Characterizing Invariance

I argue that in order to apply the most common type of criteria for logicality, invariance criteria, to natural language, we need to consider both invariance of content—modeled by functions from contexts into extensions—and invariance of character—modeled, à la Kaplan, by functions from contexts of use into contents. Logical expressions should be invariant in both senses. If we do not require this, then old objections due to Timothy McCarthy and William Hanson, suitably modified, demonstrate that content invariant expressions can display intuitive marks of non-logicality. If we do require this, we neatly avoid these objections while also managing to demonstrate desirable connections of logicality to necessity. The resulting view is more adequate as a demarcation of the logical expressions of natural language.

1. Introduction

The standard semantic definition of logical consequence, due to Tarski (1936), relies upon selecting in advance a fixed set of logical expressions. We usually do this by enumeration, treating the logical constants in a typical logic textbook as logical and those not not. Since this suffices for applications, many rest content here. However, when we turn to accurately characterizing the concepts of logical truth and logical consequence as they appear in the informal background language with which we actually do mathematics, things get more complicated. Philosophers and logicians have thus attempted to give a philosophically adequate account of why ‘and’ and ‘all’ are logical, why ‘denounced’ and ‘Deniz’ are not, and that settles in a reasonable way disputed cases like = and ‘uncountably many’. The most widespread such account takes logical expressions to be those whose meaning does not depend on the characteristics of particular objects.

The best precise development of this account of logical expressions identifies them with expressions whose extensions are always invariant under some class of transformation such as permutations (Tarski 1986), isomorphisms (Sher 1991), strong homomorphisms (Feferman 1999), potential isomorphisms (Bonnay 2008), etc. The extension of an expression φ is invariant, under some transformation T, just in case applying T to the extension of φ in a context to does not “change”—in some sense—the extension of φ. This type of approach, dating back to Tarski (1986) and Mautner (1946), and in some sense, to Kant’s view of logic as an abstraction from the content of judgments (MacFarlane 2002; Tolley 2013; Stang 2014), has a number of useful connections with definability (McGee 1996), and seems to capture at least one sense in which the meaning of logical expressions are immune to the particular features of the objects they are applied to.

Though this approach is commonly applied to formal languages, it is subject to intuitive counterexamples, due to Willian Hanson (1997) and Timothy McCarthy (1981), when it is attempted as an analysis for natural language expressions. Their examples show that the meaning of a natural language expressions can stay constant while its T-invariant content intuitively varies, in a sense, from context to context. Though there are problems with their examples, exposed recently by Gil Sagi (2015), we can reconstruct versions of their examples which do work to demonstrate their point. In light of this, I argue that applying invariance criteria to expressions of natural language, as opposed to formal languages, requires distinguishing two ways in which the meaning of expressions can be immune to the particular nature of the objects they are applied to. Doing so not only solves a number of problems, such as the reconstructed versions of McCarthy and Hanson’s examples, but also highlights important connections between logicality and necessity.

2. Sagi’s Response to McCarthy and Hanson

Sagi distinguishes two distinct versions of McCarthy and Hanson’s arguments, the modal and the epistemic, which focus on different ways to motivate and understand the third premise. On the modal interpretation, it claims that if the extension of an expression is fixed by non-logical materials, then it is possible for logical truths to be false, contra the extremely plausible claim that logical truths are necessary.[15] On the epistemic interpretation, the third premise claims that if we fix the extension of logical expressions by a posteriori materials, then there will be logical truths which we could not know a priori since we could not know, in the absence of some a posteriori knowledge, that the logical truth is true.

McCarthy’s and Hanson’s arguments are about the logical properties of expressions in natural language. This is clear from context; if it were simply an abstruse model-theoretic property at issue, arguments involving the intuitive necessity of ‘logical’ expressions and the a priori status of certain truths involving them would be out of place. So when we distinguish between the metalanguage with which we define certain constants and the object language in which they are defined, we are really distinguishing between two ways of taking our own language or some rigorized version thereof, at least in some salient cases. We are distinguishing between the language with which we introduce an expression (call it PseudoEnglish₁) and the resulting language under study in which it has been introduced (PseudoEnglish₂). I shall understand McCarthy’s and Hanson’s examples this way in what follows. One reason for this, suggested above, is that identifying logical expressions only for a rarified collection of formal languages is of significantly less interest than identifying logical expressions in natural language or close analogues thereof.[16] Note, briefly, that we have said little at this point about the details of PseudoEnglish₁—it may contain non-extensional expressions or it may not—and the character of legitimate definitions in PseudoEnglish₁—these may contain non-extensional expressions or they may not.

