This is the author version of the following article: Podlaskowski, A. (2015). "Giving Up On 'The Rest Of The Language'." Acta Analytica, September, 30 (3), pp. 293–304. The final publication is available at www.springerlink.com. More specifically: http://link.springer.com/article/10.1007/s12136-014-0250-8 Giving up on "the rest of the language" Adam C. Podlaskowski apodlaskowski@fairmontstate.edu Abstract In this essay, the tension that Benacerraf identifies for theories of mathematical truth is used as the vehicle for arguing against a particular desideratum for semantic theories. More specifically, I place in question the desideratum that a semantic theory, provided for some area of discourse, should run in parallel with the semantic theory holding for the rest of the language. The importance of this desideratum is also made clear by means of tracing out the subtle (though significant) implications of its rejection. Keywords Meaning, Semantics, Benacerraf, Truth Conditions, Pluralism Introduction It is often held that natural language deserves a homogeneous semantic theory, so that some central concept (such as truth-condition, inferential role, or expressed non-cognitive attitude) is taken to be key to directly or indirectly explaining the nature of meaning as it applies to the entire language.1 One could go so far as to say that it is a desideratum on a semantic theory for some area of discourse that it run in parallel with the semantic theory holding for the rest of the language. Paul Benacerraf, for instance, invokes just such a desideratum (the first 'concern') in his famous discussion of mathematical truth: 1For examples of truth-conditional semantics, see Davidson (1967); Lewis (1970); Millikan (1987), and Fodor (1990). For examples of inferential role semantics, see Brandom (1998) and Tennant (1997). And for examples of expressivist semantics, see Price (2011) and Schroeder (2008). 1 It is my contention that two quite distinct kinds of concerns have separately motivated accounts of the nature of mathematical truth: (1) the concern for having a homogeneous semantical theory in which semantics for the propositions of mathematics parallel the semantics for the rest of the language, and (2) the concern that the account of mathematical truth mesh with a reasonable epistemology. It will be my general thesis that almost all accounts of the concept of mathematical truth can be identified with serving one or another of these masters at the expense of the other. (Benacerraf 1973, p. 661) To be fair, Benacerraf admits that a semantic theory for the rest of the language has yet to be offered. But this does not appear to detract from his case: I am indulging here in the fiction that we have semantics for "the rest of language," or, more precisely, that the proponents of the views that take their impetus from this concern often think of themselves as having such semantics, at least for philosophically important segments of the language. (Benacerraf 1973, p. 661, note 1) In this essay, I argue that it is also a piece of fiction to suggest that there is even a "rest of the language" to be contrasted with controversial areas of discourse such as mathematics. That is, I place in question the desideratum on a semantic theory, for some area of discourse, that it run in parallel with the semantic theory holding for the rest of the language. So, while I do not target here the prospects for a specific homogeneous semantic theory, I do aim to undermine the suggestion that any such approach is the default kind. Benacerraf's problem and those like it My case begins by reminding the reader that the sort of tension identified by Benacerraf arises for other important areas of discourse as well. As a preliminary step, then, consider in more detail the principal vehicle of this tension. Recall the quotation from Benacerraf, above. Much as we regard a sentence like "The cat is sitting on the mat" as depicting a particular state-of affairs by way of terms referring to a particular cat and mat and the predicate 2 that they fall under, we should expect a similar story to hold for sentences such as "2+2=4". This is the first "master" mentioned by Benacerraf. Or, as Benacerraf expands the matter: Perhaps the applicability of this requirement to the present case amounts only to a plea that the semantical apparatus of mathematics be seen as part and parcel of that of the natural language in which it is done, and thus that whatever semantical account we are inclined to give of names or, more generally, of singular terms, predicates, and quantifiers in the mother tongue include those parts of the mother tongue which we classify as mathematese. (Benacerraf 1973, p. 666) Whatever semantic account is accepted must be compatible with a plausible explanation of how mathematical knowledge is possible-this is the second "master" mentioned by Benacerraf. However, devising an epistemically plausible truth-conditional semantics for mathematical discourse proves problematic owing to the fact that (unlike, say, cats sitting on mats) we fail to bear causal