Philosophical Topics

INTRODUCTION

That logic has something to do with necessity is one of the longest-running ideas in the history of philosophy. According to Aristotle a “deduction (συλλογισμὸς) is a discourse in which, certain things having been supposed, something different from the things supposed results of necessity (ἐξ ἀνάγκης)” (1989, 24b18–20, 2). On the other side of the historical spectrum, we are, nowadays, often told of how Saul Kripke discovered that there is metaphysical, in addition to logical, necessity; a claim that obviously presupposes that logic is connected with necessity. Does the Tractatus agree with this tradition? That’s the central question of this essay.¹

I will approach this question by considering two of the main opponents in the recent debate over “resolute” approaches to reading the Tractatus. On one side, Peter Hacker advances a positive answer. He takes one of Wittgenstein’s principal criticisms of Frege and Russell to be that they don’t have a coherent explanation of the necessity of logic, and he holds that the Tractatus’s “most significant achievement” is its “account of logical necessity,” which overcomes the incoherence of the Frege-Russell view (Hacker 1986, 42). On the other side, Cora Diamond seems to return an unequivocally negative answer. In “Throwing Away the Ladder,” the founding document of resolution, she takes 6.37, “Es gibt nur eine logische Notwendigkeit,”² to be “ironically self-destructive,” since “[i]n so far as we grasp what Wittgenstein aims at, . . . we shall not imagine necessities as states of affairs at all. We throw away the sentences about necessity; they really are, at the end, entirely empty” (Diamond 1988, 198). On this reading it seems the Tractatus espouses, to put it in contemporary parlance, an unqualified modal eliminativism.

I will argue against resting with either of these interpretations. In section I I show that Hacker’s account suffers from two significant drawbacks: it rests on a misreading of Frege’s and Russell’s views, and, in the context of his overall interpretation of the Tractatus, the conception of logical necessity he attributes to Wittgenstein falls prey to the same problems that he finds in Frege and Russell. It is not altogether clear what Diamond takes to be the considerations leading to modal eliminativism in the Tractatus. However, in section II I reconstruct a line of argument from the fundamental Tractarian logical distinction between names and sentences to the incoherence of at least the grounds of the view that the propositions of logic—tautologies and contradiction—are necessary in virtue of the special way in which they are determined as true or false. But, in this argument and in Hacker’s reading, logic is conceived of as tautologies and contradictions, and necessity is conceived of as a special mode of truth. These conceptions are not mandatory. In section III I will urge an alternative conception of necessity and contingency in the terms of 6.124, “manches an den Symbolen, die wir gebrauchen, wäre willkürlich, manches nicht.”³ That is, necessity lies in what is not arbitrary (nicht willkürlich), and contingency in what is up to our arbitrary choice (willkürlich), in the symbols we use, in how we picture or model the world. Necessity here is not a mode of truth of propositions, but lies in the requirements of the intelligibility of propositions. [End Page 290] In section IV I flesh out this conception by showing that it is implicit in Diamond’s and James Conant’s account of the Tractarian view of nonsense. I also argue that it is equally implicit in an account of the Tractatus’s treatment of Russell’s Paradox due to two critics of the resolute approach: Ian Proops and Peter Sullivan. These accounts are committed to certain logical features of language or thought, patterns of symbolizing constitutive of intelligibility of propositions, that are not up to us to institute or alter. Of the many notions of logic or the logical in the Tractatus, the one most closely involving this conception of necessity is logical syntax; as I understand it, Tractarian logical syntax consists of the non-arbitrary features of modeling the world with propositions. This constitutes no more than the beginnings of a defense of the claim that logic in the Tractatus does involve necessity. If the defense is to have any substance, one has to show how, in the text, the idea of the non-arbitrary is connected with the idea of patterns of symbolizing. In section V I outline an account of this connection, based on an attempt to understand some puzzling remarks in the 5.47s and 5.51s. The central idea of this account, developed by considering some texts of Frege from a Tractarian perspective, is a view of what it is for a proposition to be a truth-function: it is for two things that are done with propositional-signs—affirming and denying them—to be subject to standards of correctness that fall in specific patterns. These are the patterns of logical syntax, features of propositions in which the necessities and impossibilities of logic show forth.⁴

I. HACKER ON THE NECESSITY OF TAUTOLOGIES

Hacker’s writings suggest that he regards Frege and Russell as, at best, philosophical village idiots with more or less indistinguishable views. In the case of logic, they painted themselves into a corner with two commitments. First, the “apodeictic status” of “laws of logic” “was clear enough” to them (Hacker 1986, 44), in the sense that for them the propositions of logic are true in a special way, true “come what may, under all conditions” (ibid., 47). But, second, they took the propositions of logic to have “a subject-matter,” namely, “the most general features of the world” or “the properties and relations of logical objects or constants” (ibid., 38). In order to render these two commitments consistent, they made the subject matter of logic into a special, necessary, kind of facts: facts about the maximally general features of the world, which cannot fail to be a part of any possible world, or facts about logical objects, which have the properties they have and stand in the relations they stand no matter how else the world might be.⁵

But through this maneuver “neither Frege nor Russell could coherently explain” “what gives [logical laws] their necessitarian status” (ibid., 44). The reason is that, by giving logical propositions a subject matter, they “bargained away the necessity of such propositions (since completely general propositions may nevertheless be only accidentally valid)” (ibid., 38). So far as I can tell, the parenthetical remark just [End Page 291] quoted is the only explanation Hacker ever gives for the incoherence of the Frege-Russell account of logical necessity. A contrast between accidental and essential of course figures prominently in the Tractatus’s remarks on logic (see, e.g., 2.012, 6.031(2), and 6.1232). It seems that for Hacker what the contrast amounts to is obvious, perhaps because he takes it as obvious that facts, however general, remain what happens to be the case in the world, since we can always conceive of the world as containing a different set of facts.⁶ If all facts are contingent, no type of facts can ground the necessity of any truth. Hence the Frege-Russell account of logical necessity is patently inadequate.

Wittgenstein’s alternative rests on characterizing the propositions of logic as tautologies. Hacker’s interpretation of the Tractatus is a representative instance of the familiar view of that book as containing a “metaphysics of logical atomism” (1986, 55) and an account of “the essential nature of representation” (ibid., 32). The notion of tautology is a consequence of the nature of propositional representation of the world. All propositions are uniquely analyzable into logically independent elementary “propositional-sign[s]” (ibid.) Each of these “is essentially a description of a (possible) state of affairs. If the state of affairs described obtains, the proposition is true, otherwise it is false” (ibid.). So “[t]ruth and falsity . . . are essential properties of ” elementary propositions. Moreover, these propositions are “bipolar, i.e. capable of being true and capable of being false (so there are no necessary truths or necessary falsehoods among elementary propositions)” (ibid.). Hacker doesn’t specify what non-elementary propositions describe. But he does offer a gloss on Tractatus 5—Der Satz ist eine Wahrheitsfunktion der Elementarsätze: the “truth-value of a proposition . . . is a function of (is dependent on) the truth-value(s) of ” elementary propositions (ibid., 44). What Hacker has in mind may then be this. Non-elementary propositions are expressed by signs in which occur logical connectives joining propositional-signs. Wittgenstein rejects Frege’s view of propositions as names of truth-values, and of logical connectives as names of functions from truth-values to truth-values (ibid., 41). So non-elementary propositions are not descriptions of possible states of affairs consisting of combinations of truth-values and truth-functions. What they depict, rather, are the possibilities of existence and nonexistence of the states of affairs pictured by elementary propositions (4.2). The world is the totality of states of affairs that exist, so a possibility of existence and nonexistence of the states of affairs is a possible state of the world. That is, a non-elementary proposition represents a set of possible states of the world. If any of those possible states exist, then the proposition is true; otherwise it’s false. Since the truth or falsity of an elementary proposition corresponds to, i.e., is determined by, the existence or nonexistence respectively of the state of affairs it represents (4.25), a possible state of the world corresponds to a possible combination of truth and falsity of elementary propositions. Thus what a proposition represents corresponds to a set of truth-possibilities of elementary propositions (4.4), and a proposition is true if any truth-possibility of elementary propositions with which it agrees obtains, [End Page 292] false otherwise. That’s the reason why the truth-value of a proposition is determined by, i.e., is a function of, the truth-values of elementary propositions, which are the arguments of that function.

Given this account of what non-elementary propositions represent, and so of how they are determined as true or false, it follows that they are not essentially bipolar. For given any set of elementary propositions, there is a function whose value is truth for every possible combination of truth-values of those elementary propositions. The non-elementary propositions whose truth-values are determined by that function of truth-values of elementary propositions are tautologies, and they are true in every truth-possibility of elementary propositions, i.e., every possible state of the world. A tautology is thus “true . . . under all conditions” (ibid., 47), i.e., a necessary truth. But this failure of bipolarity means that a tautology “has no truth-conditions,” and is not a genuine proposition describing possible states of affairs, but rather a “limiting” or “degenerate” proposition which “describe[s] nothing” (ibid., 51).