Sagi objects to the modal interpretation by pointing out that Nφ ↔ ¬φ, a statement of PseudoEnglish₂, actually is necessarily true. Yes, if K were false, the statement expressed by the sentence ‘Nφ ↔ ¬φ’ would not have been necessarily true, but this would be a case in which the PseudoEnglish₁ definition of N yielded an expression with a different meaning, resulting in a new language PseudoEnglish₃. The PseudoEnglish₃ statement N ↔ ¬φ is false, but what of it? PseudoEnglish₃ is a different language. In brief, Sagi argues, correctly in my view, that if we distinguish the sentence ‘Nφ ↔ ¬φ’ from the interpreted statement Nφ ↔ ¬φ, we can see that the former could have been false, but not the latter. But the possible falsity of the former is not of any interest. Against the epistemic interpretation, she again notes that the sentence ‘Nφ ↔ ¬φ’ can have its meaning determined by semantic rules involving a posteriori materials without obviously transferring the a posteriori status onto the statement so determined. Sagi refrains from pursuing the epistemic objection on grounds that it would require saying far more about the a priori.

I agree with both of Sagi’s worries. Her objection to the modal interpretation is obviously cogent and her worry about the transference of a posteriori materials from the account of the meaning of an expression to the a posteriori status of logical truths involving this expression is apt. We can, however, give an alternative version of the argument which skirts Sagi’s objections. Hanson is best understood as claiming that Sher’s account allows that the logical status of the extension of an expression may crosscut the logical status of its meaning (Hanson 1991: 393–4). And McCarthy puts the upshot of his argument as follows:

So, again, if the relevant sense of meaning for the logical status of expressions is not exhausted by general extensions, then Sagi’s defense is incomplete when viewed as a defense of simple invariance as a criterion for logicality for expressions of natural language.

3. Contingency and Invariance

A secondary aim of McCarthy and Hanson’s examples is emphasizing that necessity, in some form, is itself a criterion of logicality. Logical expressions, intuitively and traditionally, should yield truths which are necessary (Hanson 1997). The necessity of logical truths is, presumably, supposed to be a consequence of the insensitivity of logical expressions to the natures of objects they are applied to. At least, a criterion of the logicality of expressions which entailed that all truths composed only of logical expressions were necessary would be all the more plausible. This because it would capture one important criterion of logicality on the basis of a plausibly more fundamental criterion of logicality. However, there is a problem here. We now have two senses of meaning—content and character—in play and, correspondingly, two senses of insensitivity in play. Invariance of character, even intuitively understood, does not guarantee necessity in the usual truth-at-all-worlds sense. This can easily be seen by noting that ‘I am here now’ is not necessary in the usual sense—possibly, I’m not. From the other direction, assuming the Kaplanian framework, we can lay down semantic rules for expressions like Ni whose content will invariably be invariant, but where which invariant content is denoted is determined, in a context of use, by contingent matters at the context of use; unlike ‘I am here now’, whether or not such claims will be true depends on which context we utter them from.

As a result, there are two senses in which a claim like Ni [∃x x = x] ↔ ¬[∃x x = x][22] can be necessary: a sense corresponding to truth at all contexts of evaluation and a sense corresponding to truth at all contexts of use. These two senses are familiar from the distinction between deep and shallow necessity drawn by Evans (1979) and Davies and Humberstone (1980). Slightly simplified, a sentence φ is shallowly necessaryinv if and only if:

$$\textrm{$\forall c$ $\phi_@(c)= {\bf T}$}$$

That is, when the content of φ determined at the actual context @ is the constant function from contexts to true. A sentence φ is deeply necessary if and only if:

$$\textrm{$\forall c$ $\phi_c(c)= {\bf T}$}$$

that is, when, for every context c, the content of φ determined at context c is true at c. Note that this distinction may, but need not, track any fancy view about actuality, content, and contexts as possible worlds. In order to make the relevant distinction, we merely need that there is a way to track contexts of use which fix the content of a notion and contexts of evaluation for that content.