relationships to the (prima facie) truth-makers for mathematical statements: namely, numbers qua abstract objects. If knowledge requires bearing at least some sort of causal relationship to the objects of knowledge, it becomes entirely unclear how we can have knowledge of numbers to which we bear no causal relationships whatsoever.2 So, if numbers are the sorts of things to which we refer, it remains a mystery how we can have any knowledge of them. But if we ignore this representational feature when providing a plausible epistemology for mathematics, it is unclear that we have the materials in place to say that such sentences are in the business of expressing truths in the same way as ordinary claims. Hence, any attempt to provide an account of mathematical truth (and its corresponding semantic basis) cannot obviously meet both demands made on it. This is what shall be referred to as Benacerraf 's problem. Much as for mathematical discourse, there is a deep tension facing modal discourse, whose discussants include Richards (1975), Lewis (1986), and Hawthorne (1996). The first source of the tension is the desideratum that 2While Benacerraf's criticism of Platonistic accounts of mathematics relies on a causal account of knowledge, the point can be adapted to others as well-especially any naturalistic account. For example, Field (1989) has argued that the true thrust of Benacerraf's criticism lies in our inability to explain how we are reliable mathematicians. See Melanson (2011), for instance, for further discussion of Field's emphasis on reliability. 3 it would be preferable for our talk about modality or possibility to be given a semantic theory in parallel with that provided for the rest of the language (e.g., a truth-conditional approach). The second source of the tension for modal discourse springs from the desideratum that, for any semantic account of modal talk, it should allow for a plausible view of modal knowledge. To appreciate the difficulty of meeting both demands, consider the problem as posed for Lewis's (1986) theory of modal realism.3 Given the broad appeal of possible world semantics (which is a truth-conditional account) for cashing out statements of modal truths, and given that any account of truth requires a corresponding account of truth-makers, Lewis embraces the commitment to the existence of other worlds, maintaining that they serve as the truth-makers for claims about contingency or necessity. Moreover, possible worlds, according to Lewis, are best understood as spatio-temporally and causally isolated from one another. The problem is raised by the fact that, while knowledge appears to require some causal relationship between an agent and the object of knowledge, possible worlds are causally isolated from one another. As such, we cannot, as inhabitants of one world (what we call the "actual one"), bear any causal relationships whatsoever to other possible worlds-including the sorts required for possessing knowledge.4 In short, much as for Benacerraf's problem for mathematics, we are left wondering how knowledge of possibility is possible. Just as for our talk of mathematics and modality, there is a similar tension facing ethical discourse. Once again, the first source of the tension is the desideratum that we should provide a semantic account of ethical discourse in parallel with that given for the rest of the language. Providing such a semantic theory for ethical discourse appears to involve reporting on moral truths about what is right or wrong, good or bad, or virtuous or vicious. Furthermore, whatever sorts of facts serve as the semantic values for ethical statements, it appears that they must have a particular sort of normative 3Hawthorne points out that the same concerns about the epistemology of modality arise whether possible worlds are construed as concrete or abstract (1996, pp. 184-6). 4It does not help matters to posit some sort of "special faculty" (e.g., modal intuition) that puts us in the right causal relationship to possible worlds. The positing of such faculties proves problematic in several respects, including that it is highly implausible that we would have acquired such special faculties by way of evolutionary processes (Nozick 2003, pp. 122-23). To his credit, though, Lewis makes a case for denying that causal acquaintance is always required for knowledge (1986, pp. 108-115). For a critical discussion of the available epistemic options (including a critique of Lewis's own reply), see Hawthorne's handling of the topic (1996, pp. 188-201). 