1 Introduction

F. H. Bradley famously argues that the idea of a relation combining distinct entities into a unity involves a vicious regress. Let R be a relation purportedly combining two distinct entities a and b into the unity consisting of Rab. Since R itself is an entity distinct from both a and b, the question immediately arises of what combines the distinct entities R, a, and b. The answer cannot be that some other relation R* combines the distinct entities R, a, and b into a unity, since R* is an entity distinct from R, a, and b for which the question then arises of what combines R*, R, a, and b into a unity, the answer to which cannot be that some even further relation R** combines R*, R, a, and b into a unity, since R** is an entity distinct from R*, R, a, and b, and so on ad infinitum. In Bradley’s words, “we are hurried off into the eddy of a hopeless process, since we are forced to go on finding new relations without end.” (Bradley 1995, 130) The same vicious regress applies not just to a dyadic but to any polyadic predicate. Therefore, the idea of a relation combining distinct entities into a unity is incoherent.

Bradley finds the idea of a subject possessing a quality to be equally incoherent:

Sugar is obviously not mere whiteness, mere hardness, and mere sweetness; for its reality lies somehow in its unity. But if, on the other hand, we inquire what there can be in the thing beside its several qualities, we are baffled once more. We can discover no real unity existing outside these qualities, or, again, existing within them. (Bradley 1995, 119)

Let s be a subject purportedly possessing a quality Q. Since s and Q are distinct entities, the question immediately arises of what combines s and Q into the unity consisting of Qs. It cannot be some further dyadic relation D, since the question then arises of what combines the distinct entities D, s, and Q into a unity, to which the answer cannot be some yet further triadic relation T, since the question then arises of what combines T, D, s, and Q into a unity, and so on ad infinitum.

Some contemporary metaphysicians continue to grapple with Bradley’s regress objection. A case in point is Graham Priest, who worries that if one takes the relation of instantiation to unify an object a and a universal U into the fact that a is U, “then a vicious regress does loom” (Priest 2016, 46). He proposes that what unifies a and U into the fact that a is U is an intrinsically contradictory entity or “gluon” g that both possesses and does not possess all the properties of a as well as all the properties of U. Following Leibniz, Priest defines the identity of x and y (“x = y”) as “∀(P) (Px ↔︎ Py),” where “∀(P)” is a second-order quantifier that ranges over all and only properties and “↔︎” expresses the material biconditional.¹ It follows that g = a and g = U, so that a is U is a fact, even though a ≠ U. Priest’s dialetheist proposal is technically impeccable yet philosophically thin. Someone like Bradley who is fundamentally mystified by the alleged unity of an entity with a universal or property will be no less mystified by the alleged unity of the gluon g with the properties of both a and U that g possesses (and also does not possess) than he is by the alleged unity of the object a with the universal U. On the other hand, if g’s unity with the properties it possesses can be taken for granted, then it is obscure why the original unity of a with U cannot be taken for granted. But then there is no need postulate a metaphysics of gluons to explain the real unity between a subject and its qualities.

In his account of how early analytic philosophy represented by Bertrand Russell emerged in part as a reaction to British Idealism represented by Bradley, Peter Hylton observes:

For both Bradley and Russell the question of the reality of relations is connected with issues of the nature and possibility of unity—especially the unity of diverse elements in a proposition. Bradley finds this issue intractable, and this is, I think, essential to his denial of the reality of relations. Russell finds it no less intractable, but is content to leave it as an unsolved problem within his philosophy, without letting it affect his insistence on the reality of relations. (Hylton 1992, 12)

According to Hylton, Russell simply affirms the reality of relations without refuting Bradley’s argument against them. Similarly, Russell simply takes for granted the reality of subjects and the qualities they possess. The question of whether Russell has more substantial philosophical resources available for resisting Bradley’s argument will not be pursued here. Additional aspects of Bradley’s idealist monism will also not be considered, given that Priest and other contemporary metaphysicians focus on Bradley’s regress objection as a threat to any kind of ontological plurality.²

Instead, it may be wondered what lessons regarding Bradley’s regress might be learned from Wittgenstein’s Tractatus. Wittgenstein urges that to avoid the most fundamental confusions of philosophy, “we must employ a symbolism which excludes them,” (Wittgenstein 1986, 3.325) adding that “The logical symbolism of Frege and Russell is such a language, which, however, still does not exclude all errors.” (1986, 3.325) Aspects of the Tractatus do indeed prove effective in enabling us to resist Bradley’s regress. In particular, what Wittgenstein says with regard to elementary propositions and simple objects, truth, the eliminability of the identity sign, the nature of logic, and the mystical feeling of the world as a limited whole will figure in the following investigation.