Of course, given the way we have set up the criterion of logicality, there will be many true sentences which are composed only out of logical expressions, but which are not necessary. For instance, ∃x, y x ≠ y, which expresses that there are at least two objects, is true but not shallowly necessary since there are at least two objects here, but there are contexts of evaluation containing only one object. Likewise, it is not deeply necessary since there are some contexts of use where it also fails. Yet it contains only content and character-invariant materials. This is a quite general problem, but one which we can finesse for now; our target will be to capture a sense in which claims composed only of logical materials are, if true, necessary. We will thus relativize the sense of necessity to contexts whose domains are isomorphic; i.e., we’ll say that a sentence φ is shallowly necessaryinv if and only if:

$$\forall d[\phi_@(d)={\bf T}\rightarrow \forall c\cong d\textrm{ }\phi_@(c)= {\bf T}]$$

that is, when the content of φ determined at the actual context @ is true at a context d only if it is true at every context isomorphic to d. For deep necessity, we say a sentence φ is deeply necessaryinv if and only if:

$$\forall c[\phi_c(c)={\bf T}\rightarrow \forall d\cong c\textrm{ }\phi_d(d)= {\bf T}]$$

that is, when the content of φ determined at a context @ is true at @ only if the content of φ determined at any isomorphic context d is true at d.

Failure of shallow necessity_inv undermines logical truth (Hanson 2006). If a claim is not shallowly necessary, then its truth can vary from context to context, even when these contexts contain the same number of objects. And this kind of contingency seems to sit badly with the idea that logic is topic-neutral on one disambiguation of this claim—that is, that the truth of a logical truth is immune to swapping around facts about the relations with a context. Logical truths, especially those constructed from purely logical expressions, should be true in whatever context we evaluate them from, unlike ‘there are orange cats’. However, certain statements which are not shallowly necessaryinv are still necessary in some sense, such as ‘I am here now’ or ‘I exist’. These are true when they are evaluated in the context of use and are thus deeply necessaryinv in the sense defined above.[23]

Plausibly, deep contingencyinv also undermines logical truth. If a claim is deeply contingentinv, then its content varies from context of use to context of use, even if its character stays constant. This kind of contingency sits rather badly with the idea that logic is topic-neutral on a different, but highly plausible disambiguation of this claim—that the content of a logical notion is immune to worldly facts about the context of use. Logical truths, especially those constructed from purely logical expressions, should be true wherever we assert them from, unlike ‘I am over six feet tall’ and potentially unlike ‘There are no unicorns’.

Call a claim that is necessary in both senses strongly necessaryinv. Truths constructed from character- and content-invariant materials are strongly necessaryinv. Ni [∃x x = x] ↔ ¬[∃x x = x] is not. If its content is fixed in a K context, it is true and (shallowly) necessarilyinv so. From a ¬K context, it is false and (shallowly) necessarily so. So it is shallowly necessaryinv and deeply contingentinv. On the face of it, Hanson’s and McCarthy’s intuition that Niφ ↔ ¬φ is not intuitively necessary seems probative and corresponds to the failure of strong necessityinv. True claims constructed from purely logical expressions should be always true—from any context of use, at any context we evaluate it at, modulo our above point about the size of the context—because their truth depends only on the meaning of logical expressions, no matter where the contents of these expressions are fixed.

We thus need an explication of topic neutrality which guarantees that the meaning (character) of the components of these truths are such that no matter what context we assert or define them, they turn out to be true at every point of evaluation given that context. We obtain this result by following the suggestion above, and requiring that logical expressions not only have invariant contents, but also invariant characters. (See the appendix for precise definition of these notions.) Double-Standard Invariance—claimed here only as a necessary, not a sufficient condition—again does a far more plausible job at carving the logical from the non-logical.

4. An Alternative Approach: Sher’s Rigidity Constraint

Gila Sher, in her seminal discussion of invariance criteria (1991), proposed that a conditions of being a logical constant was that it rigidly denote its extension:

Sher’s stated aim is a characterization of logical constants in logical languages (2003: 197) and, in particular, logical languages of a Tarskian stripe. This motivates Sher to obtain rigidity by requiring that constants be defined by a single extensional function and identified with their extensions. Though Sher’s condition works nicely for the languages she considers, the restriction to them seem less than perfectly motivated for a general account of logical constants. Why, for example, should we require that our metalanguage definitions of logical constants be extensional? Likewise, why only consider logical languages which can be so treated? Typical logical languages and accounts of logical expressions are often like this, but this fact should be explained. Anyways, our aim is the broader one of characterizing logicality for natural language. This may not be Sher’s project, but is one for which her claim about rigidity is useful, if not perfectly apt.[25]

Logical constants should be rigid in Sher’s sense; non-rigid expressions, even when they have invariant extensions, are not intuitively logical.[26] Sher’s point seems to be that expressions which change their meaning from world to world are intuitively non-logical expressions. Logical expressions should be rigid, but as we have seen above, once we move to the Kaplanian framework described above, there are two senses of meaning on offer. Rigidity captures the idea that the content of an expression is constant over worlds, but it does not capture the sense in which character is constant over context. ‘I’, famously, is a rigid designator of me; it denotes at every world, though in a different context, it would denote you in every world. Likewise, ‘Ni ’ is rigid, denoting either the trivial functor or negation, though which is a function of whether or not K is actually true. We seem to need here rigidity both of content and of character to capture the sense in which the meaning of logical expressions does not change.

