Initially Russell’s interpretation, or alternative, to the sentence
The king of France is bald.
seems highly counterintuitive, if not pedantic or even absurd. However, think of it as an ordinary definition – at least as basing itself on such a thing. It is, therefore, indeed the case that
normally when I define a term I give an equivalent term (73).
Despite the grandiloquent nature of Russell’s quantificational analyses of the said sentence, perhaps, after all, it is only an elongated definition of it – or of what it expresses. And as with normal definitions, this would automatically mean that Russell’s definition, as well as its parts, can only replace the original if it changes its meaning, or the meanings of its parts, or is initial truth-value. To put it another way, even if Russell’s definition is prima facie strange and complex, it must be capable of replacing the initial sentence and all its terms
in all contexts where it occurs (72).
This means that strange as it may seem, Russell’s reading of the sentence and its terms can replace it in all ‘contexts where it occurs’ (73). Even though Russell’s replacement looks strange and perhaps nothing like the original, it still fulfils all the stipulations laid down by the everyday concept of definition. And, again, the basic notion of definition, in logical symbols ‘= df’, is at the heart of Russell’s theory and his analysis.
However, Russell’s definition is what he called an ‘explicit’ definition, even if based on the ordinary implicit definition of a word. It is explicit because it
eliminates terms from our language, by substituting other terms with the same function (73).
Whereas in ordinary implicit definition the word being defined is not taken to be logically or ontologically suspect and neither are the defining words, in Russell’s ‘explicit definition the terms and the sentence being defined is suspect. We cannot simply find other ordinary words to define them. Instead the philosophy simply eliminates them because of the logical/ontological problems they bring about. But he does not simply leave an empty space where once the phrase, say, ‘the king of France’, was, but provides instead a substitute. And that substitute will be acceptable and it will also work, logically and grammatically speaking, because it has ‘the same function’ as the predicates being substituted.
Firstly, let’s take the subject-term, ‘the king of France’. With such a predicate Russell argues we cannot offer a definition such as:
The king of France = x
Presumably because it is taken apart from its predicate-term ‘is bald’. And, as Frege tells us, such a definition would not work on their own and apart from the sentence, just as the function ‘is wise’ or ‘x2’ need to be ‘saturated’ with a number, or variable or a proper name. However, isn’t ‘the king of France’ the ‘name’ in that sentence and didn’t Frege argue that they, whether ‘2’, ‘x’ or ‘Blair’, could indeed stand alone? It cannot stand alone, then, because it is not in fact a name or ‘argument’ (as Frege puts it). In that case,
the king of France = …
cannot actually be filled in with any kind of definition because of the predicate’s indeterminate nature. In other words, we cannot use the sign ‘=’ because the predicate cannot be taken to be identical to anything because of its suspect nature. Unlike
Tony Boy = Tony Blair
we cannot have a substitutional definition of the predicate that would also be equal or identical to it.
As I said, we can take the whole sentence instead. Would that help? S says that the whole sentence is in fact ‘meaningful’. And if that’s the case, the Russellian may respond: Then it ought to have a truth-value. The assumption here, of course, is that all sentences must have a truth-value. Or is it? The predicate is in fact a statement, not just any old sentence. It states or asserts that the king of France suffers from being bald. Not only that, but, as we shall see, it also must assume that the king of France actually exists (and just happens to be bald as well).
Not only does the use of ‘the’ commit us to the existence of the king of France but, in Russell’s quantificational speak, it also commits us to the fact
that there is at most one king of France (73).
However, this seems evident because the very use of ‘the’, in ‘The king of France…’, on its own implies, or even states, that there is only one such king. We could not say ‘The king of France’ if there were, or we thought there were, two or more kings of France. But, as I said, the phrase seems to commit us to the existence of only one such king, if only grammatically speaking. However, logically perhaps it does mean, implicitly,
There is at most one king.
This will be clarified later in the Russellian analysis. We can say, then, that
The king of France is bald.
commits us to three Russellian conclusions:
/) That there is a king of France.
//) That there is at most one king of France.
