Why the tuple theory of structured propositions isn't a ...

Why the tuple theory of structured propositions isn’t a theory of structured propositions Bjørn Jespersen

This paper seeks to establish a negative thesis: The tuple theory of structured propositions can’t possibly be a theory of structured propositions. Much is at stake. The notion of structured propositions – in particular structured singular propositions – is a cornerstone of the theory of direct reference, yet if the intuitive notion of structured proposition cannot possibly translate into ordered ntuples, as direct reference theory does, then that theory finds itself without a semantics in one of its key areas. This would prove an embarrassment, considering that its founding father, David Kaplan, first put forward the notion a quarter of a century ago. If the objections go through, however, it’s not only direct reference theory that will be seen to have left a central notion dangling. E.g., also Max J. Cresswell’s brand of categorial grammar relies on tuples as propositional structures (see Cresswell, 1985). It’s important to stress that the attack on tuples as propositional structures isn’t an attack on the notions of either singular or structured proposition. Indeed, both have much to be said for them, albeit they would deserve to be developed within a different framework. An historical parallel might be in place. Since Frege never came around to developing a formal theory of Sinne, it at some point became tempting to construe them as intensions à la possible worlds semantics, i.e., as functions (mappings) defined on a set of possible worlds. It’s well-known today that, and why, the construal was an aberration. Similarly, since Russell never got a firm grip on his propositions, it has become tempting to construe them as tuples. This essay tries to show why they shouldn’t be. In setting out why, we’ll see, among other, that the old problem of propositional unity afflicting Russell’s propositions rears its head again (see Jacquette 1992/93, Griffin 1993). But tuples-as-structures are open to an even more fundamental objection (Section IV).

I According to direct reference theory, a general proposition is, in the simplest case, a structure whose constituents are a condition satisfiable by an individual and a property. For instance, the sentence “The tallest spy is suspicious” is paired off with, (1)

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A singular proposition, in the simplest case, is a structure whose constituents are an individual and a property. For instance, the sentence “Lulu is suspicious” is paired off with, (2)

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As Kaplan says, Corresponding to the predicate we have the property of being suspicious; and corresponding to the subject we have either what Russell in 1903 called a denoting concept or an individual. Let us take the proposition to be the ordered couple of these two components. (1990, pp. 16-17.) Don’t think of propositions as sets of possible worlds, but rather as structured entities looking something like the sentences which express them. (1989, p. 494.) Handling a subject-predicate sentence like “Lulu is suspicious” as an ordered pair was subsequently turned into an industry of ‘structured propositions’, leading California semantics et al. down a new path. E.g., in 1986 Joseph Almog integrated the tuple theory into two-stage theory (Almog 1986). Two-stage theory owes its name to the two stages generation and evaluation. Evaluation is assignment of a truth-value at a world (or some other point of evaluation, or ‘index’, for that matter), while generation consists in concatenating elements of a domain into a set-theoretical structure. For instance, given the availability of Lulu and Being-suspicious, the singular proposition (2) may be generated. When labouring within a paradigm of set theory in general, and model theory in particular, including possible worlds semantics, tuples are the obvious, and only, category to turn to in one’s quest for structure. So, what can tuples do for a theory of structured propositions? First, a tuple has •

elements.

This ostensibly allows us to say that “Lulu is happy” is in part about Lulu, simply because she herself appears in the corresponding proposition. Besides, planting objects in propositions plays a pivotal role in various programmes of matching sentential (syntactic) and propositional (semantic) structure: “Lulu” in a sentence will match Lulu in a proposition, and so forth. Second, a tuple is a •

sequence (order).

This is enough to ensure hyperintensional individuation. While possible worlds semantics cannot distinguish between the proposition (i.e., function from possible worlds to the set of truth-values) that Canada is smaller than Andorra and the proposition that Andorra is larger than Canada, it will now be possible to distinguish between, and