So Sher’s account, even if adequate for the particular logical languages she is considering, seems less adequate for our purposes than one which takes seriously her (and others, such as Feferman, McGee, and Hanson’s) informal remarks about meaning and develops an independent criterion which underwrites her point about rigidity. Sher is aware of this limitation. She remarks in her (2001) that it matters little for her purposes, but significantly for other purposes, whether we ban empirical logical predicates—predicates which rigidly denote invariant extensions, but are defined by reference to contingent empirical materials.[27] However, adding a demand for character invariance to the account of logicality is better than banning such empirical predicates or treating them as a special case, even given her aims. It explains in a general and illuminating way why we should mark such predicates as non-logical.[28]

As any character- and content-invariant expression will be rigid in Sher’s sense, but not conversely, the double-standard I am advocating seemingly improves on Sher’s useful point above about rigidity in the context of natural languages. In fact, as the double-standard I advocate above implies rigidity in her sense,[29] we lose none of the cases Sher wants to exclude by means of her rigidity constraint, but additionally fend off intuitive counterexamples to Sher’s view like Ni.[30] We can thus see double-standard invariance as a way of making sense of Sher’s point in non-extensional contexts.[31]

5. Conclusion

Taking stock, I have argued that there are defensible versions of McCarthy and Hanson’s modal and epistemic arguments against simple invariance, arguments that strongly suggest we ought to adopt double-standard invariance instead. The modal argument goes by way of claiming that invariance of content is insufficient to guarantee deep necessityinv , but that deep necessityinv should hold for any truth containing only logical expressions. The epistemic argument goes by way of showing that grasp of the character or semantic rule for logical constants is insufficient to determine a priori that truths constructed out of them are true. But this also seems required for logicality.

Deep necessityinv and a priori discoverability from grasp of the character of an expression are, by themselves, insufficient for logicality. Consider ‘I am here now’. This is deeply necessary—from any (normal) context, our assertion of this will be true. We can infer this fact a priori from our grasp of the meaning (character) of ‘I’, ‘here’, and ‘now’. Likewise with other ‘logical’ truths of the logic of actuality and the logic of indexicals. But though ‘I am here now’ is deeply necessary, ‘Necessarily, I am here now’ is false. I might not have been. And logical truths should not be contingent in this way.[32] Shallow necessity, as I have argued above, is arguably insufficient for logicality as well. A joint demand seems preferable since all truths constructed from standard logical expressions are necessary in both senses (subject to the ‘inv ’ modifier discussed above.)

For many mathematical and logical definitions, the distinction between character and content collapses. But once we look to define expressions for natural language, which contains more complex expressions such as indexicals, we can and should separate content and character. Once we have done so, it seems clear that simple content invariance is insufficient for logicality. And this I take to be at least part of the lesson of Hanson’s and McCarthy’s examples.

6. Appendix – A Sketch of Character Invariance

We sketch here the formal details of the notion of character invariance and the connection of invariance to necessityinv, working with an unmodalized language for simplicity.[33] Let W be a non-empty set and D a function from W into a family of non-empty sets. We will call w ∈ W a world (since it is our surrogate for ways things could have been) and D^w the domain of w. Let D^∗ = ∪w_∈W D^w. We build a type-hierarchy over D^∗ in the usual way. A context is a pair ⟨w, s⟩ of a member w of W and an object s ∈ D^w (the speaker). Given a language L, we interpret expressions in L by assigning them contents according to the character of the expression, given a context. General extensions—contents in the sense we’ve been using—can be modeled by functions from W to members of the type-hierarchy built over D^∗, appropriately to semantic type and member of W. The type of a unary predicate is, for example, a function from D^∗ to {T,F}, the type of a unary and monadic quantifier being a function from functions from D^∗ to {T,F} to {T,F}, etc. A character is modeled by a function from contexts to contents.