2 Elementary Propositions and Truth

Initially, Wittgenstein’s reply to Bradley’s challenge seems straightforward. Wittgenstein holds that the combination of words in a proposition (propositional sign, or Satz) is an isomorphic picture that models the configuration of objects in a possible fact. (1986, 2.21, 2.15)³ Every proposition is a truth-functional combination of elementary propositions expressible in terms of the single connective of joint negation. (5.5) Elementary propositions are concatenations of names for simple objects (4.22) occurring in the possible atomic facts (2.014, 2.02) modeled by the corresponding elementary propositions. Wittgenstein offers no examples of elementary propositions, insisting that even though they cannot be given a priori (5.5571), the complete analysis of any complex proposition terminates at least in principle in unanalyzable elementary propositions (5.5562). Elementary propositions contain no relation symbols, and simple objects have no qualities but contain within themselves the possibility of combining to form different atomic facts (2.0141), so that simple objects in the atomic fact “hang one in another, like the links of a chain” (2.03). Since objects neither stand in real relations to one another nor possess real qualities, Bradley’s challenge appears unable to get off the ground.

Appearances can be deceiving. Wittgenstein remarks, “The elementary proposition I write as a function of the names, in the form”fx”, “ϕ(x,y), etc.” (4.24). Thus one example of an elementary proposition is

1. fa

where “a” names a simple object and “f” is a unary function symbol. It is by no means apparent how (1) can model any possible atomic fact of which the sole constituent is the simple object named by “a.” For any such fact contains only one item, whereas the concatenation of “f” and “a” in (1) contains exactly two, so that the isomorphism required for (1) to model the fact is absent. Ideally, an elementary proposition modeling the possible atomic fact should only contain a single occurrence of a name for the simple object, i.e. “a” treated as a name rather than as an open variable. But “a” is not a proposition, elementary or otherwise. Presumably Wittgenstein wants to admit a sentence like (1) as an elementary proposition in order to allow for singular truths about a simple object, such as that it occurs in an (unspecified) actual atomic fact. Yet if “f” ascribes the real quality of occurring in an actual atomic fact to the simple object named by “a,” then Bradley regains a foothold for his regress objection against the coherence of a subject possessing an ontologically distinct quality.

John Wisdom, an early commentator on Wittgenstein’s logical atomism, wrestled with this problem.⁴ Wisdom considers the possible atomic fact this is red in connection with the elliptical propositional sign “This red” from which the copula has been deleted. He concedes that the propositional sign consisting of exactly two words does not perfectly model the atomic fact containing the single simple object this. Wisdom proposes that “This red” perfectly models what he calls the first derivative fact this is characterized by red containing the simple object this and the characteristic red. Hence the same propositional sign models—albeit imperfectly and indirectly—the original atomic fact from which the first derivative fact is derived.⁵

As it stands, Wisdom’s proposal is not available to Wittgenstein, who does not countenance first derivative facts in addition to atomic facts. Moreover, since Wisdom’s proposal postulates simple objects as well as their qualities as distinct ontological elements, it remains vulnerable to Bradley’s regress objection. Nonetheless, Wisdom’s basic idea that different propositions can model the same fact more or less accurately sheds light on how Wittgenstein may be thinking of elementary propositions like (1). The key is Wittgenstein’s conception of truth.