Somewhat surprisingly, Hacker never spells out explicitly why this account of logical necessity overcomes the incoherence of the Frege-Russell account. Presumably the reason is precisely the combination of necessity and emptiness of tautologies. The requirements for a tautology to be true are invariably fulfilled, fulfilled in all possible worlds, so even though a tautology has no truth conditions, it is true in the special kind of way demanded by logic. But the reason why it is true in this special way is not because it describes a special type of invariably obtaining situations. It is, rather, because of the way in which the truth-values of non-elementary propositions are determined by the possible truth-values of elementary propositions. Thus, it is the nature of propositional representation, rather than features of the world or of all possible worlds, that makes tautologies true. So the explanation of the necessity of tautologies does not run afoul of the thesis that any worldly fact or state of affairs is contingent.

I turn now to detail two problems with Hacker’s account. First, Hacker misrepresents Frege’s and Russell’s views of the relation between modality and logic. Let me begin with the more serious misrepresentation, that of Russell. In a(n admittedly unpublished) paper, “Necessity and Possibility,” Russell writes, “the subject of modality ought to be banished from logic” (1994, 520). The reason he gives in this paper is that “there is no such comparative and superlative of truth as is implied by the notions of contingency and necessity” (ibid.), a reason echoed in Principles of Mathematics: “there seems to be no true proposition of which there is any sense in saying that it might have been false. . . . What is true, is true; what is false, is false; and concerning fundamentals, there is nothing more to be said” (1903, 454). This doesn’t quite have the ring of a commitment to logic having a special, necessary, mode of truth. Now, in “Necessity and Possibility” Russell also offers four accounts of necessity. For example, a proposition is necessary if it can be deduced from the axioms of logic.⁷ So Russell’s considered view in this period is not quite that modal notions are to be “banished” because there is no such thing [End Page 293] as necessity, but rather that they are explicable in various different ways, but always in terms of more fundamental logical notions.⁸

Frege is not quite so strident as Russell about banishing modality, but he does insist in Begriffsschrift §4 (1879, 3–4) that his notion of conceptual content does not distinguish between the traditional notions of apodeictic and assertoric judgments.⁹ Moreover, in the same way as Russell, Frege outlines accounts of traditional modal judgments in logical terms. For example, “Das apodiktische Urtheil unterscheidet sich vom assertorischen dadurch, dass das Bestehen allgemeiner Urtheile angedeutet wird, aus denen [sie] geschlossen werden kann,” so that “[w]enn ich einen Satz als nothwendig bezeichne, so gebe ich dadurch einen Wink über meine Urtheilsgründe” (ibid., 3). Frege’s account seems to be that the occurrence of a predicate of necessity in the expression of a judgment affects only what he would later call the tone or illumination of what is expressed, but not its sense. If so then perhaps there is no objective standard for assessing the correctness of an apodeictic judgment distinct from those standards applying to its assertoric counterpart. That is, there is no case where judging that p is correct but judging that necessarily p is incorrect because there are no universal judgments from which p follows.

The upshot is that Russell and Frege took logic to be philosophically fundamental, and to have no need for modal notions: the laws of logic need not be stated using modal expressions, nor does the nature of logic involve modal concepts. But, they were not wholesale eliminativists about modality. Rather, they were modal reductionists, willing to countenance modality to the extent that it is explicable in terms of the concepts of logic.¹⁰

Obviously Hacker’s misreading of Frege and Russell¹¹ doesn’t invalidate his account of the Tractatus, since Wittgenstein may have shared that misunderstanding,¹² and developed his view of logic on its basis. But there is a second and more serious problem with Hacker’s interpretation: his account of logical necessity in terms of tautology doesn’t in fact overcome the purported incoherence of the supposed Fregean-Russellian conception of logical necessity. The account rests on two claims:

(BP) Elementary propositions are bipolar.

(TF) Propositions are truth-functions of elementary propositions.

From these two assumptions it follows that there exist non-elementary propositions that are true for all truth-values of elementary propositions. What is the status of (TF)? Is it not a generalization about all non-elementary propositions? Does it not then state or describe a general feature of such propositions? If so, then the question is: is this a necessary or a contingent feature?

At least two aspects of Hacker’s views suggest that it is necessary for propositions to be truth-functions of elementary propositions. First, Wittgenstein’s solution to the problem of logical necessity is supposed to be that logical truth is “essential validity” (Hacker 1986, 47), which I take to mean truth that follows [End Page 294] from the essence of propositional representation. Now (BP), as we saw, is expressly claimed to describe an essential feature of elementary propositions. It’s hard to see how (TF) could fail to describe an essential feature of propositions, since it is also required to account for the existence of tautologies. So it seems that propositions don’t just happen to be truth-functions, but are necessarily truth-functions. Second, from a Fregean, if not exactly Frege’s, perspective, one can take (TF) to be a theory of the contents of non-elementary propositions: they express functions from truth-possibilities of elementary propositions to truth-values. Hacker, I take it, would say that there is still a fundamental difference between such a Fregean view and Wittgenstein’s account: the Fregean would take these functions to be part of the reality described by non-elementary propositions, while Wittgenstein holds that the sum total of reality is exhaustively described by elementary propositions, so that these functions are not part of the reality described but merely play a role in non-elementary descriptions of that same reality. But why should we think that this difference is more than terminological? After all, whether or not the functions are part of the reality described, they play a role in fixing the truth requirements of the propositions in question. This suggests that the significance of being a part of the reality described lies in Hacker’s assumption that whatever is described by an ordinary proposition is contingent. So Hacker must hold that it is not contingent, not a mere fact about language, that propositions are truth-functions of elementary propositions.

But now the question arises, in what sense is it necessary that propositions are truth-functions of elementary propositions? It’s hard to make out that it is logically necessary, for it’s hard to see that (TF) is (expressed by) a tautology. Since 6.37 suggests that all necessity is logical, there is, prima facie, a problem. Hacker has a way out of this initial problem, since, like Diamond, he also has qualms about 6.37, albeit not on the same grounds. He writes, “it is misleading . . . to say that all necessity is logical necessity, since after all most of the propositions of the Tractatus itself seem to state . . . metaphysical necessities about the nature and essence of reality, of any possible world” (1986, 51). So perhaps (TF) can be understood as describing a metaphysically rather than logically necessary feature of non-elementary propositions. This way out, however, runs into another problem. Central to Hacker’s interpretation of the Tractatus is the claim that Tractarian metaphysical necessities are “ineffable” (ibid.), but Hacker maintains, in addition and in criticism of the Tractatus, that these ineffable metaphysical necessities are “every bit as brutish as brute empirical facts” (ibid.). So (TF) describes an ineffable necessary fact about propositions. In order for this not to contradict the principle that all facts are contingent, Hacker would have to maintain that there are two kinds of necessity: the necessity of tautologies and the necessity of metaphysical facts about representation. Moreover, the latter undergirds the former, at least in the sense that if propositions were not necessarily truth-functions of elementary propositions, then there wouldn’t be the special kind of truth of tautologies. Now, this isn’t quite the supposed Frege-Russell view of logical necessity, but it’s [End Page 295] unclear how much it differs: logical propositions are necessarily true, not in virtue of depicting necessary features of the world, but in virtue of there being certain necessary features of language.¹³ Finally, is it clear that the fact that propositions are truth-functions of elementary propositions is any more necessary than the fact that matter is composed of elementary particles?

II. DIAMOND ON THE IMPLOSION OF MODAL TALK

As noted, Diamond apparently holds that understanding Wittgenstein results in realizing the emptiness of all sentences about necessity, including such remarks of the Tractatus as 6.37. From “Throwing Away the Ladder,” however, it’s not clear that the anti-modal stance she discerns in the Tractatus is quite so unqualified. Diamond begins her discussion of necessity in section 3 by saying that Wittgenstein consistently opposes conceptions of “necessity imaged as fact,” as what is “the case,” albeit in all possible worlds (1988, 195). This leaves it open that Wittgenstein would not be opposed to non-factual images of necessity. Now, it is of course a centerpiece of Diamond’s resolute approach precisely to reject Hacker’s ineffable Tractarian necessities, and as we have just noted, Hacker himself declares these to be (like) brute (empirical) facts. But Hacker’s ineffable necessities are metaphysical. What about the necessity of logic? Can it be conceived non-factually?

To address this question, let’s turn to Diamond’s account of tautology. She begins by sketching a view of logic Wittgenstein rejects, in which the necessity of logic is factual: “[i]t is overwhelmingly natural to say . . . that, if we fix the meaning of the sentential connectives, . . . then a sentence of the form ‘p or not p’ . . . has truth conditions which are in all possible circumstances fulfilled” (ibid., 192). Wittgenstein’s contrasting view is that there is

a rule determining the logical features of the comparison with reality of . . . genuine sentences formed from genuine sentences. That fundamental rule will really be the meaning, all rolled up into one, as it were, of all the logical constants. And if we grasp that, we shall not be tempted to think of tautologies and contradictions as saying that something or other is the case. We shall not be tempted really to think of them as sentences.

