The Hierarchy of Fregean Senses∗ Ori Simchen Abstract The question whether Frege's theory of indirect reference enforces an infinite hierarchy of senses has been hotly debated in the secondary literature. Perhaps the most influential treatment of the issue is that of Burge (1979), who offers an argument for the hierarchy from rather minimal Fregean assumptions. I argue that this argument, endorsed by many, does not itself enforce an infinite hierarchy of senses. I conclude that whether or not the theory of indirect reference can avail itself of only finitely many senses is pending further theoretical development. Consider the occurrence of 'Opus 132 is a masterpiece' in (1) Bela believes Opus 132 is a masterpiece and compare it with its occurrence in (2) Igor believes Bela believes Opus 132 is a masterpiece. Fregean doctrine tells us that in (1), 'Opus 132 is a masterpiece' refers to the ordinary sense of 'Opus 132 is a masterpiece', a mode of presentation of the truth-value of 'Opus 132 is a masterpiece' as it occurs unembedded, the thought that Opus 132 is a masterpiece. And the doctrine is often taken to suggest that in (2), 'Opus 132 is a masterpiece' refers to the sense of 'Opus 132 is a masterpiece' in (1), a mode of presentation of a mode of presentation of the truth-value of 'Opus 132 is a masterpiece' as it occurs unembedded, a mode of presentation of the thought that Opus 132 is a masterpiece. Question: Might the referent of 'Opus 132 is a masterpiece' in (2) be the same as the referent of 'Opus 132 is a masterpiece' in (1), namely, the thought that Opus 132 is a masterpiece? There is an influential argument due to Burge (1979) that is meant to show that it can't be. For any expression α, let 'sαs' refer to the sense of α and assume for ∗Published in Thought 7(2018): 255-261. 1 reductio that ssαss=sαs. (We treat single quotes as corners and assume the sense-of relation to be a one-many relation to expressions.) Call this reductio assumption SC for "Sense Collapse". Also assume a Sense Functionality principle SF that for any expressions α, β1, ..., βn of suitable type, sα(β1, ..., βn) s=sαs(sβ1 s,...sβn s); assume a Sense Identity principle SI that for any expressions α, β, if α is synonymous with β, then sαs=sβs; and assume the extensionality principle that for any expressions α, β, if α=β, then ...α...↔ ...β.... Now, the Fregean analysis of (1) is: (3) Believes(Bela, sOpus 132 is a masterpieces). And (2) is analyzed as (4) Believes(Igor, sBela believes Opus 132 is a masterpieces), from which by the synonymy of (1) and (3), SI, and extensionality we get (5) Believes(Igor, sBelieves(Bela, sOpus 132 is a masterpieces)s), from which by SF and extensionality we get (6) Believes(Igor, sBelievess(sBelas, ssOpus 132 is a masterpiecess)), from which by SC and extensionality we get (7) Believes(Igor, sBelievess(sBelas, sOpus 132 is a masterpieces)). But assuming Bela isn't believed to believe everything with the truth-value of Opus 132 being a masterpiece, if (5) is correct, then (7) isn't. For call what sBelievess refers to 'Believes' (a dyadic first-level concept), call what sBelas refers to 'Bela' (an individual), and call what sOpus 132 is a masterpieces refers to 'truth-value' (a truth-value). Let 'Believes' express sBelievess, 'Bela' express sBelas, and 'truthvalue' express sOpus 132 is a masterpieces. Then, (8) Believes(Bela, truth-value) expresses sBelievess(sBelas, sOpus 132 is a masterpieces), the second relatum for Igor's belief according to (7), purportedly a thought about what Bela believes. But then, for any δ for which 'δ=truth-value' is true we get (9) Believes(Bela, δ) by extensionality. Such a consequence does not follow from (3), which expresses sBelieves(Bela, sOpus 132 is a masterpieces)s, the second relatum of Igor's belief according to (5). So assuming the correctness of (5), SC should be given up. SC enables the further derivation of (7), which is incompatible with Igor not believing 2 Bela to believe everything with the truth-value of Opus 132 being a masterpiece. SC says that for any α, ssαss=sαs. The foregoing shows that for some α (i.e. 