Against Fregean Quantification

2.1. Assignment-Relative Semantics

The problem for the Tarskian theory arises from its commitment to explicit variables and open sentences in the syntax. Although variables and proper names are of the same syntactic type, terms, their semantic contribution is very different. Whereas a name refers to a particular object, a variable picks out different objects relative to different assignments. For any assignment $\sigma$ , let ${〚.〛}^{\sigma }$ be a function that takes an expression to its denotation on that assignment. The denotation of a constant is the same object, its referent, for any assignment. The value of a variable depends on the assignment $\sigma$ .

Terms:

• $\text{Constant}: \mathrm{If}\alpha \hspace{0.17em}\text{is a name},\hspace{0.17em}\hspace{0.17em}\text{then} {〚\alpha 〛}^{\sigma }=\text{the referent of}\alpha$ .

• $\text{Variable}: \text{If}\hspace{0.17em}\nu \hspace{0.17em}\text{is a variable},\mathrm{then}{〚\nu 〛}^{\sigma }=\sigma (\nu )$ .

Similarly, an open sentence such as ‘ $x$ smiles’ is not true or false absolutely. The open sentence is satisfied by (or true relative to) some assignments and unsatisfied by (or false relative to) others. For example, ‘ $x$ smiles’ is only satisfied by those assignment functions that assign the variable $x$ to a smiling individual. Thus, satisfaction by (or truth relative to) an assignment for atomic predication is defined as follows (where ${〚\pi 〛}^{\sigma }$ is the extension of the predicate $\pi$ ):

Predication Rule: ${〚\pi {\alpha }_1\dots {\alpha }_n〛}^{\sigma }=\{\begin{array}{l}\text{True}, \text{if} \langle {〚{\alpha }_1〛}^{\sigma },\dots ,{〚{\alpha }_n〛}^{\sigma }\rangle \in {〚\pi 〛}^{\sigma },\hfill \\ \text{False}, \text{otherwise}.\hfill \end{array}$

The truth conditions of sentences constructed using truth-functional connectives are also as they were in the base language.

Negation Rule: ${〚\neg \varphi 〛}^{\sigma }=\{\begin{array}{l}\text{True}, \text{if} {〚\varphi 〛}^{\sigma }=\text{False},\hfill \\ \text{False}, \text{if} {〚\varphi 〛}^{\sigma }=\text{True}.\hfill \end{array}$

Conjunction Rule: ${〚(\varphi \wedge \psi )〛}^{\sigma }=\{\begin{array}{l}\text{True}, \text{if} {〚\varphi 〛}^{\sigma }=\text{True}={〚\psi 〛}^{\sigma },\hfill \\ \text{False, otherwise}.\hfill \end{array}$

So far, the semantic theory resembles exactly the semantic theory for the base language. Moreover, the clauses are compositional since the semantic value of an expression at an assignment is a function of its constituents’ semantic values at that same assignment.

However, compositionality fails when the language takes quantifiers into account. The denotation of a quantified claim $\forall \nu \varphi$ at an assignment $\sigma$ is not a function of the denotations of the variable $\nu$ and the sentence $\varphi$ at $\sigma$ . The denotation of $\forall \nu \varphi$ at $\sigma$ depends on the denotations of $\nu$ and the sentence $\varphi$ at other assignments. If $\sigma$ is an assignment function and $\nu$ is a variable, then let $\sigma [\nu /o]$ be the assignment that differs from assignment $\sigma$ at most in that it assigns o to $\nu$ . A universally quantified sentence $\forall \nu \varphi$ is satisfied by an assignment $\sigma$ if for every object $o$ , $\sigma [\nu /o]$ satisfies $\varphi$ .

Quantification Rule: ${〚\forall \nu \varphi 〛}^{\sigma }=\{\begin{array}{l}\text{True}, \text{if for all} o,\hspace{0.17em}{〚\varphi 〛}^{\sigma [\nu /o]}=\text{True},\hfill \\ \text{False, otherwise}.\hfill \end{array}$

A sentence is true (simpliciter) if it is satisfied by every assignment. Thus, (4) is true if and only if the embedded open sentence ‘ $x$ smiles and $x$ waves’ is satisfied by every assignment function.

On this semantics, ${〚.〛}^{\sigma }$ assigns each expression to its denotation. The function ${〚.〛}^{\sigma }$ satisfies truth and reference centrality. As a consequence, it violates compositonality. To see this, consider quantified formulae that are constructed by application of the same syntactic rule, such as $\forall \mathrm{xRxy}$ and $\forall \mathrm{xRyx}$ :

$\forall \mathrm{xRxy}=\forall (x,\mathrm{Rxy})$

$\forall \mathrm{xRyx}=\forall (x,\mathrm{Ryx})$

Assume that $x$ and $y$ co-denote relative to assignment $\sigma$ . By compositionality, it follows that ${〚\mathrm{Rxy}〛}^{\sigma }={〚\mathrm{Ryx}〛}^{\sigma }$ . Again by compositionality, it follows that ${〚\forall \mathrm{xRxy}〛}^{\sigma }={〚\forall \mathrm{xRyx}〛}^{\sigma }$ .¹⁰ But these formulae aren’t equivalent: everything might stand in a relation $R$ to object $o$ without $o$ standing in $R$ to everything. This shows that the function ${〚.〛}^{\sigma }$ does not compositionally assign expressions to their denotations. This argument is fully general. Any denotation function ${〚.〛}^{*}$ that preserves truth and reference centrality for this language and yields ${〚\forall \mathrm{xRxy}〛}^{*}\ne {〚\forall \mathrm{xRyx}〛}^{*}$ results in failures of compositionality

compositionality is a core methodological constraint on semantic theorizing (see Partee 1984 and Dever 1999). Abandoning it would require revisiting every choice that it has previously motivated. It also has empirical content, explaining how a language user can understand an infinity of complex expressions by understanding a finite base of simple expressions and rules for forming complex expressions out of simpler expressions (see, e.g., Heim & Kratzer 1998).