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. In general, any two triples , are identical iff the first elements are identical, the second elements are identical and the third elements are identical, and similarly for pairs, etc. That is, even if two tuples contain the same members, the tuples will be distinct if their members are arranged as two different sequences. Besides, an ordered set offers the advantage over an unordered set of repetition: ≠ , while {a, a} = {a}. But we should demand more of the notion of structure. As both Pavel Tichý and Jeffrey C. King ask, What is the logical ‘glue’ that keeps the constituents together within the proposition?1 At first blush, the issue of internal cohesion is the Achilles’ heel of the tuple theory. The idea that a mere enumeration of objects underdetermines a complex isn’t new, of course. To use an example of Bernard Bolzano’s, although the ‘ideas in themselves’ 35 and 53 share the same ‘content’, as he calls it, they’re still two distinct ideas. The tuple theory, as we just saw, would be able to distinguish between and , but would still fail (as I point out below) to answer the question that Bolzano poses as to how those elements are connected with each other.2 In general, the tuple theory may deliver on the first two accounts (elements, sequence), but definitely founders on the third, •

internal cohesion (‘glue’). II

To see this, consider (2). Prima facie the intended relation between Lulu and Beingsuspicious would be one of functional application of a set (as a characteristic function) to an individual.3 But (2) is silent on that score. Any such indication would be external to (2), and the upshot is that (2) comes across as a shallow formalization of “Lulu is suspicious.” Another example, this time involving an operator: “Lulu is not suspicious” would go into, (3)

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For instance, Scott Soames says, A [structured] proposition is true relative to E [i.e., ‘truthsupporting circumstances’] iff the value of NEG at the extension of Prop S in E is truth […]. (1997, p. 948. See also Braun 1993, p. 464.) Soames, needless to say, knows that (in classical logic) negation is an unary operation on truth-values and not on propositions, whatever they may be, and the truthcondition for stated in prose is correct.4 However, the formula itself fails to indicate that negation is to be applied to the extension of S at the world of evaluation. What is absent are both indications of extensionalization of S and of application of negation to the thus obtained extension (i.e., truth-value) of S.

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Is there a way to remedy the theory? For instance, (3) might be furnished with modal and temporal parameters so as to somehow arrive at an intension and not just an extension, whereby the distinction between contingency and necessity, factual and non-factual sentences could be held in place. After all, Lulu is just suspicious at some worlds at some times. Moreover, instructions of extensionalization and application might be added as well. If so, (3) would then become something like, (4)

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Kaplan speaks of his structured propositions as ‘determining’ possible worlds propositions. For instance, (2) will determine that function F which assigns Truth to w and t iff Lulu is a member of P(w, t), where P is a property in the sense of a function from worlds and times into sets of individuals (see Kaplan 1990, p. 18). (4) now appears to have the elements required, and let’s just assume that the sequence is right. But we’re still left no wiser as to how those elements hook up with each other. All we’re told is what the elements are, not what to do with them. 5 Consequently, since the nature of determination is unclear, it’s far from obvious that the principle of compositionality is heeded. E.g., the presence of APPLY tells us that functional application of negation is an element, but not that it connects, as function and argument, the operation of negation with the truth-value obtained by applying the function of Being-suspicious to worlds, times and Lulu. In fact, (4) as it stands cannot indicate how it ‘determines’ a function from worlds and times into {0, 1}. Since the occurrences of w, t in (4) are free, (4) will determine, at most, a truth-value and not a truth-condition (possible worlds proposition). One might consider binding w, t by λ, but the outcome would begin to look suspiciously like a piece of notation culled from (some versions of) Montague’s Intensional Logic, IL (e.g., λiϕ), or Tichý’s Transparent Intensional Logic, TIL (e.g., λwλtϕ). The advocates of the tuple theory are quite likely not to wish to go into that direction. After all, their approach is a relational one, Russellian as it is, and not a functional one, as the neoFregean approaches. As an aside, this fact may explain why the tuple theorists aren’t adamant that (2) should be, (2’)

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(2’) would at least be helpful towards making it explicit that Being-suspicious matched the functor “Being-suspicious” by occurring before Lulu, the ‘semantic value’ of the argument-term “Lulu”.6 One might object that since functions and relations can be defined in terms of each other, there’s no difference between a relational and a functional approach. What dismantles the objection is that functions are amenable to abstraction and application, with no counterparts among relations. Thus functional theories of meaning have a richer logic, viz. lambda-calculi, at their disposal, and so relations and sets are more conveniently construed as functions (see, e.g., Materna 1998, pp. 164-5). III

