Relational Modality - Semantic Scholar
Relational Modality∗ Kathrin Gl¨ uer and Peter Pagin August 7, 2005
Abstract Saul Kripke’s thesis that ordinary proper names are rigid designators is supported by widely shared intuitions about the occurrence of names in ordinary modal contexts. By those intuitions names are scopeless with respect to the modal expressions. That is, sentences in a pair like (a) (b)
Aristotle might have been fond of dogs Concerning Aristotle, it is true that he might have been fond of dogs
will have the same truth value. The same does not in general hold for definite descriptions. Since the scope property is accounted for by means of the intensions of the names and the descriptions, the conclusion is that names do not in general have the same intension as any normal, identifying description. However, the difference in scope behavior between names and description can be accounted for alternatively by appeal to the semantics of the modal expressions. On the account we suggest, dubbed ‘relational modality’, simple singular terms, like proper names, contribute to modal contexts simply by their actual world reference, not by their intension. It turns out to be fully equivalent with the rigidity account when it comes to truth of modal and non-modal sentence (with respect to the actual world), and hence supports the same basic intuitions. It also turns out to be model theoretically equivalent with rigidity semantics with respect to logical consequence, within the class of reflexive models with non-empty domains. Here we give the truth definition for relational modality models, and prove the equivalence results.
Keywords: definite descriptions, logical consequence, modality, necessity, possible worlds semantics, proper names, rigid designators, truth. ∗ The present paper is part of a bigger joint project on names and modality by the two authors. The formal work of the present paper has been done by the second author. A hint of how to go about the technical task was given by Krister Segerberg (over an early morning coffee at Stockholm Arlanda airport, waiting for Saul Kripke to arrive for the Rolf Schock prize ceremonies in 2001).
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Introduction
In Naming and Necessity, Saul Kripke presented a number of soon classical arguments against the description theory of proper names. The perhaps most influential one is known as the modal argument. Kripke argued that proper names in general cannot have the same intensions as co-referring definite descriptions, since substituting the one for the other in modal contexts can change truth value. The intuitions on which this argument is based are widely shared and very robust. Kripke suggested to explain them by the doctrine of rigid designation. In a companion paper1 , we suggest an alternative explanation, one that is compatible with the description theory of names. We agree with Kripke that in ordinary modal thinking we operate with concepts of de re modality. That is, we are interested in the objects we refer to, no matter how they are designated. And we want to know what would be true of these very objects in counterfactual circumstances. The intuitions made use of in Kripke’s modal argument testify to this feature of ordinary modal reasoning; these are data to be accepted and explained by any good semantic theory. However, we do not agree that the best way of explaining them is by means of a thesis concerning the intension of names.2 The observed phenomena, we claim, are essentially due to the de re nature of ordinary modal thinking and are, therefore, better explained in terms of a semantics for modal expressions. We propose such an alternative semantics for ordinary modal expressions. It’s basic idea is that, in ordinary modal contexts, names and other simple singular terms occur referentially. The companion paper contrasts our account (mainly) with Kripke’s account, and elaborates on comparative merits. The aim of the present paper is to present our own account in more technical detail. In the next section, we shall present informally the idea of relational modality and the accompanying account of proper names in modal contexts. In section 3, we provide a formal truth definition for a quantificational language with a relational necessity operator, and prove that this language is semantically equivalent with a classical (notional) modal language with rigid singular terms, with respect to truth (in the actual world) of sentences with modal operators. In section 4 we give a definition of logical consequence for relational modality and show that it is equivalent with with standard logical consequence of the classical modality, with rigid individual constants, in the class of reflexive models. The equivalence also holds for models with constant domain, without the reflexivity requirement. As is shown, the equivalence holds for the usual systems of modal logic. 1 ‘Proper
names and relational modality’, draft. a difference between the semantic contributions of names and descriptions in modal contexts will be induced by the semantics for modal expressions, linguistic meaning on our account cannot be equated with intension. Cf the remark after Fact 1, section 3, page 11. 2 Since
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Relational modality
Relational modality is intended to account for the natural language modal intuitions that Kripke originally appealed to, without invoking the rigidity thesis about proper names. According to these intuitions the two sentences: (1)
Aristotle might not have gone into pedagogy
(2)
The teacher of Alexander might not have gone into pedagogy
are not equivalent (cf. Kripke 1980: 61-63). The basic intuition is that (1) is true, while (2) is false. According to Kripke’s account, the difference between (1) and (2) depends on a difference in intension between the name ‘Aristotle’ and and the description ‘the teacher of Alexander’. This difference induces a corresponding difference in intension between the two simple sentences (3) and (4): (3)
Aristotle did not go into pedagogy
(4)
The teacher of Alexander did not go into pedagogy
(cf. Kripke 1980: 6f). According to Kripke, the intensional difference between (3) and (4) explains the extensional difference between (1) and (2). On this view it is the semantic difference between the name and the description that is decisive, not the syntactic difference. It is true that, on Kripke’s view, there is an intensional property common to all the (semantically non-empty) members of the syntactic category of proper names, the property of having a constant function from worlds to objects as intension. But on Kripke’s view, that is not a property that exclusively belongs to names. Every rigid definite description has it, too. The syntactic difference between names and descriptions are not, on this view, relevant to modal contexts. What is relevant is only that some descriptions are not rigid. According to our account, the difference between (1) and (2) depends on a feature of ordinary modal thinking, not on the intensions of names. When, in ordinary modal thinking, we consider alternative possibilities, we are interested in alternative scenarios involving the objects we refer to. We are interested in what might have happened to these very objects, regardless of how their names are evaluated with respect to those alternative scenarios. At least, this is an empirical hypothesis of the present authors about ordinary modal thinking, and hence about the ordinary modal concepts expressed by locutions like ‘possibly’, ‘necessarily’, ‘it might have been’ and ‘it would have been’, as used in everyday discourse. In brief, the proposal is that simple singular terms, including proper names, occur referentially in the contexts of ordinary (alethic) modal expressions.3 On 3 In Kaplan’s terminology (Kaplan 1986: 230), the position of a singular term within a sentence is open to substitution if the result of replacing a term in that position by a coreferential one does not affect the truth value of the sentence. A sentential context is then referentially opaque, in Quine’s terminology (Quine 1952: 142) if any sentence (i.e. sentence occurrence) embedded in that context loses the positions open to substitution that it has
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the present account, these contexts are intensional with respect to other types of expression, in particular first order predicates. Because of this, the evaluation of (1) is as follows: (1) is true if, and only if, what ‘Aristotle’ actually refers to did not, in some possible world, go into pedagogy. (2), on the other hand, is true if, and only if, what ‘Alexander’ actually refers to is such that, in some possible world, his teacher (in that world) did not go into pedagogy (in that world). Since, intuitively, on this evaluation, (1) is true and (2) is false, the semantic difference has been accounted for without the appeal to the rigidity thesis. Moreover, the syntactic difference between the name and the description plays a role, for names, like all simple singular terms, contribute to truth and falsity in modal sentences with their actual reference, regardless of their possible worlds intension. For all we care, ‘Aristotle’ might have the same intension as ‘the teacher of Alexander’. This account is not an equivalent way of making Kripke’s original point. It is consistent with our account that proper names are not in general rigid. As far as we are concerned, it may be that some are and some aren’t. So, assume that we believe that ‘Aristotle’ has the same intension as ‘the teacher of Alexander’. Kripke does not believe that. On that assumption we differ with respect to the truth conditions of (3)
Aristotle did not go into pedagogy.
For Kripke, (3) is true with respect to a possible world w if, and only if, Aristotle did not go into pedagogy in w. For us, given the assumption, (3) is true with respect to w if, and only if, the referent of ‘the teacher of Alexander’ in w did not go into pedagogy in w. On our interpretation, (3) is not true in any world, whereas for Kripke it is true in some. There is a corresponding disagreement over the metalinguistic statement (5)
(3) might have been true.
On Kripke’s account, (5) is true (cf. Kripke 1980: 12). On our account, given the assumption, it is false. So the accounts are not equivalent. In both cases, of course, we consider what truth value (5) might have had given the meaning it actually has. Something that does hold on our account as well as on Kripke’s is the ‘scopelessness’ or scope indifference of names in relation to modal operators. In fact, on its own. In Kaplan 1986 it is argued against Quine that a position that is not open to substitution can nevertheless contain a variable that is bound by an initially placed quantifier (as in ∃xF x). When we say that proper names occur referentially in modal contexts we do not mean that they occur in positions open to substitution. A name in a modal context cannot in general be replaced salva veritate by a description or functional expression co-referential with it. So modal contexts are opaque. However, on our interpretation, all co-referential simple singular terms, including proper names and free variables, can be interchanged salva veritate in modal contexts. This is what we mean by saying that names occur referentially in modal contexts, and that modal contexts take names transparently (we might call these contexts semi-transparent). Because of this feature, Kaplan’s objection against Quine is exemplified by the interpretation we propose.
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it must hold on any account that agrees with ordinary modal intuitions. However, scope indifference has sometimes been equated with the rigidity thesis, it has even been held as an alternative way of stating it (Kripke does so himself in 1980: 12, fn. 15). This is correct only if names do not occur referentially in modal contexts, for then the equivalence of the wide and narrow scope readings such as (6)
F (Aristotle)
(7)
∃x(x = Aristotle & F x).
depends on the intensions of names. If names do occur referentially, then these equivalences hold as a matter of course, and whether names are rigid designators or not. In fact, this is precisely what we propose.4 The basic idea for effecting this, is the following interpretation of ‘necessary’: (N)
pIt is necessary that φq is true iff φ is true no matter what extensions are assigned to its non-logical predicates and functional expressions.
With this clause in a truth definition, the extension of singular terms is simply left unaffected by the evaluation, while there is a variation in extension of the non-logical predicates and functional expressions. For instance, (8)
It is possible that Plato’s father was richer than Aristotle’s father
comes out true, on this interpretation, just if in some extension assignment to the two-place predicate ‘...was richer than...’ and to the functional expression “...’s father” the embedded (9)
Plato’s father was richer than Aristotle’s father
comes out true. Of course, this is not formally precise. As stated it is also inadequate, for there is no mention of how the extension assignment to a predicate is restricted by its meaning, nor of how assignments to different expressions may be combined. Both these problems are solved by switching to the standard framework of possible worlds semantics. The question is how to formulate the intended equivalent to (N) within that framework. The answer comes in two very simple ideas. The first idea is what we call actualist evaluation. Standardly, an atomic sentence P t1 , ..., tn is evaluated as true in a possible world w just in case the n-tuple of the referents of t1 , ..., tn in w belongs to the extension of P in w. That is, where I is an interpretation function assigning referents to terms and extensions to predicates in possible worlds, (P)
True(P t1 , . . . , tn , w ) iff hI(t1 , w ), . . . , I(tn , w )i ∈ I(P, w ).
4 It
is also for this reason that we have chosen to call our account ‘relational modality’; because of the similarity with Quine’s distinction between the notional and the relational concepts of belief, in Quine 1956. This was first suggested to us by Sten Lindstr¨ om.
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In the actualist evaluation we consider instead the referents in the actual world, a. (A)
True(P t1 , . . . , tn , w ) iff hI(t1 , a), . . . , I(tn , a)i ∈ I(P, w ).
For the predicate, the extension in w matters, but for the terms only their extension in a.5 When considering different worlds, we consider different extensions of the predicate, but just the same extension of the terms. To complete the definition of actualist evaluation one adds clauses for connectives, quantifiers and modal operators, no different from the ordinary ones (see section 3). Not surprisingly, the actualist evaluation turns out to be semantically equivalent with a standard semantics with rigid singular terms (Fact 3). Since a rigid term denotes the same object in every world where that object exists, I(t, a) is bound to be the same as I(t, w), if t is a rigid term (and the object denoted exists in w ). An actualist evaluation semantics does not make use of the extension of a singular term in any other world than the actual world. That is, it doesn’t make use of intensions of singular terms. So instead of just using an actualist evaluation as your semantics, it would be better to drop the intensions and simply speak of the reference of a singular term. This would be to follow the Kaplan-Almog line of direct reference (see Almog 1986). As stated, (A) is well defined only for simple terms, for which the reference is given primitively by the interpretation function I. Since we prefer to stay neutral on the question whether there are complex singular terms, we have to take such terms into account. Suppose, then, that applied functional expressions, like ‘g(u)’, where ‘u’ again is a singular term, simple or complex, are singular, and that definite descriptions, like ‘the x such that F x’, or ‘ xF x’, are singular too. In order to accommodate these terms with the desired result, (A) needs to be replaced by ι
(A+)
True(P t1 , . . . , tn , w ) iff hV(t1 , w ), ..., V(tn , w )i ∈ I(P, w )
where the term evaluation function V is defined as follows: (V)
V(t, w) = I(t, a), in case t is simple V(g(u), w) = I(g, w)(V(u, w)) V( xF x, w) = the unique object b such that True(F x, w) with b assigned to x, and undefined if there is no such object ι
where I(g, w) is the function (in extension) assigned to g in w (for a formally more adequate statement, see section 3). By (V), simple singular terms are evaluated with respect to the actual world, while functional expressions and pre5 This makes it different from simply applying the actuality operator to the sentence, for that affects the evaluation of both terms and predicates: True(Aφ, w ) iff True(φ, a). Still, it is possible to have a semantics equivalent to the actualist evaluation by adding the actuality operator to a standard semantics with non-rigid singular terms: let each non-rigid term t occur only in the following context: ‘the x such that A(x = t)’. The result will be that only the actual world reference of terms will matter, while predicates are evaluated as usual.