This failure of compositionality does not exclusively rely on truth and reference centrality, which one may take to involve overly narrow conceptions of the semantic contributions of a sentence and term. It extends, for example, to those who think that the primary semantic function of a sentence is to express a content, a possible worlds intension or a structured Russellian or Fregean proposition. The problem would arise for any view on which two open sentences have the same denotation (relative to an assignment) but differ by the substitution of variables or by the substitution of a variable for a singular term.¹¹ Retaining compositionality would require abandoning this assumption. But many have worried that such a view “demotes” propositions to “secondary importance in semantics” because the contribution that a sentence makes to the truth-conditons of a sentence that contains it is not the proposition it expresses (King 2003: 200).¹²

3.1. Quantifiers as Predicates of Predicates

On standard presentations, the Tarskian quantifier, ‘ $\forall x$ ’, serves a double-duty: it both binds variables in its scope and generalizes. The truth-value of a quantified formula $\forall x\varphi$ relative to an assignment $\sigma$ depends on the truth-value of the embedded formula $\varphi$ relative to assignments that differ from $\sigma$ only in their interpretation of $x$ . The quantified formula is true if and only if the embedded formula is true on all assignments. The contribution of the quantifier to the truth conditions of a sentence that contains it is obscured by serving these two roles.

According to Fregean approaches, a quantifier has a single semantic role of generalizing a predicate (feature F1). A sentence that results from applying the universal quantifier ‘everything’ to a predicate is true if and only if the predicate is true of every object and a sentence that results from applying the existential quantifier ‘something’ to a predicate is true if and only if the predicate is true of some object.

(5) Everything smiles.

(6) Something waves.

Thus, (5) is true if everything smiles and (6) is true if something waves.

There are two reasons one might view the Fregean approach as preserving compositionality in contrast to the Tarskian. The first is that the Fregean explicitly assigns a meaning to the quantifier symbol ‘ $\forall$ ’. This position is articulated by Partee (2013: 120–21), who says that Frege’s development of predicate logic was carried out “more compositionally” than Tarski’s, where “the quantifier symbols $\forall$ and $\exists$ are not themselves given a semantic interpretation”. Indeed, Frege provides a denotation for his quantifier symbol ‘’. Or, rather, Frege provides the same denotation for his many synonymous quantifier symbols, each indexed by a different Gothic letter. (Looking ahead to feature F3, we will ignore this feature of Frege’s syntax.)¹⁷ However, as we have seen, nothing prevents the Tarskian from assigning a meaning to the quantifier expressions as well.

Therefore, we turn to the second potential advantage: variables are not immediate constituents of quantified formulae. Along these lines, Cresswell (1973: 80–87) cites the familiar difficulty in accounting for the semantic difference between $\forall \mathrm{xFx}$ and $\forall \mathrm{xFy}$ , if $x$ and $y$ happen to share their referent. Cresswell insists that by treating the quantifier as attaching to a complex predicate which is constructed by abstraction, one can then treat the quantifiers as straightforward predicates of predicates, and thereby avoid the problems that plague the standard syntax.

Yet there remains a major lacuna in the semantics. Not every quantified statement results from applying a quantifier phrase to a simple predicate.¹⁸ Consider (7).

(7)  Something smiles and waves.

This sentence results from applying the quantifier ‘Something’ to the predicate ‘smiles and waves’. But this latter predicate cannot be a simple predicate. In particular, we want to be able to explain the fact that (7) entails (6) and (8).

(6)  Something waves.

(8)  Something smiles.

To deliver this explanation, the predicate ‘smiles and waves’ must be syntactically derived from the simpler predicates ‘smiles’ and ‘waves’. The central challenge to the Fregean approach, then, is to provide a compositional account of the formation and semantic evaluation rules for these syntactically derived predicates.

Contemporary semanticists follow Frege by constructing quantified formulae using two formation rules: one derives a complex predicate $\widehat{x}\varphi \hspace{0.17em}\hspace{0.17em}(\mathrm{or}\hspace{0.17em}\hspace{0.17em}\lambda x\varphi )$ from a variable $x$ and an open sentence $\varphi$ and the other attaches the quantifier to the complex predicate.¹⁹

$\Lambda$ : Takes a variable $\nu$ and a sentence $\varphi$ and yields a monadic predicate. We represent this predicate as $⌜\widehat{\nu }\varphi ⌝$ .

$\Pi$ : Takes a monadic predicate $\pi$ and yields a sentence. We represent this sentence as $⌜\forall \pi ⌝$ .

The first step corresponds to the binding work (predicate abstraction, or “lambda binding”), while the second corresponds to the quantificational work. Visually, one can compare the standard Tarskian syntax on the left with this Fregean departure on the right:

A quantified formula $\forall \widehat{x}\varphi$ is true on this approach if the embedded predicate $\widehat{x}\varphi$ is true of all objects.