(Ibid., 192–93)

This fundamental rule requires elementary propositions and the operation N, “a method of construction of sentences from sentences” (ibid., 193).¹⁴ There is no explanation of the fundamental rule in this paper, but one obvious view is that it applies to the outputs of the method of construction, specifying how they are to be compared with reality in terms of agreement and disagreement with the truth possibilities of those elementary sentences that occur in their constructional history. Applied to certain outputs of the method, tautologies, the rule, by itself, determines these “sentence-like constructions” as true; applied to others, contradictions, the rule by itself determines them as false. They are not “genuine sentences” because [End Page 296] their truth or falsity is not the result of “comparison to reality” (ibid., 193) effected by elementary propositions, but fixed by the fundamental rule alone.

Diamond goes on to say that “[l]ogical necessity is that of tautologies”:

It is not that they are true because their truth conditions are met in all possible worlds, but because they have none. ‘True in all possible worlds’ does not describe one special case of truth conditions being met but specifies the logical character of certain sentence-like constructions formulable from sentences.

(Ibid., 198)

Does logical necessity then consist in the “logical character” of tautologies? If so, is this conception of logical necessity an image of necessity as fact? This passage suggest that Diamond is offering an explanation, as opposed to elimination, of the phrase ‘true in all possible worlds’, an explanation in terms of the way in which tautologies are determined as true, their logical character.¹⁵ Through this analysis, we can see that tautologies do not owe their necessity to truly describing what is the case in all possible worlds; hence the logical character of tautologies provides a non-factual image of logical necessity.

There are four parallels between this account and Hacker’s reading. First, what Diamond calls “genuine sentences” and Hacker “non-degenerate propositions” require, for their truth, comparison with reality. Call these ordinary sentences. Second, Wittgenstein rejects the idea that the sentences of logic, just like ordinary sentences, require comparison with reality for truth, and the idea that the necessity of logic consists of successful comparison with, not just actual but all possible, realities. So, third, the tautologies of logic are not ordinary propositions but proposition-like sequences of words with the property that the rule specifying the requirements for non-elementary propositions to be true, by itself, determines that these requirements are met for them, without comparison with reality. Finally, this “logical character” of tautologies constitutes their necessity. Given these parallels, it’s not clear why and how, if we stop “imagining necessities as states of affairs,” we should reach the apparently unqualified conclusion that “sentences about necessity are entirely empty.” Why, in other words, is this view of tautology merely another rung on the ladder?

How Diamond might answer this question only becomes clear in a later paper, “Truth before Tarski” (2002), where she develops further her reading of how, by thinking through doctrines apparently advanced in the Tractatus, one can see that they are empty. The particular “implosion,” to use Warren Goldfarb’s term for this form of Tractarian self-undermining (see 1997, 70), concerns Wittgenstein’s apparent view of the fundamental logical differences between propositions and names. The starting point is a much-quoted passage from the Notes on Logic:

Let us consider symbols of the form ‘xRy’; to these correspond primarily pairs of objects, of which one has the name ‘x,’ the other the name ‘y.’ The x’s and y’s stand in various relations to each other, among others the relation R holds between some, but not between others. I now determine the sense of ‘xRy’ by laying down: when the facts behave in regard to [sich verhaltern zu] ‘xRy’ so that the meaning of ‘x’ stands in [End Page 297] the relation R to the meaning of ‘y,’ then I say that [the facts] are ‘of like sense’ [‘gleichsinnig’] with the proposition ‘xRy’; otherwise, ‘of opposite sense’ [‘entgegengesetzt’]; I correlate the facts to the symbol ‘xRy’ by thus dividing them into those of like sense and those of opposite sense.

(Wittgenstein 1979, 104[6])

Following Thomas Ricketts (1996), Diamond takes the sense of ‘xRy’ to be a “rule” for comparing the way in which the signs (letters) making up this sentence are arranged with the way in which the objects named by ‘x’ and ‘y’ are related:

When the object named by the name standing to the left of ‘R’ . . . stands in relation R to the object named by the name standing to the right of ‘R’, this ‘behavior of the facts’ shall count as like in sense (gleichsinnig) with the proposition ‘xRy’; otherwise the facts count as opposite in sense to the proposition.

(Diamond 2002, 268)

Diamond adds the idea that the rule for comparison has a “direction” which can be “reversed.” We can take it that the idea is based on 4.0621—“die Zeichen ‘p’ und ‘~p’ das gleiche sagen können,” and 5.2341(3)—“Die Verneinung verkehrt den Sinn des Satzes.” To begin with, if R is an asymmetrical relation, then we can use the same sign, ‘xRy’, to say that the objects named by ‘x’ and ‘y’ stand in the converse relation of R, by adopting a rule of comparing this sign with the fact that interchanges “like sense” with “opposite sense” in the original rule. With the new rule, the facts agree with the sign just in case “the name standing to the right of ‘R’ names an object that stands in R to the object named by the name to the left of ‘R’” (ibid.). Wittgenstein then moves to the reversible directionality of all propositions by an analogy: we can use a sign ‘p’ to say what its negation, ‘~p’, says merely by adopting a rule of comparison interchanging what facts count as of like sense with what facts count as of opposite sense in the original rule. The logical criterion for being a proposition is this reversible directionality”; the use of names, in contrast, “has no intrinsic capacity for reversal[,] no directionality” (ibid., 269).

Diamond argues that this criterion of propositionhood leads to a phenomenon parallel to what is sometimes called the Kerry paradox (see Frege 1967a [1892]): because for Frege an expression, as a matter of logic, can refer to a concept only if it is used as a predicate, and not as a proper name, “there are ways of attempting to talk about a concept that are self-defeating, in that we put what we want to be an expression for a concept in the logical place for a proper name, whereupon it is no longer the expression for a concept” (2002, 269). Correspondingly, on Wittgenstein’s logical criterion for being a proposition,

attempting to speak of a proposition as one can speak of names, or attempting to treat it in other ways as a name, that is, by putting it into a slot where the subject of a predicate goes, or where the name of one of the terms of a relation goes, will result in its no longer being, logically speaking, a proposition. It will lack directionality.

(Ibid.)

But talk of ways of comparing propositions with reality requires precisely treating propositions as standing in some relation to reality. Thus, [End Page 298]

following out the implications of what Wittgenstein tells us about the relation between our propositions and reality lands us, and is meant to land us, in incoherence. . . . All such attempts . . . involve thinking of propositions as items going into a relation as its terms, that is, not as genuinely directional, not as genuinely capable of propositional sense. Such talk can be understood to dissolve when we understand that what we are trying to do is to treat a proposition as nameable, as a logical subject, and when we understand the self-defeatingness of that aim.

(Ibid., 270)

On the basis of this implosion we can see why on Diamond’s view the Tractarian account of tautology is supposed to be thrown away. Part of that account is set forth in sentences like:

Non-elementary propositions are compared with reality by agreeing or disagreeing with truth-possibilities of elementary propositions.

This sentence seems to be a generalization about propositions, each of whose instances, e.g.,

If ‘Frege was born in Wismar’ is a non-elementary proposition then ‘Frege is born in Wismar’ is compared with reality by agreeing or disagreeing with truth-possibilities of elementary propositions,

describes a proposition as standing in the relation of comparison to reality. So the apparent account of what qualifies a proposition as a tautology has in fact to speak of propositions in such a way that they are not “genuinely directional, not genuinely capable of propositional sense,” and so not really as propositions at all.

Before going on, I want to register a doubt about this implosion. It’s not clear that the account of tautology needs to be understood as attempting to speak of propositions, as opposed to propositional-signs. According to 3.12, “der Satz ist das Satzzeichen in seiner projektiven Beziehung zur Welt.” So why can’t we take the Notes on Logic passage as a description of how propositional-signs are compared with the world, a statement of those signs’ projective relation to the world? Reversible directionality of comparison would then be a feature of propositional-signs’ projective relation, not of propositions’. Similarly the clause of the theory of tautology cited above can be understood as a statement of how non-elementary propositional-signs are projectively related to the world.

As I understand Diamond, we are not meant merely to recognize the dissolution of talk about the relation of propositions to reality: “What is left, when we recognize the self-defeatingness of such talk, is our understanding and operating with the senseful sentences of our language” (ibid.). Although Diamond is not explicit about this, it seems that on her view this understanding and operating is connected with certain “logical features of the use of ordinary sentences, logical patterns of use,” and the Tractatus aims to “redirect[] our attention” to these features or patterns, through our recognition of the emptiness of semantic talk (ibid., 259). The model for the view is Frege, as interpreted by Peter Geach, and by Ricketts and Goldfarb.¹⁶ On Goldfarb’s and Ricketts’s reading of Frege, in particular, the apparent ontological [End Page 299] and semantic talk involved in the Kerry phenomenon—e.g., proper names refer to objects, not concepts—is really “supervenient” (Ricketts 1986, 65) on patterns of inferential relations that “articulate” or “redescribe” our inchoate grasp of logical features “present in ordinary informative discourse” (Diamond 2002, 257). These patterns of inference are the bases for identifying “parts and features of statements as logically functioning units” (Goldfarb 1997, 60–61); they “inflict” a logical segmentation on sentences. The criterion for a part of a sentence to be a proper name is that that sub-sentential sign plays certain roles in at least the inferential patterns of first-order universal and existential generalization, and Leibniz’s Law.¹⁷ Function names are isolated on the basis of second- and higher-order patterns of inference. In giving up such apparent claims as that the ontological category of object is determined by being the referent of proper names, then, we are left with and “redirected” back to the patterns of valid inference that are the logical basis of being a proper name.