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dicates within complex singular terms are evaluated with respect to the possible world in question.6 The second part of our proposal is to use the actualist evaluation only for the semantics of modal sentences. By contrast, if it were simply used across the board, we would have a uniform semantics for modal and non-modal sentences alike, where the actualist truth conditions of a non-modal sentence would be exactly what it contributes to the evaluation of a modal sentence containing it. Such a semantics would not have anything in particular to do with interpreting modal expressions. The relational modality proposal, however, is to use the standard (A) for the ordinary truth conditions of atomic sentences, and to use (A+) for the semantic contribution of an atomic sentence to a modal sentence containing it. The idea, then, is to have a truth definition clause for the modal expression that runs something like this: (M)
True(p It is necessary that φq, w) iff Actua-true(φ, w 0 ) at any world w 0 accessible from w
where ‘Actua-true’ just means true according to the (completed) actualist evaluation. In this way we distinguish between ordinary truth conditions and the semantic properties a sentence contributes to the truth conditions of modal sentences containing it (which is its actualist truth conditions).7 The resulting interpretation does accommodate all our basic modal intuitions. For instance, if we adapt the proposal to ‘might have’, understood as ‘not necessarily not’, using classical predicate logic and for simplicity treating ‘go into pedagogy’ as a simple predicate, we get the following result for (2): (2) is true iff there is some accessible world w such that V(‘the teacher of Alexander’, w ) does not belong to I(‘goes into pedagogy’, w ). The right hand side holds iff there is a unique object b of which ‘x teaches Alexander’ is true (with b assigned to ‘x’) in w, and b does not go into pedagogy in w. Again, this holds iff there is a unique object b such that the pair of b and V(‘Alexander’, w ) belongs to I(‘x teaches y’, w ), and b does not go into pedagogy in w. Since V(‘Alexander’, w ) 6 It should be stressed that we have relied on the assumption that any proper name (individual constant) that has reference in some possible world also has reference in the actual world. This is philosophically well motivated. Indeed, we think that Kripke is entirely right in denying that Sherlock Holmes might have existed (Kripke 1980: 157-8), since there is no actual referent with which to identify any particular individual in any particular possible world. This goes for our account, too. It is, of course, compatible with our view that ‘Sherlock Holmes’ has reference in other possible worlds, but because of our interpretation of ‘might have’ we have to deny that Sherlock Holmes might have existed. Since actual reference is all that matters to actual truth, even of modal statements, on our interpretation, we are justified in restricting attention to names that do have actual reference. 7 This spells out a difference corresponding to Dummett’s distinction between content and ingredient sense. In the semantics Kripke proposes there is no such difference, and he has been criticized by Dummett (cf. Dummett 1981: 572f, 1991: 48), Evans (1979), and Stanley (1997a,b) for not taking account of the distinction. In our semantics, the distinction corresponds to a real difference.
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is I(‘Alexander’, a), this holds iff there is a unique object b that in w teaches what ‘Alexander’ refers to in a, and b does not go into pedagogy in w. This is the desired interpretation. Not only do we get the desired results for examples like (1) and (2), once a clause for the definite description construction is included in the semantics. It is also provably true that the relational modality interpretation will give the same evaluation of any formula with respect to the actual world, as the standard interpretation with rigid singular terms (see below, Theorem 5). Since truth in the actual world simply is truth, if by ‘the actual world’ we do mean the actual world, not just some entity designated as such in a model, this means that the rigidity interpretation and the relational modality interpretation will agree with respect to simple truth and falsity of any modal and non-modal statement. Hence, they are empirically equivalent with respect to basic modal intuitions.
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Classical and relational modality: truth
The basic idea for implementing relational modality in possible worlds semantics is to do it indirectly, via a different semantic evaluation which is actualist in the sense that for individual constants only the reference in the actual world matters for evaluation. The method is then to let the actualist evaluation kick in precisely at the clause for necessity. The result will be that a model for relational modality, so defined, is actually equivalent with normal semantics with rigid designation, i.e. gives the same truth value for sentences with respect to the actual world. We shall give a version of Kripke-style possible worlds semantics where we allow the world-bound domain of individuals to vary from world to world. For this reason we have chosen a partial semantics, with truth value gaps in case of reference failure. We shall require that individual constants do have a referent in the actual world (see note 6). A model M is a structure hW, R, a, Dom, D, Ii where W is a set of entities (possible worlds), R a relation on W (the accessibility relation), a a privileged member of W (the actual world), Dom the domain of individuals of M, D a function from W to P (Dom) (giving for each world the set of individuals existing in that world) and I an interpretation function, giving for each pair of a non-logical constant e and w ∈ W the extension of e in w. We shall compare standard models Mc , with rigid individual constants, with relational modality models Mr , with non-rigid constants, for a single quantified modal language L. L is built up from predicate letters of varying arity from the set PL (where the notation ‘P n ’ indicates that n is the arity of P ), individual constants from CL and individual variables from Var L , and ¬, &, ∀ and , according to the ordinary formation rules. For both of Mc and Mr , we shall require that for a non-modal formula φ to have a truth value with respect to a possible world w, relative to a model M and an assignment µ to free variables, it must be the case for any constant c in φ that
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I(c, w) ∈ D(w) and for any free variable x in φ that µ(x) ∈ D(w). If it is not the case that I(c, w) ∈ D(w), then I is undefined for (c, w). For a formula φ, the requirement is that φ is well defined with respect to all accessible worlds. This is a strong requirement, but avoids some logical problems that can otherwise arise (see Gamut 1991: 55-6). The definedness relation Def between a triple hM, µ, wi and a formula φ will be set out recursively as part of each truth definition. Below, Iµ (t, w) = I(t, w) in case t is a constant, and Iµ (t, w) = µ(t) in case t is a variable. An assignment µ, or µM , of a model M, is a function from Var L , the set of individual variables of L, to Dom M . Also, assignment µ0 is an x-variant of assignment µ iff µ and µ0 differ at most in what value they assign to x. First, the definition of the classical definedness relation for arbitrary model M: Definition 1. The definedness relation Defc a)
Def c (hMc , µ, wi, P t1 , . . . , tn ) iff Iµ (ti , w) ∈ D(w), for 1 ≤ i ≤ n
b)
Def c (hMc , µ, wi, ¬φ) iff Def c (hMc , µ, wi, φ)
c)
Def c (hMc , µ, wi, φ & ψ) iff Def c (hMc , µ, wi, φ) and Def c (hMc , µ, wi, ψ)
d)
Def c (hMc , µ, wi, ∀xφ) iff Def c (hMc , µ0 , wi, φ) for any x-variant µ0 of µ such that µ0 (x) ∈ D(w)
e)
Def c (hMc , µ, wi, φ) iff Def c (hMc , µ, w0 i, φ) for any w0 such that wRw0
Definition 2. A classical model Mc is a structure hW, R, a, Dom, D, Ii such that a)
P t1 , . . . , tn iff hIµ (t1 , w), . . . , Iµ (tn , w)i ∈ I(P n , w) |= Mc , w, µ
b)
|= ¬φ iff Def c (hMc , µ, wi, ¬φ) and not |= φ Mc , w, µ Mc , w, µ
c)
φ & ψ iff |= φ and |= ψ |= Mc , w, µ Mc , w, µ Mc , w, µ
d)
|= ∀xφ iff Mc , w, µ |= φ for all x-variants µ0 of µ such that µ0 (x) ∈ D(w) Mc , w, µ0
e)
|= φ iff |= φ for any w0 such that wRw0 Mc , w, µ Mc , w0 , µ
f)
Extension postulate: for any predicate P n of L and any w ∈ W, I(P n , w) ⊆ D(w)n
g)
Rigidity postulate: for any individual constant c of L and any w ∈ W, if I(c, w) is defined, then I(c, w) = I(c, a).
Now we move on to defining the relational modality models Mr . We shall proceed the way, by first giving setting out the definedness definition, which
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comes in two steps. First, the actualist variety: Definition 3. The definedness relation Defa a)
Def a (hM, µ, w, i, P t1 , . . . , tn ) iff Iµ (ti , a) ∈ D(w), for 1 ≤ i ≤ n
b)
Def a (hM, µ, w, i, ¬φ) iff Def a (hM, µ, w, i, φ)
c)
Def a (hM, µ, w, i, φ & ψ) iff Def a (hM, µ, w, i, φ) and Def c (hMc , µ, wi, ψ)
d)
Def a (hM, µ, w, i, ∀xφ) iff Def a (hM, µ0 , wi, φ) for any x-variant µ0 of µ such that µ0 (x) ∈ D(w)
e)
Def a (hM, µ, w, i, φ) iff Def a (hM, µ, w0 i, φ) for any w0 such that wRw0
Second, there is the relational definedness relation, which appeals to the actualist in the fifth clause: Definition 4. The definedness relation Defr a)
Def r (hMr , µ, wi, P t1 , . . . , tn ) iff Iµ (ti , a) ∈ D(w), for 1 ≤ i ≤ n
b)
Def a (hM, µ, w, i, ¬φ) iff Def r (hMr , µ, wi, φ)
c)
Def r (hMr , µ, wi, φ & ψ) iff Def r (hMr , µ, wi, φ) and Def c (hMc , µ, wi, ψ)
d)
Def r (hMr , µ, wi, ∀xφ) iff Def r (hMr , µ0 , wi, φ) for any x-variant µ0 of µ such that µ0 (x) ∈ D(w)
e)
Def r (hMr , µ, wi, φ) iff Def a (hM, µ, w0 i, φ) for any w0 such that wRw0 We now state the actualist evaluation relation for relational models:
Definition 5. The actualist evaluation of a relation modality model Mr is a a such that relation |= Mr , w, µ a
a)
|= P t1 , . . . , tn iff hIµ (t1 , a), . . . , Iµ (tn , a)i ∈ I(P n , w) Mr , w, µ
b)
a ¬φ iff Def a (hM, µ, w, i, ¬φ) and not |= φ |= Mr , w, µ Mr , w, µ
c)
|= φ & ψ iff |= φ and |= ψ Mr , w, µ Mr , w, µ Mr , w, µ
d)
|= ∀xφ iff Mr , w, µ a |= φ for all x-variants µ0 of µ such that µ0 (x) ∈ D(w) Mr , w, µ0
e)
|= φ iff |= φ for any w0 such that wRw0 Mr , w, µ Mr , w0 , µ
a a
a
a
a
a
a
The following definition of the relation modality model appeals to the actualist evaluation in its fifth clause.
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Definition 6. A relational modality model Mr is a structure hW , R, a, Dom, D, Ii such that a)
|= P t1 , . . . , tn iff hIµ (t1 , w), . . . , Iµ (tn , w)i ∈ I(P n , w) Mr , w, µ
b)
¬φ iff Def r (hMr , µ, wi, ¬φ) and not |= φ |= Mr , w, µ Mr , w, µ
c)
|= φ & ψ iff |= φ and |= ψ Mr , w, µ Mr , w, µ Mc , w, µ
d)
|= ∀xφ iff Mr , w, µ |= φ for all x-variants µ0 of µ such that µ0 (x) ∈ D(w) Mr , w, µ0
e)
φ for any w0 such that wRw0 |= φ iff |= Mr , w, µ Mr , w0 , µ
f)
Extension postulate: for any predicate P n of L and any w ∈ W, I(P n , w) ⊆ D(w)n
a
Fact 1. For any model Mi , with i = c, r, if |= φ, then Defi (hMi , µ, wi, φ). Mi , w, µ Proof. By induction over formula complexity. We shall do it for i = r. For i = c it is similar and easier. For the atomic formulas, by clause a) of definition 6, hIµ (t1 , w), . . . , Iµ (tn , w)i ∈ I(P n , w) and by the extension postulate I(P n , w) ⊆ D(w)n , which together imply that for each ti , I(ti ) ∈ D(w), which is what’s required by clause a) of definition 4. For negation it is immediate from the b) clauses. For conjunction it follows directly by the c) clauses and the induction hypothesis. Similarly, for ∀xφ it follows directly by the d ) clauses and the induction hypothesis. For φ, it is required that Def a (hM, µ, w0 i, φ). To carry out this step we a must first prove that for any formula φ, if |=Mr , w, µ φ, then Def a (hM, µ, w, i, φ). This is done again by induction over formula complexity, just as above, but this time by definitions 3 and 5. For φ, it follows by clauses e) and the induction hypothesis. This, finally, gives us the induction hypothesis for φ and Def r . The step is immediate from clauses e) of definitions 4 and 6 and the induction hypothesis. QED Remark. For simplicity we have not included functional expressions and definite descriptions as complex singular terms. To include them, several changes would be needed. For instance, the following changes would be needed in definition 5: Instead of a)
a |= P t1 , . . . , tn iff hIµ (t1 , a), . . . , Iµ (tn , a)i ∈ I(P n , w) Mr , w, µ
we would have a 0)
a |= P t1 , . . . , tn iff hVµ (t1 , w ), . . . , Vµ (tn , w )i ∈ I(P n , w ) Mr , w, µ
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where V is defined as follows: (V)
Vµ (t, w) = I(t, a), in case t is an individual constant Vµ (t, w) = µ(t), in case t is a variable Vµ (g(t), w) = I(g, w)(V(t, w)) Vµ ( xψ, w) = the object b such that for a unique x-variant µ0 of µ, a ψ. Undefined if there is no such object b. µ0 (x) = b and |= Mr , w, µ0 ι
In this way a name and a description may have the same intension, but still not be synonymous, since they are not interchangeable salva veritate in modal contexts. Because of this, it would violate the principle of compositionality to equate intension with linguistic meaning. Rather, the linguistic meaning of a singular term can be taken to be the pair of its I-intension and its V-intension, where the V-intension is a function from possible worlds to V-values. This is then extended to formulas in the obvious way. For instance, the V-intension of an atomic formula P t1 , . . . , tn is a function from worlds to truth values in accordance with clause a0 ) above. Note that the V-intension is determined by the I-intension together with syntax, but only if syntactic distinctions between kinds of singular terms are made. End of remark.
We aim to prove that Mc and Mr give the same evaluation of all formulas with respect to the (shared) actual world. To this end we need to establish three facts. The second will be that Mc is fully equivalent with the actualist evaluation of Mr . Definition 7. Two models, M = hW, R, a, Dom, D, Ii and M0 = hW0 , R0 , a0 , Dom0 , D0 , I0 i, are equi-actual, M ∼ M0 , iff W = W0 , R = R0 , a = a0 , Dom = Dom0 , D = D0 and for all predicate letters P, for all worlds w ∈ W, I(P, w) = I0 (P, w) and for all individual constants c, I(c, a) = I0 (c, a0 ). M and M0 are atomically equal, M ≈ M0 , iff M ∼ M0 , and for all individual constant c and all worlds w , I(c, w) = I0 (c, w0 ). For the following claims, we shall assume that Mc ∼ Mr , where Mc = hW, R, a, Dom, D, Ii and Mr = hW0 , R0 , a0 , Dom0 , D0 , I0 i. The first claim says that a formula is well-defined with respect to a classical evaluation just if it is well-defined with respect to the actualist evaluation of the corresponding relational model. Fact 2. For all formulas φ ∈ L, assignments µ, and worlds w ∈ W, Defc (hMc , µ, wi, φ) iff Defa (hM, µ, w, i, φ). Proof. By induction over formula complexity.
