}_\neg $ ; the second is the following principle, stating that no negation is an application of $\circ $ :

• $(\neg \neq \circ )$ $\neg p\neq (q\circ r)$ .

There are various further weakenings which one might explore. For example, it might be argued that conjunction and disjunction are idempotent, so that $p\circ p=p$ , for $\circ $ being $\wedge $ or $\vee $ , without this trivializing the idea of conjunctive and disjunctive propositional structure. Such idempotence is again inconsistent with $\mathrm {O}^-_\circ $ : by idempotence, $((p\circ \neg p)\circ (p\circ \neg p))=(p\circ \neg p)$ , but by $\mathrm {O}^-_\circ $ , it follows from this that both p and $\neg p$ are $(p\circ \neg p)$ . The weaker principle $\mathrm {O}^{{-}\kern2pt{-}}_\circ $ can therefore also be motivated by idempotence. Interestingly, in this case, one has independent reason to reject the auxiliary principle $\neg \neq \circ $ just appealed to, as the idempotence of $\circ $ immediately entails $\neg p=(\neg p\circ \neg p)$ .