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The punch of my negative thesis is predicated so far on a notion of cohesion internal to structured propositions. Briefly, what would internal cohesion boil down to? King helpfully quotes Russell, Consider, for example, the proposition ‘A differs from B’. The constituents of this proposition, if we analyze it, appear to be only A, difference, B. Yet these constituents, thus placed side by side, do not reconstitute the proposition. The difference which occurs in the proposition actually relates A and B, whereas the difference after analysis is a notion which has no connection with A and B. (King 2001, p. 18, Russell 1903, p. 49.) The passage may sound a bit Hegelian, but the point seems to be this. Within a proposition the difference-relation glues A and B together, while without the proposition it doesn’t. Therefore, the proposition itself contributes towards cohesion, and is more than a sequence of A, difference, B. A functional theorist, unlike Russell, can point to the abstract procedure of applying a function to an argument as the logical glue which keeps, for instance, Lulu and a property together as a self-contained whole. The procedure is a sort of scaffolding, framework or engine which is indifferent as to what particular function and what particular argument are involved, since the procedure would in any event remain the procedure of functional application. The procedure would also be indifferent as to what value was taken (truth or falsity, an empty set, or whatnot), or whether a value was returned at all. A procedure displaying, whilst being distinct from, functional operations allows replacing constituents, leaving gaps or diminishing/increasing the number of constituents.7 Not so with set-theoretical objects. Remove Earth from {Earth,…}, and a numerically distinct collection emerges. Remove Lulu from , and a numerically distinct tuple emerges. What emerges aren’t the same complexes, only with a gap in them, but different, self-contained, gapless aggregates. 8 Remove Vulcan from or Ossian from , as in David Braun (1993), and the result is not, contra Braun, a gappy, two-place structure, but one gapless, onemembered sequence. Tuples can’t have gaps in them. Analysing “Vulcan is a planet” as (where {} is meant to stand, not for the empty set, but for the subject position within the tuple) faces at least two other problems. First, sentential and propositional structures are supposed to match each other (op. cit., p. 461), but “Vulcan is a planet” doesn’t have a hole in it. Second, such propositions are supposed to be what sentences say and believers believe (op. cit., p. 462), but < , Being-a-planet>, or , surely isn’t what any sentence says or anybody believes. Also, Braun acknowledges that “Vulcan is a planet” and “Ossian is a planet” express the same gappy proposition, but adds that, of course, a competent speaker might accept one sentence and reject the other (op. cit., p. 464). Of course, but ‘gappy tuples’ can’t explain that competence. It might be countered that tuples aren’t to be identified with structured propositions, but merely represent them. But, one’s tempted to say, why not give us the real thing straight away? Besides, what represents structured propositions shouldn’t be too remote from the real thing; such propositions may well be gappy, whereas tuples can’t be. The reason is plain: tuples are, by their very nature, saturated (complete) objects.

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The idea of slotting empirical objects like Lulu into tuples faces additional problems. Presumably, for instance, “Lulu is as dazzling as Lulu” should go into, , but whereas the tuple machinery permits two occurrences of Lulu, presupposes the absurdity that Lulu, flesh and blood, be able to occur twice over, once next to herself, in this ‘structure’. Remember that the direct reference theorist cannot, for philosophical reasons, have recourse to two occurrences of some individual concept of Lulu. On the other hand, if we take sort of the opposite situation, not with Lulus galore, but none at all, what becomes of any tuple at those indices where Lulu isn’t available? Almog, in his (1986), which comes with varying domains, doesn’t flinch. No such ‘singular proposition’ can be generated at those worlds (etc.) at which Lulu doesn’t exist. However, not only is it semantically odd and unwelcome that sentential meanings like may go in and out of existence relative to worlds, etc., but it’s also odd that mathematical objects like tuples should depend for their existence on the worldly antics of someone like Lulu. The problem is parallel to the predicament faced by theories with no fixed domain of what becomes of {Earth}, say, at those worlds or times with no Earth.9 Finally, on a similar note, it’s often observed that what ties Lulu and Beingsuspicious together, in some of Russell’s theories, is the fact that Lulu is suspicious. But now, what happens to that ‘proposition’ in those cases where Lulu isn’t suspicious does it perhaps disintegrate? One morale to be extracted is probably that sentential meanings shouldn’t be ontologically dependent on empirical objects. (Forbes, in (1989, pp. 135-7), voices objections similar to some of those stated in this section.) IV A set is nothing but a partition of a pool of objects into those that are in and those that are out, and is therefore a simple (non-composite) ideal object. An ordered n-tuple, unlike an unordered set, isn’t just a pool of n objects having been hurled together, but imposes an order on them. This may give rise to the lax interpretation that a set, perhaps particularly a tuple, is constituted by, or consists of, its elements and that its elements may count as constituents, or parts. But if we take it absolutely seriously that tuples are just sets, stripping them of benign philosophical readings, tuples, too, are mere partitions of a pool of objects. Don’t forget that =df {{a}, {a, b}}, and so = {{Lulu}, {Lulu, Being-suspicious}}. It thus becomes possible to put forward a more radical objection to the tuple theory. Tichý observes that, […] this is indeed how structures are treated in mathematical logic and its applications: a structure is explained as an ordered couple whose first component is a domain of objects and the second component a relation, or set of relations, over the domain. Such an ordered couple, however, does not even begin to be a complex. The second component, as any other set-theoretical object, is a simple entity. And the first component is simply expendable. I have already noted that a set is a dichotomy of a universe of discourse. Similarly, a relation is a dichotomy of the Cartesian square of a universe of discourse. Hence the extension of the universe of discourse is

