Philos Stud DOI 10.1007/s11098-012-9954-z

How many notions of necessity? Jordan Stein

 Springer Science+Business Media B.V. 2012

Abstract Evans distinguishes between superficial necessity and deep necessity in his analysis of the contingent a priori. The distinction between these two notions of necessity is formalized by Davies and Humberstone through the addition of the operator Fixedly to Actuality Modal Logic (AML, S5A), where deep necessity is represented by the combination Fixedly Actually. Wehmeier’s Subjunctive Modal Logic (SML) provides an extension of the expressive capacity of ordinary modal predicate logic alternative to AML. I add Fixedly to SML and show that in the system SML with Fixedly the distinction between deep and superficial necessity disappears. I conclude that the existence of the distinction between deep and superficial necessity, as well as the existence of the contingent a priori, cannot be asserted independently of the choice of background logic. Keywords Contingent a priori  Deep and superficial necessity  Quantified modal logic  Actuality operator  Indicative and subjunctive mood

1 Introduction: Kripke and Evans on the contingent a priori Kripke (1980) makes clear that an argument is required to establish the connection between a priori knowledge and necessary truth: The terms ‘necessary’ and ‘a priori’, then, as applied to statements are not obvious synonyms. There may be a philosophical argument connecting them, perhaps even identifying them; but an argument is required, not simply the observation that the two terms are clearly interchangeable. (38)

J. Stein (&) Department of Social Science, New York City College of Technology and Brooklyn College, 300 Jay Street, Brooklyn, NY 11201, USA e-mail: [email protected]

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Before turning to counterexamples to the coextensiveness of the these notions, Kripke sketches, in the context of possible worlds semantics, a plausible argument for the thesis that all a priori knowledge is of necessary truths1: I guess it is thought that if something is known a priori it must be necessary, because it was known without looking at the world. If it depended on some contingent feature of the actual world, how could you know it without looking? Maybe the actual world is one of the possible worlds in which it would have been false. (ibid.) Noting that the above argument depends on the ‘possibly unjustified’ assumption that ‘‘there can’t be a way of knowing about the actual world without looking that wouldn’t be a way of knowing the same thing about every possible world’’ (ibid.), Kripke argues against the intuitive connection between a priori knowledge and necessary truth. Kripke exploits two features of English in the formulation of his counterexamples. First, some referential expressions of English are rigid designators—a rigid designator is an expression that refers at all counterfactual worlds to its actual world referent.2 Proper names as they occur in English modal constructions are rigid designators. Kripke contrasted proper names with definite descriptions, which he argued in most cases do not designate rigidly. Secondly, it is possible to fix the reference of a rigid designator to be the unique object satisfying a non-rigid description.3 In a language that contains both rigid and non-rigid designators it is possible to construct examples of sentences that are both contingent and knowable a priori. Kripke’s example is the sentence. (1)

The length of stick S at time t is 1 m long

where the general term ‘one meter’ has its reference fixed by the non-rigid description ‘the length of stick S at time t’. Sentence (1) is a priori since a linguistically competent speaker of English can know it to be true without determining the length of S empirically, but also contingent since, for example, heat might have been applied to S at t, in which case it would not have been 1 meter long. Gareth Evans (1979) gives a similar example using what he calls a ‘descriptive name’—a name the reference of which is fixed by description. If we let ‘Julius’ refer to whoever invented the zip, then the sentence 1

Kripke also sketches an argument for the thesis that all necessary truths, if they can be known at all, can be known a priori. For reasons of space, I do not go into this issue here.

2

Throughout, we work with constant domain models; if instead we allowed world domains to vary, the definition of rigid designation would need to be modified to contain a clause that takes into consideration the value of the designator when its actual referent does not exist at a world. In addition, there are several distinctions to be made with respect to the concept of rigid designation, for example, the distinction between de jure and de facto rigidity; these do not concern us here.

3

Kripke acknowledges that sometimes the referent of a name is fixed in this manner, but he is careful to distinguish the notions of reference-fixing and meaning-giving. While Kripke allows that it is sometimes the case that a term is introduced into the language according to a linguistic stipulation, where a definite description is used to fix the reference of the term, Kripke’s celebrated modal argument maintains that reference-fixing does not amount to meaning-giving.

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(2)

If anyone invented the zip, Julius invented the zip

expresses a contingent a priori truth for the same reasons as (1)—it is contingent because the actual inventor of the zip might not have invented the zip, but a priori since a linguistically competent speaker of English can know that it is true by understanding the reference-fixing stipulation, without appeal to any a posteriori knowledge of the referent of ‘Julius’. In his analysis of the contingent a priori, Evans (1979) appeals to a distinction between two notions of necessity. A sentence U is superficially necessary if it is necessary in the familiar sense of ‘h’ in modal logic—true at all worlds w; a sentence U is deeply necessary if for all possible worlds w, if w were actual, U would be true. Evans argues that a sentence that is contingent and a priori is false at some counterfactual world w (superficially contingent), but true at all worlds w from the point of view of w as the actual world (deeply necessary). The distinction between these two types of necessity is meant to alleviate the feeling of paradox present in the contingent a priori, for there should be nothing puzzling about a sentence that is associated with two different functions from worlds to truth-values, representing the two ways of understanding possible worlds—as actual or counterfactual. The implication from a priority to necessity is preserved, according to Evans, but at the level of deep necessity, as opposed to the more familiar notion of truth at all worlds. Davies and Humberstone (1980) extend the work of Evans by formalizing the distinction between deep and superficial necessity in a system I will call ‘Actuality Modal Logic (AML) with Fixedly’ (FAML), which extends AML—modal predicate logic with the actuality operator, also known as ‘S5A’—through the addition of the operator Fixedly, which is an implicit quantifier over the world playing the role of the actual world of a model. Superficial necessity is represented by the familiar ‘h’ and deep necessity is represented by the operator Fixedly Actually. Kai Wehmeier’s (2004, 2005) Subjunctive Modal Logic (SML), which makes a semantically significant typographical distinction between indicative and subjunctive predicates, also overcomes the expressive limitation in ordinary modal predicate logic. In this paper, I show that the distinction between truth at all worlds w and truth at all worlds w from the point of view of w as actual does not arise when ‘Fixedly’ is added to SML. The fact that SML and AML are expressively equivalent, in the sense that for every sentence U of AML there is a sentence W of SML such that U and W are true in all of the same models (and vice versa), shows that that the distinction between superficial and deep necessity (together with the traditional examples of contingent a priori) is sensitive to the choice of background logic, and hence that their existence cannot be asserted independently of the choice of linguistic framework. This paper is organized as follows. In Sect. 2, I discuss the expressive limitations of ordinary modal predicate logic and the fixes that, respectively, AML and SML provide. Section 3 presents Evans’ distinction between deep and superficial necessity (contingency) and its application to the contingent a priori. In Sect. 4, I discuss the Davies and Humberstone interpretation of Evans in FAML. In Sect. 5, I assess Evans’ analysis of the conditions for the contingent a priori in the context of

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SML, and show that the traditional examples of the contingent a priori are not both contingent and a priori in this system. Finally, in Sect. 6 the operator ‘Fixedly’ is added to SML and the equivalence of deep and superficial necessity for formulas of non-modal predicate logic is explained. The proof of this theorem is provided in the Appendix.