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a) Atomic formulas. Def c (hMc , µ, wi, P t1 , . . . , tn ) iff Iµ (ti , w) ∈ D(w), for 1 ≤ i ≤ n, iff Iµ (ti , a) ∈ D(w), for 1 ≤ i ≤ n iff I0µ (ti , a0 ) ∈ D0 (w), for 1 ≤ i ≤ n iff Def a (hM, µ, w, i, P t1 , . . . , tn ) b) Negation. Def c (hMc , µ, wi, ¬φ) iff Def c (hMc , µ, wi, φ) iff Def a (hM, µ, w, i, φ) iff Def a (hM, µ, w, i, ¬φ)
(by def 1a) (by rigidity, def 2g) (by ∼ assum. and def 7) (by def 3a)
(by def 1b) (by ind. hyp.) (by def 3b).
c) Conjunction. Similar to negation. d) Universal quantifier. Def c (hMc , µ, wi, ∀xφ) iff Def c (hMc , µ0 , wi, φ), for all x-var. µ0 of µ s.t. µ0 (x) ∈ D(w) iff (by def 1d) Def a (hM, µ0 , wi, φ), for all x-var. µ0 of µ s.t. µ0 (x) ∈ D0 (w) iff (by ind. hyp., ∼-assum. and def 7) Def a (hM, µ, w, i, ∀xφ) (by def 3d) e) Necessity operator. Def c (hMc , µ, wi, φ) iff Def c (hMc , µ, w0 i, φ) for any w0 such that wRw0 iff (by def 1e) Def a (hM, µ, w, i, φ) for any w0 such that wR0 w0 iff (by ind. hyp., ∼-assum., and def 7) Def a (hM, µ, w, i, φ). This completes the induction.
QED
With the help of Fact 2 we can prove the equivalence of classical and actualist evaluations. Fact 3. For all formulas φ ∈ L, assignments µ, and worlds w ∈ W, a φ iff |= φ. |= Mc , w, µ Mr , w, µ Proof. By induction over formula complexity. The atomic, conjunction, quantification and modal cases are analogous to the corresponding cases in the proof of Fact 2. In the negation case we have b) Negation. ¬φ iff |= Mc , w, µ Defc (hMc , µ, wi, ¬φ) and not |= φ iff Mc , w, µ a φ iff Defa (hM, µ, w, i, ¬φ) and not |= Mr , w, µ a ¬φ |= r M , w, µ
(by def 2b) (by Fact 2 and ind. hyp.) (by def. 5b) QED
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Fact 2 also helps to establish the equivalence of classical and relational definedness: Fact 4. For all formulas φ ∈ L, assignments µ, and worlds w ∈ W, Defc (hMc , µ, wi, φ) iff Defr (hMr , µ, wi, φ). Proof. By induction over formula complexity. The atomic case is immediate, and the negation, conjunction and quantification cases are as in the proof of Fact 2. For the modal case we have e) Necessity operator. Def c (hMc , µ, wi, φ) iff Def c (hMc , µ, w0 i, φ) for any w0 such that wRw0 iff Def a (hM, µ, w, i, φ) for any w0 such that wR0 w0 iff Def r (hMr , µ, wi, φ)
(by def 1e) (by Fact 2) (by def 4e) QED
We can now prove the equivalence of classical and relational models with respect to truth (in the actual world). Theorem 5. Let Mc = hW, R, a, Dom, D, Ii and Mr = hW0 , R0 , a0 , Dom0 , D0 , I0 i. If Mc ∼ Mr , then for any φ ∈ L and any assignment µ, φ iff |= φ |= Mc , a, µ Mr , a0 , µ Proof. By induction over formula complexity. a) Atomic case. |= P t1 , . . . , tn iff Mc , a, µ hIµ (t1 , a), . . . , Iµ (tn , a)i ∈ I(P, a) iff hI0µ (t1 , a 0 ), . . . , I0µ (tn , a 0 )i ∈ I0 (P, a 0 ) iff |= P t1 , . . . , tn Mr , a0 , µ b) Negation. |= ¬φ iff Mc , a, µ Def c (hMc , µ, wi, ¬φ) and not |= φ iff Mc , a, µ φ iff Def r (hMr , µ, wi, ¬φ) and not |= Mr , a0 , µ |= ¬φ Mr , a0 , µ
(by def 2a) (By def 7, since Mc ∼ Mr ) (by def 6a)
(by def 2b) (by Fact 4 and ind. hyp.) (by def 6b)
c-d) Conjunction and quantification, straightforward. e) Necessity. |= φ iff Mc , a, µ |= φ at any world w0 s.t. aRw0 iff Mc , w0 , µ a |= φ at any world w0 s.t. aRw0 iff Mr , w0 , µ |= φ Mr , a0 , µ
(by def 2e) (by Fact 3) (by def 6e) QED
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Hence, the relational modality models give the same evaluations at the actual world as the corresponding classical models.
4
Classical and relational modality: consequence
Let’s turn now to the issues of consequence and validity. As usual, validity is just the special case of consequence when the set of premises is empty, so we shall simply speak of consequence. We are going to investigate relations between classical and relational models with respect to this notion. It will turn out that there is concept of consequence for relational modality that is almost equivalent with the concept of consequence for the classical semantics. Where the classical concept has it that φ is a consequence of Γ iff φ is true at a world w if Γ is true at w, for all worlds in all models, the relational counterpart is that φ is a consequence of Γ iff φ is true at the actual world a if Γ is true at a, in all models. These two notions are almost but not completely equivalent. Because of denotation failure, there is a divergence both with respect to quantification and with respect to modality. Consider the pair of sentences (10)
a) b)
∀x(P tx) P tt
where P tt is atomic and t an individual constant. With the requirement that any term that does have an interpretation in a model has a denotation in the actual world in that model, whenever (10a) is true at the actual world in some model, the term t has a denotation in the actual world, and then if (10a) is true in the actual world, so is (10b). So, it is a relational consequence. However, it is not a classical consequence. For consider a world w with an empty domain. The premise (10a) is vacuously true with respect to w, and the sentence also vacuously well-defined with respect to that world and relevant assignment µ. For this holds if P tx is well-defined with respect to that world for all x-variants µ0 of µ such that µ0 (x) ∈ D(w), and since D(w) = ∅, this condition is vacuously fulfilled by all assignments. But (10b) is not well-defined with respect to w, since t obviously has no denotation there. So, (10b) is not true at w, and the consequence fails. Similarly, consider the pair of sentences (11)
a) b)
P t ¬(P t & ¬P t)
where P t is atomic and t an individual constant. With the requirement that any term that does have an interpretation in a model has a denotation in the actual world in that model, whenever (11a) is true at the actual world in some model, the term t has a denotation in the actual world, and then (11b), being a tautology, is true at the actual world, since it is true at any world where the term does have a denotation. So, it is a relational consequence.
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Again, however, it is not a classical consequence. For consider a world wk in a classical model M such that (11a) is true at wk in M. This requires exactly that P t is true at all worlds w accessible from wk , and hence that t has a denotation in all of these worlds. Hence, in all those worlds, (11b) is true as well. However, it is not required that wk be accessible from itself, and so t need not have a denotation in wk . If not, (11b) will be neither true nor false in wk . So the consequence fails. Nevertheless, the equivalence between the notions does hold if it is required both that world-bound domains be non-empty and that worlds be accessible from themselves. That is, the notions are equivalent with respect to the class of reflexive models with non-empty world domains. Model theoretically, this is a limitation. Philosophically, the limitation is not severe, since both (10) and (11) are pairs where the the second member intuitively follows from the first. Moreover, we want to infer truth from necessity, i.e. p from necessarily p, and the validity of this inference is guaranteed, from a possible worlds semantics perspective, only when the accessibility relation is reflexive. It may be noted that in case we assume an invariant non-empty worldbound domain of individuals, the counter-examples do not go through. And as can easily be seen by inspecting the proof of Lemma 13, with this assumption there is equivalence between classical and relational consequence, without the reflexivity restriction. However, since this assumption makes the comparison less interesting, we shall proceed without it. We shall now establish the equivalence with respect to non-empty reflexive models. Assume that we have a model frame F = hW, a, Domi. From F we define a class of classical models and a class of relational models: Definition 8. Classes of classical and relational models on the frame F. a) Fc = {M : M is a model on the frame F that satisfies definitions 1—2 and such that RM is reflexive and for all w ∈ W, D(w) 6= ∅.} b) Fr = {M : M is a model on the frame F that satisfies definitions 3—6 and such that RM is reflexive and for all w ∈ W, D(w) 6= ∅.} Now we shall proceed with some facts about the relation between Fc and Fr . Fact 6. For any M ∈ Fc there is an M0 ∈ Fr such that M ≈ M0 , and hence, M ∼ M0 , and also vice versa. Proof. Immediate from definitions 7 and 8.
QED
Fact 7. For any φ ∈ L, any Mc ∈ Fc and Mr ∈ Fr such that Mc ∼ Mr , any w ∈ W and assignment µ, φ iff |= φ |= Mc , w, µ Mr , w, µ
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Proof. The proof is a simple generalization of case e) of Theorem 5: |= φ iff Mc , w, µ |= φ at any world w0 s.t. wRw0 iff Mc , w0 , µ a |= φ at any world w0 s.t. wRw0 iff Mr , w0 , µ |= φ Mr , w, µ
(by def 2e) (by Fact 3) (by def 6e) QED
Fact 8. Let Mc = hW, R, a, Dom, D, Ii and Mr = hW0 , R0 , a0 , Dom0 , D0 , I0 i. If Mc ≈ Mr , then for any φ ∈ L and any assignment µ, φ iff |= φ |= Mc , w, µ Mr , w, µ Proof. By induction over formula complexity. Since we have Mc ≈ Mr , and hence I = I0 for all atomic non-logical constants, the proofs of the atomic, negation, conjunction and quantification cases are straightforward. The modal case follows from Fact 7. QED Definition 9. Generalizing over worlds and assignments in model: a) |= φ iff for all w ∈ W, and all assignments µ, |= φ Mc Mc , w, µ b) |= φ iff for all w ∈ W, and all assignments µ, |= φ Mr Mr , w, µ Fact 8 facilitates the proof of the following inclusion claim: Fact 9. For all φ ∈ L, if |= φ for all Mr ∈ Fr , then |= φ for all Mc ∈ Fc . Mr Mc Proof. Assume that there is a formula φ, a world w ∈ W, and an assignment µ such that not |=Mc, w, µ φ. By fact 6 there is corresponding model Mr ∈ Fr s.t. Mc ≈ Mr . By Fact 8, |=Mc, w, µ φ iff |=Mr , w, µ φ. Hence, not |=Mc, w, µ φ. By contraposition and generalization, the claim follows. QED For non-modal sentences it does not in general hold that they are valid in a rigid classical model iff valid in a ∼-corresponding relational model. For example, assume that both I(c1 , w) and I(c2 , w) are defined for all w ∈ W. Then it holds that either c1 = c2 or c1 6= c2 is true in all w ∈ W in Fc models, but in general this does not hold in Fr . However, this is not of great significance, since we are interested in validity at a higher level: truth in all worlds for variable model. It can be shown that for any φ ∈ L that is modality free, i.e. does not contain any modal operator, it holds that for any Mc ∈ Fc and Mr ∈ Fr , such φ iff |= φ. that Mc ∼ Mr that |= Mc Mr Still, the converse of Fact 9 does not hold: Fact 10. It is not the case that: for all φ ∈ L, if |= φ for all Mc ∈ Fc , then Mc r |= φ for all M ∈ Fr . Mr 17
Proof. With φ → ψ defined as ¬(φ & ¬ψ), the sentence (12)
c1 = c2 → ( c1 = c2 )
is valid in all classical models. But it is not valid in all relational models. It is false in a model Mr s.t. for some world w, I(c1 , w) = I(c2 , w), but I(c1 , a) 6= I(c2 , a). Then the antecedent of (12) is true at w and the consequent is false at w. QED That (12) isn’t valid in relational models is not so strange. It isn’t valid in non-rigid standard models either. The converse, however, (13)
( c1 = c2 ) → c1 = c2
is valid in reflexive non-rigid standard models, but still not valid in relational models. To get a counter-example, let I(c1 , a) = I(c2 , a) and I(c1 , w) 6= I(c2 , w). Then the antecedent is true and the consequent false at w. Still, if a sentence is valid in a classical model, its necessitation is valid in the corresponding relational model: Fact 11. For any φ ∈ L, if |=Mc φ for all Mc ∈ Fc , then |=Mr φ for all Mr ∈ Fr . Proof. For Mc ∈ Fc , if |= φ, then |= φ, by def 2e. If Mc ∼ Mr , then also Mc Mc |= φ, by Fact 7. Since by Fact 6, for every model Mr ∈ Fr there is a model Mr c M ∈ Fc s.t. Mc ∼ Mr , if |= φ holds for all models in Fc , it also holds for all Mc models in Fr . QED As an example, (14)
( ( c1 = c2 ) → c1 = c2 )
is valid in all relational models. However, it is of more interest to investigate how classical validity relates to relational truth. For example, although (12) and (13) are not valid in relational models, they are both true, i.e. true in the actual world, in all relational models. We want to show that there are notions of validity, and consequence, for the relational interpretation of modality that agree with the corresponding standard definitions of validity and consequence for classical (rigidity) interpretation. In the sequel, ‘Γ’ will be used for denoting sets of formulas, and the notation ‘Γ |= φ’, for saying that for any world w ∈ W, Mc φ is true at w if all the formulas of Γ are true at w. Similarly for variants. Definition 10. Universal and actual consequence. a) φ is a universal classical [relational ] consequence of Γ, Γ |=C, U φ [Γ |=R, U φ], iff it holds for all M ∈ Fc [M ∈ Fr ], all w ∈ W and all assignments µ that Γ |= φ [Γ |= φ]. Mc , w, µ Mr , w, µ
18
b) φ is an actual classical [relational ] consequence of Γ, Γ |=C, A φ, [Γ |=R, A φ] iff it holds for all M ∈ Fc [M ∈ Fr ] and all assignments µ that Γ |=Mc, a, µ φ [Γ |= φ]. Mr , a0 , µ Γ |=C, U φ is a particular relation between sets of formulas of L and formulas of L, the universal classical consequence relation, holding between Γ and φ just in case, in every model M ∈ Fc , φ is true at every world w ∈ W where all the φ holds just in case φ is true (in members of Γ are true. Correspondingly, Γ |= C, A the actual world) in every model M ∈ Fc where all the members of Γ are true. We are going to show that Γ |=C, U φ iff Γ |=C, A φ, for any Γ and φ in L. To this end we shall first establish a lemma to the effect that for any model M ∈ Fc and world w we can find another model M0 ∈ Fc such that whatever is true at w in M is true at the actual world in M0 , and vice versa. In a sense this could be achieved by simply letting w be the actual world in M0 , but since for later claims we need to keep the actual world fixed, this cannot be done. Instead, the new model will be defined by means of a permutation, such that in M0 w and a have swapped all properties, including accessibility properties. The result will be almost symmetric. It will not be completely symmetric, for the reason that interpreted individual constants must have reference in the actual world. In the original model M, some constant c may be such that its referent I(c,a) does not exist in w, and then in M0 that object does not exist in the actual world. Then in M0 I(c,a) will be undefined. Definition 11. Den(I, φ, w) iff for every constant c in φ, I(c, w) is defined. Fact 12. For any φ ∈ L and M ∈ Fc , if Def c (hM, µ, wi, φ), then Den(IM , φ, w). Proof. The atomic case, and the induction steps for negation and conjunction are straightforward. d) Universal quantifier. Assume Def c hM, µ, wj i, ∀xφ). By Definition 1 we have Defc (hM, µ, wj i, ∀xφ) iff Defc (hM, µ0 , wj i, φ) for any x-variant µ0 of µ such that µ0 (x) ∈ D(wj ). Since M ∈ Fc , we have D(wj ) 6= ∅. Hence there is an x-variant µ00 of µ such that Defc (hM, µ00 , wj i, φ). By the induction hypothesis we have if Defc (hM, µ00 , wj i, φ), then Den(IM , φ, wj ). Hence, Den(IM , φ, wj ). Since ∀xφ has the same individual constants as φ, we can conclude that Den(IM , ∀xφ, wj ). e) Necessity. Assume Def c hM, µ, wj i, φ). By Definition 1 we have Defc (hM, µ, wi, φ) iff Defc (hM, µ, w0 i, φ) 19
for all worlds w0 ∈ W s.t. RM (wj , w0 ). By the induction hypothesis, if Defc (hM, µ, wi, φ), then Den(IM , φ, w) at all worlds w ∈ W. Hence, Den(IM , φ, w) is true at all worlds w0 ∈ W such that RM (wj , w0 ). Since M ∈ FC , it holds that RM (wj , wj ). Hence, Den(I, φ, wj ). But since for any constant c, c is in φ iff it is in φ, it also holds that Den(I, φ, wj ). This concludes the induction.