///) That whatever is a king of France is bald.
We can split the sentence up into three prime logical parts:
/) ‘The’
//) ‘The king of France’
///) ‘is bald’
and 2) commit us to whatever description follows it. And that it exists in at most one case. 3) offers us a predicate that we must apply to the subject-term because it is taken to actually exist. In fact 1), 2) and 3) give us the ‘truth-conditions’ of the whole sentence as it is recast by Russell.
Truth-Conditions
i) ‘The king of France’ – the object that is king of France.
ii) ‘is bald’ – the predicate
has as its extension the class all bald things which must include the king as a member. The truth-condition, initially, is every member of that class.
However, because of the subject-term, we can apply that to the class of bald things and by doing so single out one of its members – the king of France. In Fregean terms, when ‘is bald’ is ‘unsaturated’, when on its own, the function must apply to every member of the class of bald things. However, when we apply a name or argument to this predicate-function, and thereby ‘saturate’ it, such an operation allows us to say that something is the case about the named object, when both its function and argument are brought together to make the whole sentence capable of having a truth-value attributed to it or else shown to be ‘meaningless’ from a logical point of view.
To put the earlier three Russellian conclusions in a different way:
There is an x, such that x is a king, x is bald, and for every y, y is a king only if y is identical with x.
All that’s a matter for the whole statement. Let’s get back to the ‘denoting phrase’. What are we to make of it? Perhaps what we have said about the sentence needing to have a truth-value, as well as the phrase not having a definition of identity, will prove helpful to us here. We can do so, again, by thinking in terms of the whole sentence. We have found it hard to give the sentence a truth-value. Perhaps we can restructure the whole sentence in order to make it become a new one to which we can give a truth-value.
We already said that ‘the king of France’ cannot have a Russellian explicit definition. Then we can give it a basic implicit one instead. Perhaps that will help. It may do so if we place there an implicit definitional version in different sentences, but ones that keep the same truth-value. If the original is true, then all our substitute sentences must come out true as well. However, all these true sentences will not contain the denoting phrase ‘the king of France’, but those found by implicit definition. Despite the substitutions, as we said, such sentences will still have the same truth-value as the original.
For a proper definition of both the denoting phrase and its containing statement, there are things we must do in order for itto be an explicit definition of them. And that, in turn, is how we will try to solve the ontological and logical problems that come along with it when accepted at face-value – as grammatically but not logically correct and acceptable. We can still say that the first is meaningful. But it is logically ‘meaningless’ – as Russell said in the first place!
We will soon find that the prime guilty party in all this is the definite article – ‘the’. The simple point is that by using the word ‘the’, at the beginning of the phrase ‘The king of France’, I am making an ontological commitment, if unknowingly, to that king’s actual existence. For example, we would treat the phrase ‘a king of France’, in which we have the indefinite article, ‘a’, instead, very differently, both from a grammatical and a logical point of view. In other words, ‘a’ doesn’t commit us to some king the king’s existence, even if it does commit us to some king or other of France. That king could be a dead one. With the definite article ‘the’ we have a quantificational statement of existence. Thus:
$x(KxÙBx)
We would need to complexify the above to make it come out to be referring to an individual rather than to ‘some’ or ‘at least one’ individuals.
With the indefinite article we have instead:
"x(KxÙBx)
In other words, it is applied to all or any examples of x (i.e., a king) and saying what predicates they must have if they are in fact x – a king. Also, it can be taken as a conditional rather than as an existential expression.
I mentioned earlier that my logical translation of the statement told us that there is at least one king (i.e., $x), but not that there is only one king, and the proceeding expression does just that. In symbols:
$x(K (x) ÙB (x) Ù(y)(K (y) É (y = x)))
To translate the above expression: There is at least one thing, x, such that it is a king (K) and that it is also bald (B). And it is also that there if there is at least one thing, y, that is also a king, then this y must be identical to the original thing x. Thus there is at most only one thing, x, such that it is king of France and is bald.