'Opus 132 is a masterpiece'), ssαss 6=sαs. And so, SC is false. Let us now assume that Burge's argument supports the generalization that for any α, ssαss 6=sαs. Question: Does this argument force upon us a hierarchy of senses? Burge (1979) concludes his presentation of the argument as follows (passage adjusted to the terminology and numbering of the reconstruction above and supplemented accordingly): The argument shows that on these assumptions 'Opus 132 is a masterpiece' in [(2)] cannot be represented by a term ['sOpus 132 is a masterpieces'] denoting the [thought] that Opus 132 is a masterpiece. It is prima facie plausible to assume with Frege that the expression as it occurs in [(2)] should be represented by a term ['ssOpus 132 is a masterpiecess'] denoting the sense of the expression ['Opus 132 is a masterpiece'] that represents ['Opus 132 is a masterpiece'] as it occurs in unembedded belief contexts [e.g. (1)]. Given this assumption, the argument can be replicated to show that the sentential expression ['Opus 132 is a masterpiece'] as it occurs in doubly embedded oblique contexts must be represented by yet another term [i.e. a term other than 'ssOpus 132 is a masterpiecess'] – and so on. (272) It is often assumed that Burge's argument establishes a hierarchy of senses. Indeed, Burge writes: "In summary, the argument for a hierarchy... seems very powerful" (274). The assumption makes an appearance in a recent discussion by Salmon (2005: 1100 n. 31). And in an even more recent discussion Kripke writes: "I agree with Burge... that the hierarchy is an actual consequence of Frege's theory" (2008: 184 n. 9). This, I will now argue, is a mistake. Call an expression without s-quote marks an expression of level 0. Independently of Burge's argument it seems plausible on Fregean grounds to suppose that for any expression α of level 0, α 6=sαs. (Think here of the example of Mont Blanc in the famous exchange between Frege and Russell.1) Similar considerations could extend 1Frege (1980) writes: Mont Blanc with its snowfields is not itself a component part of the thought that Mont Blanc is more than 4,000 metres high... The sense of the word 'moon' is a component part of the thought that the moon is smaller than the earth. The moon itself (i.e. the reference of the word 'moon') is not part of the sense of the word 'moon'; for then it would also be a component part of that thought. (163) 3 to α 6=s(×n)αs(×n) for any n ≥ 1, where 's(×n)' designates n occurrences of 's'. And we assume that Burge's argument can be given for any n ≥ 1 to the effect that s(×n)αs(×n) 6=s(×n+ 1)αs(×n+ 1). But such an argument would still not launch a hierarchy of senses per α. For a hierarchy per α we would need to show that for any n ≥ 1, the following conjunction is true: (10) ∧ 1≤i≤n s(×i)αs(×i) 6=s(×n+ 1)αs(×n+ 1). Now consider the following instance of (10) for n = 2: (11) sαs 6=sssαsss ∧ ssαss 6=sssαsss. Burge's argument establishes, we suppose: (12) sαs 6=ssαss ∧ ssαss 6=sssαsss. Mont Blanc-type considerations establish: (13) α 6=sαs ∧α 6=ssαss ∧α 6=sssαsss. But we can see that the theory {(12), (13), ¬(11)} is consistent by constructing a simple model m = 〈{o, o′, o′′}, σ〉, where {o, o′, o′′} is the domain and σ is the interpretation function: σ(α) = o σ(sαs) = o′ σ(ssαss) = o′′ σ(sssαsss) = o′. It is easily verified that m |={(12), (13), ¬(11)}. So now consider the following model m∗ = 〈{o, o′, o′′}, σ∗〉 interpreting every expression of the form s...sαs...s, where n ≥ 1: σ∗(α) = o σ∗(s(×2n− 1)αs(×2n− 1)) = o′ σ∗(s(×2n)αs(×2n)) = o′′. Let the Burge set B be {s(×n)αs(×n) 6=s(×n+ 1)αs(×n+ 1) | n ≥ 1} and let the Mont Blanc set M be {α 6=s(×n)αs(×n) | n ≥ 1}. Clearly for any n ≥ 2, m∗ |= ¬(10). And for any n ≥ 1, m∗ |= s(×n)αs(×n) 6=s(×n+ 1)αs(×n+ 1) and m∗ |= α 6= s(×n)αs(×n). So m∗ |= {B,M}. I conclude that Burge's argument does not enforce a hierarchy of senses per α after all. In light of the foregoing, consider the following triply embedded occurrence of 'Opus 132 is a masterpiece': 4 (14) Zoltan believes Igor believes Bela believes Opus 132 is a masterpiece. Fregean doctrine