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Analyticity and Possible-World Semantics

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Abstract

Standard approaches to possible-world semantics allow us to define necessity and logical truth, but analyticity is considerably more difficult to account for. The source of this difficulty lies in the received model-theoretical conception of a language interpretation. In intuitive terms, analyticity amounts to truth in virtue of meaning alone, i.e. solely in virtue of the interpretation of linguistic expressions. In other words, an analytic sentence should remain true under all variations of ‘extralinguistic reality’ as long as the interpretation is kept constant. However, the received conception of an interpretation as a mapping from language to a model frame hinders keeping the interpretation constant while varying other components of the model. To make room for analyticity, the concept of an interpretation should therefore be revised. The latter should be made richer in content than it has usually been assumed. As a by-product, this revision also gives us a one-dimensional analogue of the influential two-dimensional account of a priori. We are thus able to map out the network of formal connections between the notions of analyticity, apriority, logical truth and necessity.

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Notes

  1. The approach to be presented is a simplified version of Kripke (1963). My apologies to the reader for this rehearsal of a standard material. For a much more detailed and thorough presentation of treatments of necessity and logical truth in various versions of possible-world semantics, see Lindström and Segerberg (2006), part 1 (“Alethic Modal Logic”).

  2. An assignment of objects to variables is often kept apart from the interpretation proper. (This assignment, or sometimes a pair consisting of this assignment together with the interpretation proper, is then called a ‘valuation’.) The reason for the separation is that when we change the assignment to variables, but keep everything else constant, the meaning of the language intuitively doesn’t change. Given such a separation, the truth of open sentences, which contain free variables, has to be relativized not only to a model (= a frame together with an interpretation, see below), but also to a particular assignment of objects to the variables. Here, however, for the sake of simplicity, we shall avoid this complication.

  3. For simplicity, we haven’t assumed that identity is a logical constant of L. If we did, then we would need a special clause for atomic identity sentences: i(t 1  = t 2 ) equals W or ∅ depending on whether or not i(t 1 ) = i(t 2 ).

  4. It is philosophically problematic that the truth condition for a quantified sentence requires us to consider other interpretations of the language. This problem would be avoided, however, if the assignment of objects to variables were kept apart from the interpretation proper.

  5. If the ideas of a possible world and a possible object were taken very literally, then it is arguable that the set of possible worlds should be the same in all admissible frames and the objects that exist in any given world should also be the same. Thus, in this limiting case, two admissible frames could only differ in the choice of the actual world. But modal logicians do not take these notions so literally. Normally, possible worlds and their object domains are allowed to vary from one admissible frame to another.

  6. This argument for the existence of contingent logical truths has as a premise that”Actually” is a logical constant, which of course might be questioned. (Thanks to a referee for pressing this point.) The existence of contingent logical truths thus partly depends on the decisions we make about the logical vocabulary of language L and these decisions might well be arbitrary to some extent. Luckily, nothing much in our subsequent discussion of analyticity turns on how we decide to deal with this issue. In particular, whether or not we allow for contingent logical truths, we’ll still be able to argue for the existence of contingent analytical truths.

  7. As a referee has pointed out, there might be other reasons for keeping the designated worlds even if one goes two-dimensional. (In particular, we still need the designated world if we want to define truth in a model. A sentence can be said to be true in a two-dimensional model iff it is true in the designated world of that model from that world’s own perspective.) But, in the present context, there is no need to do so.

  8. See, for example, Jackson (1998).

  9. Which is not to say, of course, that for a priori sentences the knowledge of meaning must immediately issue in the knowledge of truth. The road from the one to the other might be quite strenuous and require a great deal of intellectual effort.

  10. See also Jackson (1998, p. 51): “What we can know independently of what the actual world is like can properly be called a priori.” As a matter of fact, Chalmers takes epistemic possibilities to be ‘centered’ worlds, i.e. triples consisting of a possible world, an individual and a time. The time and the individual specify the location within a possible world from which the semantic evaluation is being made. Such localization is primarily needed for the semantic evaluation of indexical sentences. Here we ignore this complication.

  11. Along with Frank Jackson, Chalmers is probably the most eloquent modern proponent of the two-dimensionalist semantics. Early proposals on two-dimensionalist lines can be found in the work from the 70s and early 80s by Segerberg (1973), Evans (1977), Stalnaker (1978), Kaplan (1978) and Davies and Humberstone (1980). Stalnaker has nowadays become a critic of two-dimensionalism; see Stalnaker (2004). Another critic is Soames (2005).