So what are the inferential patterns that we are left with when we throw away the talk of reversible directionality? Diamond considers two proposals. One is Geach’s logic of duality (1982), which attempts to make more precise what it is for p and ‘~p’ to say the same. For every language L there exists a dual language L_D, with the same vocabulary and syntax, such that each sentence of L_D “equiform” with a sentence of L says the contradictory opposite of that sentence of L. The logical distinction between proposition and name is based on features of a compositional translation from sentences of L to their duals in LD that preserves what is said. In such a translation sentences of L are mapped to their negations, negations to the subsentence negated, conjunctions to disjunctions, etc. But names are mapped to themselves; names are self-dual. The logical criterion of sentencehood is thus non-self-duality.

Diamond’s own proposal is this. Talk about propositions agreeing or disagreeing with reality directs us to inferences from ascriptions of beliefs and what is the case to the correctness of beliefs:

A believes pp So A’s belief is correct

A believes p not-p So A’s belief is incorrect.

(Ibid., 260)

Presumably judgment and assertion can take the place of belief. What about reversible directionality? That talk directs us to a feature of those inferences: their correctness “depend[s] on the rules governing the judgment-verb” (ibid., 270). This feature is exhibited in an example about

the verb ‘to doubt’. To say in English that A doubts p means that A rather thinks that p is not the case; to say in Scots . . . that A doubts p means that A rather thinks or fears that p is the case. The difference between the two uses of ‘doubts’ is nothing but the difference in the rules for how we may infer: [End Page 300]

[In English

  A doubts we shall be late   We shall be late   So A is wrong

is valid. In Scots

  A doubts we shall be late   We shall be late,   A is right

is valid.] This is to say that there is no difference between the English and the Scots sentences except the direction of comparison. . . . The slot into which we put the proposition which A judges, after the words ‘A judges that’ or ‘A doubts that’ (etc.) is not a slot for a relatum but for a proposition, as is shown by the fact that, without there being any alteration in the meaning of any component of the proposition in question . . . , we have two opposite ways of operating logically with the resulting proposition. That is, from ‘p’ we can infer in two opposite directions to . . . the correctness or incorrectness of what [A] is said to hold.

(Ibid., 270–71)

That this inferential feature underlies reversible directionality leads to a worry about Diamond’s implosion. Diamond’s example makes it clear that from the fact that a sentence occurs in another in a position that looks like a placeholder of some relation expression it doesn’t follow that in that occurrence the sentence is logically a name rather than a proposition. So, even if the account of tautology has to contain statements in which propositions occur (as opposed to mentioning propositional-signs) it is not immediately obvious that these statements are self-defeating. It all depends on whether those occurrences function logically as propositions, and that is a matter of whether and how those statements occur in valid inferences.

This point leads to another worry. Even if this line of argument succeeds in showing that the account of tautology is self-defeating, it’s not clear that it thereby shows that all sentences (apparently) about necessity are empty. Consider especially the sentences of the second degree of modal involvement, in which a modal sentential operator is attached to what is grammatically a sentence, e.g., “it is necessary that water is H₂O.” Such sentences seem to ascribe properties to propositions, or to the states of affairs they depict. Do the embedded sentences function logically as names or as propositions? Well, if we go with Geach’s logic of duality, the answer seems to be yes. Like Diamond, Geach holds that sentences used to ascribe beliefs occur logically as propositions in the ascription sentences. His argument rests on the claim that the dual of a sentence forming operator φ is ⌜¬φ¬⌝. Where φ is ‘A believes that’, if we assume that p is not self-dual in ‘A believes that p’, we obtain as its dual, ‘A does not believe that it’s not the case that not-p’. Accepting the elimination of the double negation, we obtain ‘A does not believe that p’, which is clearly the contradictory in the original language of ‘A believes that p’ and so its dual. Consider now the case where φ is ‘necessarily’. Again assuming that p is not self-dual in ‘necessarily p’, we obtain as its dual, ‘it’s not necessary that it’s not the case that not-p’, [End Page 301] i.e., either ‘it’s not necessary that p’ or ‘possibly not-p’, both of which are clearly duals of the original sentence. An analogous argument works for ‘possibly’.¹⁸

I don’t know if a similar case can be made on Diamond’s views. One point we can make is this. As we have seen, on her view the cash value of talk about propositions agreeing with reality are inferences from ascriptions of beliefs and what is the case to the correctness of beliefs. So we might ask, why isn’t the cash value of talk about necessary and possible states of affair the validity of the following patterns of inference:

• • p, so ‘possibly p’ is true and ‘necessarily not-p’ is false.

• • not-p, so ‘necessarily p’ is false and ‘possibly not-p’ is true.

• • A believes that necessarily p, not-p, so A is wrong.

• • A believes that p is impossible, p, so A is wrong.

Of course, these patterns don’t show that sentences occur in logically reversible ways in modal contexts; for that one would have to make the metaphor of reversibility applicable to modal inferential patterns. I don’t know if that can be done, but I’m not concerned to establish that embedded sentences in modal contexts are logically sentences. I claim only that without further investigation of the logic of such modal sentences, we’re not entitled to conclude that they are empty.

III. THE NON-ARBITRARY AS A CONCEPTION OF LOGICAL SYNTAX

Let’s take stock. Neither Hacker’s explicit account of logical necessity in the Tractatus nor Diamond’s account (as I reconstruct it) of the Tractatus’s rejection of necessity tout court is fully satisfactory. On the one hand, Hacker’s account of tautology as an explanation of the necessity logic fails on its own terms. On the other, Diamond’s rejection of what seems to be essentially the same account is inconclusive. In order to make further progress, let’s begin by noting a common assumption underlying these opposed readings of the Tractatus. Both accept that taking the propositions of logic to be tautologies offers an explanation of the non-factual necessity of these propositions. This amounts to accepting that the primary notion of necessity applying to logic is that the sentences of logic are necessarily true; tautology comes in as an explanation of this special mode of truth. In this and the next section I will explore a different conception of necessity in the Tractatus, having to do, not with a mode of truth, but with requirements of intelligible discourse.

There are two anchors for this conception. First, 6.124:

Wir sagten, manches an den Symbolen, die wir gebrauchen, wäre will-kürlich, manches nicht. In der Logik drückt nur dieses aus: Das heißt aber, in der Logik drücken nicht wir mit Hilfe der Zeichen aus, was wir wollen, sondern in der Logik sagt die Natur der naturnotwendigen Zeichen selbst aus: Wenn wir die logische Syntax irgendeiner Zeichensprache kennen, dann sind bereits alle Sätze der Logik gegeben.

(Second emphasis mine) [End Page 302]

The first sentence of this passage I take to be an allusion to the second anchor, 3.342:

An unseren Notationen ist zwar etwas willkürlich, aber das ist nicht willkürlich: Dass, wenn wir etwas willkürlich bestimmt haben, dann etwas anderes der Fall sein muss. (Dies hängt von dem Wesen der Notation ab.)

(Third emphasis mine)

It is, I trust, relatively uncontroversial and relatively uninformative to read these passages as saying that in logic only what is “nicht willkürlich” in our notations expresses or asserts, where what is “nicht willkürlich” is characterized in two ways: it is what must be the case once we have arbitrarily (willkürlich) determined something in our notations, and it is “die Natur der naturnotwendigen Zeichen.” As a first step beyond the uncontroversial, I read “willkürlich” here as “up to us,” “up to our arbitrary decision.” That is, in any language there are particular ways of using signs that are open to us to choose. But once having made these choices, there are certain aspects of language that are no longer up to us.¹⁹ There are, then, necessary features of language that express or assert themselves in logic. These necessary features constitute logical syntax, and logical syntax is in some sense prior to the propositions of logic.

In order to go any further in understanding these passages, we have to address at least the following questions:

• • What are the particular ways of using signs that we choose?

• • What is not arbitrary once these choices are made?

• • How is the non-arbitrary (connected to) logical syntax?

• • In what sense is the non-arbitrary in language necessary?