So, for example, the character of ‘I’ is the function that takes a context ⟨w, s⟩ to the function from each member of W to s. The character of ‘cat’, being constant, is the function that takes a context ⟨w, s⟩ to the function from W to members of the type-hierarchy over D^∗ such that w ↦ f where:

$$\textrm{$f(a)={\bf T}\Leftrightarrow a$ is a cat in $w$.}$$

The character of =, again being constant, is the function that takes a context ⟨w, s⟩ to the function from W to members of the type-hierarchy over D^∗ such that w ↦ f where:

$$\textrm{$f(\langle a,b\rangle)={\bf T}$ $\Leftrightarrow$ $a=b$}.$$

Likewise, the character of ‘there are’, also being constant, is the function from W to members of the type-hierarchy over D^∗ such that w ↦ f where:

$$\textrm{$f(g)={\bf T}\Leftrightarrow \{a$ $| g(a)={\bf T}\}\subseteq D^w$ and $\{a$ $| g(a)={\bf T}\}\neq\varnothing$.}$$

Finally, the character of Ni is not constant, being modeled by the function:

$$g(\langle w,s\rangle)= \begin{cases} v\mapsto \neg &\qquad\text{$K$ is true at $w$}\\ v\mapsto id_{\{{\bf T,F}\}} &\qquad\text{$K$ is false at $w$} \end{cases}$$

As usual, when the function picked out is a characteristic function—i.e., one whose range is a subset of {T,F}—we will treat it as the set; a unary predicate being the subset of D^∗ , a binary relation being a subset of D∗ × D∗ , a unary quantifier being a subset of ℘(D), etc.

Since all objects occurring in a domain of a world occur in D^∗ and we’ve drawn our contents from D^∗ , we will treat bijections between domains of worlds in terms of permutations on D^∗ (that there will be many permutations of D^∗ for each bijection from Dw to Dv will cause no problem given our purposes). Given a permutation π of D^∗ , we extend it to a function π⁺ on all members of the type-hierarchy over D^∗ . We set π⁺(T) = T, π⁺(F) = F, and for w ∈ W, π⁺(w) = w. For all d in D, let π⁺(d) = π(d). For an ordered n-tuple ⟨m₁, . . . , m_n⟩ of members of the type-hierarchy,

$$\pi^+(\langle m_1,\ldots,m_n\rangle)=\langle \pi^+(m_1),\ldots,\pi^+(m_n)\rangle.$$

Given a function f on types, π⁺( f ) is the function composed of π⁺, f, and the inverse of π⁺: π⁺ ◦ ( f ◦ π^+⁻¹ ).

We say that a content co is isomorphism invariant on W and D when for any permutation π on D^∗ and pair of worlds w and w′ such that π⁺(D^w ) = D^w′:

$$\pi^+(co_w)=co_{w^\prime}$$

We say that co is simply isomorphism invariant when it is isomorphism invariant on W and D for any W and D. The content of the predicate ‘stockbroker’ is not isomorphism invariant. Given our actual world, @, the slightly better world w in which there are more revolutionaries and fewer stockbrokers and a permutation π on D^∗ such that π⁺(D^@) = Dw, π⁺ (broker_@) ≠ broker_w. On the other hand, it is easily verified that, for any permutation π, a pair ⟨s₁, s₂⟩ ∈ π ⁺(=@) just in case ⟨π⁺(s₁), π⁺(s₂)⟩ ∈ =_w, and so = (the content of identity) is isomorphism invariant; this case, however, is borderline uninteresting as the content of = is the same on every w ∈ W. A more interesting case is ‘there is’, and here it is again easily established that for any permutation π such that π⁺(Dw ) = Dv , A ⊆ D^w just in case π⁺(A) ⊆ D^v and A ⊆ D^v just in case π^+⁻¹ (A) ⊆ D^w, so the set of non-empty subsets of Dv is the same as result of applying π⁺ to the set of non-empty subsets of Dw.

We need an additional bit of apparatus for character invariance. Let an @-and-s preserving map from ⟨w, s⟩ to ⟨w′, s′⟩ be a pair π, π_ω such that

1. π is a permutation of D^∗,
2. π_ω is a permutation of W,
3. π(s) = s′ and π⁺(Dw ) = D^w′ ,
4. π_ω (w) = w′ .

So π permutes D^∗ while preserving the speaker-role, while π_ω permutes the set of worlds, preserving the actual world role. We extend the pair π, π_ω to a single function π⁺ in the obvious way.[35]

We say that a character ch, as modeled in the set of contexts drawn from W and D, is isomorphism invariant for W and D when for any pair ⟨w, s⟩, ⟨w′, s′⟩ (w, w′ ∈ W and s, s′ in D^∗ ) and @-and-s-preserving map π, π_ω:

$$\pi^+(ch_{\langle w,s\rangle})=ch_{\langle w^\prime,s^\prime\rangle}$$

That is, where applying π⁺ to the image of ch under ⟨w, s⟩—ch_{⟨w, s⟩}—results in ch_{⟨w′, s′⟩}. We say that a character for an expression φ is isomorphism invariant when no matter what W and D we choose, the resulting modeling of the semantic rule for φ as a function from contexts drawn from W to functions from W to the type-hierarchy over D^∗ is isomorphism invariant for W and D.