  12. Is there any (philosophical or technical) advantage to using my one-dimensional semantics over the standard two-dimensional approach? In response to this question posed by a referee, my answer is: No, there is not, if one only wants to account for the notion of a priori. However, the one-dimensional semantics is developed in order to account for analyticity. It takes care of a priori as a bi-product. As a result, there is no longer any need for going two-dimensional in order to account for the latter notion.

  13. For this conception of analyticity, see Quine (1951, pp. 20f): “Kant conceived of an analytic statement as one that attributes to its subject no more than is already conceptually contained in the subject. This formulation has two shortcomings: it limits itself to statements of subject-predicate form, and it appeals to a notion of containment which is left at a metaphorical level. But Kant's intent, evident more from the use he makes of the notion of analyticity than from his definition of it, can be restated thus: a statement is analytic when it is true by virtue of meanings and independently of fact.” A referee has pointed out that Quine wasn’t the first to present the intuitive idea of analyticity in this way. Apparently, the same suggestion can already be found in Carnap (1939, 1942).

  14. However, it should be noted that in the argument above I have implicitly assumed that reference isn’t meaning, not even in the case of purely referential proper names. (I am indebted to a referee for pointing this out.) For such a view, see, for example, Marcus (1961, pp. 309f): “to give a thing a proper name is different from giving a unique description…(An) identifying tag is a proper name of the thing…This tag, a proper name, has no meaning. It simply tags.” If we instead took a Millian-style postion that, for proper names and natural kind terms, their meaning just is their reference, rather than that such purely referential terms lack meaning at all, we would need to conclude that sentences such “Hesperus = Phosphorus” are analytic, after all. For, clearly, they are true in virtue of what their component terms refer to. Since knowledge of the reference of the component terms suffices for the knowledge that such sentences are true, we would then also be hard pressed to agree that these sentences are a priori. Thus, the widely shared intuition that sentences like “Hesperus = Phosphorus” are true a posteriori would have to be given up. In what follows, I shall disregard this radical possibility.

  15. The idea that analyticity should be understood as truth under all frame variations, with the interpretation kept fixed, was to my knowledge first put forward in Stig Kanger’s doctoral dissertation, Kanger (1957, Sect. 7.3), and in Kanger (1970, Sect. 4). (The latter essay originally appeared as a privately distributed pamphlet and was then re-published in Hilpinen (1970).) It should be pointed out, though, that Kanger’s frames were just sets of (individual) objects. On his approach, which predated possible-world semantics, analyticity was therefore understood as truth in all object domains under a fixed interpretation. For a presentation and discussion of Kanger’s views, see Lindström (1998).

  16. In what follows, we are going assume that this class equals Σ. But, strictly speaking, a more liberal and general approach might be preferable: we could let the domain of an interpretation be a subset of Σ. Different interpretations might then be more or less inclusive: their frame domains might be larger or smaller. (For a suggestion along these lines, see Lindström (2006).) The only condition we then need to impose on the frame domain of an interpretation is that it should be closed under all variations of the designated worlds. (This condition will be needed in what follows, in order to establish that analyticity entails aprioricity.) That the condition in question is possible for a subset of Σ to satisfy follows from the fact that Σ itself is closed under all such variations.

  17. For a suggestion that an interpretation should pick out such a designated frame, I am indebted to Lindström (2006). As we shall see, this device is needed in order to define the notion of truth under an interpretation. If the domain of i is taken to be a subset of Σ rather than Σ itself (see the preceding note), then S i must of course be an element of that subset.

  18. The two readings of an interpretation i, as a function from Σ to the ‘old style’ interpretations and as an assignment to linguistic expressions of functions from frames to intensions, are equivalent in the following sense: For each expression e of L and each frame in S, i(S)(e) on the former reading equals i(e)(S) on the latter reading.

  19. I am indebted to a referee for pressing this point.

  20. If Σ is a proper class, then it is preferable to work with interpretations that have as their frame domains subsets of Σ, as suggested in footnote 15 above.

  21. Since for every interpretation i and every frame S in Σ, there exists an interpretation i that is exactly like i except that its intended frame S i′  = S, it follows that logical truth, i.e. truth in all models, is equivalent with truth on all interpretations.

  22. However, as has already been mentioned, this example presupposes that the actuality operator is taken to belong to the logical vocabulary of the language.

  23. See Kripke (1980, pp. 54ff).

  24. For another class of contingent analytic truths, think of indexical sentences, such as “I exist”, or “I am here now”. But to deal with indexicals, the admissible frames would have to be enriched with additional elements: Apart from a designated world, they would need to contain a designated individual, a designated location in time and space, and so on.