I will begin with a set of answers that I will reject. These can be discerned in Hacker’s conception of logical syntax, advanced in criticism of a striking feature of Diamond and Conant’s interpretation of the Tractatus: in this text Wittgenstein countenances no such things as violations of logical syntax. Central to Hacker’s criticism is the claim that “the rules of logical syntax are constitutive rules” (2000, 366). Hacker’s main example consists in the rules of contract law, which don’t “prohibit something that can be done but should not be done”: “[f]ailure to follow such rules does not result in illegal contracts” (ibid., 365). Rather, “it results in invalid contracts,” which are “not a kind of contract” at all (ibid., 366). In the same way, “if we fail to comply with the rules of logical syntax the result is not the expression of a thought that is illogical (since there is no such thing), but a [sic] nonsense” (ibid., 367). That is to say, the result of non-compliance²⁰ is not the expression of any thought at all; for this reason Hacker takes the rules of logical syntax to determine the bounds of sense (ibid.). Are rules, constitutive of making sense, connected with necessity? There is a familiar answer. One must move the queen in straight paths in chess because otherwise one is not playing chess. So one expects that Hacker would accept a similar view here: one must comply with the rules of logical syntax, otherwise one would not be using language, or not be thinking, at all.²¹

Hacker discusses just one example of a Tractarian constitutive rule of logical syntax, governing the term ‘object’: [End Page 303]

To use the term ‘object’ as a variable name (formal concept) is correct (for this is the use we have assigned to it), but to use it as a proper concept-word is incorrect—for no meaning has been assigned to it as a concept-word (and to do so would generate undesirable ambiguity).

Once we have assigned a use to the sign ‘object’ as a variable, it will be incorrect to go on to use it in a form of words such as ‘A is an object’ (or ‘A is not an object’), for there it does not occur as a variable but as a genuine name—and no such use has been assigned to the term ‘object’, nor should it be, since the term already has a use.

(2000, 366)

From this example we can gather that for Hacker a rule of logical syntax is or specifies a particular type or category of use of a particular expression. “Type of use” because presumably Hacker doesn’t think that, e.g., it’s a rule of logical syntax that ‘green’ is a predicate correctly applied to green things, so that it would be nonsense, as opposed to plain old false, to say of a red object that it’s green. In the case of ‘object’ the rule of logical syntax specifies that it is to be used as a variable. It is not clear to me what exactly it is to use this word as a variable, but perhaps its use in ‘An object fell’ is a relatively uncontroversial case. Hacker is, in any event, certain that there are uses of this term that are not uses as a variable; for instance, its use in ‘the Earth is an object’ is as a genuine (concept?) name. So evidently “variable” and “proper concept word” or “genuine concept name” are distinct categories of use. (One wonders what Hacker would say about predicating ‘green’ of ideas; is this a type of use of ‘green’ governed by some rule of logical syntax?) In any event non-variable types of uses of ‘object’ don’t comply with the rule of logical syntax governing ‘object’, and so are incorrect in a nonsense-generating way. It seems that, for Hacker, any choice of a particular rule for a use of a sign gives rise to something non-arbitrary: once the choice is made it is not arbitrary what counts as correct and what counts as incorrect types of use of that sign. These non-arbitrary features determine sense and nonsense: incorrect types of use generate nonsense.

Thus the key question for Hacker’s view is: do non-compliant types of uses of signs invariably result in nonsensical pseudo-propositions? Since for Hacker the rules of logical syntax are constitutive of making sense, and, to continue with his example, the logico-syntactic rule governing ‘object’ is that it is to be used as a variable, any failure to comply would have to be nonsense. Here Diamond presses a compelling objection: “when I make a move in language that no one has ever made before, in accordance with no prior rules or stipulations, what I say may be perfectly intelligible and in order; it may be a perfectly good move” (2005, 86). Here is one of her examples of this possibility:

A child is showing off. Someone says ‘He thinks he’s something!’ Suppose that no one has ever used the word ‘something’ in that way before. Suppose (that is) that it has hitherto always been used in ordinary language in a way that goes over to quantifier-with-variable in a formal notation, and that there has never been any other sort of employment of ‘something’. [End Page 304]

What Tractatus 3.326 suggests is that we can recognize the symbol in ‘He thinks he’s something!’, in that we can observe how ‘something’ is used significantly in that proposition and could lay out its use in logically related ones. (We can see that syntactically correct use.)

(Ibid., 83)

More generally, “whatever rules there already are for the use of a word, the word may be given some other employment. Whether anything determinate is being said when it is given that new employment then needs to be looked at in the individual case” (ibid., 86). It seems that Hacker tacitly recognizes this point, for he describes non-compliant uses of ‘object’ as a “concept-word” merely as (in fact) “incorrect— for no meaning has been assigned to [‘object’] as a concept-word.” And he clearly sees that one could assign this word a concept-word use, since he objects that “to do so would generate undesirable ambiguity.” But this amounts to the admission that non-compliance with the supposed logical syntactic rules governing ‘object’ does not invariably result in nonsense. Moreover, if the nonsensicality of sentences is the basis of logical impossibility, then some logical impossibilities might not have to be logically impossible.

I take this objection to be decisive against Hacker’s account of logical syntax as constitutive rules. But it doesn’t rule out discerning in the Tractatus any conception of logical syntax as constitutive of significance. The phrase ‘Regeln der logischen Syntax’ occurs only twice in the Tractatus, in 3.334 and 3.344,²² ‘logischen Syntax’ by itself occurs in merely three other remarks—3.325,²³ 3.33, and 6.124, and the adjective ‘logisch-syntaktischen’ once in 3.327. Moreover, in none of these texts does Wittgenstein clearly specify examples of rules of logical syntax. Hence it is not mandatory to adopt Hacker’s assumption that they govern or consist in specific types of uses of particular expressions. In the next section I will develop a conception of logical syntax which rejects this assumption. I will observe two constraints. The first is to respect Diamond’s observation, that any word may be used in ways different from established usage without resulting in nonsense. Indeed, I want to add to that observation: as we see in Wittgenstein’s ‘Grün ist grün’ in 3.323, a word may be used ambiguously, in two different ways, within a single meaningful sentence. The second constraint is to make out how logical syntax is nevertheless connected to significance and nonsense. I will also sketch how the conception coheres with the occurrences of ‘logischen Syntax’ in the 3.3s.

IV. COMMITMENTS TO LOGICAL SYNTAX, AUSTERE AND OTHERWISE

The bases of my account are certain implicit commitments of Diamond’s and Conant’s “Frege-Wittgenstein” or “austere” view of nonsense. Their view is developed against two stalking-horses supposedly espousing the “natural” or the “substantial” view of nonsense: Dummett’s reading of Frege’s view of nonsense and Carnap’s criticism of Heidegger. I start with Conant’s account of Dummett. Frege, [End Page 305] according to Dummett, held that the meanings of expressions belong to various logical categories or types. These types determine whether or not the meanings belonging to them can, logically, combine with one another into a proposition or thought; they determine the “logical valency” of these meanings, and, thereby of linguistic expressions that have these meanings. The totality of facts about the logical valencies of meanings determines whether the expressions composing a sentence have meanings that can combine into a proposition or a thought. It thereby determines whether that sentence is meaningful or meaningless, significant or nonsensical. We can then take logical syntax to be descriptions or statements, based on logical valencies, of those combinations of expressions that make up meaningful sentences. A violation of logical syntax would then be a combination of expressions that falsifies these descriptions. Such violations are nonsensical, and their nonsensicality is a result of their being violations of logical syntax.²⁴

Thus Dummett explains why the sentence

(1) Chairman Mao is rare

is meaningless even if grammatically correct as follows. This sentence results from attempting to express the result of combining “the underlined portions of propositions” expressed by

(2) Chairman Mao ate only boiled rice

(3) An honest politician is rare

We attempt to combine the ‘Chairman Mao’ of (2) [the ‘Chairman Mao’ that denotes that individual] and the ‘—is rare’ of (3) [the ‘—is rare’ that denotes that second-level function], and we thus arrive at (1), which . . . is a concrete instance of a special type of meaningless sentence—one that involves a violation of logical category: we have tried to put a proper name into an argument place into which only a first-level function fits. Moreover, what we have here is (alleged to be) a case of fully determinate nonsense: (a) it is logically distinct from other fully determinate cases of substantial nonsense; (b) each of the ‘parts’ of this proposition has a fully determinate sense; and (c) though the sense of the resulting whole is flawed, it is flawed in a determinately specifiable respect . . . : it represents ‘an attempt’ to put that proper name into that argument place for a first-level function. But it will not fit—(in Frege’s words) “the parts cannot logically adhere,” “it makes no sense to fit them together,” “they will not hold together”—thus we get . . . not mere nonsense, but a special variety of nonsense that arises from attempting to do something logically impossible.