QED
Lemma 13. For any M ∈ Fc and any w ∈ W, there is a model M0 ∈ Fc such φ iff |= φ. that for all φ ∈ L, |= M, w, µ M0 , a, µ Proof. For M ∈ Fc and wi ∈ W, we shall define M0 by means of a permutation function h s.t. h(a) = wi , h(wi ) = a, and for all other w ∈ W, h(w) = w. Then we define M0 as i) DM0 (w) = DM (h(w)), for any w ∈ W ii) RM0 (w, w0 ) iff RM (h(w), h(w0 )), for any w ∈ W iii) IM0 (P, w) = IM (P, h(w)), for any predicate letter P in L and any w ∈ W iv) IM0 (c, w) = IM (c, h(w)), for any individual constant c in L and any w ∈ W, except that IM0 (c, w) is defined only if IM (c, wi ) is defined. It can be verified that M0 is in Fc . This holds just if the extension and 0 is reflexive. The extension postulate is rigidity postulates are satisfied, and RM satisfied, since IM0 (P, w) ⊆ DM0 (w) iff IM (P, h(w)) ⊆ DM (h(w)), which holds by the extension postulate for M. Also, the rigidity postulate is satisfied, since IM0 (c, w) = IM (c, h(w)) = IM (c, a), by def. of M0 and the rigidity postulate for M. But then IM (c, a) = IM (c, wi ) in case IM (c, a) ∈ DM (wi ). Otherwise, IM (c, wi ) is undefined. In case it is defined, IM (c, wi ) = IM (c, h(a)) = IM0 (c, a) by def of h and of M0 . Hence, IM0 (c, w) = IM0 (c, a), and so the rigidity postulate is satisfied. 0 Finally, RM (w, w) iff RM (h(w), h(w)) by def of M0 , and the right hand side holds by def of Fc and the assumption that M ∈ Fc . So the left hand holds as well. M0 is reflexive, and therefore in Fc . Next, we show (*)
For any φ ∈ L such that Den(IM , φ, wi ), it holds for any w that Defc (hM0 , µ, wi, φ) iff Defc (hM, µ, h(w)i, φ)
This is shown by induction over formula complexity. a) Atomic case. What matters is whether IM0 (c, w) ∈ DM0 (w) iff IM (c, h(w)) ∈ DM (h(w)) 20
for any constant c in φ s.t. Den(I, φ, wi ). And this holds just if (**)
IM0 (c, a) ∈ DM0 (w) iff IM (c, a) ∈ DM (h(w))
by the rigidity postulates for M and M0 , for any constant c in φ s.t. Den(I, φ, wi ). But DM0 (w) = DM (h(w)) by def of M0 . And IM0 (c, a) = IM (c, wi ) = IM (c, a), provided IM0 (c, a) is defined. So (**) holds if IM0 (c, a) is defined. But by the assumption that Den(IM , φ, wi ), and the definition of M0 , it holds for every constant cj in φ that IM0 (cj , a) is defined. Hence (**) holds, and the base step of the induction is completed. The induction steps are straightforward. Hence, (*) is true. Next we show by induction over formula complexity that (#)
For any φ ∈ L such that Den(IM , φ, wi ), and for any w ∈ W it holds that φ iff |= φ |= M0 , w, µ M, h(w), µ
a) Atomic case. Assume Den(IM , P t1 , . . . , tn , wi ). Thereby, IM (tj , wi ) is well-defined, and thereby also IM0 (tj , a) for 1 ≤ j ≤ n. Then |= P t1 , . . . , tn iff M0 , w, µ hIM0 ,µ (t1 , w ), . . . , IM0 ,µ (tn , w )i ∈ IM0 (P, w ) iff (by def 2) hIM,µ (t1 , h(w )), . . . , IM,µ (tn , h(w ))i ∈ IM (P, h(w )) iff (by def of M0 ) |= P t1 , . . . , tn (by def 2) M, h(w), µ b) Negation. Assume Den(IM , φ, wi ). Then |= ¬φ iff M0 , w, µ Def c (hM0 , µ, wi, ¬φ) and not |= φ, iff M0 , w, µ Def c (hM, µ, wi, ¬φ) and not |= φ, iff M0 , w, µ Def c (hM, µ, wi, ¬φ) and not |= φ, iff M, h(w), µ |= ¬φ M, h(w), µ
(by def 2) (by (*), since Den(IM , φ, wi )) (by ind. hyp.) (by def 2)
c) Conjunction. Straightforward. d) Universal quantifier. Straightforward. e) Necessity operator. |= φ iff M0 , w, µ (by def 2) |= φ at any world w0 s.t. RM0 (w, w0 ) iff M0 , w0 , µ 0 0 |= φ at any world h(w ) s.t. RM (h(w), h(w )) iff M, h(w0 ), µ (by ind. hyp. and def of M0 ) |= φ (by def 2) M, h(w), µ That concludes the proof of (#). As a corollary of (#), we have the instance
21
(##)
If Den(IM , φ, wi ), then |= φ iff |= φ M, wi , µ M0 , a, µ
Now, assume |=M, wi , µ φ. By Fact 1, Defc (hM, µ, wi i, φ). Then by Fact 12, Den(IM , φ, wi ). Hence, by (##), |= φ. Conversely, assume |= φ. Again, M0 , a, µ M0 , a, µ by Fact 1, Defc (hM0 , µ, ai, φ). And by Fact 12, Den(IM0 , φ, a). By definition of M0 by definition 11, this holds iff Den(IM , φ, wi ). Hence, by (##) again, φ. Thus, |= M, wi , µ (!)
|= φ iff |= φ M, wi , µ M0 , a, µ
Since M and wi were arbitrarily chosen, the lemma follows by generalization. QED With the help of the lemma, we can go on to prove Fact 14. For any φ ∈ L, Γ |= φ iff Γ |= φ C, U C, A Proof. Left to right: by instantiation to a. Right to left. Assume that Γ |=C, A φ, and that for M ∈ Fc , wi ∈ W, µ and every ψ ∈ Γ, |=M, wi , µ ψ. By Lemma 13 there is a model M0 ∈ Fc s.t. for any χ ∈ L, |= χ iff |= χ M, wi , µ M0 , a, µ Hence, for any ψ ∈ Γ, |=M0 , a, µ ψ. From the assumption that Γ |=C, A φ, we have that if for any ψ ∈ Γ, |= ψ, then |= φ. M0 , a, µ M0 , a, µ Hence |= φ. And hence, by the relation between M and M0 , |= φ. Since, M0 , a, µ M, wi , µ M, wi and µ were arbitrarily chosen, we conclude that Γ |= φ. QED C, U From Theorem 5 we can derive Fact 15. For any φ ∈ L, Γ |= φ iff Γ |= φ C, A R, A Proof. From Theorem 5 we have that |= φ iff |= φ, for any φ ∈ L, and Mc , a, µ Mr , a0 , µ any models Mc and Mr such that Mc ∼ Mr . Then, from left to right, assume that Γ |= φ and that for M0 ∈ Fr , and all C, A ψ ∈ Γ, |=M0 , a, µ ψ. By Fact 6, and Definition 8, there is a model M+ ∈ Fc such that M0 ∼ M+ . By Theorem 5, it holds that for all ψ ∈ Γ, |= ψ. And from M+ , a, µ the assumption that Γ |=C, A φ we then get |=M+, a, µ φ. Again, by Theorem 5, and the fact that M0 ∼ M+ , we conclude that |= φ. Since M0 was arbitrary, it M0 , a, µ follows that Γ |= φ. The other direction is symmetric. QED R, A Now we can conclude
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Theorem 16. For any Γ ⊆ L and φ ∈ L, Γ |= φ iff Γ |= φ. C, U R, A Proof. Immediate from Facts 14 and 15.
QED
This is the main result, establishing equivalence in the class of reflexive models. It is, however, also of interest to know whether the equivalence will stick under other restrictions. We shall show that the equivalence still holds for models with symmetric and transitive accessibility relations, and their possible combinations. Let the parameter k take these four combinations as values, and let ‘Fc+k ’ [‘Fr+k ’] denote the class of models characterized by the corresponding accessibility restrictions. Definition 12. K-restricted consequence relations. φ is a k-restricted classical universal consequence of Γ, Γ |=C + k, U φ iff it holds for all M ∈ Fc+k , all w ∈ W and all assignments µ that Γ |=M, w, µ φ. Mutatis mutandis for k-restricted relational and actual consequence relations. Definition 13. Accessibility permutations. R ≈h R0 iff for any w, w0 ∈ W, R0 (w, w0 ) iff R(h(w), h(w0 )). We now make the following observation, with k taking values as above. Fact 17. For any accessibility relations R and R0 and permutation h on W, if R ≈h R0 , then R has property k iff R0 has property k. Proof. Since reflexivity, symmetry and transitivity are logically independent properties, it suffices to show this for each property separately. It was already observed to hold for reflexivity in the proof of lemma 13. For transitivity, assume that R ≈h R0 , that R is transitive, and that R0 (w, w0 ) and R0 (w0 , w00 ). By the assumptions, R(h(w), h(w0 )) and R(h(w0 ), h(w00 )). By the transitivity of R, R(h(w), h(w00 )). By, the R ≈h R0 assumption, R0 (w, w00 ). The other direction is analogous. The proof for symmetry is similar QED Then, since permutations preserve symmetry and transitivity, we can add the following generalization of Lemma 13: Fact 18. For any of the four values of k, any M ∈ Fc+k , and any w ∈ W, there is model M0 ∈ Fc+k such that for all φ ∈ L, |= φ iff |= φ. M, w, µ M0 , a, µ Proof. By Fact 17 permutations preserve the k-value of the accessibility relation. The result follows then from lemma 13 adding k-valued accessibility restrictions to the models. QED Then again we get 23
Fact 19. For any Γ ⊆ L, φ ∈ L, and any k-value, Γ |= φ iff Γ |= φ. C + k, U C + k, A Proof. By Fact 18, analogous to the proof of Fact 14.
QED
Similarly, Fact 20. For any Γ ⊆ L, φ ∈ L, and any k-value, Γ |= φ iff Γ |= φ. C + k, A R + k, A Proof. Analogous to the proof of Fact 15, given the observation that RM = RM0 if M ∼ M0 . QED And finally, Theorem 21. For any Γ ⊆ L, φ ∈ L, and any k-value, Γ |=C + k, U φ iff Γ |= φ.. R + k, A Proof. Immediate from Facts 19 and 20.
QED
M is a model for the modal system T just in case RM is reflexive, for S4 just in case RM is reflexive and transitive, for B just in case RM is reflexive and symmetric, and for S5 just in case RM is reflexive, symmetric and transitive. It follows from Theorem 21 that φ is a classical T, S4, B, S5 consequence of Γ just in case φ is a relational T, S4, B, S5 consequence of Γ, respectively. And again, if we assume an invariant non-empty domain across worlds, we can also drop the reflexivity condition. In that case, the equivalence also holds for models of K.
Department of Philosophy, Uppsala University Department of Philosophy, Stockholm University
References Almog, J., 1986, ‘Naming without necessity’, Journal of Philosophy 83:210–42. Dummett, M., 1981, The Interpretation of Frege’s Philosophy, Harvard University Press, Cambridge, Mass. Dummett, M., 1991, The Logical Basis of Metaphysics, Harvard University Press, Cambridge, Mass. Evans, G., 1979, ‘Reference and contingency’, The Monist 62:161–89. Reprinted in Evans 1985. Evans, G., 1985, Collected Papers, Clarendon Press, Oxford.
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Gamut, L. T. F., 1991, Logic, Language and Meaning. Volume 2. Intensional Logic and Logical Grammar., University of Chicago Press, Chicago, Ill. Kaplan, D., 1986, ‘Opacity’, in P. A. Schilpp and L. E. Hahn (eds.), The Philosophy of W.V. Quine, Open Court, La Salle, Ill. Kripke, S., 1980, Naming and Necessity, Harvard University Press, Cambridge, Mass. Quine, W. V. O., 1952, Methods of Logic, Routledge and Kegan Paul, London. Quine, W. V. O., 1956, ‘Quantifiers and propositional attitudes’, Journal of Philosophy 53:177–87. Reprinted in Quine 1976. Quine, W. V. O., 1976, The Ways of Paradox and Other Essays, Harvard University Press, Cambridge, Mass., 2 edn. Stanley, J., 1997a, ‘Names and rigid designation’, in B. Hale and C. Wright (eds.), A Companion to the Philosophy of Language, Blackwell, Oxford. Stanley, J., 1997b, ‘Rigidity and content’, in R. Heck (ed.), Language, Thought and Logic: Essays in Honor of Michael Dummett, Oxford University Press, Oxford.