2 Expressive limitations in modal predicate logic 2.1 Expressive limitations There are modal statements of English that cannot be expressed in the language that results from adding ‘h’ and ‘’ to first-order logic. There is a reading of (3), for example, not expressible in the conventional language of modal predicate logic.4 (3)

It is possible for everything that is red to be shiny5

There are at least three natural readings of this sentence; paraphrasing in the idiom of possible worlds: (3a) There is a world w such that everything red at w is shiny at w (3b) For everything that is red there is a world w such that it is shiny at w (3c) There is a world w such that everything that is red at the actual world is shiny at w As there are only two semantically relevant ways to arrange the possibility operator ‘’ and the universal quantifier ‘V’, only translations of (3a) and (3b) are available: (3a0 ) Vx(Red(x) ? Shiny(x)) (3b0 ) Vx(Red(x) ? Shiny(x)) Interpretation (3c) cannot be formalized in the language of modal predicate logic, for if ‘’ falls outside of the scope of the universal quantifier, the predicate ‘Red’ falls within the scope of ‘’, and thus its interpretation will be the extension of ‘Red’ at the possible world activated by ‘’. If, on the other hand, as in (3b0 ), ‘’ is placed in front of the predicate ‘Shiny’, the truth-condition is such that possibly different worlds are activated for different objects falling within the actual extension of ‘Red’.6 It is clear that what is needed to be able to express the interpretation (3c) of sentence (3) is a device that allows the extension of a predicate to be determined according to how things actually are, even when the predicate occurs within the scope of a modal operator.

4

For a proof of this fact see Wehmeier (2003).

5

This example is taken from Crossley and Humberstone (1977).

6

Since we’re working with constant domain models (3a0 ) and the sentence Vx(Red(x) ? Shiny(x)) have the same truth-conditions, but even with respect to varying domain models there is no sentence expressing the truth-conditions of (3c).

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2.2 Actuality Modal Logic (AML) One way to extend the expressive capacity of the language of modal predicate logic is to add the operator ‘Actually’, or ‘A’ for short, which is governed by a particular clause in the definition of truth at a world of a model. Our notion of a modal model will be the same between all of the modal systems we discuss. A model K is a tuple consisting of a non-empty set W whose elements we call worlds, a distinguished element of that set, @, which we regard as the actual world of the model, an interpretation function I assigning world-relative extensions to the predicate symbols, and a non-empty set D over which the quantifiers are to range. The new operator ‘Actually’ is understood according to the following clause: •

(Actually): K, w k AU [r] iff K, @ k U [r]

where w is a world of K and r is a variable assignment. Truth in a model is here understood to be truth at the distinguished world in that model. We refer to the resulting modal system as ‘Actuality Modal Logic’, or ‘AML’ for short.7 Intuitively, the operator ‘A’ serves to shield (Humberstone 1982; 2004) what occurs within its scope from an outlying modal operator.8,9 Interpretation (3c) of sentence (3) can be expressed with the help of this operator as follows: (3c0 )

Vx(ARed(x) ? Shiny(x))

It is easy to see that the truth-conditions for (3c0 ) are that everything that is actually red is shiny together at someone counterfactual world w: K, w k Vx(ARed(x) ? Shiny(x)) [r] iff there is a w0 [ W such that K, w0 k Vx(ARed(x) ? Shiny(x)) [r] iff there is a w0 [ W such that for all x-variants r0 of r: K, w0 k ARed(x) ? Shiny(x) [r0 ] iff there is a w0 [ W such that for all x-variants r0 of r: NOT K, w0 k ARed(x) [r0 ] or K, w0 k Shiny(x) [r0 ] iff there is a w0 [ W such that for all x-variants r0 of r: NOT K, @ k Red(x) [r0 ] or K, w0 k Shiny(x) [r0 ] iff

7

In the following we only consider S5 models.

8

A similar limitation on expressiveness has been diagnosed in first-order tense logic. The interpretation of 1 day, everyone alive will be dead analogous to (3c) cannot be expressed, and analogous to the actuality operator the operator ‘Now’ has been added, with the help of which the interpretation 1 day, everyone Now alive will be dead can be expressed (Kamp 1971).

9

The addition of the operator ‘Actually’ requires that careful attention be paid to the notion of logical validity, for within such a modal system two notions of logical validity may be distinguished. Following traditional terminology, we refer to these as ‘real world validity’ and ‘general validity’, respectively. A formula / is real world valid in a class of models if it is true at the actual world of every model of the class. A formula / is generally valid (in a class of models) if it is true at every world in every model in the class. To see that the set of real world valid formulae does not coincide with the set of generally valid formulae in the modal system AML consider the schema: A/ ? /. It is easy to see that this schema is true at @ for every model, and is hence real world valid, but since / may be false at a counterfactual world and yet true at @, this formulae is not generally valid. General validity implies real world validity, and if no ‘Actuality’ operator is added to the language the two notions of validity coincide.

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there is a w0 [ W such that for all x-variants r0 of r: r0 (x) 62 I(Red, @) or r0 (x) [ I(Shiny, w0 ) iff there is a w0 [ W such that every a in the interpretation of Red at @ is in the interpretation of Shiny at w0 . 2.3 Subjunctive Modal Logic (SML) Modal predicate logic does not distinguish typographically between predication in the indicative and subjunctive mood, representing both predicates in (4) and (5), for example, by the symbol F: (4) (5)

Ruth Ginsberg is not a justice of the US Supreme Court Under certain circumstances, Ruth Ginsberg would not have been a justice of the US Supreme Court (40 ) :F(a) (50 ) :F(a) Standard modal logic represents mood as a matter of scope. If a predicate falls within the scope of a modal operator, it is interpreted as in the subjunctive mood; if it falls outside the scope of a modal operator it is interpreted in the indicative mood. If this way of capturing the indicative-subjunctive distinction is correct, then as Wehmeier observes: [The] scope theory of mood also explains why we do not find any typographical distinction between ‘indicative’ and ‘subjunctive’ predicate symbols in modal logic: Its canonical notation is set up in such a way that no scope ambiguities can arise. There is no leeway in word order, as there is in ordinary English, and hence no need for additional scope indicators such as mood. (Wehmeier 2004, 7–8) The reading (3c) of (3) can be represented in English with the help of mood distinctions: (6)

Under certain circumstances, everything that is red would be shiny

If the scope theory of mood were correct, the truth-conditions that we claim are possible for (6) (i.e. (3c)) would be expressible in modal predicate logic. However, it is known that reading (3c) is not expressible, and hence that the scope theory of mood cannot be correct. We have seen that enriching the language to contain the operator ‘Actually’ overcomes this expressive limitation by inhibiting a formula that falls within its scope from being activated by any outlying modal operators (Humberstone 1982; 2004). Wehmeier (2004, 2005) develops a modal logic according to which ‘‘mood is a semantically significant syntactic feature of individual predicates, determining their orientation towards actual or counterfactual situations, and not just a scopal element pertaining to the logical structure of whole sentences’’ (2004, 15). Following Wehmeier, we refer to this logic as ‘Subjunctive Modal Logic’, or ‘SML’ for short.