To recap. What interested Russell the most in all this analysis is the nature of the definite article ‘the’. More specifically, ‘the’ when it comes just before a description, such as in ‘the golden mountain’, ‘the Morning Star’, and so on. Clearly, then, they are not actually proper names, even if that is how they appear to us in a grammatical context. In addition, they are, if anything, ‘empty names’ that don’t refer (although some do – e.g., ‘The Prime Minister of the UK’, etc.) Russell does, therefore, accept that they are descriptions of some kind (how could he have done otherwise?). However, Russell calls them ‘definite descriptions’ primarily because of the definite article that proceeds them. Then a description like ‘a fat man’ would not be such a thing. It could be called an indefinite description, or it may not be a description at all!
Logical Form
Russell gives the meaning of a sentence by
stating the conditions for its truth (74).
Clearly, then, this definition is not unlike the logical positivist one:
The meaning of ‘S’ is its means of verification.
However, such a Russellian definition is not epistemic in nature, as in the positivist case. Instead, with its bare reference to S’s truth-conditions, we could even call it a metaphysical definition of meaning, at least as it stands. Or, alternatively, it is semantically uncommitted to a metaphysic, as is:
The sentence ‘S’ is true iff if p.
Tarski’s T-sentence, then, relates a sentence ‘S’ to its truth-conditions that come after the biconditional ‘iff’. Again, this is purely semantic in nature, and devoid of epistemic and metaphysical content, as with Russell’s own definition. Because it is so bare and purely semantical in nature that it is often taken, and can be taken, as a schema that expresses the nature of the correspondence theory of truth. However, Tarski himself argued that precisely because it is devoid of epistemic and metaphysical predicates that it couldn’t function as a definition or scheme for the correspondence theory of truth. In fact it is not a theory of truth at all. It is, instead, only a semantic definition of what must be the minimal case in a natural language for it to qualify as ‘true’ in the first place. As Tarski says, it could still be used as a basis for any kind of truth-theory because most candidates for that role would fit it and accept it, as it were. Instead Tarski’s T-schema and Russell’s definition show us what form any sentence must have for it to qualify itself for a predicated truth-value. The way we find out its truth, say by its truth-conditions and their causal relations to the utterers of S, is left open by both Russell and Tarski. It is just the case that S must have truth-conditions of some kind and that they must match up with what S states is the case. The causal relations between S and the ‘iff…’, or any kind of relation, are left out and must be filled in by the philosopher or epistemologist. How does ‘S’ relate or connect with S, when the disquoted ‘S’ – S are taken to state its truth-conditions.
Russell’s analysis, however, is not of the subject-predicate sentence-form we know so well. This will not be surprising if we also remember that it is precisely the suspect sentence’s subject-predicate form that leads us towards various logical and ontological problems; just as the early logical positivists, for example, said about the seemingly innocuous subject-predicate and therefore grammatical form of metaphysical assertions that led us to accept them as genuine and even believe in their strange ontological commitments. Instead Russell’s analysis is existential in form. It makes a claim, in this instance, about an individual’s existence, unlike the original – at least not if taken at prima facie level. In addition, taking the analyzed sentence in a purely grammatical way also leads us to believe that it has no truth-value at all (as Frege believes), and/or that the subject-term, ‘the king of France’, does actually refer to something (Meinong’s position).
To some up. The original sentence’s grammatical form is actually very misleading, at least from a logical, ontological and non-grammatical point of view. And that’s precisely what Wittgenstein, Russell, and the logical positivists argued. However, I still think that none of them should have ever used the equally misleading term ‘meaningless’, as Strawson, for example, was later to argue.
In terms of the logical forms of sentences, they become apparent to us when we use their role in inferences. We are given the logical form of S when we see what other sentences we can legitimately infer from it (as Brandom now does). In addition, such inferences will also help us determine the actual truth-value of ‘S’. How we derive such inferences from S and also how they determine its meaning and truth-value are another matter.