Now, given an @-and-s preserving map π, π_ω between ⟨@, me⟩ and ⟨@, Fred⟩, π⁺(me)=Fred. I _{⟨@, me⟩} is the constant function from any w ∈ W to me, so π⁺ of I _{⟨@, me⟩}is the constant function from any w ∈ W to Fred. But, of course, this is exactly I _{⟨@, Fred⟩}. The invariance of the character of ‘I’ is borderline trivial given our definition. But this is as it should be—invariance of character tracks the fact that the semantic function of an invariant expression only depends on structural features of the context, such as the designated speaker. It does not matter who the speaker is, merely that there is one. We need @-and-s preserving maps here, not simply permutations of D^∗ , since setting character invariance up in terms of the latter would eradicate the possibility of invariance for notions like ‘I’ whose content differs depending on the choice of speaker.[36] The definition given above, of course, generalizes for larger indexes, though we will need a corresponding generalized notion of an index-preserving map. Here we will stick with the simple speaker-only index.

Ni has variant character. Let K be false at w and true at @ and π, π_ω an @-and-s preserving map between ⟨@,s⟩ and ⟨w,s⟩. Ni_⟨@,s⟩ is the function such that Ni_⟨@,s⟩ (v) = ¬ for all v ∈ W. But π⁺(Ni_⟨@,s⟩) is Ni_⟨@,s⟩ which is not identical to the function Ni_⟨w,s⟩—rather, Ni_⟨w,s⟩ is the constant function from w ∈ W to id _{T,F} .But both Ni_⟨@,d⟩ and Ni_⟨w,d⟩ are (trivially) content invariant. In contrast, ‘I’ has invariant character, but variant content. I_⟨@,me⟩ = g where g is the constant function from W to me. Now, given π swapping only myself and Fred,

$$\pi^+(g_v)=\pi^+(\textrm{me})=\textrm{Fred}\neq g_v$$

A little work shows that the usual logical constants, as well as standard cardinality quantifiers, have both invariant character and invariant content.

Call a statement shallowly stable just in case if it is actually true, it is shallowly necessarilyinv true, and if actually false, shallowly necessarilyinv false. Being constructed entirely from content-invariant expressions implies shallow stabilityinv.

Call an expression deeply stable just in case if it is actually true, it is deeply necessarilyinv true and if actually false, deeply necessarilyinv false. Being constructed entirely from character-invariant expressions implies deep stability:

So, stringing these identities together, we have

$$\phi_{\langle w,s\rangle}(w) = \phi_{\langle v,t\rangle}(v)={\bf T}$$

and similarly for F.

So if logical expressions are both character and content invariant—that is, if they are strongly invariant—then sentences composed entirely of them are both deeply and shallowly stable and hence strongly stable. Extending these methods to mixed sentences and to modalized language awaits another opportunity.[37]

Acknowledgements

Thanks to Corine Besson, Denis Bonnay, Tim Button, Catharine Diehl, Paul Egré, William Hanson, Eliot Michaelson, Julien Murzi, Beau M. Mount, Gil Sagi, Florian Steinberger, and a couple of helpful referees for very valuable discussion. Thanks also to audiences at Leeds University, the School of Advanced Study at the Institute of Philosophy, University of Western Ontario, and the Munich Center for Mathematical Philosophy for very valuable feedback. The roots of this paper grew out of discussions with William Hanson during my time at the University of Minnesota. Thanks to him for stimulating my initial interest in this issue and being an early bulwark of sensibility against my more farfetched ideas.

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Notes

1. For details and extensive further discussion, see Woods (2014).

2. I do not have space here to discuss the relationship between semantic criteria and proof-theoretic criteria for logicality. My aim here is to make more plausible the best going semantic criteria, not to argue against entirely alternative approaches.

3. Although Woods (2014) and Dutilh Novaes (2014) further argue that permutation invariance is not even necessary for logicality. The former argues that the criterion can be patched. The latter does not.

4. In this context, an isomorphism between domains is simply a bijection. The reader is cautioned to keep this unfortunately well-established terminological choice in mind below and when reading the literature on invariance criteria generally.