analyzes it as (15) Believes(Zoltan, sIgor believes Bela believes Opus 132 is a masterpieces), which by the synonymy of (2) and (5), SI, and extensionality gets us (16) Believes(Zoltan, sBelieves(Igor, sBelieves(Bela, sOpus 132 is a masterpieces)s)s), which by repeated applications of SF and extensionality gets us (17) Believes(Zoltan, sBelievess(sIgors, ssBelievesss(ssBelass, sssOpus 132 is a masterpiecesss))). Now, according to interpretation σ∗ above, σ∗(sssαsss) = σ∗(sαs). Let us consider such an interpretation for 'Opus 132 is a masterpiece' and assume, accordingly, that (18) sssOpus 132 is a masterpiecessssss = sOpus 132 is a masterpiecess. From (17), (18), and extensionality, we get (19) Believes(Zoltan, sBelievess(sIgors, ssBelievesss(ssBelass, sOpus 132 is a masterpieces))). Can we say here, as we did in the case of Burge's original argument, that assuming Bela isn't believed to believe everything with the truth-value of Opus 132 being a masterpiece, if (16) is correct, then (19) isn't? Call what ssBelievesss refers to 'Believes' (a mode of presentation of the relational concept Believes), call what ssBelass refers to 'Bela' (a mode of presentation of the individual Bela), and call what sOpus 132 is a masterpieces refers to 'truth-value' as before. Let 'Believes' express ssbelievesss, 'Bela' express ssBelass, and 'truth-value' express sOpus 132 is a masterpieces as before. Then (20) Believes(Bela, truth-value) expresses ssBelievesss(ssBelass, sOpus 132 is a masterpieces)), which is the second relatum for Igor's belief according to Zoltan's belief according to analysis (19). Now, it does follow from (20) by extensionality that (21) Believes(Bela, δ) for any δ that agrees in truth-value with 'Opus 132 is a masterpiece'. But notice that (20) does not purport to express any thought about what Bela believes. With (20) we are in pretheoretically unfamiliar territory. What (20) expresses is a complex consisting of a mode of presentation of a mode of presentation of a relational concept, a mode of presentation of a mode of presentation of an individual, and the thought that Opus 132 is a masterpiece. To simply insist that while (21) follows from (20), 5 it does not follow from whatever would express ssBelievesss(ssBelass, sssOpus 132 is a masterpiecesss), the second relatum for Igor's belief according to Zoltan's belief according to (17), and so to conclude that the latter complex is distinct from the second relatum of Igor's belief according to Zoltan's belief according to (19), is to beg the issue at hand. There might well be other theoretical reasons to resist (18), and more generally reasons to resist the collapse of third-level senses to first-level senses, fourth-level senses to second-level senses, fifth-level senses to first-level senses, sixth-level senses to second-level senses, and so on. For example, under the present scheme the usual Fregean assumption whereby for each n > 1, s(×n)αs(×n) refers to s(×n− 1)αs(×n− 1), entails that for any even number k, s(×k)αs(×k) refers to and is referred by s(×k − 1)αs(×k − 1), whereas for any odd number m, s(×m)αs(×m) will refer to both s(×m+ 1)αs(×m+ 1) and the direct referent of α (however this is to be modeled, ultimately). The present point isn't that we in fact need only two levels of sense. That is a question pending further theoretical development. The present point is that the familiar argument for the hierarchy fails to establish an infinite hierarchy of senses. In an important paper on Frege's theory of indirect reference, Parsons (1981) argues that a hierarchy of senses is not an inevitable outcome of basic Fregean doctrine. He considers a Carnapian version that resorts to a single level of sense by revisiting the commonly held assumption that the referent of a complex expression is invariably a function of the referents of its parts. On this version, where X( ) is an indirect sentential context, the referent of X(Y ) is the application of the referent of X to sY s rather than the application of the referent of X to the referent of Y .2 (The Sense Functionality principle SF remains unaltered.) This