25
Abstract Saul Kripke’s thesis that ordinary proper names are rigid designators is supported by widely shared intuitions about the occurrence of names in ordinary modal contexts. By those intuitions names are scopeless with respect to the modal expressions. That is, sentences in a pair like (a) (b)
Aristotle might have been fond of dogs Concerning Aristotle, it is true that he might have been fond of dogs
will have the same truth value. The same does not in general hold for definite descriptions. Since the scope property is accounted for by means of the intensions of the names and the descriptions, the conclusion is that names do not in general have the same intension as any normal, identifying description. However, the difference in scope behavior between names and description can be accounted for alternatively by appeal to the semantics of the modal expressions. On the account we suggest, dubbed ‘relational modality’, simple singular terms, like proper names, contribute to modal contexts simply by their actual world reference, not by their intension. It turns out to be fully equivalent with the rigidity account when it comes to truth of modal and non-modal sentence (with respect to the actual world), and hence supports the same basic intuitions. It also turns out to be model theoretically equivalent with rigidity semantics with respect to logical consequence, within the class of reflexive models with non-empty domains. Here we give the truth definition for relational modality models, and prove the equivalence results.
Keywords: definite descriptions, logical consequence, modality, necessity, possible worlds semantics, proper names, rigid designators, truth. ∗ The present paper is part of a bigger joint project on names and modality by the two authors. The formal work of the present paper has been done by the second author. A hint of how to go about the technical task was given by Krister Segerberg (over an early morning coffee at Stockholm Arlanda airport, waiting for Saul Kripke to arrive for the Rolf Schock prize ceremonies in 2001).
1
1
Introduction
In Naming and Necessity, Saul Kripke presented a number of soon classical arguments against the description theory of proper names. The perhaps most influential one is known as the modal argument. Kripke argued that proper names in general cannot have the same intensions as co-referring definite descriptions, since substituting the one for the other in modal contexts can change truth value. The intuitions on which this argument is based are widely shared and very robust. Kripke suggested to explain them by the doctrine of rigid designation. In a companion paper1 , we suggest an alternative explanation, one that is compatible with the description theory of names. We agree with Kripke that in ordinary modal thinking we operate with concepts of de re modality. That is, we are interested in the objects we refer to, no matter how they are designated. And we want to know what would be true of these very objects in counterfactual circumstances. The intuitions made use of in Kripke’s modal argument testify to this feature of ordinary modal reasoning; these are data to be accepted and explained by any good semantic theory. However, we do not agree that the best way of explaining them is by means of a thesis concerning the intension of names.2 The observed phenomena, we claim, are essentially due to the de re nature of ordinary modal thinking and are, therefore, better explained in terms of a semantics for modal expressions. We propose such an alternative semantics for ordinary modal expressions. It’s basic idea is that, in ordinary modal contexts, names and other simple singular terms occur referentially. The companion paper contrasts our account (mainly) with Kripke’s account, and elaborates on comparative merits. The aim of the present paper is to present our own account in more technical detail. In the next section, we shall present informally the idea of relational modality and the accompanying account of proper names in modal contexts. In section 3, we provide a formal truth definition for a quantificational language with a relational necessity operator, and prove that this language is semantically equivalent with a classical (notional) modal language with rigid singular terms, with respect to truth (in the actual world) of sentences with modal operators. In section 4 we give a definition of logical consequence for relational modality and show that it is equivalent with with standard logical consequence of the classical modality, with rigid individual constants, in the class of reflexive models. The equivalence also holds for models with constant domain, without the reflexivity requirement. As is shown, the equivalence holds for the usual systems of modal logic. 1 ‘Proper
names and relational modality’, draft. a difference between the semantic contributions of names and descriptions in modal contexts will be induced by the semantics for modal expressions, linguistic meaning on our account cannot be equated with intension. Cf the remark after Fact 1, section 3, page 11. 2 Since
2
2
Relational modality
Relational modality is intended to account for the natural language modal intuitions that Kripke originally appealed to, without invoking the rigidity thesis about proper names. According to these intuitions the two sentences: (1)
Aristotle might not have gone into pedagogy
(2)
The teacher of Alexander might not have gone into pedagogy
are not equivalent (cf. Kripke 1980: 61-63). The basic intuition is that (1) is true, while (2) is false. According to Kripke’s account, the difference between (1) and (2) depends on a difference in intension between the name ‘Aristotle’ and and the description ‘the teacher of Alexander’. This difference induces a corresponding difference in intension between the two simple sentences (3) and (4): (3)
Aristotle did not go into pedagogy
(4)
The teacher of Alexander did not go into pedagogy
(cf. Kripke 1980: 6f). According to Kripke, the intensional difference between (3) and (4) explains the extensional difference between (1) and (2). On this view it is the semantic difference between the name and the description that is decisive, not the syntactic difference. It is true that, on Kripke’s view, there is an intensional property common to all the (semantically non-empty) members of the syntactic category of proper names, the property of having a constant function from worlds to objects as intension. But on Kripke’s view, that is not a property that exclusively belongs to names. Every rigid definite description has it, too. The syntactic difference between names and descriptions are not, on this view, relevant to modal contexts. What is relevant is only that some descriptions are not rigid. According to our account, the difference between (1) and (2) depends on a feature of ordinary modal thinking, not on the intensions of names. When, in ordinary modal thinking, we consider alternative possibilities, we are interested in alternative scenarios involving the objects we refer to. We are interested in what might have happened to these very objects, regardless of how their names are evaluated with respect to those alternative scenarios. At least, this is an empirical hypothesis of the present authors about ordinary modal thinking, and hence about the ordinary modal concepts expressed by locutions like ‘possibly’, ‘necessarily’, ‘it might have been’ and ‘it would have been’, as used in everyday discourse. In brief, the proposal is that simple singular terms, including proper names, occur referentially in the contexts of ordinary (alethic) modal expressions.3 On 3 In Kaplan’s terminology (Kaplan 1986: 230), the position of a singular term within a sentence is open to substitution if the result of replacing a term in that position by a coreferential one does not affect the truth value of the sentence. A sentential context is then referentially opaque, in Quine’s terminology (Quine 1952: 142) if any sentence (i.e. sentence occurrence) embedded in that context loses the positions open to substitution that it has
3
the present account, these contexts are intensional with respect to other types of expression, in particular first order predicates. Because of this, the evaluation of (1) is as follows: (1) is true if, and only if, what ‘Aristotle’ actually refers to did not, in some possible world, go into pedagogy. (2), on the other hand, is true if, and only if, what ‘Alexander’ actually refers to is such that, in some possible world, his teacher (in that world) did not go into pedagogy (in that world). Since, intuitively, on this evaluation, (1) is true and (2) is false, the semantic difference has been accounted for without the appeal to the rigidity thesis. Moreover, the syntactic difference between the name and the description plays a role, for names, like all simple singular terms, contribute to truth and falsity in modal sentences with their actual reference, regardless of their possible worlds intension. For all we care, ‘Aristotle’ might have the same intension as ‘the teacher of Alexander’. This account is not an equivalent way of making Kripke’s original point. It is consistent with our account that proper names are not in general rigid. As far as we are concerned, it may be that some are and some aren’t. So, assume that we believe that ‘Aristotle’ has the same intension as ‘the teacher of Alexander’. Kripke does not believe that. On that assumption we differ with respect to the truth conditions of (3)
Aristotle did not go into pedagogy.
For Kripke, (3) is true with respect to a possible world w if, and only if, Aristotle did not go into pedagogy in w. For us, given the assumption, (3) is true with respect to w if, and only if, the referent of ‘the teacher of Alexander’ in w did not go into pedagogy in w. On our interpretation, (3) is not true in any world, whereas for Kripke it is true in some. There is a corresponding disagreement over the metalinguistic statement (5)
(3) might have been true.
On Kripke’s account, (5) is true (cf. Kripke 1980: 12). On our account, given the assumption, it is false. So the accounts are not equivalent. In both cases, of course, we consider what truth value (5) might have had given the meaning it actually has. Something that does hold on our account as well as on Kripke’s is the ‘scopelessness’ or scope indifference of names in relation to modal operators. In fact, on its own. In Kaplan 1986 it is argued against Quine that a position that is not open to substitution can nevertheless contain a variable that is bound by an initially placed quantifier (as in ∃xF x). When we say that proper names occur referentially in modal contexts we do not mean that they occur in positions open to substitution. A name in a modal context cannot in general be replaced salva veritate by a description or functional expression co-referential with it. So modal contexts are opaque. However, on our interpretation, all co-referential simple singular terms, including proper names and free variables, can be interchanged salva veritate in modal contexts. This is what we mean by saying that names occur referentially in modal contexts, and that modal contexts take names transparently (we might call these contexts semi-transparent). Because of this feature, Kaplan’s objection against Quine is exemplified by the interpretation we propose.
4
it must hold on any account that agrees with ordinary modal intuitions. However, scope indifference has sometimes been equated with the rigidity thesis, it has even been held as an alternative way of stating it (Kripke does so himself in 1980: 12, fn. 15). This is correct only if names do not occur referentially in modal contexts, for then the equivalence of the wide and narrow scope readings such as (6)
F (Aristotle)
(7)
∃x(x = Aristotle & F x).
depends on the intensions of names. If names do occur referentially, then these equivalences hold as a matter of course, and whether names are rigid designators or not. In fact, this is precisely what we propose.4 The basic idea for effecting this, is the following interpretation of ‘necessary’: (N)
pIt is necessary that φq is true iff φ is true no matter what extensions are assigned to its non-logical predicates and functional expressions.
With this clause in a truth definition, the extension of singular terms is simply left unaffected by the evaluation, while there is a variation in extension of the non-logical predicates and functional expressions. For instance, (8)
It is possible that Plato’s father was richer than Aristotle’s father
comes out true, on this interpretation, just if in some extension assignment to the two-place predicate ‘...was richer than...’ and to the functional expression “...’s father” the embedded (9)
Plato’s father was richer than Aristotle’s father
comes out true. Of course, this is not formally precise. As stated it is also inadequate, for there is no mention of how the extension assignment to a predicate is restricted by its meaning, nor of how assignments to different expressions may be combined. Both these problems are solved by switching to the standard framework of possible worlds semantics. The question is how to formulate the intended equivalent to (N) within that framework. The answer comes in two very simple ideas. The first idea is what we call actualist evaluation. Standardly, an atomic sentence P t1 , ..., tn is evaluated as true in a possible world w just in case the n-tuple of the referents of t1 , ..., tn in w belongs to the extension of P in w. That is, where I is an interpretation function assigning referents to terms and extensions to predicates in possible worlds, (P)
True(P t1 , . . . , tn , w ) iff hI(t1 , w ), . . . , I(tn , w )i ∈ I(P, w ).
4 It
is also for this reason that we have chosen to call our account ‘relational modality’; because of the similarity with Quine’s distinction between the notional and the relational concepts of belief, in Quine 1956. This was first suggested to us by Sten Lindstr¨ om.
5
In the actualist evaluation we consider instead the referents in the actual world, a. (A)
True(P t1 , . . . , tn , w ) iff hI(t1 , a), . . . , I(tn , a)i ∈ I(P, w ).
For the predicate, the extension in w matters, but for the terms only their extension in a.5 When considering different worlds, we consider different extensions of the predicate, but just the same extension of the terms. To complete the definition of actualist evaluation one adds clauses for connectives, quantifiers and modal operators, no different from the ordinary ones (see section 3). Not surprisingly, the actualist evaluation turns out to be semantically equivalent with a standard semantics with rigid singular terms (Fact 3). Since a rigid term denotes the same object in every world where that object exists, I(t, a) is bound to be the same as I(t, w), if t is a rigid term (and the object denoted exists in w ). An actualist evaluation semantics does not make use of the extension of a singular term in any other world than the actual world. That is, it doesn’t make use of intensions of singular terms. So instead of just using an actualist evaluation as your semantics, it would be better to drop the intensions and simply speak of the reference of a singular term. This would be to follow the Kaplan-Almog line of direct reference (see Almog 1986). As stated, (A) is well defined only for simple terms, for which the reference is given primitively by the interpretation function I. Since we prefer to stay neutral on the question whether there are complex singular terms, we have to take such terms into account. Suppose, then, that applied functional expressions, like ‘g(u)’, where ‘u’ again is a singular term, simple or complex, are singular, and that definite descriptions, like ‘the x such that F x’, or ‘ xF x’, are singular too. In order to accommodate these terms with the desired result, (A) needs to be replaced by ι
(A+)
True(P t1 , . . . , tn , w ) iff hV(t1 , w ), ..., V(tn , w )i ∈ I(P, w )
where the term evaluation function V is defined as follows: (V)
V(t, w) = I(t, a), in case t is simple V(g(u), w) = I(g, w)(V(u, w)) V( xF x, w) = the unique object b such that True(F x, w) with b assigned to x, and undefined if there is no such object ι
where I(g, w) is the function (in extension) assigned to g in w (for a formally more adequate statement, see section 3). By (V), simple singular terms are evaluated with respect to the actual world, while functional expressions and pre5 This makes it different from simply applying the actuality operator to the sentence, for that affects the evaluation of both terms and predicates: True(Aφ, w ) iff True(φ, a). Still, it is possible to have a semantics equivalent to the actualist evaluation by adding the actuality operator to a standard semantics with non-rigid singular terms: let each non-rigid term t occur only in the following context: ‘the x such that A(x = t)’. The result will be that only the actual world reference of terms will matter, while predicates are evaluated as usual.