4 import: as Mackie (1977) puts it, they must possess a sort of authoritative prescriptivity. Such facts must intrinsically motivate rational agents to pursue or not pursue particular aims, so that reasons for action are somehow built into them. And it is precisely this normative feature (if it is represented by talk about ethics) which problematizes accounting for our knowledge of ethical matters-a point serving as the second source of the tension. As Harman puts the matter, observation does not obviously play the same role in confirming or disconfirming ethical principles that it does for scientific principles (Harman 1977, pp. 3-10). Mackie makes a stronger version of the point as part of his case against the existence of objective moral values. He cautions that, "none of our ordinary accounts of sensory perception or introspection or the framing and confirming of explanatory hypotheses or inference or logical construction or conceptual analysis, or any combination of these," (Mackie 1977, pp. 38-39) could yield knowledge of such values. So, while Mackie did not express his point as a tension like Benacerraf's problem (so much as supporting an error-theoretic approach to ethics), the materials are clearly in place for doing so. Taken on their own, the three highlighted areas of discourse (just discussed) make up a significant stretch of the language. However, it is likely that additional areas of discourse-many, perhaps-also face Benacerraf-style concerns. To cite one class of examples, other areas of discourse taking abstract objects as their subject matter (e.g., set theory) face the same sort of concern as Benacerraf discussed regarding our talk of numbers. To cite another class, there are additional (and quite pervasive) modal locutions in our language for expressing subjunctives and dispositional properties; and it is perfectly plausible that the kind of problem facing possible world semantics (within Lewis's modal realist system) could afflict these other locutions as well. To cite a final class of examples, other (also quite pervasive) areas of discourse expressing agent-motivating norms are likely to face the same sort of tension facing our talk of ethical matters. So, while at least three major areas of discourse face a distinctive sort of tension (each involving a contrast with the rest of the language), there is also a potential glut of additional tensions. In the next section, an implication is drawn from facing this host of problems simultaneously, rather than in their usual fashion-considered each, in isolation, in contrast to the rest of the language. 5 The argument from compounding tensions Before proceeding, notice that Benacerraf's problem is often framed by commentators as a dilemma, where each of the "masters" to which Benacerraf alludes constitutes a separate horn (e.g., Hale and Wright 2002). But by avoiding framing the matter as a dilemma here, it is emphasized that each "master" serves instead as desiderata to be satisfied by any area of discourse. And by focusing on only a single tension (as Benacerraf does for mathematics), it is natural to contrast that particular area of discourse with the rest of the language. Benacerraf's problem, for instance, shows that there is something peculiar about the language of mathematics-unlike the rest of the language for which both desiderata (or "masters") appear to be satisfied uncontroversially. But, again, several additional parts of the language-and perhaps many, many more-are subject to problems like the one Benacerraf identifies for mathematics-they are equally peculiar. Each problem-on its own-appears plausible, due in part to the strong appeal of providing an account of meaning that parallels that provided for the rest of the language. But while striking the contrast between one area of discourse and the rest of the language is plausible, repeatedly striking the same sort of contrast puts pressure on the idea of the rest of the language with which Benacerraf initially contrasts mathematics. In short, I do not aim to merely the remind the reader that problems like Benacerraf's arise for many areas of discourse; but rather to grasp the implication of facing many such problems simultaneously for the notion of the "rest of the language" featured in each such problem. To better see this, consider an analogy with trying to identify what rule is in force, if too many (would be) exceptions are cited. Recall the old adage "The exception proves the rule." The basic idea is that the stated exception to a rule (e.g., "Free parking on weekends") implies that a general rule (permitting of exceptions) is in place (e.g., "Parking is not free"). Notice, though, that with too many cited exceptions, there is little left to which a general rule can apply. For instance, if the exceptions cited are "Free parking on weekends", and "Free parking on Mondays", and "Free parking on Tuesdays and Thursdays", there are few days of the week left to which a general rule such as "Parking is not free" can apply, much less enough to deem it a general rule (that permits of exceptions). So, while exceptions can prove that a rule is in place, too many (would be) exceptions can place in doubt whether there is any sort of general rule in force in the first place. The situation is similar for maintaining that there is a notion of the rest of 6 the language to be contrasted with any particular Benacerraf-style problem. Each such problem presupposes that the rest of the language behaves in a particular way, such that it deserves the same sort of semantics. But now it looks like there are many parts of the language for