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The syntactic feature unique to SML is a semantically significant typographical distinction between predication in the indicative mood and predication in the subjunctive mood. The language thus contains infinitely many n-place predicates F in the indicative mood, and for each such n-place indicative predicate, a predicate symbol Fs in the subjunctive mood.10 If we let the indicative predicate ‘is a justice of the US Supreme Court’ be symbolized by the predicate F, then the corresponding subjunctive predicate ‘would be a justice of the US Supreme Court’ is symbolized as Fs. The corresponding translations of sentences (4) and (5) into SML are: (400 ) (500 )

:F(a) :Fs(a)

The clauses in the recursive definition of truth with respect to a world distinctive of SML are the two base clauses. Let K be a model, w a world, and r a variable assignment: (7) (8)

K, w  F(t1,…,tn) [r] iff hr(t1),…,r(tn)i [ I(F, @) K, w  Fs(t1,…,tn) [r] iff hr(t1),…,r(tn)i [ I(F, w)

(7) States that an atomic formula containing a predicate in the indicative mood is true at an arbitrary world w just in case the values of the terms of the formula are in the extension of the predicate at the actual world (so the only world that is relevant to the truth-conditions of an atomic formula containing an indicative predicate is the actual world); (8) states that an atomic subjunctive predication is true at an arbitrary world w just in case the values of the terms are in the extension of the predicate at w . In characterizing the notion of truth in a model for SML two parameters must be discussed. The first parameter is familiar from predicate logic. Without knowing the referent of ‘he’ in the sentence ‘he is winning’ it’s not possible to evaluate this sentence for truth; accordingly, not every formula of predicate logic can be assigned a truth-value outright. In order to determine the truth of the predicate logic translation of ‘he is winning’ we must assign an object in the domain to the variable representing the pronoun ‘he’. This gives us the auxiliary notion of truth under an assignment in virtue of which the notion of truth simpliciter is defined. A sentence (i.e., closed formula) is true under some variable assignment r if and only if it is true under every variable assignment—that is, the truth of a sentence does not depend on the initial choice of variable assignment. Analogously, a formula containing a free subjunctive predicate symbol (i.e., a subjunctive predicate not in the scope of a modal operator) cannot be evaluated for truth outright. Consider the following sentence: ‘Aristotle would have been an athlete’. It is not possible to determine whether this sentence is true or false, for it is true in some circumstances and false in others. Accordingly, the formula of SML that represents this English sentence cannot be evaluated for truth independently of specifying a salient possible world w. This leads to the following definition. A 10 If we are dealing with varying domain models we will need to add the subjunctive quantifier As read ‘there would have been’. In the case of AML, we would similarly need to add an actuality existential quantifier AA, read ‘there exists at the actual world’.

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formula is subjunctively closed if it does not contain an occurrence of a subjunctive predicate that is not within the scope of a modal operator. A subjunctively closed formula is true with respect to some variable assignment r and world w if and only if it is true with respect to r and every world w. A sentence of SML is thus defined to be a subjunctively closed formula without free variables; SML sentences are true or false independently of assignments and worlds. The reading (3c) of (3), inexpressible in the language of modal predicate logic, is expressed by the following formula of SML: (9)

Vx(Red (x) ? Shinys(x))

It is clear that this sentence is true if and only if every object in the extension of the predicate ‘Red’ at the actual world is in the extension of ‘Shiny’ at the counterfactual world introduced by the possibility operator: K  Vx(Red(x) ? Shinys(x)) iff for some world w (equivalently all worlds) and some variable assignment r (equivalently all variable assignments) K, w  Vx(Red(x) ? Shinys(x)) [r] iff there is a w0 [ W such that K, w0  Vx(Red(x) ? Shinys(x)) [r] iff there is a w0 [ W such that for all x-variants r0 of r: K, w0  Red(x) ? Shinys(x) [r0 ] iff there is a w0 [ W such that for all x-variants r0 of r: NOT K, w0  Red(x) [r0 ] or K, w0  Shinys(x) [r0 ] iff there is a w0 [ W such that for all x-variants r0 of r: r0 (x) 62 I(Red, @) or r0 (x) [ I(Shiny, w0 ) iff there is a w0 [ W such that every a in the interpretation of Red at @, is in the interpretation of Shiny at w0 . 2.4 The expressive equivalence of AML and SML Both AML and SML are independently motivated logics, each overcoming the same limitation of expressiveness in the language of ordinary modal predicate logic. AML and SML are, moreover, expressively equivalent in the sense that for each sentence U of AML there is a sentence W of SML such that U and W are true in all of the same models, and vice versa (Wehmeier 2004).11 To see that this is the case, we define a translation between AML and SML. In translating between AML and SML we make use of the fact that every occurrence of an actuality operator can be positioned so that it only occurs in front of atomic subformulae. The following pairs, that is, are provably equivalent in AML (we let F and G be meta-variables for atomic formulae): • • •

A:F and :AF A(F ^ G) and (AF ^ AG) AF and F12

11 In contrast to AML, for SML there is no distinction between truth at the actual world of a model and truth at all worlds of the model. 12

Because we are only considering S5 models.

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• •

AAF and AF AAxF and AxAF.13

To find a sentence W of SML that is true in the same models as a sentence U of AML first apply the above equivalences in U until ‘A’ only occurs in front of atomic formulas. Then, put a superscripted ‘s’ after every occurrence of a predicate in U that is in the scope of a modal operator but not in the immediate scope of an occurrence of ‘A’. Finally, delete every occurrence of ‘A’ in U. The resulting sentence W of SML will be true in all of the same models as U. There is, in addition, a translation from SML into AML. To find a sentence U of AML that translates a sentence W of SML, first prefix every occurrence of an indicative predicate in W with an occurrence of ‘A’, and then delete every occurrence of ‘s’.14 AML and SML are not, however, trivial notational variants (Wehmeier 2004). We elaborate on some differences between the two in the remaining sections of this paper.