In terms of what all this tells us about the nature of logic, or specifically of S’s logical form, it is because the truth-values of all sentences in logic are as important as names and the logical constants are in, say, Wittgenstein’s Tractatus and in predicate logic generally. The parts of S also required extensions or references to make them logically legitimate, as well as to be a function of the reference and truth-value of S as a whole. We can immediately see that the original sentence did not have a reference for its subject-term and therefore perhaps didn’t have an extension qua class for its predicate-term or for the concept it refers to before it gets to its class. In other words, we saw that ‘The king of France’ is an ‘empty name’, but we also saw that it is, in fact, a definite description.
Thus in Russell’s analysis, or in his definition, the elementary operations, such as truth-functions, variables, are made good use of by Russell and that they also help us understand the precise logical and ontological nature of the given sentence. All parts of S have a ‘logical role’ to play in it, as does the whole of S itself. They also show us the extensional and referential commitments of all the predicates in a sentence as well as the truth-value of the whole of a sentence, which is in turn a ‘function’ of the said parts or of its logical operators and names. Again, what matters with S in predicate logic is the extensions and references of its names or descriptions, the logical functions of its constants, and the truth-value of the whole of S, which could impact on other sentences if they were top be inferred or connected by another logical connective to the original.
The end result of Russell’s analysis, if we accept it, is quite remarkable in the sense of what we can claim about it. For example, like Russell we can now say that
logic takes precedence over natural grammar (75).
In addition, such logic
tells us what we really mean by our ordinary language (75)
despite the fact that such a claim, which implies that we do not know what we mean or really mean, seems extremely arrogant, extreme and absurd at a prima facie level. However, it should now be clear to us why the first Russellian claim is acceptable and makes sense because we have clearly seen how subject-predicate grammar can seriously mislead us, if only logically and ontologically. As for the last claim about our original lack of understanding of such sentences, this would only apply, of course, to those of us who are interested in philosophy or who are philosophically minded. And in such cases, the strong claim really only tells us
What we ought to mean.
when we use such-and-such a logically problematic subject-predicate sentence. In everyday life, or in fiction, or if we practice metaphysics for the hell of it, we can indeed ignore such logical animadversions.
Even philosophers today are prone to accuse us that we do not understand the sentences of ordinary language, or it at least some of them. In terms of Davidson, for example, he offers us a new start on the matter by saying that we do not understand the
truth-conditions of the thoughts expressed (75).
by such sentences. Whereas Russell only talks in terms of truth-conditions giving us the conditions for a sentence’s truth or falsehood, or to determine such a thing, Davidson talks in terms of our understanding, orlack of, the truth-conditions of the thoughts expressed by statemental or declarative sentences. We now need to know what understanding truth-conditions actually means. And we will come to see if this is a real or big difference when compared to what Russell said and argued.
In terms of truth-conditions, our previous remarks about Tarski’s T-sentences and about object-languages and meta-languages comes in handy here. When
assigning truth-conditions, we should use an ideal language (a logical language) whose operations are entirely understood (75).
So Tarski’s schema or T-sentence thus:
The sentence ‘S’ is true iff if p.
This is taken from just an ideal logical language. And it is a meta-language because it cannot use terms from the object-language, as, in this case, we don’t we the disquoted ‘S’. The truth-conditions of the above are simply symbolized by S – which is a disquotation of ‘S’, or, really, just a restating of ‘S’ but taking away its quotation marks and taking it to symbolize the sentences truth-conditions which we need to assign a truth-value to it.
However, we said earlier that the Tarskian T-sentence does not itself offer us a theory of truth. But now S says that for Davidson what makes a language ‘ideal’ is that
there exists what he calls a “theory of truth” for it (75).
Does that mean, then, that Davidson is bringing in metaphysics or epistemology to an otherwise Tarskian semantics? Does his ‘theory of truth’ as it were come alongside Tarskian semantics? Davidson himself says that he means that such a theory
which shows us, in a systematic way, how all the sentences of the language can be evaluated, given an evaluation of their parts (75).
