From the ALWS archives: A selection of papers from the International Wittgenstein Symposia in Kirchberg am Wechsel

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Wittgenstein, in the Tractatus Logico-Philosophicus, states that we do not require a sign for identity in an ideal logical notation. (TLP 5.533) His intention is not to do away with the notion of identity, but rather to show that we may do without the sign for identity as an expression of our notion of identity. Instead of using an identity sign, we are to express sameness of object by way of a sameness of sign; difference of object would then be expressed by way of a difference of sign. A motivation for this change in conventions is that, according to Wittgenstein, the use of the identity sign can lead to philosophical confusion. One of the confusions Wittgenstein endeavours to dispel is the view that identity can be a relation between an object and itself. If self-identity is to present a genuine relation, such that it is universally and trivially true of any object that it is self-identical, then self-identity may be used as a universal necessary condition for objecthood. The use of such a criterion would allow us to refer to objects qua objects, in disregard of their specific properties. This criterion is used by Russell in his Axiom of Infinity, and Frege in his definition of the number zero. Moreover, the use of a criterion of self-identity to refer to the totality of objects, qua objects, stands opposed to Wittgenstein’s claim that the world is a totality of facts, not objects, presented at the outset of the Tractatus. In the following, I will show why, for Wittgenstein, identity is not a genuine relation between an object and itself. This will involve showing that assertions of self-identity are nonsense. These assertions are neither true nor false, and a fortiori, not trivially or universally true.

In question is whether assertions of identity between an object and itself are simply without sense, or whether they are nonsense. If these statements are senseless, as all mathematical propositions are for the early Wittgenstein, then they are true but trivially true. That is, it is trivially true that an object is identical to itself, and so it is universally true as well. These statements are senseless for being trivial and uninformative, but they are nonetheless true statements. If they are nonsense, however, then they are neither true nor false; they are meaningless assertions. Thus, if these assertions of self-identity are nonsense, then it is not true, and hence not trivially and universally true, that an object is identical with itself. The difference between whether statements of self-identity are senseless versus nonsense is a difference between whether self-identity is true of all objects (and thus provides a criterion of objecthood that allows us to refer to objects qua objects, in disregard of any other properties) or not true of any object (in which case self-identity is not a criterion for objecthood).

At first sight, Wittgenstein seems to be ambiguous on this point. On the one hand, he seems to conclude that such alleged identity statements are nonsense when, in referring to the identity of objects, he states: “So all problems disappear which are connected with such pseudo-propositions.” (TLP 5.535) It does not seem that the philosophical problems associated with self-identity would disappear if the assertion of self-identity was merely senseless for this still upholds that the assertion is true (albeit trivially true, just as with mathematical propositions). Indeed, it is, and was during Wittgenstein’s early period, the accepted view that the assertion of the self-identity of an object is trivially and universally true (and thus senseless under Wittgenstein’s rendering). Thus, when Wittgenstein concludes that the problems associated with self-identity will “disappear”, it seems this should be on finding assertions of self-identity to be nonsense rather than senseless. On the other hand, Wittgenstein suggests otherwise, also in the Tractatus: “…to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing.” (TLP 5.5303). In this remark, the assertion of the self-identity of an object is not upheld as nonsense, but rather as saying “nothing”. This passage suggests that the assertion of self-identity is something different from nonsense, and thus that “nothing” should be read here as uninformative or trivial (i.e., senseless).

Turning to the Philosophical Investigations does not help much. Wittgenstein herein states, “ ‘A thing is identical with itself’ - There is no finer example of a useless proposition.” (PI §216). “Useless” may be interpreted as trivial or uninformative, and hence senseless. However, if we interpret “useless” as meaningless (as per the statement of “meaning is use” in PI §43) then an assertion of self-identity is meaningless, and hence presumably nonsense. In the least, the matter is ambiguous if we are left to these remarks on self-identity from the Tractatus and the Investigations.

A correspondence between Wittgenstein and Ramsey provides some clarification. Ramsey interpreted identity in a way he thought consonant with the Tractatus: He upheld true identity statements to be tautologies and false ones to be contradictions. “In reply”, Hans-Johann Glock conveys, “Wittgenstein insisted that a false identity statement involving logically proper names is nonsensical rather than contradictory, and that the same holds for true identity statements, since the negation of nonsense is itself a nonsense.” (Glock 1996, 167) Thus, despite the noted ambiguity in the Tractatus, the early Wittgenstein did affirm that an assertion of self-identity is nonsense. I will now explain this.

Consider this passage from Friederich Waismann (the content of which, he remarks, is largely drawn from Wittgenstein):

If it makes sense to ask whether the [two] armchairs can be distinguished, then they are two armchairs; if this question makes no sense, then it is one chair. In other words, the question whether two things are identical is not the question whether they can be distinguished, but whether it makes sense to ask whether they can be distinguished. (Waismann 1977, 26)

According to Waismann, the question to consider concerning the identity of objects is not whether the objects can be distinguished, but whether it even makes sense to ask whether they can be distinguished. On Waisman’s reading, if we affirm that an object is identical with itself, it is not because we cannot in fact distinguish an object from itself, but rather because we cannot conceive or make sense of what it would be to distinguish an object from itself. The truth of asserting the self-identity of an object is a result, not of the impossibility of denying self-identity, but rather the non-sensibility of denying self-identity. That is, Waismann conveys it is correct to assert that an object is self-identical, and this truth is the result of nonsense: the nonsense of distinguishing an object from itself.