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part and parcel of what makes a relation the relation it is. Thus the first component of the domain/relation couple can be read off the second component and considerations of elegance would seem to dictate that it be dropped. (Tichý 1995, pp. 176-7.) That is, not only is ungainly, but the one-place relation Being-suspicious already says all there’s to say about Lulu’s and everyone else’s being suspicious. If it’s true that ∈R, then it’s mathematically true and can be read off R in and by itself, because will be a member of R and sets are individuated in terms of their members. (2), and (2’), reduces to, (2’’)

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Notice that this applies even in the presence of modal and temporal parameters. For an intension such as Being-suspicious (a function from pairs of worlds and times into sets of individuals) is just the set of all the sets of all the suspicious individuals at all worlds and times. Complexes behave altogether differently. E.g., a complex is something that lends itself to one unique, ultimate analysis. Both King and Tichý demand that the constituents of a complex should be retrievable from rather than ‘lost’ in it. Retrievability would, among other, allow a piecemeal approach to a complex, focusing on one constituent at a time, thus making it possible to look upon complexes as manystep procedures. Tichý’s objection to tuples as structures is a damning one, for it cripples the attempt to plant, e.g., individuals within such ‘structured’ propositions, and is also detrimental to, for instance, Cresswell’s and the direct reference theorists’ project of mimicking sentential structure. In sum, what passes for structure in today’s logic and mathematics just doesn’t make it as genuine structure in semantics.10 Where does this leave the tuple theory? Tuples may or (more likely) may not qualify as propositions, but structured they aren’t. Interestingly, in 1986 Almog said, I don’t actually think propositions are sequences, in the set theoretic sense. But, lacking a full understanding of the matter at present, I use the n-tuple notation as a temporary measure. (1986, p. 233.) There are basically two courses possible at this point, I suppose. One is simply to eschew structured propositions, or propositions altogether, as Russell eventually did. The other is to work out a regular theory of structured propositions compatible with the demands that direct reference theorists and others make of the concept. Either way the ‘temporary measure’ calls for cancellation.11