The challenge for the Fregean in implementing this proposal is to semantically characterize complex predicates in terms of the expressions from which they derive. In contemporary formal semantics, this problem is relocated to the semantics of abstraction. Consider predicates that are constructed by applying the same syntactic rule, such as the following:

$\widehat{x}\mathrm{Rxy}=\Lambda (x,\mathrm{Rxy})$

$\widehat{x}\mathrm{Ryx}=\Lambda (x,\mathrm{Ryx})$

If the terms $x$ and $y$ co-denote, then $〚\mathrm{Rxy}〛=〚\mathrm{Ryx}〛$ , and so it follows—by compositionality—that $〚\widehat{x}\mathrm{Rxy}〛=〚\widehat{x}\mathrm{Ryx}〛$ . But, of course, these predicate abstracts should denote distinct properties, the properties of being an  $x$ that stands in $R$ to $y$ and being an $x$ that $y$ stands in $R$ to, respectively. So separating quantification from abstraction is no advance in terms of violations of compositionality. Abstraction does not save compositionality—it instead locates precisely where it fails (given the Fregean assumptions).²⁰

We can, however, provide a compositional semantics for this syntax by again making anti-Fregean assumptions about denotation. We replace the semantic clause for quantification with the following two clauses:²¹

Abstraction Rule: $〚\widehat{\nu }\varphi 〛=f_{\Lambda }(〚\nu 〛,\hspace{0.17em}\hspace{0.17em}〚\varphi 〛)=\lambda \sigma .\lambda o.〚\varphi 〛\hspace{0.17em}\hspace{0.17em}(\sigma [\nu /o])$

Quantification Rule: $〚\forall \pi 〛=f_{\Pi }(〚\pi 〛)=\lambda \sigma .\{\begin{array}{l}\text{True}, \text{if for all} o,\hspace{0.17em}〚\pi 〛(o)=\text{True},\hfill \\ \text{False, otherwise}.\hfill \end{array}$

While an approach that employs abstraction may be advantageous in terms of providing an explicit meaning to the quantifier, the approach described here clearly does not avoid the dilemma because it relies on the assumption that the denotation of a variable is a function from assignments to objects and the denotation of a sentence is a function from assignments to truth-values.

3.2. Recursion on Names and Closed Sentences

An apparently distinctive feature of Fregean syntax is its avoidance of “open” sentences and variables. Whereas the Tarskian syntax constructs quantified sentences from open sentences which contain variables, the Fregean syntax first constructs a closed sentence, then constructs a predicate by a process of name removal, and finally combines the resulting predicate with a quantifier. This is feature F2 of Frege’s program. We argue that direct recursion on names and closed formulae in this manner forces the same uncomfortable choice faced by Tarski.

Syntactically, Fregean approaches defined by F2 differ from the Tarskian approach in two respects: (i) the language includes only terms drawn from the beginning of the Latin alphabet $a,b,c,\dots$ rather than additionally containing terms from the end of the alphabet $x,y,z\dots$ and (ii) the abstraction rule is replaced with the following rule:²²

$\Delta$ : Takes a name $\alpha$ and a sentence $\varphi$ and yields a monadic predicate. We represent this predicate as $⌜\widehat{\alpha }\varphi ⌝$ .

Rule $\Delta$ differs from rule $\Lambda$ only by appealing to letters at the beginning rather than end of the alphabet. And, of course, merely avoiding the use of lower case letters from the end of the alphabet cannot in itself be advance on the Tarskian approach. Nothing prohibits using the beginning of the alphabet for variables. This choice of alphabetic location is arbitrary.

But the Fregean doesn’t just avoid use of certain parts of the alphabet, they use the same set of terms—the names—to regiment ordinary proper names and to derive complex predicates or quantified formulae. In contrast, a textbook Tarskian system (e.g., Kalish & Montague 1964) has two syntactic species within a broader genus of terms—names and variables—and the category of names regiments proper names of ordinary language, while the category of variables are used to derive quantified formulae. For the Fregean, all terms are names, but names enter into the construction rules in two different ways.

The use of a unified category of terms rather than a bifurcated category does not interestingly differentiate the Fregean from the Tarskian approach. In fact, using two categories of terms is not essential to the the Tarskian approach—the languages defined in, e.g., Tarski (1935/1956a) and Tarski and Vaught (1957) didn’t actually have a category of names in addition to the variables. For us, the question of whether an expression is a variable is tantamount to the question of whether it can play the role of variables in a quantified sentence. That is, expressions that can be non-trivially bound by operators are variables. The fact that an expression may also perform a different function while not bound is irrelevant.

Our position regarding Frege’s syntax agrees with Carnap (1934/1959: §54), when he considers the similar position that quantified sentences might be constructed using constants rather than variables. (He attributes this possibility to Quine.) Thus, rather than ‘ $\forall x(x=x)$ ’, one might have ‘ $\forall 0(0=0)$ ’ or ‘ $\forall 3(3=3)$ ’, where ‘ $0$ ’ and ‘ $3$ ’ are numerical constants. Carnap argues that even if this language lacks free variables (because he treats open sentences as having universal force), the proper names are nonetheless variable-expressions, since they play the relevant role in the construction of quantified formulae.

F2 is a form of variabilism, names and variables constitute a unified semantic category. It differs from more recent forms of variablism only by using letters from the beginning of the alphabet rather than the end to regiment names and variables.²³ Both contemporary and Fregean approaches use the same sort of expression to regiment (referential) proper names and in the derivation of complex predicates or quantified sentences. Thus, there is no interesting syntactic difference between the Fregean who forms complex predicates by applying a construction rule to names and closed sentences and the Tarskian who does so by applying a construction rule to variables and open sentences. The syntactic difference lies only in the alphabetic location of the “variables”.