(Conant 2002, 397–98)

I pause to register a point that is, by now, familiar.²⁵ This conception of violations of logical syntax does not imply that a meaningless sentence has sense or expresses a thought, only one that is logically flawed. Nothing in the idea of logical valencies implies that the result of combining expressions having incompatible meanings can’t be a sentence that fails to have any sense or express any thought at all. What is crucial to the idea, and crucial to Conant’s subsequent argument, is that it is possible for all the component expressions of substantially meaningless sentences to be meaningful. [End Page 306]

The austere conception of nonsense opposes, against this crucial idea, the claim that if a sentence is meaningless, then at least one, perhaps all, of the expressions that compose it also lack meaning. Part of the case for the austere as against the substantial conception is defensive: one has to remove the main obstacle to accepting the austere conception, the fact that, prima facie, all the words of (1) are words of English with established meanings. So, if (1) is indeed meaningless, it is hard to see that the explanation of its meaninglessness can advert to the meaninglessness of any or all of its component expressions. The argument against this obstacle begins with an observation: the mere fact that (tokens of) a word (type) occur in distinct sentences does not imply that (those occurrences of) that word (type) have the same meaning in those sentences. For example, intuitively, there is a way of understanding the sentence

(4) Trieste is no Vienna

So that the word ‘Vienna’ in this sentence does not mean the same as the word ‘Vienna’ in

(5) Vienna is the capital of Austria

That is, we can take ‘Vienna’ in (4), as Conant suggests, to “mean something like ‘metropolis’ (or perhaps even beautiful or majestic metropolis)” (2002, 399), so that in (4) this word expresses a property that (4) represents Trieste as not having. In contrast, in (5) ‘Vienna’ refers to a city, rather than expresses a property.

This observation can be couched in terms of the distinction between sign and symbol drawn in Tractatus 3.31–3.322. Conant characterizes the distinction thus:

sign  an orthographic unit, that which the perceptible expressions for propositions have in common (a sign design, inscription, icon, grapheme, etc.)

symbol  a logical unit, that which meaningful propositions have in common (i.e., an item belonging to a given logical category: proper name, first-level function, etc.)

(Ibid., 400; emphasis mine)

In these terms, (4) and (5) “have the sign ‘Vienna’ in common [but] have no symbol in common” (ibid., 401).

This observation clearly doesn’t by itself constitute a full response to the obstacle. It tells us no more than that “there will always be room for a question as to whether a given sign, when it occurs in two different sentences of ordinary language, is symbolizing the same way in each of those occurrences” (ibid., 403). But what the austere conception requires is that a sign which symbolizes in some way in one sentential occurrence, e.g., ‘___ is rare’ in (3), fails to symbolize altogether in another sentential occurrence, e.g., ‘___ is rare’ in (1). Are there any such sign occurrences? If there are, what accounts for such divergences in their symbolizing?

The answers to these questions lie in the Tractatus’s version of Frege’s Context Principle. One expression of it is 3.3: “Nur der Satz hat Sinn; nur im Zusammenhang des Satzes hat ein Name Bedeutung.” As Conant reads it, one aspect of this Tractarian [End Page 307] context principle is a view of how we settle the question whether a sign symbolizes in the same way in distinct occurrences, e.g., how do we come to see that ‘Vienna’ in (4) is not the same symbol as ‘Vienna’ in (5)? The view is formulated thus:

Wittgenstein says: “In order to recognize the symbol in the sign we must consider the context of significant use” (§3.326). We must ask ourselves on what occasion we would utter this sentence and what, in that context of use, we would then mean by it. . . . In asking ourselves this, we still rely upon our familiarity with the way words (signs) ordinarily occur (symbolize) in propositions to fashion a segmentation of the propositional sign in question.

(Ibid., 403–4)

One reading of this formulation is that sentence meaning is prior to sub-sentential meaning in the sense that the way a sentence is used in a context, its meaning in that context, determines its division into sub-sentential signs and the ways in which those signs symbolize. Another reading is more overtly epistemic: it is by understanding or making sense of a sentence that we come to see what its (logical) component expressions are and to understand what those components mean. Let’s call the view, in either of these guises, the Tractarian Context Principle, TCP for short. In each case the view opposes the idea that sub-sentential expressions have meanings or can be understood independently of the sentences in which they occur and these independent meanings determine the senses of the sentences in which they occur.

Given the TCP there seems to be a straightforward argument against the substantial conception of nonsense:

To perceive a Satz as sinnvoll is to be able to perceive the propositional symbol in the propositional sign. To perceive a Satz as Unsinn is to be unable to recognize the symbol in the sign. . . . [I]n the case of a piece of nonsense—that is, in the absence of the provision of a context of sinnvollen Gebrauch determining a possible logical segmentation of the Satz—we have no basis upon which to isolate the logical roles played by the working parts of a propositional sign; for, ex hypothesi, there are no working parts of ‘the proposition’.

(Ibid., 404; emphases mine)

It follows then that no meaningless sentence can be composed of meaningful sub-sentential expressions, if such expressions are taken to be signs occurring in that sentence and symbolizing in those occurrences. It also follows that no sentence is an example of a violation of logical syntax in the sense set out above. To be such a violation, a sentence would have to be composed of meaningful expressions whose meanings don’t have the logical valencies that enable them to combine into a proposition. But such a sentence would have no meaning, and so, by the Wittgensteinian view, would not be composed of meaningful expressions. As Conant puts it,

there are no examples of the sort Dummett was looking for—examples of putting a proper name where a concept word belongs—for if one can properly make out that what belongs in that place is a concept word, then that is a sufficient condition for treating whatever is in that place as a concept word. There is not anything . . . that corresponds to [End Page 308] a proposition’s failing to make sense because of the meaning that the parts already have taken in isolation.

(Ibid.)

Let me note in passing that Conant sometimes seems to make a stronger claim: on the Tractarian view of nonsense “we can[not] so much as try to put a logical item into an argument place in which it does not fit” (ibid., 398; emphasis mine).

We are now in a position to see that implicit in this treatment of instances of the priority of sentential to sub-sentential symbolization is a conception of logical valencies of symbols—Tractarian symbols, not signs. These valencies concern what logical segmentations of sentences in use are possible, i.e., what types of sub-sentential symbols can be discerned in meaningful propositions.

It’ll be useful to divide the argument into two steps. The first isolates a crucial assumption in Conant’s view of how, on the TCP, we understand “novel” sentences. Let’s observe, to begin with, that the priority of making sense of a proposition over discerning the (types of) meanings of its component symbols does not imply that we understand sentences as wholes, entirely independently of our understanding of words. Although Conant insists that we begin by asking when we would utter, and what we would mean by, a sentence we are trying to understand, he acknowledges that “we still rely upon our familiarity with the way words (signs) ordinarily occur (symbolize) in propositions to fashion a segmentation of the propositional sign in question.” Indeed, he states in a footnote that

In the absence of any familiarity with the way words (signs) ordinarily occur (symbolize) in propositions, we would have no basis upon which to fashion possible segmentations of propositional signs, and hence no way to recognize . . . the symbol in the sign.

(Ibid., 445, note 84)

What exactly is this “reliance on familiarity with ordinary ways in which words symbolize in propositions”? This becomes clearer if we look at Conant’s account of how we in fact can makes sense of

(1) Chairman Mao is rare,

Dummett’s example of nonsense from violation of logical syntax:

If . . . we attempt to provide a context of significant use for (1), it becomes possible to see the symbol in the sign in ways which Dummett does not consider. There are two equally natural ways to segment this string: (a) to construe ‘Chairman Mao’ as symbolizing a first-level function (on the model of ‘You’re no Jack Kennedy’) [then the sentence might mean something like ‘The kind of exemplary statesmanship Chairman Mao exhibited is rare’], (b) to construe ‘rare’ as symbolizing a first-level function [as in the established English usage ‘That piece of meat is rare!’] These are ‘natural’ ways of ‘reading’ the string because each reading segments the string along lines dictated by an established usage (i.e., an established method of symbolizing by means) of signs. The expression ‘—is rare’ has an established use in the language (in sentences such as ‘An honest politician is rare’) as a second-level function; the expression ‘Chairman Mao’ has an established use in the language (in sentences such as ‘Chairman Mao ate only boiled rice’) as a proper name. Each of [End Page 309] these established uses dictates a possible segmentation of the string—each of which excludes the other. There is not anything that is simultaneously segmenting the string along both lines at once. Segmenting it either way, we supply a possible context of significant use and thus confer upon the string ‘Chairman Mao is rare’ a sense. According to the Tractatus, until we have done this, we have yet to confer any method of symbolizing on any of the signs that make up the string.

(Ibid., 404; emphases mine)

Making sense of sentence (1) in either of these two ways seems to require knowing an “established usage” of some of the component signs of (1). This evidently means knowing the ways those signs symbolize in meaningful sentences. Or, if one wants to avoid the reifying overtones of “ways,” one might put it thus: making sense of (1) requires knowing how these signs symbolize in other meaningful sentences. Thus, the second way of making sense of (1) requires knowing how the words ‘Chairman Mao’ symbolizes in clearly meaningful sentences such as

(2) Chairman Mao ate only boiled rice

Making sense of (1) then proceeds through attempting to understand the sign ‘Chairman Mao’ as it occurs in (1) as symbolizing in the same way as it does in (2).