Wittgenstein denies that truth and falsity are properties of propositions:

Theories which make a proposition of logic appear substantial are always false. One could e.g. believe that the words “true” and “false” signify two properties among other properties, and then it would appear as a remarkable fact that every proposition possesses one of these properties. (1986, 6.111)

Wittgenstein maintains that “the proposition is the expression of its truth-conditions” (4.431). As he elaborates in 4.063, “in order to be able to say ‘p’ is true (or false) I must have determined under what conditions I call ‘p’ true, and thereby I determine the sense of the proposition.” If truth were a property of propositions, then to determine the sense of a proposition p would require determining when some other proposition q ascribing truth to p would be true—i.e., determining the sense of q—which in turn would require determining when some other proposition r ascribing truth to q would be true—i.e., determining the sense of r—and so on ad infinitum so that, contrary to what is in fact the case, the sense of original proposition p would never be determined. Therefore, truth is not a property of propositions.

Bradley of all people should be highly receptive to Wittgenstein’s argument against truth as a property of propositions. It is merely a special instance of Bradley’s more general critique that any plurality of particulars together with properties and relations leads to a vicious regress. This receptivity enables Wittgenstein to construct, for any possible atomic fact with a single simple object as its sole constituent, a propositional sign that models the fact without insinuating there to be a metaphysical plurality consisting of the simple object somehow combined with an ontologically distinct quality.

Suppose that the proposition (1) is true. Then there is also the following proposition:

2. “fa” is true

where “fa” names the proposition (1). Since truth is not a property of propositions, the occurrence of the predicate “is true” in (2) does not ascribe the property of truth to (1). Nor does (2) contain any expression designating a real quality. The name ‘ “fa” ’ is the only designative expression that (2) contains. Yet (2) models the same possible fact that (1) models; otherwise, truth would be a real property ascribed by (2) to (1). Hence the possible fact that contains no quality but is confusingly modeled by the subject-predicate proposition (1) is more accurately modeled by the single-name proposition (2) with a dummy predicate that does not ascribe any real quality to what is named.

A possible objection is that (2) cannot model the same fact as (1). For (2) contains a name that denotes the proposition (1) while (1) contains a name that denotes a non-propositional simple object. Wittgenstein’s idea of propositional signs as models or pictures blunts this objection. A picture of a picture of X that preserves the structure of the latter picture is also a picture of X. For example, a compositionally accurate photograph of a compositionally accurate photograph representing the Eiffel Tower also represents the Eiffel Tower. By accurately picturing the potentially misleading proposition (1) in a way that keeps (1)’s syntactic structure intact via quotation, proposition (2) also pictures the possible fact pictured by (1) while more perspicuously “showing” that this possible fact does not contain any real quality but only a single simple object as constituent.

This intuition is buttressed by Wittgenstein’s Bradley-style regress argument that truth is not a property of propositions. Like (1), (2) is a perfectly intelligible proposition asserting some fact to obtain. The fact that (2) asserts to obtain is obviously not an atomic fact entirely different from the atomic fact that (1) asserts to obtain. Nor, in light of Wittgenstein’s regress argument, does (2) assert that (1) possesses the ontologically distinct property of truth. Therefore, the fact that (2) asserts to obtain is the same atomic fact that (1) asserts to obtain—only (2) does so in a manner without suggesting that there is a simple object denoted by “a” plus a real quality expressed by “f.”

Another possible objection is that Wittgenstein is merely speaking loosely when he gives (1) as an example of an elementary proposition. More precisely, a complete analysis of (1) would reveal it to be a truth-functional combination of truly elementary propositions, none of which contain any unary function symbols like “f” which might be mistaken to designate qualities, but only a configuration of two or more names for simple objects that mirrors their configuration in the corresponding possible atomic fact, as in the truly elementary proposition

3. bac

modeling the possible atomic fact in which the simple object named by “a” is between the simple objects named by “b” and “c.” Unfortunately, Wittgenstein never indicates how a subject-predicate proposition like (1) can be analyzed as a truth-functional combination of multi-name proposition like (3). In the absence of such an analysis, Bradley can point to the unanalyzed (1) as a proposition that tries to ascribe a real quality to an ontologically distinct object and thus falls prey to the regress. Wittgenstein’s deflationary view of truth provides him with an insurance policy against this metaphysical risk: the availability of a proposition like (2) for any proposition like (1) “shows” that no proposition like (1) ascribes a quality to an object.