If Fregeans want to offer a genuine alternative to the Tarskian approach, they must offer a substantively different semantics for abstraction from that offered by the Tarskian. In particular, Fregeans must offer a semantics that evades the dilemma. But the only compositional implementation of Frege’s idea gives up truth and reference centrality.

Suppose that the Fregean wishes to maintain truth and reference centrality. Consider predicates that are constructed by applying the same syntactic rule, such as abstracting the name $a$ from $\mathrm{Rab}$ and $\mathrm{Rba}$ :

$\widehat{a}\mathrm{Rab}=\Delta (a,\mathrm{Rab})$

$\widehat{a}\mathrm{Rba}=\Delta (a,\mathrm{Rba})$

If the names $a$ and $b$ refer to the same object, then, by compositionality, it follows that $〚\mathrm{Rab}〛=〚\mathrm{Rba}〛$ , and then again by compositionality it follows that $〚\widehat{a}\mathrm{Rab}〛=〚\widehat{a}\mathrm{Rba}〛$ . But, of course, these predicate abstracts should denote distinct properties, the properties of being an  $x$ that stands in $R$ to $b$ and being an $x$ that $b$ stands in $R$ to, respectively. So, compositionality fails for the Fregean approach.²⁴ The appeal to letters from the beginning rather than the end of the alphabet has in no way changed the underlying semantic picture.

Dummett (1973: 16–18) argued that there was “no real contrast” between the Tarskian account of the formation of quantified sentences deploying variables and Frege’s account that deploys names. Dummett’s reason is that

. . . a free variable is treated exactly as if it were a proper name at every stage in the step-by-step construction of a given sentence up to that at which a quantifier is to be prefixed which will bind that variable. (1973: 17)

To a certain extent, we agree with Dummett here. In both approaches, a quantified sentence $\forall \mathrm{xRxb}$ is first formed by applying the predication rule $\rho$ to a predicate and some terms. The relevant complex predicate is then constructed by applying an abstraction rule, before applying the quantification rule. Thus, the Tarskian and Fregean syntactic derivations are structurally isomorphic.

Tarskian Derivation: $\Pi (\Lambda (x,\rho (R,x,b)))$

Fregean Derivation: $\Pi (\Delta (a,\rho (R,a,b)))$

Dummett (1981: 284–86), however, went on to argue that the Tarskian semantics gives up the advantages of the Fregean approach, namely, that the denotation of a complex sentence can be understood in terms of the truth conditions of the closed atomic sentences from which it is derived. However, we think the equivalence goes the other way. Given that the complex predicate $\widehat{a}\mathrm{Rab}$ is derived from the name $a$ and the sentence $\mathrm{Rab}$ , the denotation of the name $a$ must include more than its referent and the denotation of the sentence $\mathrm{Rab}$ must include more than its truth-value (or truth-conditions). Otherwise, $\widehat{a}\mathrm{Rab}$ will be identical to $\widehat{a}\mathrm{Rba}$ , should $a$ and $b$ happen to have the same denotation.

3.3. Identification of Alphabetic Variants

We now turn to the third, and final, element of Frege’s approach to quantification: F3. As Dummett remarked, Frege’s syntactic rule removes or excludes a name such as ‘Annabel’ from a sentence such as ‘Annabel waves’, leaving a gap or mark of incompleteness $\xi$ . The resulting predicate abstract can be written as ‘ $\xi$ waves’. Removing a different name, ‘Hazel’, from a corresponding sentence, ‘Hazel waves’, results in the same predicate abstract, ‘ $\xi$ waves’. Generalizing, if predicate abstract ${\pi }_1$ results from the application of the removal operation $\Delta$ to a name $\alpha$ and a sentence ${\varphi }_{\alpha }$ and ${\pi }_2$ results from the application of the removal operation $\Delta$ to a name $\beta$ and a sentence ${\varphi }_{\beta }$ differing from φα only by the total proper substitution of $\alpha$ for $\beta$ , then ${\pi }_1$ and ${\pi }_2$ are syntactically identical. On this view, alphabetically-variant complex predicates are strictly and literally identical:

$\Delta (a,\mathrm{Rab})=\Delta (c,\mathrm{Rcb})$

So, contrary to a more standard syntax (§3.1), there are multiple ways to syntactically derive one and the same complex predicate. The metaphor often used here is “removal” of a name (or as Frege says “einen Eigennamen ausschließen”) from the sentence—the names that contribute to their derivation do not show up in the finished product. This is why the resulting predicate abstract ‘ $R\_b$ ’ has a gap or mark of incompleteness.

$\Delta (a,\mathrm{Rab})=\text{'}R\_b\text{'}$

$\Delta (c,\mathrm{Rcb})=\text{'}R\_b\text{'}$

The name $a$ doesn’t occur in the resulting predicate abstract ‘ $R\_b$ ’, but it does figure in a syntactic derivation of the predicate abstract.

For many formal languages one can read a syntactic derivation off of the part-whole structure of a derived expression. There is therefore a temptation to conflate the notions of parthood and syntactic derivation. However, Frege’s rule for constructing complex predicates is a transformation rule (in the sense of Chomsky 1957) requiring us to delete all embedded occurrences of a name from the output string. One may specify this string as result of applying the construction rule $\Delta$ to $a$ and $\mathrm{Rab}:\Delta (a,\mathrm{Rab})$ . Or, one may specify this string intrinsically as a certain sequence of marks (e.g., ‘ $R\_b$ ’, ‘ $R\xi b$ ’, ‘ $R\mathrm{\dots }b$ ’ or ‘ $R•b$ ’) or as a Gödel number. But the latter is an issue for the science of calligraphy not logic.²⁵

In Frege’s own semantics, there remains a hint of “variables” in the typography of quantified formulae because these result from saturating a dispersed quantifier sign which contains a Fraktur letter ‘’ with a complex predicate such as ‘ $R\xi b$ ’, yielding ‘’. The apparent variable ‘a’ is not a genuine expression of the language but only a typographic feature of the dispersed quantifier sign. Some have seen this as a sign that variables—insofar as they show up in Frege’s approach to quantification—are mere punctuation.