It is inessential to this account that it’s the very same sign that occurs in the sentence one is trying to understand—call it the “target sentence”—and in sentences one already understands, and equally inessential that the sign occurring in the already understood sentence has a known established use. For example, one might try to make sense of (1) by taking ‘Chairman Mao’ as it occurs in (1) to symbolize in the same way as the sign ‘毛澤東’ symbolizes in

(6) 毛澤東在一九七六年去世

Nor, for that matter, does it seem essential to this account that the established use be what one might call a “conventionally” established use in some natural language. Suppose I use the sign ‘Chairman Mao’ in

(7) Chairman Mao is fond of chasing cats

to describe the behavior of my dog. Nothing, it seems, prevents me from trying to make sense of (1) by hypothesizing that ‘Chairman Mao’ in (1) symbolizes in the same way as that very sign does in sentence (7), as I use that sentence. This last claim is in fact ambiguous. My hypothesis could be that (1) is also about my dog; one might say that in this case that I’m taking these two sign-occurrences to be the same symbol. But I could also suspend judgment over whether (1) is about my dog, or indeed be quite sure that it’s not, and yet hypothesize that just as I use ‘Chairman Mao’ in (7) to describe my dog, so this sign is used in (1) to describe something. Either way, it seems I could thereby succeed in making sense of (1) using just this hypothesis. We could put the point this way: neither partial understanding nor misunderstanding is failure of understanding. So, symbolizing in the same way is implied by but distinct from being the same symbol.²⁶ The upshot of the first step of the argument is that it is crucial, for understanding sentences we have not [End Page 310] encountered before, that there is such a thing as two signs, occurring in distinct sentences, symbolizing in the same way (as opposed to being the same symbol). Call these “co-symbolizing” sign(-occurrence)s.

The second step of the argument establishes the notion of logical valency. The critical point is this: once one adopts a hypothesis about how a component sign of the target sentence symbolizes, the logical segmentation of the target sentence is no longer (fully) up to us. That hypothesis “dictates a possible segmentation of the” target sentence. Thus, once one adopts the hypothesis that the sign ‘Chairman Mao’ in (1) symbolizes in the same way as it does in (2), one cannot take ‘is rare’ in (1) to symbolize in the same way as that very string of letter symbolizes in

(3) An honest politician is rare

That is to say, one cannot take ‘___ is rare’ to function in (1) as a second-level predicate. This is why the two natural segmentations of (1) exclude one another. Of course this exclusion leaves other things open. For instance, it doesn’t prevent one from taking ‘___ is rare’ in (1) to symbolize in the same way as 在一九七六年去世 symbolizes in (6), since that sign functions as a first-level predicate in (6).

It should be fairly clear now in what sense there is a conception of logical valency in the view of the TCP developed by Diamond and Conant. Logical valency in this sense is not a property of meanings, and so of expressions bearing those meanings, that can be possessed independently of the use of sentential signs to express propositions. Logical valency depends, rather, on what logical segmentations of significant sentences are possible. So, for example, proper names have the right logical valency to combine with first-level predicates because, if one discerns or hypothesizes that a part of a sentential sign functions logically as a proper name, then the rest of that sentential sign, which, following Diamond, I’ll call the leftover part (1984, 134), would have to function as a first-level predicate if the entire sentential sign is to make sense. Similarly, proper names don’t have the right logical valency to combine with second-level predicates because, if one discerns or hypothesizes that a part of a sentential sign functions logically as a proper name, then the leftover part of that sentential sign cannot function as a second-level predicate if one is to make sense of the entire sentential sign. Note the direction of explanation. The claim is not that a sentence is meaningful or nonsensical because its component expressions have certain logical valencies specifiable independently of the meaningfulness or the nonsensicality of the sentences in which they occur. It is, rather, because a sentence is meaningful or meaningless when one attempts to take its parts as symbolizing in the same ways as they do in meaningful sentences that expressions symbolizing in these ways have the right or the wrong valencies to combine into a meaningful sentence.

The idea of combinations of symbols allowed by logic is very close to the surface in Diamond’s discussion of the other major stalking-horse of resolution, Carnap’s supposedly nonsensical sentence

(8) Caesar is a prime number [End Page 311]

Diamond spells out how one could turn (8) from nonsense to sense:

Suppose we accept that [(8)] makes no sense, because only of a number can it truly or falsely be said that it is a prime number. In that case it would follow, on the view of nonsense I am explaining, that one kind of Logical Element is: term-for-a-number. That sort of Logical Element can be combined with the predicate of

(9) 53 is a prime number

If ‘Caesar’ is defined as a number term, [(8)] can be regarded as a logical combination of that number term and the predicate term we have in [(9)] understood as it normally is. But unless ‘Caesar’ is defined in such a way, the last four words of [(8)] do not mean what they do in [(9)]; we do not have that numerical predicate, that Logical Element, any more than we have the proper name ‘Parkinson’ in ‘Smith has Parkinson’s disease’.

(1981, 101–2; emphases mine)

Here the idea is that the sign ‘___ is a prime number’, as it occurs in (9), is a particular type of Logical Element, i.e., it symbolizes in a particular way; let’s call this type of Logical Element numerical-predicate. In order for this sign, as it occurs in (8), to be this very Logical Element, i.e., for its occurrence in (8) to co-symbolize with its occurrence in (9), ‘Caesar’ in (8) cannot be the Logical Element that ‘Caesar’ is in

(10) Caesar crossed the Rubicon in 49 BC

as (10) is normally understood. This established usage of ‘Caesar’-in-(10) is as a proper-name-of-a-person Logical Element, so unless ‘Caesar’-in-(8) is given a new (i.e., not usual or established) definition so as to make it a term-for-a-number Logical Element, ‘___ is a prime number’-in-(8) is a different Logical Element from ‘___ is a prime number’-in-(9). That is to say, terms-for-a-number symbols can be combined with numerical-predicate symbols in a meaningful sentence, but proper-name-of-a-person symbols cannot be combined with numerical-predicate symbols in a meaningful sentence. Or rather, to put the point so that its conformity with the TCP is more explicit: in any meaningful sentence in which we can discern a numerical-predicate symbol (logically) combined with another symbol, that other symbol can be a term-for-a-number and cannot be a proper-name-of-a-person. (Clearly the other symbol also could be, e.g., a quantifier Logical Element.)

Just as in the case of (1), there is an alternative to this method of making (8) meaningful. We could take ‘Caesar’-in-(8) to symbolize in the same established way as its occurrence does in (10), i.e., to co-symbolize with ‘Caesar’-in-(10), as a proper name, and define the remainder of (8) so that it could be the kind of symbol that can combine with a personal proper name, for instance, what Diamond calls a “personal predicate.” In either case, however, one has to make a “new assignment of meaning” to make (8) meaningful. At this point Diamond writes,

If we make no such new assignments of meaning, the sentence is simply one which has some superficial resemblance to sentences of two distinct logical patterns; it has a word but no Logical Element in common with some sentences about Caesar, sentences of the pattern: [End Page 312]

(11) proper name of a person combined with personal predicate,

[such as

(10) Caesar crossed the Rubicon in 49 BC]

and it has words but no Logical Element in common with sentences [of the pattern:

term for a number combined with numerical predicate

such as

(12) 53 is a prime number]

It fails to be of . . . pattern [(11)], because a meaning of a certain sort has been determined for ‘is a prime number’; it fails to be of [pattern (12)], because a meaning of a certain sort has not been given to ‘Caesar’. Being of neither pattern (nor of any other), it is nonsense. It is our not having made certain determinations of meaning that we could make that is responsible for its being nonsense.

(Diamond 1981, 102; emphases mine; (11) and (12) my interpolations)

I can now characterize more fully a notion of logical syntax based on the implicit commitments of the austere conception I’ve been making explicit. Logical syntax comprises Diamond’s logical patterns, combinations of ways of symbolizing discernible in meaningful sentences, as well as the impossibilities for sentences to be meaningful through combinations of signs co-symbolizing with their occurrences in meaningful sentences. Logical syntax in this sense, like the logical valencies discussed earlier, is ultimately no more than the possibilities, necessities, and impossibilities in making sense of propositional signs.

On this view the austere conception allows for two kinds of nonsense involving only meaningless signs. There are nonsensical sentences some of whose component signs do not occur in any significant sentence. And there are nonsensical sentences p such that all component signs of p occur in significant sentences, but have no meaning as they occur in p because p cannot exemplify a logically possible pattern if its component signs symbolize in p in the same way that they do in the significant sentences in which they do occur.

That this conception satisfies the two constraints mentioned at the end of the last section is not hard to see. It is consistent with Diamond’s observation, because logical syntax doesn’t dictate any particular type of use of any sign, on pain of nonsense. ‘Something’ can be used intelligibly as a first-level predicate even if hitherto it has always been used as a quantifier phrase. Nor does logical syntax dictate that a sign may not be used in two types of ways in a single sentence, on pain of nonsense: ‘Green is green’ is perfectly intelligible. All that syntax dictates are such things as that there is no significant sentence in which ‘something’ occurs, is used in the same way as ‘He thinks he’s something’, and yet the leftover part of that sentence is also used as a first-level predicate. For instance, ‘Is green is something’ is not intelligible if ‘is something’ is used in the same way as it is used in ‘He thinks he’s something’, and ‘is green’ is used in the same way as it is used in ‘Green is green’. [End Page 313]

That on the present conception logical syntax is connected with sense and nonsense is obvious, and it shows that Hacker’s mistake lies not in the claim that logical syntax is in some way constitutive of making sense, but in looking at the wrong place—specific types of uses of individual expressions—for what is genuinely constitutive of intelligibility. Diamond’s observation exposes this mistake, but note that in the course of making the observation she also claims that “[w]hether anything determinate is being said when [a word with an established use] is given [a] new employment then needs to be looked at in the individual case” (2005, 86). That is, not just any new way of using a sign results in intelligible acts of saying. What, then, if anything, determines whether a new use is intelligible? The present answer is: the patterns of logical syntax. It is these patterns, then, and not types of use of individual expressions, that constitute the bounds of sense.