The same syntactic strategy does not work for multi-name propositions like (3). The result of concatenating a name for (3) with Wittgenstein’s dummy truth predicate would be a proposition like

4. “bac” is true

which does not model more accurately the possible atomic fact modeled by (3). For the latter fact is a configuration of three simple objects denoted by the three names contained in (3), whereas (4) contains just one name denoting the proposition (3). Wittgenstein’s dummy truth predicate retains its utility as a tool for resisting Bradley’s idea that propositions like (1) try to ascribe qualities to objects. Additionally, the absence in elementary propositions like (3) of any symbols for relations short-circuits Bradley’s regress objection to the possibility of real relations combining distinct entities into a unity, since there simply are no real relations.⁶ However, the Tractatus suggests a deeper diagnosis of what has gone wrong in Bradley’s thinking about relations. To see why, let us turn to Wittgenstein’s elimination of the identity sign from his logical notation.

3 Logical Syntax and Relation Symbols

In 5.6 Wittgenstein notes, “The limits of my language mean the limits of my world.” We cannot think about the world without using language to do so. The same point applies to relations: to formulate Bradley’s regress objection that a relation cannot combine distinct entities into a unity, we must be able to think about some relation and the distinct entities it is supposed to combine. Since we cannot think of anything without using some language to describe it, in trying to think about Bradley’s regress objection against relations it is appropriate to relax Wittgenstein’s strictures against relation symbols in an ideal logical notation. In 5.53, Wittgenstein dispenses with the identity sign: “Identity of the object I express by identity of the sign and not by means of a sign of identity. Difference of the objects by difference of the signs.” The distinct objects the relation is supposed to relate are indicated just by assigning different names or variables to those objects. So we may provisionally use a relation symbol as well as different names or variables for the non-identical relata the relation is supposed to relate. Wittgenstein himself gives some examples. Where “f” here is a two-place relation symbol and “a” and “b” are different names, rather than “f (a, b) . a ≠ b” he writes just “f (a, b)” (5.531). And rather than “∃x ∃y (f(x, y) . x ≠ y” he simply writes “∃x ∃y f(x, y)” (5.532).

Wittgenstein’s convention for expressing identity or difference of objects by identity or difference of signs admits of two interpretations.⁷ Each interpretation can be illustrated in connection with the second example from 5.532:

5. ∃x ∃y f(x, y)

On weakly exclusive interpretation, the range of the quantifier binding a variable must be restricted to exclude whatever value is assigned to any variable that occurs free within the scope of that quantifier. Since the variable “x” occurs free within the scope of the quantifier “∃y” binding the variable “y,” the range of “∃y” must then be restricted to exclude whatever value is assigned to “x.” Outside the scope of “∃y,” “x” is bound by the quantifier “∃x.” Thus by working from the inside out, so to speak, the difference between the object assigned to “x” and the object assigned “y” in (5) can be expressed without using the identity sign “=” with the negation sign (“≠”).

The strongly exclusive interpretation moves in the opposite direction, working from the outside in. It stipulates that no object in the range of a quantifier can be assigned to any variable that occurs within scope of that quantifier but is unbound by it. In (5), the variable “x” bound by the quantifier “∃x” does not occur within the scope of some other quantifier that does not bind “x.” But since the variable “y” bound by the quantifier “∃y” occurs unbound within the scope of the quantifier “∃x,” no object in the range of “∃x” can be assigned to “y” and hence to the range of the quantifier “∃y.” Once again, the difference between the object assigned to “x” and the object assigned to “y” in (5) can be expressed without using “≠.”