5. Many theorists, such as Sher (1991), use invariance as only one necessary criterion in a list of jointly sufficient criteria for being a logical expression. I discuss Sher’s view extensively below, but it is worth noting at the outset that isomorphism invariance is clearly the principal criterion on the list and has been taken as more-or-less a necessary and sufficient condition (perhaps with the addition of Sher’s point about rigidity, described below) by many after her.

6. Recent work by Tim Button and Sean Walsh (in press) suggests that this defense will require more than might initially be thought; in particular, one needs to argue in detail that the sort of non-dependence picked out by isomorphism invariance (or a close cousin) really does capture an intuitive non-dependence property characteristic of the meanings of logical expressions. I remain hopeful that this can be done and hope, in particular, to address their objections and their relation to the work here in another venue.

7. I am here privileging what Bonnay (2008: 6) calls the formality argument. I do not mean to suggest that this is the only way into the invariance criterion; see Bonnay’s discussion of the generality argument and the relationship between the two of them. The issue of what these joint arguments support is aimed at picking which transformations matter rather than at my question below about whether we should be concerned with invariance of extensions, meanings, or both, so I will bracket it.

8. Few have seriously criticized the invariance criterion in its original Tarskian usage as a way of discriminating a set of logical (set-theoretic) objects from non-logical objects, though see (Button & Walsh in press: Chapter 8). See Sher (2001; 2003) for a useful reminder that we ought distinguish the two projects. I offer some reasons to broaden the Tarskian criterion in Woods (2014), but even these are meant as a modification, not a rejection of Tarski’s approach.

9. I have switched McCarthy’s talk of structures for talk of contexts throughout to generalize the point and emphasize that little depends on how we represent the intuitive idea.

10. I believe that Hanson’s argument, as opposed to McCarthy’s, is best interpreted in the way that I will suggest. However, as my point is not to explicate McCarthy or Hanson’s prose, I shall leave interpretive issues at that.

11. Dealing with modalized language adds an entire layer of complexity; I hope to return to this complexity in future work.

12. Nota Bene: ‘contexts of evaluation’ are not the same as the modern-day relativist’s contexts of assessment; rather, these are what Kaplan calls circumstances. Moving to a third role for contexts, assessment in the relativist’s sense, opens the possibility for a third type of invariance. I hope to return to this issue elsewhere. Thanks to Julien Murzi for discussion.

13. See Sher’s (1991: 54–56) discussion of condition (B).

14. I use Kaplanian characters for simplicity, but the approach could be adapted to different semantic pictures. Bonnay, in his dissertation, suggests a similar approach using a Stalnakerian approach to content; I hope to discuss the relation between these two approaches elsewhere.

15. The objection is not that classical “logical” truths like ∃x x = x, which are intuitively contingent, come out necessary—this is unavoidable given the assumption that domains are non-empty and, perhaps, a pragmatic justification of this could be given. It is rather that some logical truths will come out as contingent on this way of specifying logical expressions even if we hold cardinality facts fixed. For discussion of truths constructed out of logical materials which are equivalent to facts about the size of the domain, see Woods (2014: 302–303).

16. For relevant discussion, see Section 4 below.

17. For Kaplan, strictly speaking contents model the notion of a propositional constituent (the sort of thing figuring in a proposition), whereas characters model the notion of a semantic rule. For our purposes here, we can occasionally speak with the vulgar and identify what is modeled with what models it. They will be distinguished where it matters. Thanks to Eliot Michaelson for discussion.

18. Nota Bene: (N) is not an abbreviative definition in PseudoEnglish₁ . Sagi is right to distinguish that case from the case of a metalanguage definition and it is in this latter sense I intend both interpretations of (N).

19. (N) does not explicitly make use of any non-extensional materials; that is, abstracting from the quantification over contexts necessary to make clear which expressions are being used and which are being mentioned, none of the content of the semantic rule is non-extensional. We could, of course, use non-extensional materials in laying down the definition, without changing the overall point in the slightest.

20. ‘Speaker’ could also be interpreted as elliptical for ‘speaker at c’. This is not the intended reading, so put it aside.

21. I have argued elsewhere that we should recognize arbitrary choice operators like ϵ as logical operators in more-or-less the fullest sense and shown that it is possible to set up a principled account of the denotations of expressions which does so (Woods 2014). The character/content distinction, however, provides a fallback position for those, like Breckinridge and Magidor (2012) who treat such operators having a standard, though arbitrarily specified, denotation. They can treat the choice of an arbitrary representative for each predicate as part of the context and let ϵ have invariant character and wildly variant content.

22. I have switched our leading example here to one constructed entirely out of logical expressions so as to connect this discussion cleanly with the results proved in the appendix.