The advantages of Fregean over Tarskian predicate logic are due to the former’s treating variables not as meaningful lexical items, but as mere marks of punctuation, similar to parentheses. (Wehmeier 2018: 1)

It has been pointed out that even this modest appearance of variables is unnecessary, cf. Potter (2020: 38). There is a tradition issuing independently from Quine and Bourbaki of typographically writing quantified formulae without explicit variables at all (see Quine 1940/1981: 69–70 and Bourbaki 1954).²⁶ Rather than devices such as ‘ $a$ ’ marking the positions bound to the quantifier, this approach manually connects the bound positions to the quantifier sign. Kaplan motivates the position as follows:

We need no variables. We could permit gaping formulas (as Frege would have had it) and use wiring diagrams to link the quantifier to its gaps and to channel in values. (Kaplan 1986: 244).

The suggestion is to render a quantified formula such as $\forall x\exists y(\mathrm{Ryx}\to \mathrm{Rxy})$ as the following wire diagram:

This approach fully identifies alphabetic variant quantified formulae and not merely predicate abstracts. This has been taken by many to be an ultimate vindication of the Fregean approach.²⁷

. . . if we adopt the Quine-Bourbaki notation, then we will not even be able to ask whether typographically distinct variables like ‘ $x$ ’ and ‘ $y$ ’ have different ‘semantic roles’. (Button & Walsh 2018: 14)

. . . variables could be completely eliminated from Fregean predicate logic in favor of the graphical “bonds” once proposed by Quine as a means to indicate the dependence of argument positions on outlying quantifiers. (Wehmeier 2018: 215)

However, we see things differently. Variables do not reside in the typographic presentation of quantified formulae or complex predicates, but in the syntactic derivations of those quantified formulae or complex predicates. The identification of alphabetic variant quantified formulae or complex predicates does nothing to relieve the tension between truth and reference centrality and compositionality.

In particular, Fregeans must still offer a syntactic derivation of quantified formulae and complex predicates. For example, in the derivation of a quantified formula such as

the derivation of the complex predicate ‘ $R\_b$ ’ still runs by way of applying a syntactic operation $\Delta$ to a name and a sentence (e.g., $a$ and $\mathrm{Rab}$ , or $c$ and $\mathrm{Rcb}$ , or $e$ and $\mathrm{Reb}$ , etc.). And then Π is applied to the result as follows:

For such an operation to be compositional, the name cannot denote its referent and the sentence cannot denote its truth-value for all of the reasons we have already (perhaps tediously) enumerated. The reasons are essentially the same as we saw in §3.2. The syntactic identification of alphabetic variants doesn’t help. For example, the left-hand sides of the following ought to be distinct, but their right-hand sides would be equivalent, if $a$ and $b$ co-refer.²⁸

A critic might suggest that we are missing the significance of the identification of alphabetic variants on the Fregean approach, because we wrongly suppose that compositionality requires that the semantic value of a complex expression is determined by the semantic values of the expressions that occur in its derivation rather than by the semantic values of its “parts”.²⁹ We follow the standard characterization of compositionality, which demands that the denotation of a derived expression is determined by the denotations of the expressions from which it is syntactically derived (cf. Pagin & Westerståhl 2010). The critic would argue that the denotation of a derived expression must be a function of the denotations of the parts of the derived expression. Because the name $a$ and the sentence $\mathrm{Rab}$ are not parts of quantified sentence the critic insists that one cannot draw conclusions about the denotations of $a$ or $\mathrm{Rab}$ from the denotation of the quantified sentence.

However, the conception of compositionality that appeals to parthood in this sense (i.e., parts of derived expressions) is insufficiently general to capture Frege’s approach. The outputs of Frege’s syntactic theory, the Begriffsschrift expressions, are (two-dimensional) arrays of symbols. But, since it may happen that one and the same Begriffsschrift expression can be constructed in more than one way, the components of the array do not reflect the syntactic derivation or semantic evaluation of the expressions that Frege offers. Moreover, they simply lack sufficient structure to calculate their semantic values without appealing to their syntactic derivations.³⁰ To give one example: Frege syntactically derives a formula that we might represent as ‘ $\forall x(x=x)$ ’ by first deriving ‘ $a=a$ ’ from the dyadic predicate ‘ $\xi =\zeta$ ’ and a name ‘ $a$ ’. He then removes occurrences of ‘ $a$ ’ to yield the monadic predicate ‘ $\xi =\xi$ ’, which is fed into the quantifier ‘ $\forall x(\dots x\dots )$ ’, resulting in ‘ $\forall x(x=x)$ ’.³¹ In the resulting formula, it is not clear whether the monadic predicate ‘ $\xi =\xi$ ’, the dyadic predicate ‘ $\xi =\zeta$ ’, or both are present as parts. Thus, it’s not clear how to apply the principle of compositionality on the basis of the orthographic parts of the formula ‘ $\forall x(x=x)$ ’. But this is not a problem. What matters to the semantic evaluation of a formula is its syntactic derivation.³²

So on the common, general conception of compositionality we are concerned with, the semantic value of a complex expression need not be determined by the semantic values of its literal parts and their arrangements (after all, formulae may be Gödel numbers). Rather, the semantic value of a complex expression is determined by the semantic values of the expressions from which it is derived and the semantic significance of the derivation rule. This conception of compositionality is common in contemporary formal semantics (cf. Pagin & Westerståhl 2010).