Although these two interpretations yield the same result for (5), for other sentences they do not. Consider the sentence “∀y (∀x Px → Py).” On the weakly exclusive interpretation, since the sentence contains no variable that occurs free within the scope of any quantifier, the ranges of the quantifiers “∀y” and “∀x” are not restricted to exclude any objects. So construed, the sentence says anything is such that if everything has P then it has P and thus the sentence is logically true. On the strongly exclusive interpretation, since the variable “x” (though bound by the quantifier “∀x”) occurs within the scope of the quantifier “∀y” by which “x” is unbound, no object in the range of “∀y” can be assigned to “x” and hence to the range of “∀x.” The original sentence is then equivalent to “∀y (∀x (x ≠ y → Px) → Py)” which says anything is such that if everything other than it has P then it has P. So construed, the sentence is not true in a case where something does not have P but everything else does, and thus the sentence is not logically true. ⁸

Frank Ramsey, who discussed the Tractatus extensively with Wittgenstein, rejects the strongly exclusive interpretation of the convention for eliminating the identity sign in favor of the weakly exclusive interpretation as Wittgenstein’s considered opinion.⁹ The reason has to do with sentences like

6. ∀y (∀x R(x, x) → R (y, y))

where “R” is a relation symbol. On the strongly exclusive interpretation, the variable “x” and hence the range of the quantifier “∀x” must be restricted to exclude any object in the range of the quantifier “∀y.” Yet when the sentence “∀x R(x, x)” occurs alone, “∀x” is not restricted to exclude any object in the range of “∀y,” since then “∀x” does not occur within the scope of “∀y.” Thus the sense “∀x R(x, x)” has when it occurs alone is not the same as the sense it has when it occurs within the scope of “∀y” in (6), since in the latter case its range is restricted to exclude any object in the range of “∀y” while in the former case it is not.¹⁰ But then one cannot understand (6) by understanding its constituent parts—including the constituent part “∀x R(x, x)” when it occurs alone. This consequence conflicts with Wittgenstein’s dictum that “One understands it [a proposition] if one understands its constituent parts” (4.042).¹¹ No such consequence follows on the weakly exclusive reading, according to which the range of “∀x R(x, x)” is unrestricted both when it occurs alone and when it occurs in (6).

One might be tempted to regard the weakly exclusive interpretation of (6) as nothing more than a purely arbitrary stipulation excluding any objects in the range of the quantifier “∀x” from the range of the quantifier “∀y.” On the contrary, the weakly exclusive interpretation of (6) and other quantified sentences flows from Wittgenstein’s commitment to the determination of the sense of a proposition by the senses of its parts and the determination of reference by sense. If the range of the quantifier “∀x” in “∀x R(x, x)” when the latter occurs in (6) were restricted to exclude any objects in the range of the quantifier “∀y” in (6), then the sense and reference of the whole proposition would determine the sense and reference of one of its component expressions rather than vice versa. But combining the complete expression “∀x R(x, x)” with the incomplete expression “∀y ( ______→ R (y, y))” and restricting the range of the quantifier “∀y” in the resulting sentence to exclude any objects in the range of the quantifier “∀x” in in “∀x R(x, x)” when the latter occurs alone ensures that the sense and the reference of the whole proposition is determined by the sense and reference of its component expressions.

Philosophically, Ramsey’s rejection of the strongly exclusive interpretation in favor of the weakly exclusive interpretation based on his conversations with Wittgenstein has significant repercussions for Bradley’s regress objection against relations. In thinking about relations and the entities they are supposed to relate, we think in terms of a provisional logical notation featuring a relation symbol along with different names or variables for different objects yet lacking the identity sign. Letting our notation be our guide, we see that there is indeed a legitimate issue regarding how a relation symbol can be combined with other expressions to form propositions depicting possible facts in accordance with the requirement that one understands a proposition by understanding its constituent parts. But we also see that the same issue can be resolved by implementing the convention for eliminating the identity sign in accordance with the weakly exclusive interpretation. To put the point in a way that Carnap and other members of the Vienna Circle impressed by the Tractatus would find congenial, the seemingly intractable riddle of relations becomes a manageable problem of logical syntax.