  25. I wonder what Kripke would have said of sentences such as “Actually A → A”. Would he deny them the analytic status in view of their being only contingently true?

  26. Proof: We need to show F and G have the same ultraintension on an interpretation i iff it is analytically true on i that □ (∀x (F(x) ↔ G(x))). (1) Suppose that i ascribes to F and G different ultraintensions. That is, in some model based on i, F and G have different intensions. This in turn means that for some possible world w in that model and some individual x, x belongs to the extension of F in w but not to the extension of G, or vice versa. Consequently, in the designated world of that model, it is false that □(∀x (F(x) ↔ G(x))). (Note, however, that the last step in this derivation depends on our account of necessity as truth in all possible worlds. If we instead had worked with frames containing an accessibility relation that corresponds to the necessity operator, then the proof would not go through: the world w might not be accessible from the designated world.) (2) Suppose □(∀x (F(x) ↔ G(x))) is not analytic on i. Then this sentence is false in some model based on i, which means that there is some world in this model in which F and G have different extensions. Consequently, the intensions of F ad G differ in the model in question, which implies that i ascribes to these predicates different ultraintensions.

  27. As for the term “ultraintension”, I have chosen it to indicate the connection between the notion we are after and the so-called ‘ultraintensional’ sentential operators. While in the scope of intensional operators (such as □ or Actually) co-intensional sentences are freely substitutable salva veritate, free substitution in the scope of ultraintensional operators (such as ‘S believes that …” or “S hopes that …”) requires not just co-intensionality but synonymy. Consequently, the meanings that are supposed to be identical in the case of synonymy deserve being called “ultraintensions”.

  28. Cf. Boghossian (1996, p. 364). (See also Boghossian (1997)). Gilbert Harman makes the observation Boghossian quotes in Harman (1967, p. 128). For an elaboration of this point, see Williamson (2007, ch. 3).

  29. As a referee has pointed out, the reference to the class of admissible frames Σ as an additional meaning-determining factor, along with an interpretation, is strictly speaking unnecessary. Since the class of admissible frames is the domain of an interpretation function, by specifying the interpretation we ipso facto specify Σ.

  30. Can these semantic notions of analyticity and a priori, as defined above, be explicitly introduced into the object language L itself, as sentential operators? This does seem possible; cf. Lindström (2006).

  31. Explanation: As we know, an interpretation assigns semantic values to linguistic expressions only with respect to frames that belong to Σ. Thus, if a sentence is true in a model M, it wouldn’t be possible for that sentence to have truth-valuea in models that only differ from M with respect to the choice of a designated world if the frames of these models didn’t belong to Σ. For the same reason, if the frame domain of an interpretation is restricted to a subset of Σ, this subset must itself satisfy the closure condition.

    Given the closure condition it is easy to show that our one-dimensional account of a priori is equivalent to the standard two-dimensional account. To see this, note that if we restrict the domain of i to the class of frames that consists of S i  = < W, ω, I, D > together with all the frames that differ from S i only with respect to the choice of the designated world, we get a function that coincides with the set of two-dimensional world-indexed interpretations i w from the language L to the reduced frame < W, I, D > , where index w varies over all members of W. Conversely, any such set of two-dimensional interpretations i w from L to a reduced frame < W, I, D > can be seen as the interpretation of L with respect to a class of frames < W, w, I, D > , where W, I and D are drawn from the reduced frame, while w varies over W. This interpretation can be assigned an arbitrary element of its domain as its intended frame. Clearly any such interpretation of L can then be extended to an interpretation that takes as its domain the whole of Σ.

  32. In fact, as is easily seen, analyticity on a given interpretation is equivalent with aprioricity in all models based on that interpretation. Proof: Suppose a sentence A is analytic on i. Consider any model M = (S, i), with SΣ. Let M′ be a model that only differs from M in the choice of the designated world of M′. We have to show that A is true in M′. But this immediately follows from the analyticity of A on i. For the other direction, suppose A is a priori in every model M = (S, i), where SΣ. But then it immediately follows that A is true in every such model, which means that A is analytic on i.

  33. Another example of synthetic a priori has been suggested in Lindström (2006). Consider the sentence “Water is composed of hydrogen and oxygen → □(Water is composed of hydrogen and oxygen)”. Again, a metaphysician could argue that this sentence is true in all possible worlds: If water actually has this composition (which is for the natural science to decide), then having this composition is essential to water. Furthermore, it is arguable that the truth of the sentence in question (unlike that of its antecedent) does not depend on the choice of the designated world: From the perspective of every possible world, the physical composition of water (whatever that composition might be from the perspective of that world) is its essential property. Consequently, the sentence comes out as a priori. But, at the same time, it might be meaningful to consider how things would be like in other frames in which physical composition wouldn’t be essential to water. If such frames are entertainable, the sentence is not analytic.