3 Evans’ distinction between deep and superficial necessity Evans (1979) emphasizes that the contingent a priori has no special connection with the concept of reference by pointing out that examples of the contingent a priori can be constructed with help of the operator ‘Actually’. Indeed, examples of the contingent a priori can be formulated at the level of Propositional AML. For example, any sentence of the form (10)

AP ? P

is knowable a priori by anyone who understands the semantics for ‘Actually’ and ‘?’, but will be contingent when the embedded sentence P is contingent. Since there are examples of the contingent a priori that do not involve the concept of reference, the explanation of the contingent a priori, according to Evans, is not to be found by an analysis of the concept of reference. Evans proposes that the proper analysis of the contingent a priori is to be found in the modal notion of necessity (contingency): There is no paradox in the existence of statements which are both contingent and a priori, at least, not in the sense in which the problematical statements may be claimed to be contingent. There are two quite different conceptions of what it is for a statement to be contingent; statements may be, as we might say, deeply contingent or superficially contingent. Whether a statement is deeply contingent depends upon what makes it true; whether a statement is superficially contingent depends upon how it embeds inside the scope of modal operators. (Evans 1979, 179) A sentence U is superficially contingent, if both U and :U are true; a sentence U is deeply contingent if it depends on a contingent feature of the actual world. Evans 13

Because all models are constant domain.

14

Humberstone (1982) gives a similar translation for the case in which the subjunctive mood is represented as an operator rather than a feature of atomic predicates.

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(1979, pp. 211–212) argues that (i) the notion of contingency that applies to a contingent a priori statement is not the notion captured by the phrase of Kripke ‘‘dependent upon some contingent feature of the actual world’’. In addition, Evans argues (ii) the notion ‘‘dependent upon some contingent feature of the actual world’’ (ibid.) is not expressible in AML. Although linguistic stipulation seems to suggest that there can be a way of knowing something about the actual world without looking that does not amount to verifying this state of affairs at every possible world, the state of affairs in question is not a contingent feature of reality. The criterion for superficial contingency (necessity) is the pattern of embedding of a sentence under a modal operator, which is determined by the model-theoretic notion truth at a world (truew). This notion, Evans (1979, p. 199) writes, ‘‘is specifically designed to account for the way sentences embed inside modal contexts. If (Q) is true, for example, then the truew relation for (Q) must be so characterized that there exists a world w such that (Q) is truew’’. The sentence ‘if anyone invented the zip, Julius did’ and every sentence of the form AU ? U (where U is a superficially contingent truth) is superficially contingent: both are true at the actual world, but false at some counterfactual world w. It is possible, according to Evans, for the pattern of embedding of two sentences under a modal operator not to reflect the fact that it is the same state of affairs of the actual world that makes both of them true. Evans (1979) calls the content of a sentence the condition under which it is true. If two sentences have the same content, then they are both made true by the same feature of the actual world. This is the case, for example, with the pair of sentences ‘Julius is F’ and ‘The inventor of the zip is F’. When the world of evaluation is the actual world both sentences have the same truth-value, since it is the same feature of the actual world that is relevant for their evaluation. The sentences, however, have different truthw conditions, that is, they embed differently under the modal operators. Each sentence in the language of modal logic is associated with a condition that an arbitrary world must satisfy in order for the sentence to be true at that world. The sentence ‘grass is green’, for example, requires for its truth at an arbitrary world w that grass be green at w, whereas the ‘Actually’-embedded ‘grass is actually green’ requires for its truth at an arbitrary world w that grass be green at the actual world of the model. The set of worlds satisfying the condition required for the truth of a sentence is sometimes called a proposition, which is best understood to be a function that determines for each sentence and possible world whether the sentence is true at that world. Two sentences may be made true by the same feature of the actual world (no matter how the actual world happens to be) and yet have different counterfactual truth-conditions, that is, be associated with different functions from possible worlds to truth-values. The model-theoretic relation truew does not reflect the existence of a way of understanding the alethic modalities according to which two sentences made true by the same feature of the world have the same modal properties. Evans proposes a notion of truth to underwrite the fact that if two sentences have the same content, then they embed identically under the modal operators. Evans calls this notion truth in a world, understood as if w were actual, U would be true. This notion adequately reflects the fact that two sentences made true by the same feature of the world

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cannot have different truth in a world conditions. If a world w where grass is orange were actual, grass would be orange in w if, and only if, the sentence ‘grass is actually orange’ would be true in w. A sentence is deeply necessary if it depends on no contingent feature of reality, where a contingent feature of reality is one for which there is no guarantee that this feature obtain in the actual world. If despite variation in the world designated as actual a sentence remains true, what makes the sentence true is not a contingent feature of reality. A deeply necessary statement is true in all worlds w where w is the actual world. A sentence of the form Au ? u, although a priori, is not deeply contingent; it is true in all worlds w from the point of view of w as actual. The same is the case for the contingent a priori sentence ‘if someone uniquely invented the zip, Julius did’. No matter which world happens to be the actual world, if the zipper is not invented at this world (or if a group of people invent the zip), then the antecedent is false, in which case the sentence is true. If someone uniquely invents the zip, then the consequent is true, since by definition ‘Julius’ refers to whoever actually invented the zip, which is the inventor of the zip in w, when w is the actual world. There is thus no contingent feature of reality on which the truth of this sentence depends, for a feature that makes it true is guaranteed to obtain no matter which world turns out to be the actual world.

4 The logic of ‘Fixedly Actually’ As determined by the semantics for ‘Actually’ the schema Au ? hAu is generally valid in AML. This consequence of how the actuality operator behaves may give rise to suspicion, since when ‘Actually’ combines with ‘h’, sentences prefixed by ‘Actually’ that are true at the actual world will be necessary truths—which intuitively we understand to mean that no matter which world is actual the sentence would be true—yet we can understand, for example, that it might have been the case from the point of view of the actual world that grass would be orange. The interaction of ‘Actually’ and ‘h’ thus results in necessary truths that are made true solely in virtue of how the actual world is, without taking into consideration all other non-actual possibilities. Crossley and Humberstone (1977) address this suspicion by the addition of the operator ‘Fixedly’ to the language of AML, defined in terms of the relation ‘is a variant of’ (symbolized ‘&’) on the class of AML models, where two models are variants if they differ at most over which world is designated as actual. In terms of this relation the operator Fixedly is defined: •

Fixedly (F): K, w k F(U) [r] iff for all K0 & K: K0 , w k U [r].

The operator ‘Fixedly’ is thus an implicit quantifier over the world playing the role of the actual world of a model. The operator ‘Fixedly Actually’ (which I sometimes abbreviate to ‘FA’) follows from the clauses in the inductive definition of truth at a world in a model for ‘Fixedly’ and ‘Actually’: •

Fixedly Actually (FA): K, w k FA(U) [r] iff for all K0 & K, if w0 [ W is the actual world of K0 , then K0 , w0 k U [r].