6
dicates within complex singular terms are evaluated with respect to the possible world in question.6 The second part of our proposal is to use the actualist evaluation only for the semantics of modal sentences. By contrast, if it were simply used across the board, we would have a uniform semantics for modal and non-modal sentences alike, where the actualist truth conditions of a non-modal sentence would be exactly what it contributes to the evaluation of a modal sentence containing it. Such a semantics would not have anything in particular to do with interpreting modal expressions. The relational modality proposal, however, is to use the standard (A) for the ordinary truth conditions of atomic sentences, and to use (A+) for the semantic contribution of an atomic sentence to a modal sentence containing it. The idea, then, is to have a truth definition clause for the modal expression that runs something like this: (M)
True(p It is necessary that φq, w) iff Actua-true(φ, w 0 ) at any world w 0 accessible from w
where ‘Actua-true’ just means true according to the (completed) actualist evaluation. In this way we distinguish between ordinary truth conditions and the semantic properties a sentence contributes to the truth conditions of modal sentences containing it (which is its actualist truth conditions).7 The resulting interpretation does accommodate all our basic modal intuitions. For instance, if we adapt the proposal to ‘might have’, understood as ‘not necessarily not’, using classical predicate logic and for simplicity treating ‘go into pedagogy’ as a simple predicate, we get the following result for (2): (2) is true iff there is some accessible world w such that V(‘the teacher of Alexander’, w ) does not belong to I(‘goes into pedagogy’, w ). The right hand side holds iff there is a unique object b of which ‘x teaches Alexander’ is true (with b assigned to ‘x’) in w, and b does not go into pedagogy in w. Again, this holds iff there is a unique object b such that the pair of b and V(‘Alexander’, w ) belongs to I(‘x teaches y’, w ), and b does not go into pedagogy in w. Since V(‘Alexander’, w ) 6 It should be stressed that we have relied on the assumption that any proper name (individual constant) that has reference in some possible world also has reference in the actual world. This is philosophically well motivated. Indeed, we think that Kripke is entirely right in denying that Sherlock Holmes might have existed (Kripke 1980: 157-8), since there is no actual referent with which to identify any particular individual in any particular possible world. This goes for our account, too. It is, of course, compatible with our view that ‘Sherlock Holmes’ has reference in other possible worlds, but because of our interpretation of ‘might have’ we have to deny that Sherlock Holmes might have existed. Since actual reference is all that matters to actual truth, even of modal statements, on our interpretation, we are justified in restricting attention to names that do have actual reference. 7 This spells out a difference corresponding to Dummett’s distinction between content and ingredient sense. In the semantics Kripke proposes there is no such difference, and he has been criticized by Dummett (cf. Dummett 1981: 572f, 1991: 48), Evans (1979), and Stanley (1997a,b) for not taking account of the distinction. In our semantics, the distinction corresponds to a real difference.
7
is I(‘Alexander’, a), this holds iff there is a unique object b that in w teaches what ‘Alexander’ refers to in a, and b does not go into pedagogy in w. This is the desired interpretation. Not only do we get the desired results for examples like (1) and (2), once a clause for the definite description construction is included in the semantics. It is also provably true that the relational modality interpretation will give the same evaluation of any formula with respect to the actual world, as the standard interpretation with rigid singular terms (see below, Theorem 5). Since truth in the actual world simply is truth, if by ‘the actual world’ we do mean the actual world, not just some entity designated as such in a model, this means that the rigidity interpretation and the relational modality interpretation will agree with respect to simple truth and falsity of any modal and non-modal statement. Hence, they are empirically equivalent with respect to basic modal intuitions.
3
Classical and relational modality: truth
The basic idea for implementing relational modality in possible worlds semantics is to do it indirectly, via a different semantic evaluation which is actualist in the sense that for individual constants only the reference in the actual world matters for evaluation. The method is then to let the actualist evaluation kick in precisely at the clause for necessity. The result will be that a model for relational modality, so defined, is actually equivalent with normal semantics with rigid designation, i.e. gives the same truth value for sentences with respect to the actual world. We shall give a version of Kripke-style possible worlds semantics where we allow the world-bound domain of individuals to vary from world to world. For this reason we have chosen a partial semantics, with truth value gaps in case of reference failure. We shall require that individual constants do have a referent in the actual world (see note 6). A model M is a structure hW, R, a, Dom, D, Ii where W is a set of entities (possible worlds), R a relation on W (the accessibility relation), a a privileged member of W (the actual world), Dom the domain of individuals of M, D a function from W to P (Dom) (giving for each world the set of individuals existing in that world) and I an interpretation function, giving for each pair of a non-logical constant e and w ∈ W the extension of e in w. We shall compare standard models Mc , with rigid individual constants, with relational modality models Mr , with non-rigid constants, for a single quantified modal language L. L is built up from predicate letters of varying arity from the set PL (where the notation ‘P n ’ indicates that n is the arity of P ), individual constants from CL and individual variables from Var L , and ¬, &, ∀ and , according to the ordinary formation rules. For both of Mc and Mr , we shall require that for a non-modal formula φ to have a truth value with respect to a possible world w, relative to a model M and an assignment µ to free variables, it must be the case for any constant c in φ that
8
I(c, w) ∈ D(w) and for any free variable x in φ that µ(x) ∈ D(w). If it is not the case that I(c, w) ∈ D(w), then I is undefined for (c, w). For a formula φ, the requirement is that φ is well defined with respect to all accessible worlds. This is a strong requirement, but avoids some logical problems that can otherwise arise (see Gamut 1991: 55-6). The definedness relation Def between a triple hM, µ, wi and a formula φ will be set out recursively as part of each truth definition. Below, Iµ (t, w) = I(t, w) in case t is a constant, and Iµ (t, w) = µ(t) in case t is a variable. An assignment µ, or µM , of a model M, is a function from Var L , the set of individual variables of L, to Dom M . Also, assignment µ0 is an x-variant of assignment µ iff µ and µ0 differ at most in what value they assign to x. First, the definition of the classical definedness relation for arbitrary model M: Definition 1. The definedness relation Defc a)
Def c (hMc , µ, wi, P t1 , . . . , tn ) iff Iµ (ti , w) ∈ D(w), for 1 ≤ i ≤ n
b)
Def c (hMc , µ, wi, ¬φ) iff Def c (hMc , µ, wi, φ)
c)
Def c (hMc , µ, wi, φ & ψ) iff Def c (hMc , µ, wi, φ) and Def c (hMc , µ, wi, ψ)
d)
Def c (hMc , µ, wi, ∀xφ) iff Def c (hMc , µ0 , wi, φ) for any x-variant µ0 of µ such that µ0 (x) ∈ D(w)
e)
Def c (hMc , µ, wi, φ) iff Def c (hMc , µ, w0 i, φ) for any w0 such that wRw0
Definition 2. A classical model Mc is a structure hW, R, a, Dom, D, Ii such that a)
P t1 , . . . , tn iff hIµ (t1 , w), . . . , Iµ (tn , w)i ∈ I(P n , w) |= Mc , w, µ
b)
|= ¬φ iff Def c (hMc , µ, wi, ¬φ) and not |= φ Mc , w, µ Mc , w, µ
c)
φ & ψ iff |= φ and |= ψ |= Mc , w, µ Mc , w, µ Mc , w, µ
d)
|= ∀xφ iff Mc , w, µ |= φ for all x-variants µ0 of µ such that µ0 (x) ∈ D(w) Mc , w, µ0
e)
|= φ iff |= φ for any w0 such that wRw0 Mc , w, µ Mc , w0 , µ
f)
Extension postulate: for any predicate P n of L and any w ∈ W, I(P n , w) ⊆ D(w)n
g)
Rigidity postulate: for any individual constant c of L and any w ∈ W, if I(c, w) is defined, then I(c, w) = I(c, a).
Now we move on to defining the relational modality models Mr . We shall proceed the way, by first giving setting out the definedness definition, which
9
comes in two steps. First, the actualist variety: Definition 3. The definedness relation Defa a)
Def a (hM, µ, w, i, P t1 , . . . , tn ) iff Iµ (ti , a) ∈ D(w), for 1 ≤ i ≤ n
b)
Def a (hM, µ, w, i, ¬φ) iff Def a (hM, µ, w, i, φ)
c)
Def a (hM, µ, w, i, φ & ψ) iff Def a (hM, µ, w, i, φ) and Def c (hMc , µ, wi, ψ)
d)
Def a (hM, µ, w, i, ∀xφ) iff Def a (hM, µ0 , wi, φ) for any x-variant µ0 of µ such that µ0 (x) ∈ D(w)
e)
Def a (hM, µ, w, i, φ) iff Def a (hM, µ, w0 i, φ) for any w0 such that wRw0
Second, there is the relational definedness relation, which appeals to the actualist in the fifth clause: Definition 4. The definedness relation Defr a)
Def r (hMr , µ, wi, P t1 , . . . , tn ) iff Iµ (ti , a) ∈ D(w), for 1 ≤ i ≤ n
b)
Def a (hM, µ, w, i, ¬φ) iff Def r (hMr , µ, wi, φ)
c)
Def r (hMr , µ, wi, φ & ψ) iff Def r (hMr , µ, wi, φ) and Def c (hMc , µ, wi, ψ)
d)
Def r (hMr , µ, wi, ∀xφ) iff Def r (hMr , µ0 , wi, φ) for any x-variant µ0 of µ such that µ0 (x) ∈ D(w)
e)
Def r (hMr , µ, wi, φ) iff Def a (hM, µ, w0 i, φ) for any w0 such that wRw0 We now state the actualist evaluation relation for relational models:
Definition 5. The actualist evaluation of a relation modality model Mr is a a such that relation |= Mr , w, µ a
a)
|= P t1 , . . . , tn iff hIµ (t1 , a), . . . , Iµ (tn , a)i ∈ I(P n , w) Mr , w, µ
b)
a ¬φ iff Def a (hM, µ, w, i, ¬φ) and not |= φ |= Mr , w, µ Mr , w, µ
c)
|= φ & ψ iff |= φ and |= ψ Mr , w, µ Mr , w, µ Mr , w, µ
d)
|= ∀xφ iff Mr , w, µ a |= φ for all x-variants µ0 of µ such that µ0 (x) ∈ D(w) Mr , w, µ0
e)
|= φ iff |= φ for any w0 such that wRw0 Mr , w, µ Mr , w0 , µ
a a
a
a
a
a
a
The following definition of the relation modality model appeals to the actualist evaluation in its fifth clause.
10
Definition 6. A relational modality model Mr is a structure hW , R, a, Dom, D, Ii such that a)
|= P t1 , . . . , tn iff hIµ (t1 , w), . . . , Iµ (tn , w)i ∈ I(P n , w) Mr , w, µ
b)
¬φ iff Def r (hMr , µ, wi, ¬φ) and not |= φ |= Mr , w, µ Mr , w, µ
c)
|= φ & ψ iff |= φ and |= ψ Mr , w, µ Mr , w, µ Mc , w, µ
d)
|= ∀xφ iff Mr , w, µ |= φ for all x-variants µ0 of µ such that µ0 (x) ∈ D(w) Mr , w, µ0
e)
φ for any w0 such that wRw0 |= φ iff |= Mr , w, µ Mr , w0 , µ
f)
Extension postulate: for any predicate P n of L and any w ∈ W, I(P n , w) ⊆ D(w)n
a
Fact 1. For any model Mi , with i = c, r, if |= φ, then Defi (hMi , µ, wi, φ). Mi , w, µ Proof. By induction over formula complexity. We shall do it for i = r. For i = c it is similar and easier. For the atomic formulas, by clause a) of definition 6, hIµ (t1 , w), . . . , Iµ (tn , w)i ∈ I(P n , w) and by the extension postulate I(P n , w) ⊆ D(w)n , which together imply that for each ti , I(ti ) ∈ D(w), which is what’s required by clause a) of definition 4. For negation it is immediate from the b) clauses. For conjunction it follows directly by the c) clauses and the induction hypothesis. Similarly, for ∀xφ it follows directly by the d ) clauses and the induction hypothesis. For φ, it is required that Def a (hM, µ, w0 i, φ). To carry out this step we a must first prove that for any formula φ, if |=Mr , w, µ φ, then Def a (hM, µ, w, i, φ). This is done again by induction over formula complexity, just as above, but this time by definitions 3 and 5. For φ, it follows by clauses e) and the induction hypothesis. This, finally, gives us the induction hypothesis for φ and Def r . The step is immediate from clauses e) of definitions 4 and 6 and the induction hypothesis. QED Remark. For simplicity we have not included functional expressions and definite descriptions as complex singular terms. To include them, several changes would be needed. For instance, the following changes would be needed in definition 5: Instead of a)
a |= P t1 , . . . , tn iff hIµ (t1 , a), . . . , Iµ (tn , a)i ∈ I(P n , w) Mr , w, µ
we would have a 0)
a |= P t1 , . . . , tn iff hVµ (t1 , w ), . . . , Vµ (tn , w )i ∈ I(P n , w ) Mr , w, µ
11
where V is defined as follows: (V)
Vµ (t, w) = I(t, a), in case t is an individual constant Vµ (t, w) = µ(t), in case t is a variable Vµ (g(t), w) = I(g, w)(V(t, w)) Vµ ( xψ, w) = the object b such that for a unique x-variant µ0 of µ, a ψ. Undefined if there is no such object b. µ0 (x) = b and |= Mr , w, µ0 ι
In this way a name and a description may have the same intension, but still not be synonymous, since they are not interchangeable salva veritate in modal contexts. Because of this, it would violate the principle of compositionality to equate intension with linguistic meaning. Rather, the linguistic meaning of a singular term can be taken to be the pair of its I-intension and its V-intension, where the V-intension is a function from possible worlds to V-values. This is then extended to formulas in the obvious way. For instance, the V-intension of an atomic formula P t1 , . . . , tn is a function from worlds to truth values in accordance with clause a0 ) above. Note that the V-intension is determined by the I-intension together with syntax, but only if syntactic distinctions between kinds of singular terms are made. End of remark.
We aim to prove that Mc and Mr give the same evaluation of all formulas with respect to the (shared) actual world. To this end we need to establish three facts. The second will be that Mc is fully equivalent with the actualist evaluation of Mr . Definition 7. Two models, M = hW, R, a, Dom, D, Ii and M0 = hW0 , R0 , a0 , Dom0 , D0 , I0 i, are equi-actual, M ∼ M0 , iff W = W0 , R = R0 , a = a0 , Dom = Dom0 , D = D0 and for all predicate letters P, for all worlds w ∈ W, I(P, w) = I0 (P, w) and for all individual constants c, I(c, a) = I0 (c, a0 ). M and M0 are atomically equal, M ≈ M0 , iff M ∼ M0 , and for all individual constant c and all worlds w , I(c, w) = I0 (c, w0 ). For the following claims, we shall assume that Mc ∼ Mr , where Mc = hW, R, a, Dom, D, Ii and Mr = hW0 , R0 , a0 , Dom0 , D0 , I0 i. The first claim says that a formula is well-defined with respect to a classical evaluation just if it is well-defined with respect to the actualist evaluation of the corresponding relational model. Fact 2. For all formulas φ ∈ L, assignments µ, and worlds w ∈ W, Defc (hMc , µ, wi, φ) iff Defa (hM, µ, w, i, φ). Proof. By induction over formula complexity.