It might be protested that this Tractarian response to Bradley’s regress objection against relations comes at the high price of abandoning the idea that there are real relations in favor of the purely linguistic expedient of a logical syntax containing relation symbols combined with names in accordance with the weakly exclusive interpretation. Here, we do well to return to Wittgenstein’s remark that the limits of my language mean the limits of my world. Any thinking about relations must be conducted in a language featuring expressions for relations and the objects they supposedly relate. Bradley’s regress objection itself is formulated in just such a language. A Tractarian logical notation governed by the weakly exclusive interpretation allows us to construct true sentences containing relation symbols and names for objects without engendering a vicious regress. Since the limits of our language are the limits of our world, we have no purchase on what “realism” about relations is supposed to be, other than the availability of true relation-sentences formulated in a logically perspicuous language governed by the weakly exclusive interpretation. This internal realism eschews any external realism that attempts to step back from our language in search of a metaphysical foundation undergirding it.¹²

A further concern is that even if the weakly exclusive interpretation shows how true sentences containing relation symbols and names for objects are immune to Bradley’s regress, Wittgenstein’s logical notation is committed to metaphysically problematic relations in another way. A sentence like “R(a, b)” containing the relation symbol “R” and the names “a” and “b” is ultimately analyzable in Tractarian terms as a truth-functional combination of elementary propositions using only the truth-functional connective “|” of joint negation. (Wittgenstein 1986, 5, 5.51) Although Wittgenstein provides no example of an ultimate analysis, for the sake of simplicity suppose that the ultimate analysis of “R(a, b)” is

(7) | (fc, gd)

Where “fc” and “gd” are elementary propositions containing the unary function symbols “f” and “g” with the names of “c” and “d” for simple objects. Now let “λ” symbolize the following relation:

(8) λ (x, y) ≡ | (fx, gy).

That is, λ is the relation that obtains between any two simple objects x and y just in case they occur in the atomic fact modeled by the elementary proposition containing names for x and y concatened with “|.” Thus the relation λ obtains between the simple objects c and d which are named in the elementary propostion (7). In particular, the sentence “λ (c, d)” mirrors the relational fact modeled by (7).¹³ But then Bradley’s regress objection gains a toehold with the relation λ.

The foregoing concern assumes that the truth-functional connective “|” in (7) represents a relation that is mirrored by the relation λ between c and d. However, Wittgenstein denies that “|” and the other logical constants represent anything at all. (4.0312) His basis for the denial is made apparent by the last line of the truth-table (9), which he deems to be a logically more perspicious propostional sign for the elementary fact modeled by (7):

The truth-table (9)—specifically, its final column “FFFT”—displays in a most perspicious manner the logical form of (7). (4.442) Neither “|” nor any relation symbol occurs in that final column of (9). Therefore, the logical form of (7) as perspicuously depicted by (9) does not include any relation that is mirrored by the relation λ. Of course, once can always introduce “λ” defined as a symbol for an external relation between simple objects. Yet any such introduction is entirely extraneous to the most perspicuous version of Wittgenstein’s Tractarian logical syntax. Nothing in the syntax itself affords a toehold for Bradley’s regress objection.

Unfortunately, at times Wittgenstein himself does seem to succumb to the metaphysical temptation of external realism. In doing so, he opens the door to a potentially powerful Bradleyan critique.

4 Necessity, Impossibility, and the Mystery of the World as a Limited Whole

Let us approach the Bradleyan critique by considering Ishiguro’s argument for why the simple objects of the Tractatus are not sense-data. Citing Wittgenstein’s remarks that if all objects are given then all states of affairs are given (2.0124), that the whole of logical space is already given with a proposition (3.42), and that if the objects are given then at the same time all objects are given, (5.524) Ishiguro argues that “Such claims would be nonsense if we were to understand the given as something we are immediately aware of, or anything like Moore or Russell’s sense-data, or even objects of phenomenological reflection.” (2001, 40) Identifying simple objects with sense-data would have the absurd consequence that all visual, auditory, olfactory, gustatory, and tactile sense-data are directly perceived simultaneously.

It is what Ishiguro says earlier about simple objects that provides a toehold for the Bradleyan critique:

Even when we cannot get to the simplest propositions, we assume that there must be propositions whose truth or falsity does not depend on any other propositions. There is no reason to think that these logically simple propositions are all about the same kind of things, but whatever these propositions are about, these are called objects. (2001, 30)

Wittgenstein does claim that “If we know on purely logical grounds, that there must be elementary propositions, then this must be known by everyone who understands propositions in their unanalyzed form.” (5.5562) The question is how the claim that there must be simple objects corresponding to elementary propositions is to be understood as a claim of logical necessity.