The Davidson explanation of his truth-theory starts off purely Tarskian in manner in that like Tarski’s T-sentences it too systematizes all its sentences. But as we have already said, Tarski’s T-sentence is just as important for what it leaves out, as for what it includes. At the end of his explanation Davidson brings in the word ‘evaluation’, which is missing from the T-sentence and does not have an equivalent. More specifically, we not only evaluate all the sentences of an ideal language, but we do so via the evaluation of the parts of each sentence under scrutiny. And here we seem to be entering Russellian territory in the sense that we have already said of the sentence in his ‘theory of descriptions’ that we must evaluate not just the sentence in terms of its truth-value, but also it parts in terms of their extensions and their references. Thus Russellian logic also gains logical content by doing so. A Tarskian T-sentence, on the other hand, is purely formal in the way that the propositions of pure formal logic are purely formal and without any content. In the cases where formal logic’s symbols or predicates are purely auto-referential – they refer to themselves and nothing else. In that sense we can also say that a T-sentence is purely formal in that what matters is its shape, as it were. It has no, as we have said, ontological content and therefore cannot be used as a genuine truth-theory. Instead it is not only purely semantic in nature. It is also an example of formal semantic – that is, semantics with little or no philosophical commitments or even philosophical relevance. However, despite saying that, we can still say that Tarski’s schema does indeed provide us with a starting-point in our philosophizing. And it does so by formally showing us what we mean, or what we must mean, when we use and apply the truth-predicate, as well as what we mean by a sentence’s ‘truth-conditions’ and their relations to that sentence. But like the propositions of formal logic, they have no empirical or any other kind of content; they ‘say nothing’, to use Wittgenstein’s phrase. And just as Wittgenstein’s logical tautologies can show us ‘the form of the world’, so Tarski’s T-sentence can show us the necessary form or structure of all our truth-theories and indeed of the metaphysical truth-claims we make within them or elsewhere. Just as Wittgenstein’s Tractatus can be taken as a prolegomena for future work, so too can Tarski’s semantical explication of the truth-predicate ‘is true’.
As we have said earlier, Davidson’s ‘truth-theory’ is similar to what Russell has already said in his theory of descriptions. Davidson also says that a ‘theory of truth’ cannot be given for a natural language, though evidently he does not think the same about an ‘ideal’ language or, perhaps, a Tarskian meta-language. Although the T-sentences of Tarski’s own meta-language do show us the form or structure, as we put it, of our truth-valued sentences and the procedure we must follow when assigning truth-values to our sentences via the relation between truth-conditions (S) and sentence (‘S’) shown by his shema, it does not, as we have said, provide us with a ‘theory of truth’ for the sentences in the object-language, just as Davidson’s own ‘theory of truth’ would not be applicable to ordinary language sentences or the sentences of an object-language. Instead Davidson
seems at times to agree with Russell in thinking the “predicate calculus” to be the best language that we have for representing reality (75).
In consequence of this, we can conclude that all the resultant expressions of Russell’s theory of descriptions are just that – examples either of the predicate calculus or derivations from it. And we must also conclude that expressions in the predicate calculus as well as Russell’s own previous expressions are not, and cannot be, given in ordinary language sentences. Such expressions then have a meta-language or ideal language status, just as with Tarski’s own that includes T-sentences. And, again like Tarski, even though the predicate calculus and its expressions can taken an object-language or a natural language as its subject-matter, its consequent expressions cannot also be expressed in them (just as Tarski says we cannot use the terms of an object-language). Instead, if we want to come closer to both knowing and expressing ‘reality’, we must do so, according to early Russell and Davidson, by learning the predicate language in order to ‘represent reality’ better or more faithfully. No wonder, then, logic is so important to so many parts of 20th century and contemporary analytic philosophy. Such uses of logic are simply a matter of both truth and the adequate expression of the world’s true nature. Metaphysical statements, on the other hand, had the oppositeeffect: they distorted ‘reality’ and ‘systematically mislead’ numerous philosophers because of their lack of a logical and ontological form and their parallel ostensibly acceptable grammatical form.