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Davies and Humberstone (1980) suggest that Evans’ distinction between deep and superficial necessity is characterized formally in the modal system FAML. They argue that the notion of superficial necessity is formally represented as the familiar ‘h’, and the notion of deep necessity is represented by the combination ‘FA’. The operator ‘Fixedly Actually’ varies across the actual world variants of a model K, and then directs the evaluation of the formula prefixed by ‘Actually’ to the world that plays the role of the actual world. The a priori schema AU ? U is superficially contingent: it is true at the actual world of every model, but when U is contingent, h(AU ? U) is false. The schema FA(AU ? U), on the other hand, is generally valid in the modal system FAML. The operator ‘Fixedly Actually’ serves to accomplish a means by which to target the reading of a sentence that takes into consideration possible variation in the world designated as actual, precisely what is required in the case of a sentence containing a rigid expression such as ‘Actually’ and ‘Julius’. The notion of deep contingency is characterized by Evans according to Kripke’s phrase ‘‘dependent on some contingent feature of reality’’, and the operator Fixedly Actually seems to represent the notion: not dependent on a contingent feature of the actual world (reality); if U is dependent on a contingent feature of the actual world, then U is false at a world w from the point of view of w as actual, and so FA(U) is false.

5 The contingent a priori and SML The contingent a priori arises, according to Evans (1979, p. 204), when ‘‘to formulate the property of an arbitrary world which is associated with a sentence, one must make reference to what is actually the case’’. A sentence of the form AU, for example, when evaluated at an arbitrary world w will depend for its truth on how things stand with respect to U at @. A sentence containing a rigid name such as ‘Julius is F’ is true at a counterfactual world w if the referent of ‘Julius’ at @ is F at w. The contingent a priori sentence ‘if anyone invented the zip, Julius did’, for example, is true at an arbitrary world w if and only if the actual inventor of the zip invented the zip at w. However, as I will argue, the particular modal framework Evans was working with, namely AML, obscures what is required for the construction of the contingent a priori. A tacit assumption of the possibility of the contingent a priori is that the modal language in which such sentences are formulated be so constructed as to force the truth-value of formulas (sentences) to be tied to the actual world in two ways: (i)

The formula contains a rigidified designator or an actuality operator (or any other device that takes us back to the actual world when evaluating a formula in which it occurs). (ii) The formula doesn’t occur within the scope of a modal operator (so that it is evaluated at the actual world of the model by default). A formula of AML can vary its extension depending on the world of evaluation. It is only the extension of a formula or descriptive singular term prefixed with the operator ‘Actually’ that depends invariably on how things are with respect to the

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actual world of a model. The sentence ‘If anyone uniquely invented the zip, Julius did’ is contingent because when it is evaluated at an arbitrary world w the extension of the predicate invented the zip is the individual who invented the zip at w, possibly someone other than the actual inventor of the zip. SML represents a predicate as explicitly in either the indicative or subjunctive mood, the mood of a predicate remaining fixed according to the design of the language. The extension of an indicative predicate is always determined relative to the actual world of a model, whereas the extension of a subjunctive predicate is always determined relative to the salient (generally non-actual) world of evaluation. Although in SML there are sentences whose truth at an arbitrary world w depends on how things stand at the actual world, there is no contingent a priori in this modal system. To substantiate this last claim we must settle on the interpretation of necessity for SML. A necessary truth, according to the possible worlds analysis, is truth at all worlds w. Let U be a sentence of non-modal predicate logic. If we define necessary truth for SML according to the definition •

(Vacuous Necessity): U is necessary, if hU is true

It would render the notion of contingent truth incoherent. According to the proposed definition, U is possible if U is true, and a contingent truth if both U and :U are true. Thus, if U is a contingent truth, then for some world w we have: K, w  :U. But since U is a truth every predicate of which is in the indicative mood, we also have: K, w  U, and so we have: K, w  (U ^ :U), which is not possible. When U is true and only contains predicates in the indicative mood, hU is also true, but counterfactual worlds play no role in the truth-conditions of U. This notion is better understood as ‘vacuous necessity’. Instead, we define necessary truth in SML as follows: •

(Genuine Necessity)15: U is a necessary truth, if hUs is true, where Us results from U by attaching a subjunctive marker ‘s’ to every predicate in U.

For the genuine necessities all worlds play a role in the obtaining of the truthconditions of U. Moreover, the genuine modalities allow for the coherent expression of contingency for SML. We say that a sentence U is genuinely possible, if Us is true, where Us results from U by attaching a subjunctive marker to every predicate in U. A sentence U is a contingent truth for SML, if both U and :Us are true. It is reasonable to not attribute necessity to formulas containing free subjunctives. This is because we want to predicate necessity of the kinds of sentences typically formalized in first-order logic. Formulas that contain free subjunctives are just like formulas that contain free variables: in each case it does not make sense to attribute truth (qualified or not) to such formulas. Formulas containing free subjunctives are not truth-evaluable without specifying a salient world of evaluation, and hence are not candidates for necessary truth. If we allow necessity to be predicated of

15 As in the case of the definition of vacuous necessity, we restrict U to be a sentence of non-modal predicate logic. See footnote 18 for the connection between the definition of necessity for SML and the equivalence of deep and superficial necessity.

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formulas containing free subjunctives, what we end up with a notational variant of AML, which is not the point of SML. Having thus defined necessity we are now in a position to show that the sentence ‘if anyone invented the zip, Julius invented the zip’ is not both contingent and a priori in SML (since the definite description ‘the inventor of the zip’ is a rigid designator in SML it makes no difference whether we use the description or the name ‘Julius’16). Sentence (11) is put into the form of a disjunction so as not to obscure the analysis of the contingent a priori with the possibly contentious issue of the logic of subjunctive conditionals: (11)

Either nobody uniquely invented the zip or the individual who invented the zip invented the zip

In (11) each predicate is in the indicative mood. It is a necessary truth because h(11)s is true, that is, for all worlds w it is the case that either nobody would have uniquely invented the zip at w, or whoever would have uniquely invented the zip at w would have invented the zip at w. Because (11) is necessary it is not an instance of the contingent a priori.17 There are other English sentences (and their corresponding representation in SML) to consider, but none are candidates for the contingent a priori: (11.1) (11.2) (11.3) (11.4) (11.5) (11.6)

Either nobody would have uniquely invented the zip or the individual who invented the zip invented the zip. Either nobody would have uniquely invented the zip or the individual who would have invented the zip invented the zip. Either nobody would have uniquely invented the zip or the individual who would have invented the zip would have invented the zip. Either nobody uniquely invented the zip or the individual who would have invented the zip invented the zip. Either nobody uniquely invented the zip or the individual who would have invented the zip would have invented the zip. Either nobody uniquely invented the zip or the individual who invented the zip would have invented the zip.