A cardinal Tractarian tenet is that “Certainty, possibility or impossibility of a state of affairs are not expressed by a proposition but by the fact that an expression is a tautology, a significant proposition or a contradiction.” (5.525) Specifically, the only kind of necessity countenanced in the Tractatus is the logical tautology (“certainty”) of a complex proposition that comes out true under any assignment of truth-values to its constituent propositions. The sentence “There are simple objects corresponding to elementary propositions” appears to affirm a substantive metaphysical fact regarding the ontological foundations of all propositional discourse. From the vantage point of the Tractatus, the only way for the same sentence to express a necessity is for it to be a logical tautology. Yet a logical tautology does not express any fact, let alone a substantive metaphysical fact about simple objects. It is, as Wittgenstein emphasizes, sinnlos.¹⁴ Bradley agrees: “A bare tautology (Hegel has taught us this, and I wish we could all learn it) is not even so much as a poor truth or a thin truth. It is not a truth in any way, in any sense, or at all.” (1995, 214) The Bradleyan critique is the Tractarian reductio that Wittgenstein’s own explication of necessity as tautology strips his metaphysical thesis about simple objects of any substantive content.

Interestingly, the same may be said of the opposing metaphysical thesis Bradley accepts:

The whole reality is so immanent and so active in every partial element, that you have only to make an object of anything short of the whole, in order to see this object pass beyond itself. The object visibly contradicts itself and goes on to include its complementary opposite in a wider unity. And this process repeats itself as long as and wherever the whole fails to express itself in the object Hence the two principles of coherence and contradiction are one. And not only are they one but they include also the principle of non-contradiction. (Bradley 1995, 314)

From the impossibility that the world is a plurality of objects standing in relations and having qualities, we are supposed to infer that the world is a single, coherent, all-encompassing reality—the Absolute—possessing the same necessity as the principle of non-contradiction it includes. In Tractarian terms, the only kind of impossibility is the logical contradiction of a complex proposition that comes out false under any assignment of truth-values to its constituent propositions. As sinnlos, a logical contradiction says nothing at all and ipso facto says nothing about how the world cannot be (i.e., a plurality of objects, relations, and properties), just as a sinnlos logical tautology says nothing about how the world must be (i.e., the sole ultimate reality of the Absolute). Bradley’s monism is no less vulnerable to the reductio than Wittgenstein’s pluralism is.

The situation is reminiscent of Kant’s antinomies, where diametrically opposite metaphysical theses like “The world is finite” and “The world is infinite” are equally susceptible to philosophical “proof.”¹⁵ For Kant, these pseudo-disputes can be avoided by limiting the operation of reason to judgments in which concepts are applied to empirical or pure intuitions. Similarly, if we try to step back and state what cannot be true of the world a whole as well as what must be true of the world as a whole in order for our propositional discourse to make sense, the result is opposing “claims” like “The world cannot be a single reality but must consist of simple objects” and “The world cannot consist of simple objects but must be a single reality” which are equally devoid of content. For Wittgenstein, “The contemplation of the world sub specie aeterni is the contemplation of it as a limited whole,” (6.45) where the emptiness of such claims becomes obvious when attributions of impossibility and necessity are limited to the perspicuous Tractarian elucidation of them as, respectively, logical contradiction and logical tautology. We may be mystified that there is any world at all rather than nothing: “Not how the world is, is the mystical, but that it is.” (6.44) But if we are to avoid becoming entangled in empty antinomies, this mystical feeling should not serve as an impetus for further metaphysical speculation.

We are then left with a purely internal realism in which an expanded Tractarian logical notation of propositional signs, truth-functional connectives, monadic predicates, relation symbols, and quantifiers governed by the weakly exclusive interpretation can be used to formulate true relation-sentences. Perhaps it is no surprise that in his later writings, Wittgenstein drops all talk about simple objects, elementary propositions, and the correspondence between them. Such talk can all too easily be misconstrued as a “transcendental realism” that purports to lay bare the ultimate ontological foundations of our language.