The two ways of talking about compositionality are not actually opposed. Rather, the conception of compositionality in terms of the parts of an expression agrees with the derivational conception of compositionality, whenever the parts of an expression mirror its syntactic derivation. Thus, the part-whole conception of compositionality is a special case of the general, derivational conception of compositionality. Its range of application is restricted to languages where the output formula has constituents that do match its derivation.³³ Evaluated by the standard of compositionality appropriate to Frege’s syntactic theory of Basic Laws §30, we believe—contrary to Dummett, Evans, etc.—that the Fregean approach does not reconcile compositionality with truth and reference centrality.

4.1. Rejecting compositionality

Frege syntactically analyzes a quantificational statement—which we notate as $\forall x{\varphi }_x$ —as deriving from a quantifier $\forall$ and a complex predicate. The complex predicate is derived by applying a syntactic operation to a name $\alpha$ and sentence ${\varphi }_{\alpha }$ —Frege would represent the output of the operation as ${\varphi }_{\xi }$ . In Grundgesetze (§31), Frege offers suggestive rules for determining the denotation of the complex predicate and in turn the quantified statement. In particular, Frege proposes that $\forall x{\varphi }_x$ is true if ${\varphi }_{\xi }$ is true for every argument. He then suggests that ${\varphi }_{\xi }$ is true for every argument if and only if ${\varphi }_{\beta }$ is true for every name $\beta$ . Such passages suggest a substitutional rendering of the abstraction rule and quantification more broadly. For example, Dummett says that on Frege’s view “[i]n the case of a complex predicate, the notion of the predicates’ being true or false of an object is derivative from that of the truth or falsity of the sentence which results from filling the argument place of the predicate with some name of the object” (Dummett 1973: 405, cf. 521ff.; see also Evans 1977 and Heck 2012: 64).

These glosses on Frege’s view have been taken to suggest a substitutional reading where the clause for quantification (building in abstraction) would be as follows:

Substitutional Quantification Rule:

$〚\forall \nu \varphi 〛=\{\begin{array}{l}\text{True}, \text{if for call names} \beta ,\hspace{0.17em}\hspace{0.17em}〚\varphi [\beta /\nu ]〛=\text{True},\hfill \\ \text{False, otherwise}.\hfill \end{array}$

Although certain passages of Frege point towards such a clause, Frege’s intentions here have been an issue of controversy (see, e.g., Stevenson 1973).³⁴ Interpreting Frege substitutionally is in tension with Frege’s insistence that functions have a value for every possible argument—unless it is assumed that every object has a name (Heck 2012: 56–57; Dummett 1973: 17–19). We won’t weigh in on the inter- pretative issue, but it seems clear that insofar as a Fregean acknowledges unnamed objects they ought not advocate for the simple substitutional quantification rule.³⁵

Benson Mates’s textbook treatment of first-order logic from the 1960’s develops Frege’s approach in a modern, model-theoretic setting (Mates 1965); cf. Evans (1977: 473–77), Bostock (1997: 84–86), and Heck (2012: 53–64). While model- theoretic semantics is absent from, and possibly even in conflict with, Frege’s own program, contemporary semantic theories in a Fregean tradition often make use of models.³⁶ A model $\mathfrak{M}$ is an interpretation of the basic expressions of the language. The interpretation of a name $\alpha$ is an individual $\mathfrak{M}(\alpha )$ and the interpretation of an atomic $n$ -ary predicate $\pi$ is a set of $n$ -tuples, $\mathfrak{M}(\pi )$ . Each model $\mathfrak{M}$ gives a model-relative denotation function ${〚.〛}^{\mathfrak{M}}$ for every expression of the language.

Mates offers a simple model-theoretic semantics for the quantifier-free sentences. He then offers recursive truth conditions of quantified sentences based on the interpretations assigned ultimately to the quantifier-free sentences. The approach is broadly Fregean because the semantic evaluation of quantified sentences reduces to the semantic evaluation of closed sentences rather than open sentences. It thus appears that these approaches “run their recursion directly on truth” (Evans 1977: 475) and thereby preserve truth and reference centrality.

Mates’s treatment of predicate logic has a standard Tarskian syntax with variables, where quantifiers, such as ‘ $\forall x$ ’, are sentential connectives (see §2). Yet, unlike Tarski, Mates does not appeal to variable assignments in the definition of truth for quantified sentences. In fact, Mates’s semantics doesn’t interpret the variables and open sentences at all. Instead, Mates’s strategy provides the semantics of quantified sentences via the reinterpretation of names and closed sentences. The semantics of the quantified sentences are given in terms of reinter- pretations of certain quantifier-free sentences. Namely, those that are related to the (open) formulae embedded under the quantifier by substitution of a name for the occurrences of a variable (see Mates 1965: 54–63). This is carried out in terms of what Mates calls $\beta$ -variant models:

Definition. A model $\mathfrak{M}$ is a $\beta$ -variant of $\mathfrak{M}$ ′ iff $\mathfrak{M}$ and $\mathfrak{M}$ ′ differ at most in what they assign to the name $\beta$ .