None of (11.1)–(11.6) are truth-evaluable without specifying a salient possible world. The SML representations of (11.1)–(11.6) are not candidates for necessary truth since for SML necessary truth is only predicated of sentences of non-modal predicate logic. In discussing the schema AU ? U (or rather (:AU _ U), for reasons as above) we focus on a particular instance, the conclusion we draw intended to generalize to every sentence of this form. The AML sentence (:AR(a, b) _ R(a, b)) translates to the SML sentence (:R(a, b) _ R(a, b)). The latter sentence is a necessary truth in 16 ‘Julius’ has the same semantic content as the description ‘the inventor of the zip’ that is used to fix its referent. The description must be in the indicative mood to plausibly be a synonymy candidate for ‘Julius’, and indicative descriptions in SML are rigid designators. 17

(11) is of course also necessary in the sense that h(11) is true—that is (11) is vacuously necessary.

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SML, since for all worlds w (:Rs(a, b) _ Rs(a, b)) is true at w—that is, h(:Rs(a, b) _ Rs(a, b)) is true. A formula such as (:R(a, b) _ Rs(a, b)) in which there is a free occurrence of a subjunctive predicate is not a sentence of SML and so not a candidate for truth, in particular necessary truth, without specifying a salient world of evaluation. We find that the interaction of (i) and (ii)—the interaction of rigidity and forcing the extension of a non-rigid formula to be the extension at the actual world when the actual world is the world of evaluation by removing the formula from the scope of any modal operators—together with the definition of necessity for SML are responsible for the existence of the contingent a priori in AML. Removing a formula containing a subjunctive predicate from the scope of a modal operator results in a formula that is no longer truth-evaluable and hence no longer a candidate for necessary truth. We thus find that the existence of the contingent a priori is sensitive to the background modal logic. Kripke (1980, p. 38) is correct to point out that in the context of ordinary modal predicate logic (and AML as well) there is a way of knowing something about the world without looking that does not amount to knowing ‘‘the same thing about every possible world’’. We have shown in the present paper that in SML it is plausible to regard knowing something without looking as amounting to the same thing as knowing that the feature obtains at every possible world.

6 The equivalence of deep and superficial necessity in FSML The distinction between truth at all worlds (superficial necessity) and truth at all worlds considered as actual (deep necessity) does not exist in SML with Fixedly (FSML). Indeed, the following is a theorem about the modal system FSML, the logic that results from adding ‘Fixedly’ to SML: Theorem 118 For every purely indicative sentence U (i.e. U contains no subjunctive predicates and no modal operators), every S5 model K, and world w [ W: K, w  hUs [r] iff K, w  FU [r] (where Us is the result of subjunctivizing every predicate in U). If we restrict the predicate symbols in U to the indicative mood, Fixedly represents Fixedly Actually. Fixedly Actually, recall, varies the actual world across the class of actual world variants of a model K, and then directs the evaluation of the 18 A more general version of the theorem holds that shows the equivalence of h and Fixedly for formulas already containing modal operators. Formulating the theorem, however, requires a more complicated definition of necessity than the definition of necessity given in the paper. This complication is required to prevent unintended capture of subjunctive markers by modal operators, as in the formula: h(Gsa ? Ga). Subjunctivizing the second occurrence of ‘G’ results in the predicate being bound by what becomes the innermost occurrence of ‘h’, but this does not fit the original definition of genuine necessity. To deal with this complication we must add indexed subjunctive makers and modal operators. Enriching the language to include indexed modal operators is not required only for SML, however, as adding a single actuality operator to ordinary modal predicate logic is not sufficient for the purposes of formalizing the entire modal fragment of English. See (Wehmeier 2010) for sentences of modal English not expressible in AML and Peacocke (1978) and Forbes (1985) for more on indexed actuality and modal operators.

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formula to which it is prefixed to the actual world of each model variant; Fixedly in SML, when it occurs in front of a formula all of the predicates of which are in the indicative mood, has the same effect. Since all of the semantic facts remain the same as we move between variant models, whenever a formula Us is true at a world w relative to a variable assignment, U is true at any variant K0 of K where w is now the actual world of K0 , relative to that same variable assignment. In the case of interest, when Us is true at all worlds w relative to a variable assignment, then U is true at all variants K0 of K and all worlds w such that w is the actual world of K0 (relative to the same variable assignment). The reason why a sentence such as AR(a, b) ? R(a, b) is deeply necessary (Fixedly Actually true) in AML, but not superficially necessary, is because when the world of evaluation is the actual world w0 of any K0 variant of K both AR(a, b) and R(a, b) have the same truth-value at w0 , but when AR(a, b) ? R(a, b) is embedded in the context of ‘h’ the actuality operator directs the evaluation of R(a, b) back to w0 , whereas ‘h’ activates counterfactual worlds for the evaluation of R(a, b). The SML translation R(a, b) ? R(a, b) is a necessary truth, that is, Rs(a, b) ? Rs(a, b) is truew for all worlds w; when we consider a model variant K0 of K, were a counterfactual world of K is now the actual world of K0 , R(a, b) ? R(a, b) remains true at w. Thus, since R(a, b) ? R(a, b) is true at all w, it is true at all worlds considered as actual. The addition of the rigidifying operator ‘Actually’ to overcome an expressive limitation in modal predicate logic gives rise to intuitively necessary truths that cannot be recognized as such by the modal operator ‘h’. A sentence of the form AU ? U, for example, is made true by the same feature of the world that makes a sentence of the form U ? U true; this feature is a necessary feature, and thus obtains at every metaphysically possible world. However, whereas the latter sentence is necessary, the former sentence is contingent. This may lead us to wonder whether the notion of necessity expressed by this operator captures a robust notion within the possible worlds framework: Once an ‘Actuality’-operator is introduced it is the idea of ‘h’ as capturing an intuitive notion of necessary truth for sentences that stands in need of defense. In contrast, the idea that a necessarily true sentence is one that is true no matter what strikes us immediately as being right. (Davies 2006, 151) Though the notion of necessity expressed by ‘h’ may ‘‘stand in need of defense’’ in AML, it is not exactly for the reason that Davies suggests. For though not representing actuality by means of an operator, the modal system SML still contains something very much like it in virtue of the fact that an indicative predicate always pertains to how things are with respect to @, even when it occurs within the scope of a modal operator. The problem is due to the two ways of forcing the truthconditions of a sentence tied to the actual world. However, we believe that Davies is correct in his observation that in the context of ‘Actually’ the operator ‘h’ ‘‘stands in need of defense’’ as an expression of necessity. This claim is supported for FSML by the following theorem:

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Theorem 2 Let K be a model, w a world, and r a variable assignment. For every sentence of non-modal predicate logic U such that (i) K, w k hU [r] and (ii) NOT K, w k FA(U) [r] there is a sentence of non-modal predicate logic W such that (iii) U and W are true in all of the same models (iv) K, w  hW [r], and (v) NOT K, w  hWs (Here k is truth in a model with respect to FAML-formulas, and  is truth in a model for FSML-formulas.) The intuitive idea behind the proof can be sketched as follows. Every AML sentence meeting conditions (i) and (ii) must contain at least one occurrence of ‘A’. This follows from the fact that for sentences that are ‘A’-free the two notions of necessity coincide. We then know how to find an appropriate translation in the language of SML meeting condition (iii). It follows from (i) that U is true, and hence that the SML translation is true, thus satisfying (iv). Lastly, it is shown that K, w  F(W) [r] is not consistent with condition (ii), whence by Theorem 1, NOT K, w  hWs [r]. The superficially necessary but deeply contingent sentences of AML are properly seen as merely vacuously necessary.