We can conclude by re-stating the three vital ideas which come out of Russell’s theory of descriptions. They are:
i) That the truth-conditions of various problematic sentences are of vital consequence. And we must also explicate the relation between such truth-conditions and the thoughts expressed by sentences.
ii) These truth-conditions can be extracted by giving the logical form of the said sentences. And we can only do this by using a logically precise and transparent logic such as that found in the predicate calculus or extensionalist and quantificational logic.
iii) More fundamentally, and perhaps controversially, we must conclude that there is ‘one and only one correct analysis (or logical form)’ (76) of the aforementioned sentences.
In terms of the huge claim in 3), we can say that precisely because what it says is in fact correct that even today philosophers still refer to Russell’s theory of descriptions as a classic example of brilliant logical analysis. And also the reason why philosophers still often refer to it and still use its logical forms and analyses in their own analyses of new philosophical problems that may also be brought about by philosophically suspect but grammatically acceptable and correct sentences. Indeed Davidson himself is just one example of such a philosopher in that he believes the predicate calculus, as used by Russell, is still
the best language that we have for representing language (75).
On a smaller scale, 2) above shows us how vitally important the notion of logical form is in all our philosophical analyses. In addition, 1) shows us the parallel importance we must give to the truth-conditions of sentences, or to the very notion of a truth-condition itself. And today we can still see how important the notion of truth-conditions are in that they have been variously interpreted as, for example, in the verificationist way of the logical positivists, and perhaps also the anti-realists, and also as the determinants of truth and meaning in other analyses; and, finally, in Davidson’s ‘theory of truth’ in which truth-conditions are systematically assigned to all sentences and also to the constituent parts of them in terms of their extensions and references and the way truth-functional operators act on them to determine a sentence’s truth-value and thereby also enable us to create other sentences which are related to the original via inference or some other kind of logical relation. So just as in Frege the ‘Thought’ or ‘sense’ of a sentence determined its truth-value and reference, and was also a function of the senses and references of its parts, so truth-conditions in Russell and Davidson’s schemes determine the thought expressed by them but which are distinct from them, just as Fregean thoughts are both distinct from their expressions and whatever role truth-conditions play in his scheme.
Note:
The definite article ‘the’ clearly distinguishes some definite descriptions from proper names. We are making a “uniqueness claim” with them, which is not the case with proper names (i.e., we don’t say ‘The Tony Blair’). With definite descriptions we are saying that there is only and only one person named (e.g., ‘the King of France’). There is probably more than one person with the name ‘Tony Blair’, whereas there can’t be two individuals described as ‘the first man on the moon’. However, these are examples of definite descriptions that do apply to more than one individual. For example, ‘the Prime Minister’ could refer to different individuals at different times and places. Of course the context of utterance is important for these kinds of definite description. Whereas the name ‘Tony Blair’ is meant to refer to a unique individual in all situations. There could be, of course, another person named ‘Tony Blair’. So context is relevant for the use of proper names too. If Cheri Blair, however, uses the name ‘Tony Blair’ she will be referring to her Tony Blair, whereas a person married to a Tony Blair in South Africa will possibly not be referring to the Prime Minister of Great Britain (at least not on every occasion).
However, in Lepore’s example of
The tallest man is happy.
the domain of its application is not specified. Does he mean the tallest man in his house? On the street? Or in the world? If the domain were literally universal, then perhaps the speaker would not know who the tallest man actually is, let alone whether or not he is also happy. Also, there may be two tallest men if both are of an identical height. Then the original locution would need to be changed to ‘The tallest men are happy’. But then, of course, the happiness of one of the tallest men would not guarantee the happiness of the other.
Of course when we use a definite description we are saying something about the object or person described. This does not seem to be the case with proper names. However, if we use the name ‘Tony Blair’ aren’t we describing Tony Blair as the person who has the name ‘Tony Blair’? We have the definite description
The man who has the name ‘Tony Blair’.
Here we have a proper name embedded within a definite description. The definite description uniquely describes because the embedded proper name uniquely refers. If we had the description
The man who has a name.
this wouldn’t in fact describe a unique individual because billions of men and objects have names. Indeed the definite article is not warranted in such a pseudo-description.