which it is an open question how best to capture the relevant semantic properties, yielding the compounding result that there was never much to the notion of the rest of the language to begin with. So, again, while striking the contrast between one area of discourse and the rest of the language is plausible, repeatedly striking the same sort of contrast becomes increasing untenable. In light of so many problems like the one identified by Benacerraf, we should give-up the expectation that there is much to the notion of the rest of the language with which we began. The same conclusion can also be made evident using diagrams. For simplicity's sake, only three areas of discourse will be represented here. Consider first the following representation of the contrast between mathematical discourse ('ma') and the rest of the language ('r '): Figure 1: Contrasting ma with r r ma Now, consider the respective contrasts between ethical discourse ('et ') and the rest of the language, and modal discourse ('mo') and the rest of the language ('r '): et r mo r Figure 2: Contrasting et with r Figure 3: Contrasting mo with r Finally, notice the compounding effect of repeatedly striking the same sort of contrast with the "rest" of the language: 7 ma et mo r Figure 4: The compounding result for r, after contrasting it with ma, et, and mo Again, the three areas of discourse featured here make up a significant stretch of the language. Though there is no clear means of measuring how much of the language remains to be placed in contrast with each featured area of discourse, it nevertheless becomes less convincing that "the rest of the language" is something to be placed in strong contrast to each such discourse. Furthermore, this result is strengthened once we acknowledge the range of additional areas of discourse that might also face problems akin to the one identified by Benacerraf. Therefore, it is entirely unclear that there is any sense of "the rest of the language" to deserve such a semantic theory. As such, we should give up on the desideratum that we provide a semantic theory for any area of discourse in parallel with that holding for the rest of the language. In short, the fiction Benacerraf (1973, p. 661, note 1) helps himself to (that a semantic theory has already been given for the rest of the language) takes on the status of one fiction embedded within another: namely, the fiction that there is a "rest of the language" in the first place! Implications of compounding tensions If the case made thus far proves convincing, there is little left of the notion of the rest of the language. Where does this leave us? An immediate implication is that labels such as "Benacerraf's problem" and "Benacerraf-style problem" turn out to be misnomers. After all, the troublesome areas of discourse are no longer so troublesome, if we are freed from the desideratum to provide a homogeneous semantic theory. Of course, there remains a deep tension between a truth-conditional approach to many areas of discourse (on the one hand)-e.g., mathematics, modality, and ethics-and furnishing a plausible epistemology (on the other). But the initial motivation for providing a truth-conditional approach across the language is now lacking. Framing Benacerraf's concern about mathematics as a dilemma, an implication of the 8 case made thus far is that the dilemma is no longer convincing, since its first horn has been revealed to be little more than a nub. By itself, this is a significant result, given the importance of Benacerraf's problem for the philosophy of mathematics and the importance of similar problems for modal and ethical discourse (and perhaps many others). To illustrate the import of our conclusion, consider first Lewis's reaction to Benacerraf's problem (framed here as a dilemma and addressed with an eye towards its impact on modal realism): I think it is very plain which horn of Benacerraf's dilemma to prefer. To serve epistemology by giving mathematics some devious semantics would be to reform mathematics. Even if verbal agreement with mathematics as we know it could be secured-and that is doubtful-the plan would be to understand those words in a new and different way. (Lewis 1986, p. 109) The idea seems to be that we should not abandon a homogeneous semantic theory, for anything short of assimilating an area of discourse with the semantic account "obviously" holding for the rest of the language counts as some variety of reform effort. Lewis's misgivings about providing a "devious semantics" highlights the deep-seated appeal of providing a single, uniform semantic theory for the entire language. Indeed, those misgivings motivate, without much in the way of additional support, his drawing the conclusion that any attempt to deviate from a truth-conditional approach involves reforming what we mean by the targeted expressions. To be fair, some efforts to provide a non-truth-conditional semantics aim at reforming the relevant practice. However, we should avoid overgeneralizing this point.5 For example, inferentialists characteristically maintain that the