12
a) Atomic formulas. Def c (hMc , µ, wi, P t1 , . . . , tn ) iff Iµ (ti , w) ∈ D(w), for 1 ≤ i ≤ n, iff Iµ (ti , a) ∈ D(w), for 1 ≤ i ≤ n iff I0µ (ti , a0 ) ∈ D0 (w), for 1 ≤ i ≤ n iff Def a (hM, µ, w, i, P t1 , . . . , tn ) b) Negation. Def c (hMc , µ, wi, ¬φ) iff Def c (hMc , µ, wi, φ) iff Def a (hM, µ, w, i, φ) iff Def a (hM, µ, w, i, ¬φ)
(by def 1a) (by rigidity, def 2g) (by ∼ assum. and def 7) (by def 3a)
(by def 1b) (by ind. hyp.) (by def 3b).
c) Conjunction. Similar to negation. d) Universal quantifier. Def c (hMc , µ, wi, ∀xφ) iff Def c (hMc , µ0 , wi, φ), for all x-var. µ0 of µ s.t. µ0 (x) ∈ D(w) iff (by def 1d) Def a (hM, µ0 , wi, φ), for all x-var. µ0 of µ s.t. µ0 (x) ∈ D0 (w) iff (by ind. hyp., ∼-assum. and def 7) Def a (hM, µ, w, i, ∀xφ) (by def 3d) e) Necessity operator. Def c (hMc , µ, wi, φ) iff Def c (hMc , µ, w0 i, φ) for any w0 such that wRw0 iff (by def 1e) Def a (hM, µ, w, i, φ) for any w0 such that wR0 w0 iff (by ind. hyp., ∼-assum., and def 7) Def a (hM, µ, w, i, φ). This completes the induction.
QED
With the help of Fact 2 we can prove the equivalence of classical and actualist evaluations. Fact 3. For all formulas φ ∈ L, assignments µ, and worlds w ∈ W, a φ iff |= φ. |= Mc , w, µ Mr , w, µ Proof. By induction over formula complexity. The atomic, conjunction, quantification and modal cases are analogous to the corresponding cases in the proof of Fact 2. In the negation case we have b) Negation. ¬φ iff |= Mc , w, µ Defc (hMc , µ, wi, ¬φ) and not |= φ iff Mc , w, µ a φ iff Defa (hM, µ, w, i, ¬φ) and not |= Mr , w, µ a ¬φ |= r M , w, µ
(by def 2b) (by Fact 2 and ind. hyp.) (by def. 5b) QED
13
Fact 2 also helps to establish the equivalence of classical and relational definedness: Fact 4. For all formulas φ ∈ L, assignments µ, and worlds w ∈ W, Defc (hMc , µ, wi, φ) iff Defr (hMr , µ, wi, φ). Proof. By induction over formula complexity. The atomic case is immediate, and the negation, conjunction and quantification cases are as in the proof of Fact 2. For the modal case we have e) Necessity operator. Def c (hMc , µ, wi, φ) iff Def c (hMc , µ, w0 i, φ) for any w0 such that wRw0 iff Def a (hM, µ, w, i, φ) for any w0 such that wR0 w0 iff Def r (hMr , µ, wi, φ)
(by def 1e) (by Fact 2) (by def 4e) QED
We can now prove the equivalence of classical and relational models with respect to truth (in the actual world). Theorem 5. Let Mc = hW, R, a, Dom, D, Ii and Mr = hW0 , R0 , a0 , Dom0 , D0 , I0 i. If Mc ∼ Mr , then for any φ ∈ L and any assignment µ, φ iff |= φ |= Mc , a, µ Mr , a0 , µ Proof. By induction over formula complexity. a) Atomic case. |= P t1 , . . . , tn iff Mc , a, µ hIµ (t1 , a), . . . , Iµ (tn , a)i ∈ I(P, a) iff hI0µ (t1 , a 0 ), . . . , I0µ (tn , a 0 )i ∈ I0 (P, a 0 ) iff |= P t1 , . . . , tn Mr , a0 , µ b) Negation. |= ¬φ iff Mc , a, µ Def c (hMc , µ, wi, ¬φ) and not |= φ iff Mc , a, µ φ iff Def r (hMr , µ, wi, ¬φ) and not |= Mr , a0 , µ |= ¬φ Mr , a0 , µ
(by def 2a) (By def 7, since Mc ∼ Mr ) (by def 6a)
(by def 2b) (by Fact 4 and ind. hyp.) (by def 6b)
c-d) Conjunction and quantification, straightforward. e) Necessity. |= φ iff Mc , a, µ |= φ at any world w0 s.t. aRw0 iff Mc , w0 , µ a |= φ at any world w0 s.t. aRw0 iff Mr , w0 , µ |= φ Mr , a0 , µ
(by def 2e) (by Fact 3) (by def 6e) QED
14
Hence, the relational modality models give the same evaluations at the actual world as the corresponding classical models.
4
Classical and relational modality: consequence
Let’s turn now to the issues of consequence and validity. As usual, validity is just the special case of consequence when the set of premises is empty, so we shall simply speak of consequence. We are going to investigate relations between classical and relational models with respect to this notion. It will turn out that there is concept of consequence for relational modality that is almost equivalent with the concept of consequence for the classical semantics. Where the classical concept has it that φ is a consequence of Γ iff φ is true at a world w if Γ is true at w, for all worlds in all models, the relational counterpart is that φ is a consequence of Γ iff φ is true at the actual world a if Γ is true at a, in all models. These two notions are almost but not completely equivalent. Because of denotation failure, there is a divergence both with respect to quantification and with respect to modality. Consider the pair of sentences (10)
a) b)
∀x(P tx) P tt
where P tt is atomic and t an individual constant. With the requirement that any term that does have an interpretation in a model has a denotation in the actual world in that model, whenever (10a) is true at the actual world in some model, the term t has a denotation in the actual world, and then if (10a) is true in the actual world, so is (10b). So, it is a relational consequence. However, it is not a classical consequence. For consider a world w with an empty domain. The premise (10a) is vacuously true with respect to w, and the sentence also vacuously well-defined with respect to that world and relevant assignment µ. For this holds if P tx is well-defined with respect to that world for all x-variants µ0 of µ such that µ0 (x) ∈ D(w), and since D(w) = ∅, this condition is vacuously fulfilled by all assignments. But (10b) is not well-defined with respect to w, since t obviously has no denotation there. So, (10b) is not true at w, and the consequence fails. Similarly, consider the pair of sentences (11)
a) b)
P t ¬(P t & ¬P t)
where P t is atomic and t an individual constant. With the requirement that any term that does have an interpretation in a model has a denotation in the actual world in that model, whenever (11a) is true at the actual world in some model, the term t has a denotation in the actual world, and then (11b), being a tautology, is true at the actual world, since it is true at any world where the term does have a denotation. So, it is a relational consequence.
15
Again, however, it is not a classical consequence. For consider a world wk in a classical model M such that (11a) is true at wk in M. This requires exactly that P t is true at all worlds w accessible from wk , and hence that t has a denotation in all of these worlds. Hence, in all those worlds, (11b) is true as well. However, it is not required that wk be accessible from itself, and so t need not have a denotation in wk . If not, (11b) will be neither true nor false in wk . So the consequence fails. Nevertheless, the equivalence between the notions does hold if it is required both that world-bound domains be non-empty and that worlds be accessible from themselves. That is, the notions are equivalent with respect to the class of reflexive models with non-empty world domains. Model theoretically, this is a limitation. Philosophically, the limitation is not severe, since both (10) and (11) are pairs where the the second member intuitively follows from the first. Moreover, we want to infer truth from necessity, i.e. p from necessarily p, and the validity of this inference is guaranteed, from a possible worlds semantics perspective, only when the accessibility relation is reflexive. It may be noted that in case we assume an invariant non-empty worldbound domain of individuals, the counter-examples do not go through. And as can easily be seen by inspecting the proof of Lemma 13, with this assumption there is equivalence between classical and relational consequence, without the reflexivity restriction. However, since this assumption makes the comparison less interesting, we shall proceed without it. We shall now establish the equivalence with respect to non-empty reflexive models. Assume that we have a model frame F = hW, a, Domi. From F we define a class of classical models and a class of relational models: Definition 8. Classes of classical and relational models on the frame F. a) Fc = {M : M is a model on the frame F that satisfies definitions 1—2 and such that RM is reflexive and for all w ∈ W, D(w) 6= ∅.} b) Fr = {M : M is a model on the frame F that satisfies definitions 3—6 and such that RM is reflexive and for all w ∈ W, D(w) 6= ∅.} Now we shall proceed with some facts about the relation between Fc and Fr . Fact 6. For any M ∈ Fc there is an M0 ∈ Fr such that M ≈ M0 , and hence, M ∼ M0 , and also vice versa. Proof. Immediate from definitions 7 and 8.
QED
Fact 7. For any φ ∈ L, any Mc ∈ Fc and Mr ∈ Fr such that Mc ∼ Mr , any w ∈ W and assignment µ, φ iff |= φ |= Mc , w, µ Mr , w, µ
16
Proof. The proof is a simple generalization of case e) of Theorem 5: |= φ iff Mc , w, µ |= φ at any world w0 s.t. wRw0 iff Mc , w0 , µ a |= φ at any world w0 s.t. wRw0 iff Mr , w0 , µ |= φ Mr , w, µ
(by def 2e) (by Fact 3) (by def 6e) QED
Fact 8. Let Mc = hW, R, a, Dom, D, Ii and Mr = hW0 , R0 , a0 , Dom0 , D0 , I0 i. If Mc ≈ Mr , then for any φ ∈ L and any assignment µ, φ iff |= φ |= Mc , w, µ Mr , w, µ Proof. By induction over formula complexity. Since we have Mc ≈ Mr , and hence I = I0 for all atomic non-logical constants, the proofs of the atomic, negation, conjunction and quantification cases are straightforward. The modal case follows from Fact 7. QED Definition 9. Generalizing over worlds and assignments in model: a) |= φ iff for all w ∈ W, and all assignments µ, |= φ Mc Mc , w, µ b) |= φ iff for all w ∈ W, and all assignments µ, |= φ Mr Mr , w, µ Fact 8 facilitates the proof of the following inclusion claim: Fact 9. For all φ ∈ L, if |= φ for all Mr ∈ Fr , then |= φ for all Mc ∈ Fc . Mr Mc Proof. Assume that there is a formula φ, a world w ∈ W, and an assignment µ such that not |=Mc, w, µ φ. By fact 6 there is corresponding model Mr ∈ Fr s.t. Mc ≈ Mr . By Fact 8, |=Mc, w, µ φ iff |=Mr , w, µ φ. Hence, not |=Mc, w, µ φ. By contraposition and generalization, the claim follows. QED For non-modal sentences it does not in general hold that they are valid in a rigid classical model iff valid in a ∼-corresponding relational model. For example, assume that both I(c1 , w) and I(c2 , w) are defined for all w ∈ W. Then it holds that either c1 = c2 or c1 6= c2 is true in all w ∈ W in Fc models, but in general this does not hold in Fr . However, this is not of great significance, since we are interested in validity at a higher level: truth in all worlds for variable model. It can be shown that for any φ ∈ L that is modality free, i.e. does not contain any modal operator, it holds that for any Mc ∈ Fc and Mr ∈ Fr , such φ iff |= φ. that Mc ∼ Mr that |= Mc Mr Still, the converse of Fact 9 does not hold: Fact 10. It is not the case that: for all φ ∈ L, if |= φ for all Mc ∈ Fc , then Mc r |= φ for all M ∈ Fr . Mr 17
Proof. With φ → ψ defined as ¬(φ & ¬ψ), the sentence (12)
c1 = c2 → ( c1 = c2 )
is valid in all classical models. But it is not valid in all relational models. It is false in a model Mr s.t. for some world w, I(c1 , w) = I(c2 , w), but I(c1 , a) 6= I(c2 , a). Then the antecedent of (12) is true at w and the consequent is false at w. QED That (12) isn’t valid in relational models is not so strange. It isn’t valid in non-rigid standard models either. The converse, however, (13)
( c1 = c2 ) → c1 = c2
is valid in reflexive non-rigid standard models, but still not valid in relational models. To get a counter-example, let I(c1 , a) = I(c2 , a) and I(c1 , w) 6= I(c2 , w). Then the antecedent is true and the consequent false at w. Still, if a sentence is valid in a classical model, its necessitation is valid in the corresponding relational model: Fact 11. For any φ ∈ L, if |=Mc φ for all Mc ∈ Fc , then |=Mr φ for all Mr ∈ Fr . Proof. For Mc ∈ Fc , if |= φ, then |= φ, by def 2e. If Mc ∼ Mr , then also Mc Mc |= φ, by Fact 7. Since by Fact 6, for every model Mr ∈ Fr there is a model Mr c M ∈ Fc s.t. Mc ∼ Mr , if |= φ holds for all models in Fc , it also holds for all Mc models in Fr . QED As an example, (14)
( ( c1 = c2 ) → c1 = c2 )
is valid in all relational models. However, it is of more interest to investigate how classical validity relates to relational truth. For example, although (12) and (13) are not valid in relational models, they are both true, i.e. true in the actual world, in all relational models. We want to show that there are notions of validity, and consequence, for the relational interpretation of modality that agree with the corresponding standard definitions of validity and consequence for classical (rigidity) interpretation. In the sequel, ‘Γ’ will be used for denoting sets of formulas, and the notation ‘Γ |= φ’, for saying that for any world w ∈ W, Mc φ is true at w if all the formulas of Γ are true at w. Similarly for variants. Definition 10. Universal and actual consequence. a) φ is a universal classical [relational ] consequence of Γ, Γ |=C, U φ [Γ |=R, U φ], iff it holds for all M ∈ Fc [M ∈ Fr ], all w ∈ W and all assignments µ that Γ |= φ [Γ |= φ]. Mc , w, µ Mr , w, µ
18
b) φ is an actual classical [relational ] consequence of Γ, Γ |=C, A φ, [Γ |=R, A φ] iff it holds for all M ∈ Fc [M ∈ Fr ] and all assignments µ that Γ |=Mc, a, µ φ [Γ |= φ]. Mr , a0 , µ Γ |=C, U φ is a particular relation between sets of formulas of L and formulas of L, the universal classical consequence relation, holding between Γ and φ just in case, in every model M ∈ Fc , φ is true at every world w ∈ W where all the φ holds just in case φ is true (in members of Γ are true. Correspondingly, Γ |= C, A the actual world) in every model M ∈ Fc where all the members of Γ are true. We are going to show that Γ |=C, U φ iff Γ |=C, A φ, for any Γ and φ in L. To this end we shall first establish a lemma to the effect that for any model M ∈ Fc and world w we can find another model M0 ∈ Fc such that whatever is true at w in M is true at the actual world in M0 , and vice versa. In a sense this could be achieved by simply letting w be the actual world in M0 , but since for later claims we need to keep the actual world fixed, this cannot be done. Instead, the new model will be defined by means of a permutation, such that in M0 w and a have swapped all properties, including accessibility properties. The result will be almost symmetric. It will not be completely symmetric, for the reason that interpreted individual constants must have reference in the actual world. In the original model M, some constant c may be such that its referent I(c,a) does not exist in w, and then in M0 that object does not exist in the actual world. Then in M0 I(c,a) will be undefined. Definition 11. Den(I, φ, w) iff for every constant c in φ, I(c, w) is defined. Fact 12. For any φ ∈ L and M ∈ Fc , if Def c (hM, µ, wi, φ), then Den(IM , φ, w). Proof. The atomic case, and the induction steps for negation and conjunction are straightforward. d) Universal quantifier. Assume Def c hM, µ, wj i, ∀xφ). By Definition 1 we have Defc (hM, µ, wj i, ∀xφ) iff Defc (hM, µ0 , wj i, φ) for any x-variant µ0 of µ such that µ0 (x) ∈ D(wj ). Since M ∈ Fc , we have D(wj ) 6= ∅. Hence there is an x-variant µ00 of µ such that Defc (hM, µ00 , wj i, φ). By the induction hypothesis we have if Defc (hM, µ00 , wj i, φ), then Den(IM , φ, wj ). Hence, Den(IM , φ, wj ). Since ∀xφ has the same individual constants as φ, we can conclude that Den(IM , ∀xφ, wj ). e) Necessity. Assume Def c hM, µ, wj i, φ). By Definition 1 we have Defc (hM, µ, wi, φ) iff Defc (hM, µ, w0 i, φ) 19
for all worlds w0 ∈ W s.t. RM (wj , w0 ). By the induction hypothesis, if Defc (hM, µ, wi, φ), then Den(IM , φ, w) at all worlds w ∈ W. Hence, Den(IM , φ, w) is true at all worlds w0 ∈ W such that RM (wj , w0 ). Since M ∈ FC , it holds that RM (wj , wj ). Hence, Den(I, φ, wj ). But since for any constant c, c is in φ iff it is in φ, it also holds that Den(I, φ, wj ). This concludes the induction.