Let’s now come back to the questions I raised about how to understand 3.342 and 6.124. It is open to us to choose, arbitrarily, to use any sign in any way we want; indeed, nothing prevents us from using signs ambiguously (even if certain confusions may arise more easily if we do). Nothing prevents us from using ‘object’ as a variable, and also or instead as a first-level predicate. What is not up to us, not arbitrary, are the patterns of logical syntax, the ways in which signs symbolize in a meaningful sentence. Thus, once we have chosen to use ‘object’ as a variable, it is no longer up to us whether ‘object’ could be used in this way in a meaningful sentence whose leftover part is used as a proper name. That the leftover part has to comprise a part symbolizing as a quantificational expression and another symbolizing as an incomplete expression, and that the leftover part cannot symbolize as a proper name, are some of the patterns that are logical syntax. These necessities and impossibilities for the symbolizing of signs, given the uses of signs we have arbitrarily settled on, are “die Natur der naturnotwendigen Zeichen” that expresses or asserts itself in our attempts to make meaningful use of sentential signs, determining the successes and failures of those attempts independently of whether we think we have succeeded or failed in making sense.²⁷ Note the necessity of logic, on this conception, is rather different from Hacker’s explanation in terms of constitutive rules. Logical impossibility, on his view, is explained in terms of sentences’ being nonsensical, and nonsense, in turn, is explained in terms of uses of expressions that count as incorrect according to the rules of logical syntax. But on the present view, it should be clear, the modalities of logic are intrinsic to logical syntax since, as noted above, it just consists of the necessities, possibilities, and impossibilities of making sense of propositional signs.

I end this section by showing that this idea of patterns of symbolizing constitutive of significant propositions is implicit also in the work of some critics of Diamond and Conant’s resolute approach, specifically, Sullivan’s (2000) and Proops’s (2001a) account of symbolizing in the Tractatus, based on the passages from Notes on Logic and the Moore Notes:²⁸

Symbols are not what they seem to be. In ‘aRb’, ‘R’ looks like a substantive, but is not one. What symbolizes in ‘aRb’ is that ‘R’ occurs between ‘a’ and ‘b’.

(Wittgenstein 1979, 98) [End Page 314]

[T]he symbol of a property e.g. ψx, is that ψ stands to the left of a name form.

(Ibid., 116).

The reason why, e.g., it seems as if ‘Plato Socrates’ might have a meaning, while ‘Abracadabra Socrates’ will never be suspected to have one, is because we know that ‘Plato’ has one, and do not observe that in order that the whole phrase should have one, what is necessary is not that ‘Plato’ should have one, but that the fact that ‘Plato’ is to the left of a name should.

(Ibid., 116)

I will consider mainly Proops. On his account, in order for a sign to symbolize, facts about its occurrences in sentences have to have significance. Thus the occur-rence of ‘Plato’ in the string ‘Plato Socrates’ fails to symbolize because “the fact that a token of ‘Plato’ occurs to the left of a token of a name” “has not been given meaning” (Proops 2001a, 168). Similarly,

(13) is red is green

is a “nonsense string” because what “symbolizes in ‘predicative’ occurrences of the English expression ‘is green’ is the fact that a token of the sign ‘is green’ stands to the right of a token of a name,” but this string “exemplifies no fact that a token of ‘is green’ occurs to the right of a token of a ‘name’, since, obviously, ‘is red’ is not an English name” (ibid., emphasis mine).

Of course, it isn’t entirely obvious that ‘is red’ is not an English name, since nothing prevents, say, an idiosyncratic landlord from naming her pub (whose exterior is painted green) with just those two words. I doubt that Proops would disagree, and the reason is his account of understanding novel sentences:

When we are presented with a string of phonemes we first have to find the significant facts, if any, and then grasp their significance. This does not mean that understanding a novel sentence is a purely creative act, unconstrained by features of that sentence, for our interpretation of a novel sentence is guided by our recognition of certain familiar signs that crop up again in that sentence. In making sense of a novel sentence we will (tacitly) hypothesize that these signs play host to certain ‘symbolizing facts’ to which we have given meaning in the language. In a nonsense ‘sentence’ this preliminary hypothesis fails: there turn out to be no symbolizing facts in the string. But in a meaningful sentence, once we have recognized the relevant symbolizing facts in the presented string, we will be able to deploy our knowledge of their significance to make sense of the string.

(Ibid., 172)

It is clear that this account, like the Diamond-Conant view, rests on a notion of co-symbolizing. We start, and indeed have to start, with hypotheses that facts about sign-occurrences in the target sentence have the same significance as the same facts about occurrences of the same signs in sentences whose meanings we already understand.²⁹ In the case of ‘is red is green’ our initial hypotheses that ‘is green’ in this sentence symbolizes by occurring to the right of a token of a name fails only because our other initial hypothesis is that ‘is red’ symbolizes in the way it does in sentences we already understand, through the fact it stands to the left [End Page 315] of a name. But we could surely learn, from some locals, say, that in their English idiolect the string ‘is red’ symbolizes in certain sentences through the fact that in those sentences it stands to the left of predicates, and then we would be in a position to understand that sentence as perfectly meaningful. What this shows is that whether “there turns out to be symbolizing facts in the string” depends on whether it is possible for our co-symbolizing hypotheses to fit together. If ‘is green’-in-(13) symbolizes by occurring to the right of a token of a name, then in order for (13) to be meaningful it is impossible for the remaining signs, ‘is red’-in-(13) to symbolize by occurring to the left of a predicate.

The essential involvement of modality in this account of symbolizing is explicit in Wittgenstein’s use of it in the Moore Notes to dissolve Russell’s Paradox. The argument focuses on a sign that is part of the Principia sentence resulting from eliminating all the contextually defined terms from a sentence that purports to state that some class contains itself as a member: φ{ψ!û}. Wittgenstein writes that φ “cannot possibly stand to the left of (or in any other relation to) the symbol of a property. For the symbol of a property, e.g., ψx, is that ψ stands to the left of a name form, and another symbol φ cannot possibly stand to the left of such a fact” (Wittgenstein 1979, 116; second emphasis in original). Now, Proops’s explanation of this argument is, “for the predication ‘φ(ψx)’ to have significance—given the fact . . . that ‘φ’ and ‘ψ!’ symbolize in the same way—there would have to be significance to the fact that the letter ‘φ’ stands immediately to the left of another fact, and of course there are no such facts” (2001a, 171). I take it that Proops is claiming something stronger than: it just happens that there are no such facts. He is surely claiming that there can be no such facts.

V. PROPOSITIONS AS TRUTH-FUNCTIONS AND PATTERNS OF LOGICAL SYNTAX

We have now seen that a view of patterns of symbolizing constitutive of intelligibility of propositions is (largely tacitly) discerned in the Tractatus by resolute interpreters and some of their critics. Is there basis in the Tractatus for connecting this idea to the idea of what is not arbitrary in the symbols we use, explicitly stated in 3.342 and 6.124? In this section I argue that there is. I begin in 5.1 below by tracking down two particularly puzzling sets of remarks in the Tractatus, occurring after 3.342, that link notations with necessity. After outlining the interpretive difficulties they pose, I show, in 5.2, that on the basis of the kind of metaphysical semantics that underlies Hacker’s account of tautology one can make considerable progress on the interpretive difficulties, and formulate an account of logical necessity more satisfying than Hacker’s. However, this account still, like Hacker’s, grounds the necessity of logic on semantic features of propositions that seem to have to be necessary, but whose necessity is unclear. Moreover, in face of the arguments against ascribing commitment to semantic and metaphysical theories to [End Page 316] the Tractatus, it’s worthwhile to see if our interpretive puzzles can be resolved in a different way. In 5.3 and 5.4 I sketch an alternative to the semantic approach. The central idea of this alternative, developed in 5.3 by considering some texts of Frege from a Tractarian perspective, is an account of what it is for a proposition to be a truth-function: it is for two things that are done with propositional-signs— affirming and denying them—to be subject to standards of correctness that fall in specific patterns. These are the patterns of logical syntax; conformity to these patterns is constitutive of the intelligibility of picturing the world with propositions. In 5.4 I complete the sketch of this alternative by arguing, again by drawing on Frege, that the patterns of logical syntax are not facts about how we do or must reason. Moreover, I argue that although, in a sense, they are features of propositions, they are features in which the necessities and impossibilities of logic show forth.