23. This is to disagree with those who hold that logical truths can be shallowly contingent (Zalta 1988). The matter is too detailed for this paper, but see Hanson (2006) for what strike me as conclusive reasons to reject Zalta’s claim.

24. This is essentially the complaint raised by Hanson (1997) against Sher’s account. It also potentially explicates a sense in which logical truths are analytic.

25. Though it is not always entirely clear what her project is. In recent work, she claims that something loosely analogous to Simple Invariance gives a necessary and sufficient answer to the question “Which choice of logical constants will give rise to a logical system whose consequences transmit truth from premises to conclusion with an especially strong modal force in all fields of knowledge?" (2014: 182) But, if that is the question, then the restriction to extensional languages is very implausible since it is not at all obvious that we can formulate adequate logical systems for all fields of knowledge in extensional terms. Double-Standard Invariance, absent this restriction, would seem to be a better answer to this question though perhaps not a sufficient condition.

26. It is clear from her later discussion that Sher doesn’t think that counterexamples involving constants which have different meanings on domains of different cardinality, such as McGee’s (1996) funky disjunction, are non-rigid in her sense. Similar problems affect content- and character-invariant expressions, as discussed above (Section 3). As a referee points out, once we have identity and the existential quantifier as logical constants, finite cardinality quantifiers will be logical so these counterexamples don’t have much force. Of course, this leaves non-finite cases, but anyways, it strikes me that so much as there is a problem here, it is one which should be solved by choosing a more appropriate class of transformation—such as those of Bonnay (2008) or Feferman (1999).

27. Her concessive remark is directed at the inability to know, a priori, the meaning of certain empirical logical predicates, not at the case of empirical materials playing a role in specifying the character of an expression.

28. Feferman has also argued, in two ways, for two-pronged approaches to logicality. In his (2010), he argues that logical notions should be set-theoretically absolute in order to avoid being beholden to a background account of set theory. In his later (2015), he argues that there should be a proof-theoretic analogue to the semantic notion of invariance. I am sympathetic with both claims, viewing them as arising similar dissatisfaction with pure invariance criteria, but space precludes discussing them here. I hope to compare my approach with them elsewhere.

29. Though, like expressions rigid in her sense, not sameness of denotation in every world unless we use a transformation other than isomorphism.

30. However, this is not to say that we can exclude all the cases which might be troublesome. See endnote 37 below.

31. Sher also has a programmatic reason to aim only at logical languages and to not care about empirical predicates as she is primarily interested in worldly formal structure. Space is too limited to discuss this here, but note that even if logical constants are formal in this sense, it doesn’t follow that all formal notions can be grasped using only extensional resources. See, though, McGee (1996) for technical reasons to think that all standard logical objects can be defined using only extensional resources.

32. For a sensible discussion of logical truth in a Kaplanian context, see Michaelson (2014: 530–531).

33. Various simplifications employed here make it simpler to give an idea of how to proceed, but the general suggestion is clearly portable to different semantical pictures.

34. The reader will note that this means that ‘a = b’ can be true at w even if a does not exist at w. Such is life in modal semantics.

35. If we were to work with a modalized language, we would need to complicate this definition to accommodate additional modal expressions such as modal operators or the actuality operator @. As MacFarlane (2000) has pointed out, this requires taking a stand on how much modal structure—such as the accessibility relation—we should hold fixed. We ignore such complications here for expository purposes.

36. Note that the invariance of a term like ‘I’ depends on us allowing any member of the domain to count as a designated speaker or, alternatively, restricting the set of contexts to proper contexts in which the speaker of a context is, say, a person. The former is preferable; otherwise terms like ‘I’ carry significant non-logical information along with them—such as the information that the speaker is a person. Such terms are not intuitively logical. We will thus assume here that anything can serve as the designated “speaker” of a domain.

37. Note that notions like (NW)

    $$(N^W)\textrm{ }\forall c[`N^W\textrm{$\phi$' holds at $c\leftrightarrow$ (`$\phi$' holds at $c$ iff Water$\neq H_2O$})]$$

    where the substitute for K (Water≠H₂O) is necessary a posteriori are also plausibly not character invariant. This allows us to explain away some troublesome like cases mooted by MacFarlane (2009). But it does not help with another troublesome set of cases, due to Gómez-Torrente (2003). Consider an analytically empty predicate like ‘male widow’. Such expressions plausibly are both content- and character-invariant, as defined, yet are intuitively not logical. It is some consolation that they are complex constructions, composed of character-variant materials, but the cases remain troublesome. I hope to address this in future work.