QED
Lemma 13. For any M ∈ Fc and any w ∈ W, there is a model M0 ∈ Fc such φ iff |= φ. that for all φ ∈ L, |= M, w, µ M0 , a, µ Proof. For M ∈ Fc and wi ∈ W, we shall define M0 by means of a permutation function h s.t. h(a) = wi , h(wi ) = a, and for all other w ∈ W, h(w) = w. Then we define M0 as i) DM0 (w) = DM (h(w)), for any w ∈ W ii) RM0 (w, w0 ) iff RM (h(w), h(w0 )), for any w ∈ W iii) IM0 (P, w) = IM (P, h(w)), for any predicate letter P in L and any w ∈ W iv) IM0 (c, w) = IM (c, h(w)), for any individual constant c in L and any w ∈ W, except that IM0 (c, w) is defined only if IM (c, wi ) is defined. It can be verified that M0 is in Fc . This holds just if the extension and 0 is reflexive. The extension postulate is rigidity postulates are satisfied, and RM satisfied, since IM0 (P, w) ⊆ DM0 (w) iff IM (P, h(w)) ⊆ DM (h(w)), which holds by the extension postulate for M. Also, the rigidity postulate is satisfied, since IM0 (c, w) = IM (c, h(w)) = IM (c, a), by def. of M0 and the rigidity postulate for M. But then IM (c, a) = IM (c, wi ) in case IM (c, a) ∈ DM (wi ). Otherwise, IM (c, wi ) is undefined. In case it is defined, IM (c, wi ) = IM (c, h(a)) = IM0 (c, a) by def of h and of M0 . Hence, IM0 (c, w) = IM0 (c, a), and so the rigidity postulate is satisfied. 0 Finally, RM (w, w) iff RM (h(w), h(w)) by def of M0 , and the right hand side holds by def of Fc and the assumption that M ∈ Fc . So the left hand holds as well. M0 is reflexive, and therefore in Fc . Next, we show (*)
For any φ ∈ L such that Den(IM , φ, wi ), it holds for any w that Defc (hM0 , µ, wi, φ) iff Defc (hM, µ, h(w)i, φ)
This is shown by induction over formula complexity. a) Atomic case. What matters is whether IM0 (c, w) ∈ DM0 (w) iff IM (c, h(w)) ∈ DM (h(w)) 20
for any constant c in φ s.t. Den(I, φ, wi ). And this holds just if (**)
IM0 (c, a) ∈ DM0 (w) iff IM (c, a) ∈ DM (h(w))
by the rigidity postulates for M and M0 , for any constant c in φ s.t. Den(I, φ, wi ). But DM0 (w) = DM (h(w)) by def of M0 . And IM0 (c, a) = IM (c, wi ) = IM (c, a), provided IM0 (c, a) is defined. So (**) holds if IM0 (c, a) is defined. But by the assumption that Den(IM , φ, wi ), and the definition of M0 , it holds for every constant cj in φ that IM0 (cj , a) is defined. Hence (**) holds, and the base step of the induction is completed. The induction steps are straightforward. Hence, (*) is true. Next we show by induction over formula complexity that (#)
For any φ ∈ L such that Den(IM , φ, wi ), and for any w ∈ W it holds that φ iff |= φ |= M0 , w, µ M, h(w), µ
a) Atomic case. Assume Den(IM , P t1 , . . . , tn , wi ). Thereby, IM (tj , wi ) is well-defined, and thereby also IM0 (tj , a) for 1 ≤ j ≤ n. Then |= P t1 , . . . , tn iff M0 , w, µ hIM0 ,µ (t1 , w ), . . . , IM0 ,µ (tn , w )i ∈ IM0 (P, w ) iff (by def 2) hIM,µ (t1 , h(w )), . . . , IM,µ (tn , h(w ))i ∈ IM (P, h(w )) iff (by def of M0 ) |= P t1 , . . . , tn (by def 2) M, h(w), µ b) Negation. Assume Den(IM , φ, wi ). Then |= ¬φ iff M0 , w, µ Def c (hM0 , µ, wi, ¬φ) and not |= φ, iff M0 , w, µ Def c (hM, µ, wi, ¬φ) and not |= φ, iff M0 , w, µ Def c (hM, µ, wi, ¬φ) and not |= φ, iff M, h(w), µ |= ¬φ M, h(w), µ
(by def 2) (by (*), since Den(IM , φ, wi )) (by ind. hyp.) (by def 2)
c) Conjunction. Straightforward. d) Universal quantifier. Straightforward. e) Necessity operator. |= φ iff M0 , w, µ (by def 2) |= φ at any world w0 s.t. RM0 (w, w0 ) iff M0 , w0 , µ 0 0 |= φ at any world h(w ) s.t. RM (h(w), h(w )) iff M, h(w0 ), µ (by ind. hyp. and def of M0 ) |= φ (by def 2) M, h(w), µ That concludes the proof of (#). As a corollary of (#), we have the instance
21
(##)
If Den(IM , φ, wi ), then |= φ iff |= φ M, wi , µ M0 , a, µ
Now, assume |=M, wi , µ φ. By Fact 1, Defc (hM, µ, wi i, φ). Then by Fact 12, Den(IM , φ, wi ). Hence, by (##), |= φ. Conversely, assume |= φ. Again, M0 , a, µ M0 , a, µ by Fact 1, Defc (hM0 , µ, ai, φ). And by Fact 12, Den(IM0 , φ, a). By definition of M0 by definition 11, this holds iff Den(IM , φ, wi ). Hence, by (##) again, φ. Thus, |= M, wi , µ (!)
|= φ iff |= φ M, wi , µ M0 , a, µ
Since M and wi were arbitrarily chosen, the lemma follows by generalization. QED With the help of the lemma, we can go on to prove Fact 14. For any φ ∈ L, Γ |= φ iff Γ |= φ C, U C, A Proof. Left to right: by instantiation to a. Right to left. Assume that Γ |=C, A φ, and that for M ∈ Fc , wi ∈ W, µ and every ψ ∈ Γ, |=M, wi , µ ψ. By Lemma 13 there is a model M0 ∈ Fc s.t. for any χ ∈ L, |= χ iff |= χ M, wi , µ M0 , a, µ Hence, for any ψ ∈ Γ, |=M0 , a, µ ψ. From the assumption that Γ |=C, A φ, we have that if for any ψ ∈ Γ, |= ψ, then |= φ. M0 , a, µ M0 , a, µ Hence |= φ. And hence, by the relation between M and M0 , |= φ. Since, M0 , a, µ M, wi , µ M, wi and µ were arbitrarily chosen, we conclude that Γ |= φ. QED C, U From Theorem 5 we can derive Fact 15. For any φ ∈ L, Γ |= φ iff Γ |= φ C, A R, A Proof. From Theorem 5 we have that |= φ iff |= φ, for any φ ∈ L, and Mc , a, µ Mr , a0 , µ any models Mc and Mr such that Mc ∼ Mr . Then, from left to right, assume that Γ |= φ and that for M0 ∈ Fr , and all C, A ψ ∈ Γ, |=M0 , a, µ ψ. By Fact 6, and Definition 8, there is a model M+ ∈ Fc such that M0 ∼ M+ . By Theorem 5, it holds that for all ψ ∈ Γ, |= ψ. And from M+ , a, µ the assumption that Γ |=C, A φ we then get |=M+, a, µ φ. Again, by Theorem 5, and the fact that M0 ∼ M+ , we conclude that |= φ. Since M0 was arbitrary, it M0 , a, µ follows that Γ |= φ. The other direction is symmetric. QED R, A Now we can conclude
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Theorem 16. For any Γ ⊆ L and φ ∈ L, Γ |= φ iff Γ |= φ. C, U R, A Proof. Immediate from Facts 14 and 15.
QED
This is the main result, establishing equivalence in the class of reflexive models. It is, however, also of interest to know whether the equivalence will stick under other restrictions. We shall show that the equivalence still holds for models with symmetric and transitive accessibility relations, and their possible combinations. Let the parameter k take these four combinations as values, and let ‘Fc+k ’ [‘Fr+k ’] denote the class of models characterized by the corresponding accessibility restrictions. Definition 12. K-restricted consequence relations. φ is a k-restricted classical universal consequence of Γ, Γ |=C + k, U φ iff it holds for all M ∈ Fc+k , all w ∈ W and all assignments µ that Γ |=M, w, µ φ. Mutatis mutandis for k-restricted relational and actual consequence relations. Definition 13. Accessibility permutations. R ≈h R0 iff for any w, w0 ∈ W, R0 (w, w0 ) iff R(h(w), h(w0 )). We now make the following observation, with k taking values as above. Fact 17. For any accessibility relations R and R0 and permutation h on W, if R ≈h R0 , then R has property k iff R0 has property k. Proof. Since reflexivity, symmetry and transitivity are logically independent properties, it suffices to show this for each property separately. It was already observed to hold for reflexivity in the proof of lemma 13. For transitivity, assume that R ≈h R0 , that R is transitive, and that R0 (w, w0 ) and R0 (w0 , w00 ). By the assumptions, R(h(w), h(w0 )) and R(h(w0 ), h(w00 )). By the transitivity of R, R(h(w), h(w00 )). By, the R ≈h R0 assumption, R0 (w, w00 ). The other direction is analogous. The proof for symmetry is similar QED Then, since permutations preserve symmetry and transitivity, we can add the following generalization of Lemma 13: Fact 18. For any of the four values of k, any M ∈ Fc+k , and any w ∈ W, there is model M0 ∈ Fc+k such that for all φ ∈ L, |= φ iff |= φ. M, w, µ M0 , a, µ Proof. By Fact 17 permutations preserve the k-value of the accessibility relation. The result follows then from lemma 13 adding k-valued accessibility restrictions to the models. QED Then again we get 23
Fact 19. For any Γ ⊆ L, φ ∈ L, and any k-value, Γ |= φ iff Γ |= φ. C + k, U C + k, A Proof. By Fact 18, analogous to the proof of Fact 14.
QED
Similarly, Fact 20. For any Γ ⊆ L, φ ∈ L, and any k-value, Γ |= φ iff Γ |= φ. C + k, A R + k, A Proof. Analogous to the proof of Fact 15, given the observation that RM = RM0 if M ∼ M0 . QED And finally, Theorem 21. For any Γ ⊆ L, φ ∈ L, and any k-value, Γ |=C + k, U φ iff Γ |= φ.. R + k, A Proof. Immediate from Facts 19 and 20.
QED
M is a model for the modal system T just in case RM is reflexive, for S4 just in case RM is reflexive and transitive, for B just in case RM is reflexive and symmetric, and for S5 just in case RM is reflexive, symmetric and transitive. It follows from Theorem 21 that φ is a classical T, S4, B, S5 consequence of Γ just in case φ is a relational T, S4, B, S5 consequence of Γ, respectively. And again, if we assume an invariant non-empty domain across worlds, we can also drop the reflexivity condition. In that case, the equivalence also holds for models of K.
Department of Philosophy, Uppsala University Department of Philosophy, Stockholm University
References Almog, J., 1986, ‘Naming without necessity’, Journal of Philosophy 83:210–42. Dummett, M., 1981, The Interpretation of Frege’s Philosophy, Harvard University Press, Cambridge, Mass. Dummett, M., 1991, The Logical Basis of Metaphysics, Harvard University Press, Cambridge, Mass. Evans, G., 1979, ‘Reference and contingency’, The Monist 62:161–89. Reprinted in Evans 1985. Evans, G., 1985, Collected Papers, Clarendon Press, Oxford.
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Gamut, L. T. F., 1991, Logic, Language and Meaning. Volume 2. Intensional Logic and Logical Grammar., University of Chicago Press, Chicago, Ill. Kaplan, D., 1986, ‘Opacity’, in P. A. Schilpp and L. E. Hahn (eds.), The Philosophy of W.V. Quine, Open Court, La Salle, Ill. Kripke, S., 1980, Naming and Necessity, Harvard University Press, Cambridge, Mass. Quine, W. V. O., 1952, Methods of Logic, Routledge and Kegan Paul, London. Quine, W. V. O., 1956, ‘Quantifiers and propositional attitudes’, Journal of Philosophy 53:177–87. Reprinted in Quine 1976. Quine, W. V. O., 1976, The Ways of Paradox and Other Essays, Harvard University Press, Cambridge, Mass., 2 edn. Stanley, J., 1997a, ‘Names and rigid designation’, in B. Hale and C. Wright (eds.), A Companion to the Philosophy of Language, Blackwell, Oxford. Stanley, J., 1997b, ‘Rigidity and content’, in R. Heck (ed.), Language, Thought and Logic: Essays in Honor of Michael Dummett, Oxford University Press, Oxford.
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