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References Almog, Joseph, 1986, “Naming Without Necessity”, Journal of Philosophy, Vol. 83, pp. 210-42. Almog, J., Wettsten, H. and Perry, J. (eds.), 1989, Themes From Kaplan, Oxford U.P., Oxford. Bealer, George, 1998, “Propositions”, Mind, Vol. 107, pp. 1-32. Bolzano, Bernard, 1837, Wissenschaftslehre, Jan Berg (ed.), Friedrich Frommann Verlag, Stuttgart - Bad Cannstatt (1987). Braun, David, 1993, “Empty Names”, Noûs, Vol. 27, No. 4, pp. 449-69. Cresswell, Max J. , 1985, Structured Meanings, MIT Press, Cambridge, MA. Forbes, Graeme, 1989, Languages of Possibility, Aristotelian Society Series, Vol. 9, Oxford: Basil Blackwell. Griffin, Nicholas, 1993, “Terms, Relations, Complexes”, in Irvine and Wedeking (eds.), pp. 159-192. Irvine, A.D., Wedeking, G.A. (eds.), 1993, Russell and Analytic Philosophy, University of Toronto Press, Toronto. Jacquette, Dale, 1992/93, “Wittgenstein’s Critique of Propositional Attitude and Russell’s Theory of Judgment”, Brentano Studien, Vol. 4, pp. 193-220. Jespersen, Bjørn, 2000, “Singular Propositions in Two-Stage Theory”, in O. Majer (ed.), Topics in Conceptual Analysis and Modelling, Filosofia, Czech Academy of Sciences, Prague, pp. 196-218. Kaplan, David, 1990, “Dthat”, in P. Yourgrau (ed.), pp. 11-33. Originally appeared in P. Cole (ed.), Syntax and Semantics, Academic Press, New York (1978).  1989, Demonstratives, Draft #2, in Almog et al. (eds.), pp. 481-563. King, Jeffrey C., 2001, “Structured Propositions”, Stanford Encyclopaedia of Philosophy, http://plato.stanford.edu/entries/propositions-structured/. Ludlow, Peter (ed.), 1997, Readings in the Philosophy of Language, MIT Press, Cambridge, pp. 9221-62. Materna, Pavel, 1998, Concept and Object, Acta Philosophica Fennica, Vol. 63, Societas Philosophica Fennica, Helsinki. DePauli-Schimanovich, W. et al. (eds.), 1995, The Foundational Debate, Kluwer, Dordrecht. Russell, Bertrand, 1903, Principles of Mathematics, New York, Norton. Soames, Scott, 1997, “Direct Reference, Propositional Attitudes, and Semantic Content”, in Ludlow (ed.), pp. 921-62. Originally appeared in Philosophical Topics, Vol. 15 (1987). Tichý, Pavel, 1986, “Constructions”, Philosophy of Science, Vol. 53, pp. 514-34.  1988, The Logical Foundations of Frege’s Logic, de Gruyter, Berlin, New York.  1995, “Constructions as the Subject-Matter of Mathematics”, in W. DePauliSchimanovich et al. (eds.), pp. 175-85. Yourgrau, Palle (ed.), 1990, Demonstratives: Oxford Readings in Philosophy, Oxford University Press, New York.

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See Tichý, 1988, §7ff. King 2001, p. 10. Bolzano, 1837, §56. Bolzano, idem, first argues that many ‘ideas in themselves’ (Vorstellungen an sich) are composed of parts (aus Theilen zusammengesetzt) and that the ‘sum’ (Summe) of the parts of an idea in itself forms the latter’s ‘content’ (Inhalt). He then adds, “Da unter diesem Inhalte nur die Summe der Bestandtheile, aus denen die Vorstellung [an sich] bestehet, nicht aber die Art, wie diese Theile untereinander verbunden sind, verstanden wird: so wird durch die bloße Angabe ihres Inhaltes eine Vorstellung [an sich] noch nicht ganz bestimmt, sondern es können aus einerlei gegebenem Inhalte zuweilen zwei und mehr verschiedene Vorstellungen hervorgehen.” (I owe the reference to Pavel Materna.) 3 What’s certain is that the relation isn’t merely set membership, for then the truth that Lulu is suspicious comes out a mathematical truth (viz. that Lulu is a member of a set which has Lulu as a member) and the falsity that Lulu is not suspicious, a mathematical falsity (viz. that Lulu isn’t a member of a set of which Lulu is a member). 4 Applying negation to classical propositions would be appropriate only in the case where there were but two propositions, the True and the False, but Soames entertains, to my knowledge, no such ‘extensional propositions’. 5 One might run the regress argument here. Since tuples aren’t themselves structures, they require something external to them that interprets them as structures, e.g., as procedures. But those external props will, for their part, similarly demand interpretation - and so forth. The regress is to do with the fact that there’s no sense in which tuples can be said to induce, or ‘generate’, functions. As for functional construction in TIL, see Tichý 1986, 1988. 6 At least the tuple theorist must be careful not to move the factors freely around within the same theory, for when = , then a = b. 7 Tichý’s theory of constructions is a large-scale project devoted to complex logical objects (‘framework’, etc., in the main text), which evidently steps beyond set theory and harks back to turn-of-the-century Gegenstandstheorie (see his 1988). 8 See Tichý, 1995, p. 176. 9 My (2000) elaborates on the point of propositional ‘generation gaps’. See Tichý (1986) for gappy complexes. 10 It’s interesting to note that, although the tuple propositions are meant to be ‘richer’ than the possible-world propositions, Bealer ranks the tuple theory among the reductionistic propositional theories (see Bealer 1998, §2). 11 This work was supported by Grant No. 55-00-0711, Danish Research Council for the Humanities. I’m indebted to the following elements, in no particular order, for making valuable comments to this note: {Arianna Betti, Marie Duzí, Pavel Materna, Göran Sundholm}. 1 2

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