Waismann provides a step in the right direction in interpreting Wittgenstein, but only a step. Consider that, according to Waismann, it makes no sense to even ask whether a chair can be distinguished from itself. The point is not that it is impossible in fact for me to distinguish a chair from itself but rather, as Waismann conveys, it impossible for me to even conceive of what it would be to do so. That is, it is not that the question of the negation of self-identity lacks a positive answer, rather it is that the question cannot be properly understood (such that we can even begin to consider an answer). Waismann is right about this much, but wrong to convey that this implies the truth of the self-identity of objects. If it is simply the case that it is universally false to distinguish an object from itself, then we may affirm that self-identity is universally true. However, if it is nonsense to assert that an object can be distinguished from itself – if the question of distinguishing an object from itself does not make sense as a question – then it is also nonsense to assert that an object is identical with itself; it is nonsense because, as Glock summarizes above, the negation of nonsense is still nonsense. In contrast, the negation of an arithmetical truth is a mistake, and not nonsense (for people do make intelligible arithmetical mistakes, and these can be understood and corrected). Asking whether 1+1=3 is not a nonsense question, even if it is not a bright question. But with the self-identity of objects, the question of negation cannot be sensibly considered; it is nonsense, according to Wittgenstein, and thus so is its assertion.

This same conclusion may be arrived at a little differently. Consider that a proposition that expresses a genuine relation is a molecular proposition. It is a proposition with constituent parts that are atomic, and these should be able to be conceived independently. That is, each item related – each relata – should be able to stand independently; each expression related should have its own sense. Roger White affirms, “…if the identity sign were a relational expression, each of these propositions or phrases would have to make sense, even if they merely expressed obvious logical truths or logical falsehoods.” (White 1977-8, 169) To explain further, if “a=b” expresses a relation of identity between a and b (presuming these are names of objects), then “a=a” should likewise express a relation of identity (for it follows by way of substitution). However, “a=a” is not a similar case. As Glock observes, “The ‘partners’ of the apparent relationship are not independent.” (Glock 1996, 168) That is to say, a does not stand apart from itself as it may from b, which is to say the sense of a does not stand separate from itself as it may from b. Thus, “a=a” is not a molecular proposition, and hence, does not present a genuine relation. (Glock 1996, 165) In an expression of self-identity, the items on either side of the identity sign do not stand independently; they do not carry independent sense and hence, do not express a genuine relation according to this analysis. However, it is precisely because the items related cannot stand separately, or express independent sense, that the relation of self-identity of an object is presumed to be trivially and necessarily true. But this analysis implies that this is mistaken for it cannot be a trivially true relation if it is not a genuine (molecular) relation. Once again we see that the assertion of self-identity is not a meaningful assertion. When Wittgenstein says in the Tractatus that the assertion of self-identity says “nothing” (see TLP 5.5303 above), we may now interpret this to mean it says nothing meaningful, as opposed to saying nothing in the sense of saying something trivial.

With the repudiation of self-identity as a genuine relation between an object and itself, certain philosophical confusions – confusions in Wittgenstein’s view at least – can be cleared. For instance, if self-identity is not true of all objects, then we cannot use self-identity as a universal criterion of objecthood. If there is no other criterion we may apply to an object qua object, and there does not appear to be one, then we cannot speak of an object in disregard of any properties and we cannot refer to or identify the universe of objects qua objects. Russell’s Axiom of Infinity, for instance, does just this in asserting the infinity of objects in the universe. This is an assertion about how many objects there are rather than an assertion about how many objects of a particular kind there are. (Glock 1996, 167) Again, the repudiation of identity as a genuine relation between an object and itself means that we cannot use self-identity as a way of speaking of an object qua object. To speak of or refer to an object we must do so in terms of some property or other. This is a reason why Wittgenstein states, at the outset of the Tractatus, that the world is a totality of facts, not objects (TLP 1.1). In addition, if the assertion of self-identity is not a basis for speaking of a universe of objects qua objects (by reason of nonsense), then the denial of self-identity is not a basis for speaking of a universe or set that is empty of objects qua objects (again by reason of nonsense). This means that the negation of self-identity cannot be used to define the empty or null set or, as Frege does, to define the number zero. In short, Frege’s logical derivation of the numbers is put in jeopardy if the null class of objects cannot be defined as the group of objects that are not self-identical.

The presumption that identity constitutes a genuine relation between and object and itself is a tenet of more than one philosophical position. These positions are undermined by Wittgenstein’s case for the nonsense, as opposed to senselessness, of assertions of self-identity. While the interpretive case seems clear, despite the initial ambiguity raised, the following observations can also be made in favour of reading Wittgenstein as upholding that assertions of self-identity are nonsense (and so neither true nor false), rather than merely senseless (and so trivially true): Wittgenstein admonished Russell’s Axiom of Infinity; in addition, at the outset of the Tractatus, he denied that the world was a totality of objects. These positions are supported if assertions of self-identity are deemed nonsense, but are belied if deemed senseless and trivially true. As noted at the beginning, according to Wittgenstein, in an ideal logical notation identity would be conveyed or shown by sameness of sign, rather than asserted or said through a special sign for identity. That is, an identity sign is an attempt to express what is better shown through sameness of sign. Showing that the aforementioned philosophical positions are built on a philosophical confusion is at least one motivation, for Wittgenstein, for adopting this ideal notation.

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