  34. Some examples of perspective-independent sentences have already been provided above: “For all events x, y and z, if x is later than y and y is later than z, then x is later than z”, “Water is composed of hydrogen and oxygen → □(Water is composed of hydrogen and oxygen)”, “All bachelors are unmarried”. These are all a priori, but it is not difficult to give examples of a posteriori sentences that are perspective-independent, say, “There is an event that is not later than any other event” or “Some bachelors have children”. As for perspective-dependent sentences, “Water = H2O” is just one example among many. This one is necessarily true and a posteriori, but contingent and/or a priori perspective-dependent sentences are clearly easy to find. (“This is water”, “I exist”, “The standard metre in Paris is one metre long”, etc.).

  35. For a good presentation of Quine’s views on this matter, see Hylton (2002).

  36. Frege (Frege 1960, p. 4): “The problem becomes, in fact, that of finding the proof of the proposition, and of following it up right back to the primitive truths. If, in carrying out this process, we come only on general logical laws and on definitions, then the truth is an analytic one, bearing in mind that we must take account also of all propositions upon which the admissibility of any of the definitions depends. If, however, it is impossible to give [such a] proof […], then the proposition is a synthetic one.”

  37. See Carnap (1952), reprinted as a supplement in Carnap (1956, pp. 222-9).

  38. Cf. Montague (1974); see especially chapters 7 and 8, and Richmond Thomason’s introduction (pp. 53f). For earlier proposals along the same lines, see Kemeny (1952, 1956). (Note, though, that Kemeny and Montague use the term “interpretation” as synonymous with “model”. That is, an interpretation in the standard sense, i.e. a function from the language to a frame, is only one component in what they call an “interpretation,” along with the frame itself.).

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Acknowledgments

I am particularly grateful to Sten Lindström for his helpful comments (cf. also Lindström (2006)). I have also been helped by discussions with John Cantwell, Erik Olsson, and the participants of philosophy seminars at Lund University and at the Research School of Social Sciences at the Australian National University in Canberra, in which this paper has been presented. I am much indebted to the anonymous referees of Erkenntnis for their very useful comments and suggestions and to Hans Rott, the editor of Erkenntnis, for his kindness and encouragement. Last but not least, I’d like to mention a paper by Kathrin Glüer and Peter Pagin, in which they amicably but critically discuss my proposal and propose a solution of their own (Glüer & Pagin 2007). I regret I haven’t responded to their criticisms here; to do it properly would require a paper of its own.

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Correspondence to Wlodek Rabinowicz.

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An earlier version of this paper, entitled “Analyticity: An Unfinished Business in Possible-World Semantics”, was published in H. Lagerlund, S. Lindström and R. Sliwinski (eds), Modality Matters: Twenty-Five essays in Honour of Krister Segerberg, Uppsala Philosophical Studies 53, Uppsala 2006, pp. 345–58. The present version is considerably revised.

Appendix: Perspective-Independence

Appendix: Perspective-Independence

We want to prove the following claims:

  1. (1)

    For every model M and every perspective-independent sentence A in M,

if A is apriori in M, then □A is apriori in M.

  1. (2)

    For every interpretation i and every strictly perspective-independent sentence A on i,

if A is analytic on i, then □A is analytic on i.

  1. (3)

    For every logically perspective-independent sentence A,

if A is logically true, then □A is logically true.

Proof of (1): Suppose that A is both perspective-independent and a priori in M = (< W, ω, I, D > ,i). Consider any model M′, which differs from M only in its choice of the designated world. To establish that □A is a priori in M, we need to show that □A is true in every such M′. Consider therefore an arbitrary world w in M′. We need to show that (i) A is true at w in M′. By the perspective-independence of A in M, (i) is the case iff (ii) A is true at w in M, which—again by the perspective-independence of A—is the case iff (iii) A is true at w in M″ = (< W, w, I, D > ,i). But (iii) must be the case if A is a priori in M.

Proof of (2): Since strict perspective-independence on i is perspective-independence in all models based on i, while analyticity on i is equivalent to aprioricity in all models based on i, (2) follows from (1).

Proof of (3): Since logical perspective-independence is equivalent to strict perspective-independence on every interpretation, while logical truth is equivalent to analyticity on every interpretation, (3) follows from (2).

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Rabinowicz, W. Analyticity and Possible-World Semantics. Erkenn 72, 295–314 (2010). https://doi.org/10.1007/s10670-010-9216-4

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