allows us to maintain that for any α and any n ≥ 1, sαs=s(×n)αs(×n). And it blocks the transition from (8) to (9) in the above reconstruction of Burge's argument. In response, however, it might be claimed that extensionality is too fundamental to the Fregean outlook to be amenable to revision as required by such an approach. The argument presented here, by contrast, does not depend on tinkering with extensionality in any way. In a more recent paper on the hierarchy, Parsons (2009) considers a semantic framework that avoids a hierarchy of senses by introducing a function symbol '↑' into the metalanguage (and a new governing logical principle) that stands for the function that for any sentence α takes sαs to sthat αs. But such enrichment of the metalanguage, with the presupposed object-level/meta-level distinction, is difficult to motivate within a 2For the original version of the view in terms of the Carnapian notions of extension and intension, see Carnap (1947): §§13-15. It is interesting to note in this context that Carnap also claims that Frege's theory of indirect reference leads to a hierarchy of senses but without providing an explicit argument for the claim. See Carnap (1947): §30. 6 more orthodox Fregean outlook. Whether Frege's theory of indirect reference is committed to an infinite hierarchy of senses is a question that is usually handled under the tacit assumption that the theory's pronouncements are to be taken in a realistic spirit. This is certainly Frege's original approach to the topic. What the assumption amounts to, at the very least, is the idea that the theory of indirect reference is to reveal the very meaning of attitude reports (as they are "in the wild", so to speak). By extension, the theory is often also presumed to reveal the nature of the reported cognitive facts themselves, the facts of belief for example. But such large methodological assumptions are not inevitable. When thinking of the theory in a more instrumentalist spirit we can ask what explanatory work the theory can be expected to achieve in the first place and how much of the pretheoretical situation surrounding the meaning of attitude reports should be captured. Under such instrumentalism, it is at least not obvious that finitely many levels of sense could not suffice for the theory's explanatory success. Perhaps, for example, we need as many levels of sense per α as there are humanly parsable embeddings of α and no more. The foregoing discussion aimed to show that there is no principled barrier to such finitism.3,4 References Burge, T. (1979) 'Frege and the Hierarchy', Synthese 40: 265-81. Carnap, R. (1947) Meaning and Necessity: A Study in Semantics and Modal Logic (Chicago: University of Chicago Press). Frege, G. (1980) Philosophical and Mathematical Correspondence, G. Gabriel, H. Hermes, F. Kambartel, C. Thiel and A. Veraart (eds.), abridged by B. McGuinness and translated by H. Kaal (Chicago: University of Chicago Press). Kripke, S. (2008) 'Frege's Theory of Sense and Reference: Some Exegetical Notes', Theoria 74: 181-218. Parsons, T. (1981) 'Frege's Hierarchies of Indirect Senses and the Paradox of Analysis', Midwest Studies in Philosophy 6: 37-57. Parsons, T. (2009) 'Higher-Order Senses', in Almog, J. and Leonardi, P. (eds.), The Philosophy of David Kaplan (New York: Oxford University Press): 45-59. 3For further discussion of such choice points between realist and instrumentalist attitude towards theoretical representations in various areas of philosophical pursuit, see Simchen (2019) and (unpublished). 4Thanks to Gurpreet Rattan for discussion of an earlier version of this paper and to two anonymous referees for their useful suggestions. 7 Salmon, N. (2005) 'On Designating', Mind 114: 1069-1133. Simchen, O. (2019) 'Instrumentalism About Structured Propositions', in Tillman, C. (ed.), The Routledge Handbook of Propositions (New York: Routledge) (available at: http://faculty.arts.ubc.ca/osimchen/drafts/IASP.pdf). Simchen, O. (unpublished) 'Realism and Instrumentalism in Philosophical Explanation' (available at: http://faculty.arts.ubc.ca/osimchen/drafts/RIPE.pdf).