notion of inferential role (i.e., the use of symbols in reasoning) is explanatorily basic to semantic accounts. Those following Dummett (1973, 1977) conclude that accepting inferentialism also requires reforming our logical and mathematical practices by replacing classical logic with intuitionistic logic (which differs from classical logic in that it lacks the law of excluded middle). Drawing this conclusion, though, is not inevitable. Brandom (1985, 2008), for example, is a prominent inferentialist who also accepts classical logic. It is not essential to one's providing an alternative semantic framework that one distorts how the subject matter is understood-only 5See also Burgess and Rosen's (1999) discussion of "hermeneutic" approaches in the philosophy of mathematics. 9 that one captures the semantic features of our talk about the subject matter by way of a different explanatory model. Just as expressivists about ethics, for instance, should be charitably understood as attempting to provide the semantic account that best underwrites how we actually make ethical judgments, we should expect the same to hold for many accounts of mathematical discourse. So, while there are noteworthy efforts to reform our practices (e.g., Dummettian inferentialists favoring intuitionistic logic), not every alternative semantic account is designed for such a task. A reply like Lewis's appears to beg the question against alternative semantic frameworks, by presuming that some variety of truth-conditional semantic theory must be on the right track for the rest of the language-and so, the target language is obviously in the business of representing states-of-affairs.6 But that is precisely what is at issue. Admittedly, many areas of discourse might appear to be in the business of representing matters. As such, any attempt to explain the meanings of the targeted expressions by non-truth-conditional theories might seem to ignore important semantic appearances (as in the semantic function that they appear to perform). An initial counter, inspired by (non-cognitivist) expressivists, is that semantic appearances can be deceiving. According to such advocates, while ethical judgments appear to be in the business of depicting ethical states-of-affairs, they actually perform an altogether different function. Now, whatever the virtues or vices of expressivism, the point about semantic appearances is well taken. Indeed, this point suggests a related one: while truth conditional approaches often respect some of the semantic appearances of the expressions of interest, they also ignore others-indeed, those which usually serve as the footholds for alternative semantic theories. For example, (non-cognitivist) expressivists about ethics often cite the motivational feature of ethical statements-that ethical judgments such as "Murder is wrong" motivate rational agents-as something for which truthconditional approaches cannot provide an adequate account since beliefs (as the mental states that possess truth-conditions) are, by themselves, motivationally inert. At least since Hume, it has been hard to ignore that ethical judgments perform a motivational role for us-something that many nontruth-conditional approaches capture but often seems lacking from compet6Cf. Frank Jackson's comment that "Although it is obvious that much of language is representational, it is occasionally denied... I surmise that it is through conflating the obviously correct view that much of language is representational with various controversial views." (Jackson 1997, p. 270). 10 ing truth-conditional accounts. But, of course, it is also difficult to forget the retort that ethical discourse nevertheless appears to be in the business of conveying ethical truths. What we can gather from this exchange is that it is difficult to save all of the semantic appearances: often, one can only save some semantic appearances at the expense of saving others-a point not lost on anyone wrestling with Benacerraf-style problems. This point helps to highlight another implication of the case made in the previous section: without the desideratum in hand that any area of discourse should be given a semantics in parallel for the rest of the language, we do not have the antecedent motivation for privileging some semantic appearances over others. That is, grounds are now lacking for the insistence that some sorts of semantic appearances (such as the appearance of possessing truth-conditions) must always be saved at the expense of others. This point promises to have a subtle, but significant, effect on how we theorize about the nature of meaning. To see this, notice first that it is often seen as a prima facie strike against a semantic theory (for an area of discourse) that it fails to accord with the dominant (truth-conditional) semantic tradition. This is certainly the case for expressivists about ethics (Blackburn 1993; Gibbard 2003) and modal discourse (Blackburn 1993).7 And it is often seen as a prima facie advantage of a non-truth conditional semantic theory that it can capture the appearance of the