Given the notion of a $\beta$ -variant model, a quantified sentence $\forall \nu \varphi$ is true in a model $\mathfrak{M}$ if and only if for the first name $\beta$ that doesn’t occur in $\varphi$ the sentence $\varphi [\beta /\nu ]$ is true in every $\beta$ -variant model $\mathfrak{M}$ ′. Or to keep in line with the notation we have been employing, where $\beta$ is the first name that doesn’t occur in $\varphi$ , Mates’s rule is as follows:

Mates’s Rule:  ${〚\forall \nu \varphi 〛}^{\mathfrak{M}}=\{\begin{array}{l}\text{True}, \text{if for all} o,\hspace{0.17em}\hspace{0.17em}{〚\varphi [\beta /\nu ]〛}^{\mathfrak{M}[\beta /o]}=\text{True}\hfill \\ \text{False, otherwise}.\hfill \end{array}$

Although this semantics may lay claim to preserving truth and reference centrality, it obviously fails to preserve compositionality. In particular, the semantic evaluation of a quantified sentence such as $\forall \mathrm{xRxa}$ does not proceed by way of the denotation of $\mathrm{Rxa}$ from which it is derived. Indeed, this formula is given no interpretation in the model.³⁷

While Mates’s particular implementation of his approach is obviously non-compositional, perhaps the appeal to model relativity can be used to restore compositionality.³⁸ Maybe Mates went wrong by assuming that quantified sentences are syntactically derived along Tarskian lines from variables and open sentences while he proposed to semantically evaluate them in terms of related closed sentences. The Fregean syntactic strategies discussed above based on F1, F2, and F3, might aid us in rehabilitating the Fregean semantics.

This approach analyses a quantified sentence as derived from a quantifier $\forall$ and a complex predicate $\widehat{\alpha }\varphi$ . The complex predicate $\widehat{\alpha }\varphi$ syntactically derives from an abstraction rule applied to name $\alpha$ and closed sentence $\varphi$ . A semantic approach similar to that offered by Mates could be developed to offer a model-relative interpretation of the complex predicate $\widehat{\alpha }\varphi$ in terms of model-relative interpretations of the closed sentence $\varphi$ . The model-relative denotation of a predicate abstract $\widehat{\alpha }\varphi$ is the function that takes an object o and returns true if and only if $\varphi$ is true relative to a model $\mathfrak{M}[\alpha /o]$ differing from $\mathfrak{M}$ only by assigning object o to the name $\alpha$ .

Model-Relative Abstraction Rule: ${〚\widehat{\alpha }\varphi 〛}^{\mathfrak{M}}=\lambda o.{〚\varphi 〛}^{\mathfrak{M}[\alpha /o]}$

This semantic clause recursively assigns denotations to all predicate abstracts relative to a model in terms of model relative denotations of the basic expressions. One might hope that there is no difficulty in preserving compositionality while holding onto truth and reference centrality.

Closer inspection dashes this hope. Since we have introduced explicit model relativity, we must talk in terms of the model-relativized versions of the principles at play in the dilemma.

model-relative truth and reference centrality: The denotation in a model of a term is an object and the denotation in a model of a sentence is a truth-value.

We must also speak of a language being compositional relative to a model. If the language is compositional relative to a model, then the denotation of a derived expression relative to model  $\mathfrak{M}$ is a function of the denotations of the expressions from which it is immediately derived relative to model  $\mathfrak{M}$ . Thus, where $\eta$ is a syntactic rule, compositionality relative to a model says the following:

model-relative compositionality: $\text{If}\hspace{0.17em}\hspace{0.17em}\alpha =\eta \hspace{0.17em}\hspace{0.17em}({\beta }_1,\dots {\beta }_n)\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}\gamma =\eta ({\delta }_1,\dots {\delta }_n)$

$\mathrm{then}\mathrm{if}{〚{\beta }_i〛}^{\mathfrak{M}}={〚{\delta }_i〛}^{\mathfrak{M}}\hspace{0.17em}(\mathrm{for}\mathrm{all}i),\hspace{0.17em}\hspace{0.17em}\mathrm{then}{〚\alpha 〛}^{\mathfrak{M}}={〚\gamma 〛}^{\mathfrak{M}}$

If the Fregean semantics on offer accepts model-relative truth and reference centrality, it violates model-relative compositionality. To see this compare the model relative denotations of the two abstracts considered above, $\widehat{a}\mathrm{Rab}$ and $\widehat{a}\mathrm{Rba}$ . These have different model-relative denotations according to the present Fregean semantics, as can be seen from (i).

(i) ${〚\widehat{a}\mathrm{Rab}〛}^{\mathfrak{M}}=\lambda o.{〚\mathrm{Rab}〛}^{\mathfrak{M}[a/o]}\ne \lambda o.{〚\mathrm{Rba}〛}^{\mathfrak{M}[a/o]}={〚\widehat{a}\mathrm{Rba}〛}^{\mathfrak{M}}$

But supposing that $\mathfrak{M}(a)=\mathfrak{M}(b)$ , it would follow that (ii).

(ii) ${〚\mathrm{Rab}〛}^{\mathfrak{M}}={〚\mathrm{Rba}〛}^{\mathfrak{M}}$

Thus, $\widehat{a}\mathrm{Rab}$ and $\widehat{a}\mathrm{Rba}$ differ only by the substitution of expressions— $\mathrm{Rab}$ and $\mathrm{Rba}$ —with the same model-relative denotation. Even when placed in the model-theoretic setting, the Fregean still faces the choice between truth and reference centrality on the one hand and compositionality on the other—when those are stated in the appropriate model-relative way.

4.2. Rejecting truth and reference centrality

We have argued that neither Mates’s $\beta$ -variant approach nor model-theoretic approaches which might look to Mates for their inspiration allows the Fregean to preserve compositionality. Some Fregeans regard this cost as too high. For example, Wehmeier acknowledges that “[t]he principal challenge in developing Fregean predicate logic is the construction of a compositional semantics” (2018: 214).