7 The linguistic relativity of the distinction between deep and superficial necessity We have observed that AML and SML are expressively equivalent. Though expressively equivalent, they are not, as Wehmeier (2004) stresses, trivial notational variants. There are two ways to express indicativity in AML—either in the form of a formula prefixed by an occurrence of ‘Actually’ or by removing the formula from the scope of all modal operators; by contrast, indicativity is always expressed through an indicative predicate in SML. AML allows non-rigidified predicates to go double-duty as either in the indicative or subjunctive mood depending on whether they fall within the scope of a modal operator, whereas SML marks mood as a semantically stable feature of predicates. This difference results, to put the point in terms of a distinction of Evans (1979, p. 201), in epistemically equivalent but modally distinct statements in AML. The apparatus of two-dimensional modal semantics has been deployed to deflate the significance of the contingent a priori. We have shown that the two-dimensional deflation of the contingent a priori, according to the distinction between deep and superficial necessity, cannot be divorced from a linguistic framework. More generally, we have shown that the contingent a priori and the distinction between deep and superficial necessity are not robust between expressively equivalent formal frameworks. When we consider the question which is the title of this paper we should conclude that there is no fact of the matter, and thus that the distinction between deep and superficial necessity and the contingent a priori cannot be maintained independently of the choice of formal framework. Acknowledgments Special thanks Kai Wehmeier for his encouragement and for a number of very helpful conversations. I also want to thank Aldo Antonelli, Michael Hicks, Lloyd Humberstone, Kent Johnson, Brian Skyrms, and an anonymous referee for helpful comments on earlier drafts.

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Appendix Syntax for AML Definition [Language of AML]: The language of AML is characterized by the following symbols: • • • • • • • • •

The truth-functionally complete set of connectives :, ^ Symbols of punctuation (,), and, Existential Quantifier A19 Denumerably many individual variables x1, x2, x3,… Denumerably many constant symbols c1, c2, c3,… Denumerably many predicate symbols of arity n (for all n) F1, F2,… The equality symbol = The possibility operator  The one-place operator A, read ‘Actually’

The universal quantifier, the necessity operator ‘h’, and the other truth-functional connectives are defined in the usual way. Definition [Term]: The terms of AML are of the following form: • •

Individual variables are terms Constant symbols are terms

We use the letters ‘s’ and ‘t’ with or without subscripts as metalinguistic variables for terms. Note also that we choose to treat descriptions according to Russell’s quantifier analysis and so don’t recognize descriptions as terms of the language. Definition [Formula]: The formulae of AML are defined inductively by the following clauses: • • • • • •

If F is an n-ary predicate symbol and t1,…,tn terms, then F(t1,…,tn) is a formula. If s and t are terms, then (s = t) is a formula If U and W are formulae, then so are :U and (U ^ W) If U is a formula and x is an individual variable, then AxU is a formula If U is a formula, then AU is a formula If U is a formula, then so is U

Definition [Sentence]: A sentence is a formula with no free individual variables. Semantics for AML Definition [Constant Domain Kripke Structure]: A Constant Domain Kripke Structure K = hW, @, D, IKi is a structure such that • •

W is a set of possible worlds @ is a distinguished element of W called the ‘actual world’

19 If the model theory allows for possible variation in the Domain at each world of the model, then in addition to the familiar existential quantifier A, the quantifier AA would be needed, understood to mean ‘there actually exists’ or ‘there exists at the actual world’.

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• •

D is a set of objects called the ‘Domain’ of K IK is an interpretation in hW, @, Di such that IK assigns to each n-place relation symbol Rn and each world w [ W an n-place relation IK(R, w) on D; and for each constant symbol c, IK(c) [ D

Definition [Variable Assignment]: Let K = hW, @, D, Ii be a constant domain model. A variable assignment r in K is an assignment to each variable x of some member of D. Definition [Variant Variable Assignment]: Let r and r0 be two variable assignments in K. We say that r0 is an x-variant of r, if r and r0 agree on all variables except possibly on the value assigned to x. Definition [Term Valuation]: Let K = hW, @, D, Ii be a constant domain model, w a world, and r a variable assignment in K. We assign a member of D to each term t, written (r * IK)(t), as follows: • •

It t is a variable x (r * IK)(x) = r(x) If t is a constant symbol c (r * IK)(c) = IK(c)

Next we define the relation K, w k U [r], understood to mean U is true under variable assignment r at world w of constant domain model K. Definition [Truth in a Model]: Let K be a constant domain model. For each w [ W and valuation r: • • • • • • •

If R is an n-place relation symbol, K, w k R(t1,…,tn) iff hr * IK(t1),…,r * IK(tn)i [ I(R, w) K, w k AU [r] iff K, @ k U [r] K, w k :U [r] iff NOT K, w k U [r] K, w k (U ^ W) [r] iff K, w k U [r] and K, w k W [r] K, w k (t1 = t2) [r] iff (r * IK)(t1) = (r * IK)(t2) K, w k U [r] iff for some u [ W, K, u k U [r] K, w k AxU [r] iff for some x-variant r0 of r, K, w k U [r0 ]

Syntax and semantics for ‘Fixedly’ and ‘Fixedly Actually’ Definition [Model Variance]: Let K and K0 be two constant domain models. K & K0 , if K = hW, z, D, IKi and K0 = hW, w, D, IKi for some w [ W. Intuitively, K0 is a variant of K, if K and K0 differ at most over which world is designated as actual. Definition [Fixedly]: K, w k FU [r] if and only if for all K0 & K, K0 , w k U [r]. Definition [Superficial Necessity]: K, z k hU [r] iff Vw [ W, K, w k U [r]. [Deep Necessity]: K, w k FA(U) [r] iff for any K0 & K, K0 , z k U [r], where K0 = hW, z, D, IK0 i. Syntax for SML Definition [Language of SML]: The language of SML is characterized by the following elements:

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• • • • • • • •

The truth-functionally complete set of connectives :, ^ Symbols of punctuation (,), and, Existential Quantifier A20 Denumerably many individual variables x1, x2, x3,… Denumerably many constant symbols c1, c2, c3,… The possibility operator  Denumerably many indicative predicate symbols of arity n, for all n: F1, F2,… For each n-ary indicative predicate symbol F, an n-ary subjunctive predicate Fs Definition [Term]: The set of terms of SML is characterized by the following list:

• •

Variables are terms Constant symbols are terms.