targeted sentences expressing truths. This is an advertised advantage, for instance, of fictionalist approaches to mathematics (Field 1989), modality (Rosen 1990), and ethics (Kalderon 2005). As Benacerraf puts the point: Any departure from a theory thus homogeneous would have to be strongly motivated to be worth considering. Such a departure, for example, might manifest itself in a theory that gave an account of the contribution of quantifiers in mathematical reasoning different from that in normal everyday reasoning about pencils, elephants, and vice-presidents. (Benacerraf 1973, p. 662-3) But without the desideratum (tied to semantic homogeneity) in hand, no additional advantage or disadvantage should be conferred on a semantic 7For example, expressivists famously face the Frege-Geach problem (Geach 1960, 1965). See also Boghossian's (1990, pp. 165-66) case against non-factualist accounts of content in order to better appreciate the central point of contrast played by robust truth-conditional approaches for appreciating non-factualist alternatives. 11 theory (for a specific area of discourse) simply because it accords with the dominant semantic tradition (taken to hold for much of the language). Of course, other reasons might be furnished for accepting a homogeneous semantic theory.8 But those reasons deserve independent consideration, and fall outside of the aim of this essay. So, while rejecting the desideratum of interest does not conclusively undermine the possibility of providing the same sort of semantic theory (truth-conditional or otherwise) for each area of discourse, one less reason is available in support of doing so. Indeed, it appears that providing any such account might be achievable only in an altogether more piecemeal fashion. Whether this opens up an altogether more pluralistic approach to theorizing about meaning depends on the details of the distinctive efforts of any such project.9 And whether any such approach can be devised that avoids its own distinctive problems remains to be seen.10 However, one road block to any such approach has been removed. 8One such reason-likely held by Benacerraf-is that only a truth-conditional semantic can be given that any part of the language. (Of course, that sort of reason invites replies from competing semantic traditions such as inferentialism and expressivism.) Another reason, also suggested by Benacerraf (1973, p. 662-3) pertains to quantifiers and the suggestion that a semantic theory deserves a univocal account of them. I do not dispute that here. Rather, I use Benacerraf's case (and those like it) to put pressure on a particular reason for semantic homogeneity; and that reason alone, if my case succeeds, is not sufficient to insist on a univocal semantic theory for quantifiers. 9One basic strategy for a more pluralistic approach to meaning is to identify cases to which the theory, initially taken to hold for the rest of the language, does not plausibly apply. For instance, Wittgenstein aimed to show the various roles language can perform, each of which break with a descriptive function. Prominent candidates include our talk of psychological states (1953, part ii, §x-xi), ethics (1965), knowledge (1969), mathematics (1978), and necessity (1978). Quite generally, the presence of intuitive counterexamples to (or scope problems for) a targeted homogeneous theory-truth-conditional or otherwise- undermines the theory's advertised range over the entirety of language. This sort of strategy resembles one used to support alethic pluralism. One such pluralist about truth, Lynch, also comes close to providing an argument from scope problems for the case of meaning when he extends his brand of alethic pluralism to semantic matters (2009, pp. 129-158). 10One such problem is inspired by the Frege-Geach problem facing non-cognitivist expressivists (e.g., Geach 1960, 1965) and the problem of mixed compounds and mixed atoms facing alethic pluralism (e.g., Tappolet 1997; Sher 2005), namely: "How can we provide a single semantic value for mixed expressions like 'It is contingently the case that adding together numbers is evil' if the meanings of its constituents are best explained by different, distinct semantic theories? But, again, this and related questions fall well outside of the scope of this essay, especially given that we have not argued for any such pluralistic theory here, but only against one reason for semantic homogeneity. 12 Conclusion In concluding this essay, it should be stressed that no attempt here has been made to show that no semantic theory can successfully apply to the entirety of the language. Rather, the aim here has been altogether more humble: I have argued that there is little to the notion of the rest of the language to be contrasted with various areas of discourse; and, as such, there is no default stance towards theorizing about the nature of meaning. That a theory accords with the one given for other areas of discourse-or perhaps many-does not lend weight for or against its own acceptance. 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