To offer a compositional Fregean semantics, Wehmeier shifts his semantic perspective. Rather than offering a denotation relative to a model  $\mathfrak{M}$ , Wehmeier offers an absolute, but model-sensitive, denotation for every expression of the language. Thus, an atomic predication $\mathrm{Rab}$ will not receive a separate denotation ${〚\mathrm{Rab}〛}^{\mathfrak{M}}$ for each model $\mathfrak{M}$ . Instead, the denotation of the sentence will be a function which takes a model $\mathfrak{M}$ as input and outputs the truth-value of the sentence on that model.

These model-sensitive denotations allow Wehmeier to retain compositionality even for the hard case of predicate abstraction. The denotation of a complex predicate $\widehat{a}\mathrm{Rab}$ depends on the referent of $a$ and truth-value of $\mathrm{Rab}$ relative to a range of other models. For any model $\mathfrak{M}$ , the denotation of $\widehat{a}\mathrm{Rab}$ is true on $\mathfrak{M}$ of an object $o$ if and only if $\mathrm{Rab}$ is true relative to a model that differs from $\mathfrak{M}$ only by assigning $a$ to the object $o$ . In symbols,

Fregean Abstraction Rule: $〚\widehat{\alpha }\varphi 〛=\lambda \mathfrak{M}.\lambda o.〚\varphi 〛\hspace{0.17em}(\mathfrak{M}[\alpha \text{/}o])$

Since the semantics is given in terms of the richer model-sensitive values, the counterexamples to compositionality above will be blocked.

While Wehmeier’s semantics is compositional, it flagrantly violates truth and reference centrality. Rather than offering truth-values as the denotations of sentences and objects as the denotations of names, Wehmeier offers functions from models onto truth-values and onto objects, respectively.

. . . the Fregean meanings we are about to construct will . . . be functions defined on the class of all models; indeed, the Fregean meaning. . . of any linguistic item $\mathfrak{s}$ is going to be the function that maps any model $\mathfrak{M}$ to the reference ${〚\mathfrak{s}〛}^{\mathfrak{M}}$ of s in $\mathfrak{M}$ . (Wehmeier 2018: 224)

For example, the denotation of a name $a$ will be the function that maps a model $\mathfrak{M}$ to the value of $a$ in that model. In symbols: $〚a〛=\lambda \mathfrak{M}.\mathfrak{M}(a)$ . The denotation of a sentence will be a function from models to truth-values. The denotation of a name, then, will clearly not be an object, and the denotation of a sentence will not be a truth-value. Moreover, Wehmeier explicitly accepts the consequence that distinct names never have the same denotation (Wehmeier 2018: 234).³⁹

Wehmeier is correct that one can retain compositionality by abandoning truth and reference centrality. Indeed, that is what the Tarskians have done (cf. §2.2) by treating the denotation of a term as a function from assignments to objects and the denotation of a sentence as a function from assignments into truth-values. Comparing the semantic clauses for predicate abstraction on each approach, the difference appears to lie (i) in the appeal to letters from the earlier half of the alphabet rather than the latter half and (ii) in the substitution of $\mathfrak{M}$ for $\sigma$ .

Fregean Abstraction Rule: $〚\widehat{\alpha }\varphi 〛=\lambda \mathfrak{M}.\lambda o.〚\varphi 〛\hspace{0.17em}(\mathfrak{M}[\alpha \text{/}o])$

Tarskian Abstraction Rule: $〚\widehat{\nu }\varphi 〛=\lambda \sigma .\lambda o.〚\varphi 〛 (\sigma [\nu \text{/}o])$

Wehmeier’s clause for predicate abstraction seems to us to be no more than a typographic variant of the Tarskian. A model $\mathfrak{M}$ is an assignment function written in Gothic script. Thus, the model shifting approach should be construed as a form of variabilism, as discussed above.

Wehmeier (2018: 247) argues that his proposal is a more conservative rejection of truth and reference centrality. He points out that in the model-theoretic setting every approach, whether Fregean or Tarskian, involves “a dependency of meanings on models”. He argues that the denotation of an expression must be model sensitive in order to differentiate the meanings of the sentences $\forall \mathrm{xPx}$ and $\exists \mathrm{xPx}$ , where $P$ is empty. The crucial difference between models and assignment functions for Wehmeier is that models reinterpret the predicate constants whereas assignment functions do not. Thus, $\forall \mathrm{xPx}$ and $\exists \mathrm{xPx}$ will be true in different models and will not have the same model-sensitive meaning.

Wehmeier is correct that in classical presentations of Tarskian predicate logic, an assignment function reinterprets only the individual variables and not the upper case letters that act as predicates. However, the reason assignment functions do not reinterpret the predicates in these presentation of Tarskian first-order logic is that the logic is first-order. In particular, there is no binding of predicate variables. By way of contrast, standard presentations of second-order logic—for instance, Shapiro (1991: 72)—do allow assignment functions to interpret predicate letters. Assignments—so construed—can do all of the theoretical work that Wehmeier assigns to models. This is perhaps why Tarski himself appeals to assignment functions in his early definition of logical consequence.⁴⁰ Nothing prohibits one from saying with Tarski (1936/1956b) that models are assignments functions. A model is an assignment function that reinterprets not only the terms but also the predicate letters. Thus, there is no need to introduce both assignment and model sensitivity and Wehmeier’s semantics is not more conservative than Tarski’s.