We use the letters ‘s’ and ‘t’ with or without subscripts as metalinguistic variables for terms. Note also that we choose to treat descriptions according to Russell’s quantifier analysis and so don’t recognize descriptions as terms of the language. Definition [Formula]: The formulae of SML are defined inductively by the following clauses: • • • • •

If F is an n-ary predicate symbol and t1,…,tn terms, then F(t1,…,tn) and Fs(t1,…,tn) are formulae If s and t are terms, then (s = t) is a formula If U and W are formulae, then so are :U and (U ^ W) If U is a formula and x is an individual variable, then AxU is a formula If U is a formula, then so is U

Definition [Subjunctively Closed]: A formula is subjunctively closed if it does not contain an occurrence of a subjunctive predicate that does not lie within the scope of a modal operator. Definition [Sentence]: A sentence of SML is a subjunctively closed formula that contains no free occurrences of variables. Semantics for SML The definitions of constant domain model, term, term valuations, variable assignment, and variant variable assignment are the same as for AML. Definition [Truth in a Model]: Let K be a constant domain model. For each w [ W and variable assignment r: • •

If R is an n-ary predicate symbol, K, w  R(t1,…,tn) [r] iff hr * IK(t1),…,r * IK (tn)i [ IK(R, @) If R is an n-ary predicate symbol, K, w  Rs(t1,…,tn) [r] iff hr * IK(t1),…,r * IK(tn)i [ IK(R, w)

20 If the model theory allows for possible variation in the Domain at each world of the model, then in addition to the indicative quantifier A, the quantifier As would be needed, understood to mean ‘there would have been’.

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• • • • •

K, K, K, K, K,

w w w w w

    

(t1 = t2) [r] iff r * IK(t1) = r * IK(t2) :U [r] iff NOT K, w  U [r] (U ^ W) [r] iff K, w  U [r] and K, w  W [r] U [r] iff there is a u [ W such that K, u  U [r] AxU [r] iff for some x-variant r0 of r, K, w  U [r0 ]

Theorems and proofs Lemma 1 Let U be a formula of non-modal predicate logic. For every S5 constant domain model K = hW, @, D, IKi, world w, and variable assignment r, 0

K; w  Us ½r iff K ; w  U ½r where Us is the result of adding a subjunctive marker to every predicate symbol in U and K0 = hW, w, D, IKi is the variant of K with w as the actual world. Proof •

By induction on the complexity of U

Base Case: U is of the form R(t1,…,tn) K, w  R(t1,…,tn) [r] iff hr * IK(t1),…,r * IK(tn)i [ IK(R, w) iff K0 , w  R(t1,…,tn) [r]

•

U is of the form :W K, w  :Ws [r] iff NOT K, w  Ws [r] iff NOT K0 , w  W [r] (by the Induction Hypothesis) iff K0 , w  :W [r]

•

U is of the form W1 ^ W2 K, w  (Ws1 ^ Ws2) [r] iff K, w  Ws1 [r] and K, w  Ws2 [r] iff K0 , w  W1 [r] and K0 , w  W2 [r] (by the Induction Hypothesis) iff K0 , w  (W1 ^ W2) [r]

•

U is of the form AxW K, w  AxWs [r] iff For some x-variant r0 of r, K, w  Ws [r0 ] iff For some x-variant r0 of r, K0 , w  W [r0 ] (Induction Hypothesis) iff K0 , w  AxW [r].

Theorem 1 For every formula U of non-modal predicate logic, every S5 constant domain model K, world w, and variable assignment r,

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K; w  hUs ½r iff K; w  FU ½r where Us is the result of adding a subjunctive marker to every predicate symbol in U. Proof Follows as a corollary to Lemma 1. K, w  h(Us) [r] iff For all worlds z [ W, K, z  Us [r] iff For all worlds z [ W and for all K0 & K with z the actual world of K0 , K0 , z  U [r] (By Lemma 1) iff For all worlds z [ W and for all K0 & K with z the actual world of K0 , K0 , w  U [r] (Because if a formula containing only indicative predicates is true at the actual world, then it is true at all worlds) iff K, w  F(U) [r]. Lemma 2 Fix a variable assignment r. For every sentence U of non-modal predicate logic, if U is ‘A’-free, then K, w k U [r] iff K0 , w k U [r], where K0 is the variant of K with w actual. Proof •

By induction on the complexity of U

Base Case: U is of the form R(t1,…,tn) K, w k R(t1,…,tn) [r] iff hr * IK(t1),…,r * IK(tn)i [ IK(R, w) iff K0 , w k R(t1,…,tn) [r]

•

U is of the form :W K, w k :W [r] iff NOT K, w k W [r] iff NOT K0 , w k W [r] (by the Induction Hypothesis) iff K0 , w k :W [r]

•

U is of the form W1 ^ W2 K, w k (W1 ^ W2) [r] iff K, w k W1 [r] and K, w k W2 [r] iff K0 , w k W1 [r] and K0 , w k W2 [r] (by the Induction Hypothesis) iff K0 , w k (W1 ^ W2) [r]

•

U is of the form AxW K, w k AxW [r] iff For some x-variant r0 of r, K, w k W [r0 ] iff For some x-variant r0 of r, K0 , w k W [r0 ] (by the Induction Hypothesis) iff K0 , w k AxW [r].

Lemma 3 For every sentence U of non-modal predicate logic, if U is ‘A’-free, then K, w k hU [r] iff K, w k FAU [r]. Proof

Follows as a corollary of Lemma 2

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How many notions of necessity?

K, w k h(U) [r] iff For all z [ W, K, z k U [r] iff For all z [ W and for all K0 & K such that z is actual for K0 , K0 , z k U [r] iff K, w k FA(U) [r]. Theorem 2 Let K be a constant domain model. For every AML sentence U of non-modal predicate logic such that (i) K k hU and (ii) NOT K k FA(U), there is a SML sentence W of non-modal predicate logic such that (iii) U and W are true in all of the same models, (iv) K  hW, and (v) NOT K  hWs. Proof Having discerned the general structure of U (by Lemma 3) such that (i) and (ii) hold (U contains an occurrence of ‘A’) we know by the translation between AML and SML that for each such case there is a sentence W of SML that is true in all of the same models as U (iii). From (i) it follows that U is true at the actual world of K, and hence by (iii) that W is true at the actual world of K. For SML, whenever a sentence containing only indicative predicates is true at the actual world, it is true at every world, and hence (iv). From (ii) we know that for some model K0 & K, U is false at the actual world of K0 , and hence we know by the translation between AML and SML with respect to the notion of truth in a model W is false at the actual world of K0 . This means that NOT K  F(W), and hence by Theorem 1 that (v).

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