Appendix to Chapter 4
Frege, Russell, and the puzzles of reference
OriginalEla provém da consideração de que na definição da
existência do pensamento não entra em questão a mente singular que o tem, nem a
pessoa na qual ele ocorre.
Russell conceived his theory of descriptions as a way to solve the puzzles of reference. Frege’s theory of sense suggests a very different way to solve the same puzzles. While these two alternative solutions are usually assumed as irreconcilable, each of them has its own appeal. Considering this, my proposal is that the best way to deal with this contrast is not by means of dispute, but by means of reconciliation, which one can reach by salvaging the truth in each and discarding the false. More precisely, I will propose to build a bridge between Russell’s and Frege’s solutions by transforming each of them in ways that make them fully compatible. I will proceed first by removing the metaphysical load from each of these views, and then by showing that with the help of appropriate adjustments, including reading senses as semantic-cognitive rules, they can be seen as two different ways of saying the same.
1. Russell’s solutions to puzzles of reference
I will first explain Russell’s four puzzles and his solutions by means of his theory of descriptions (Russell 1905: 479-493; 1919, ch. XVI).
(i) Reference to the non-existent Consider first a statement whose grammatical subject does not refer to anything, ‘The present King of France is bald’. How can we attribute baldness to someone who does not exist? Russell’s response is that this problem only arises when we understand a definite description ‘the present King of France’ as a referential expression functioning like a proper name. We can easily show that it actually does not function in this way. Letting K abbreviate ‘…is a present King of France’ and B abbreviating the predicate ‘…is bald’, the theory of descriptions allows us to symbolize the ‘The present King of France is bald’ as ‘(Ǝx) [(Kx & (y) (Ky → y = x)) & Bx]’. Or, to use an intuitively clearer formulation, we get the following false sentence:
1. There is at least one x and at most one x, such that x is a present King of France and x is bald.
In these last formulations, one thing is clear: there is no baldness predicated for a present King of France. When the definite description ‘the present King of France’ is replaced by quantified predicates, it becomes clear that we do not need to assume the existence of any present King of France to whom we can apply the predicate baldness. Moreover, the whole statement must be false, because the first statement of the conjunction is false.
(ii) Negative Existential The second puzzle concerns the apparent impossibility of denying the existence of an object when the expression that denies the existence is about the same object. The problem assumes a striking form when we consider the following two statements:
1. The present King of France does not exist.
2. Statement (1) is about the present King of France.
Both statements seem to be true. However, they are mutually inconsistent. If statement (2) is true because it states that statement (1) is about the present King of France, (1) must be false and vice versa.
Russell solves the riddle by suggesting that statement (2) is false. In order to show this, he interprets the negation in statement (1) as possessing wide scope in relation to the definite description. The form of statement (1) is ~(Ǝx) [Kx & (y) (Ky → y = x)]; more intuitively:
2. It is not the case that there is at least one x and at most one x, such that x is a present King of France.
This is a true sentence, since it is the negation of a false conjunction. However, it does not commit us to the existence of the present King of France, since it commits us only to denying the existence of precisely one thing that has the property of being a present King of France.
(iii) Identity Statements A third puzzle is the Fregean paradox of identity. Consider the statement: (1) ‘The author of Waverley is Scott’. This contains two referential expressions, both referring to the same object. But if this is so, then statement (1) should be tautological, stating the same thing as (2) ‘Scott is Scott.’ However, as Russell notes, we definitely know that (1) is a contingent and informative statement. Why?
Russell’s solution is again to make the definite description disappear. Letting s abbreviate the name ‘Scott’, w abbreviate ‘Waverley’ and A abbreviate the two-place predicate ‘…is the author of…’, we can paraphrase the identity statement (1) as ‘(Ǝx) [(Axw & (y) (Ayw → y = x)) & (x = s)].’ More intuitively:
3. There is precisely one x who is author of Waverley, and this x is Scott.
From these formulations, it is clear that (1) is an informative statement, since there is no doubt that (3) is an informative statement.
(iv) Intentional context A final riddle that the theory of descriptions is expected to solve is that of inter-substitutability in statements of propositional attitudes. These statements express relational states connecting a mental attitude expressed by verbs like ‘believe’, ‘desire’, ‘hope’, ‘think’, ‘want’… to what I prefer to call a thought-content (s-thought, proposition). Consider, for instance, the two following statements:
(1) George IV believes that Scott is Scott.
(2) George IV believes that the author of Waverley is Scott.
Statement (1) is true, since George IV was certainly able to apply the principle of identity to a proper name. However, since the name ‘Scott’ and the description ‘the author of Waverley’ refer to the same thing, it seems that we can apply here the principle of identity substitution. It seems that we can replace the first occurrence of the word ‘Scott’ in statement (1) with the description ‘the author of Waverley’, obtaining statement (2), ‘George IV believes that the author of Waverley is Scott’, so that (2) will preserve the truth-value true. However, this does not happen: it may well be that statement (2) is false because George IV does not entertain this last belief, despite the obvious truth of (1). Why is this so?
In order to answer such objections we can use the theory of descriptions, paraphrasing (at least in the relevant cases) (2) with the following statement: ‘George IV believes that Ǝx [(Axw & (y) (Ayw = x) & (x = s)]’. More concretely, we can express (2) as:
4. George IV believes that there is at least one x and at most one x, such that x is author of Waverley and that this x is Scott.
Certainly, this is an informative belief, clearly distinct from the tautological belief that Scott is Scott. This is why George IV can believe in (1) and disbelieve (2).
2. Fregean solutions to puzzles of reference
Frege has an explicit answer to the last two puzzles of reference. As for the first two, one can suggest solutions in a reconstructive way.
(i) Reference to the non-existent Frege suggested that in a scientific language a singular term without reference could refer to an empty set. We can try to apply this suggestion to ordinary language, suggesting the falsity of a sentence like
(1) The present King of France is bald,
since the empty set isn’t bald. However, in addition to being arbitrary, this suggestion leads to absurd conclusions, such as that the statement ‘Pegasus = the present King of France’ is true, since both singular terms, ‘Pegasus’ and ‘the present King of France’ refer to the same thing, namely, the empty set.
The alternative that I would like to propose starts from the notion that we can say things about non-existents insofar as the corresponding empty singular terms still preserve their senses. Once we have these senses in mind, we are still able to say something about their objects, not as real ones, but as merely conceivable ones, like Odysseus, who has only a fictional, but still imaginable, reference. Because of this, we are still able to articulate the sense of the predicate with the sense of the singular term – in their relative dependence and independence – producing a complete thought-content like (i) ‘Odysseus, while sleeping, was set ashore in Ithaca’, which has no real reference, but only an imaginable one.
Frege also suggested that the thought-content of a sentence such as (i) should have no truth-value, since if a part of a thought (Odysseus) has no reference, the thought as a whole is also devoid of reference, devoid of truth-value (1892: 32-33). P. F. Strawson influentially supported such a view, as he considered such statements as having what some today call ‘truth-value gaps’ (see Strawson 1971: 85). This view is opposed to that of Russell’s theory of descriptions, according to which statements such as (i) must be false, as for him ‘Odysseus’ should be the abbreviation of a cluster of definite descriptions without reference. (See Russell 1912, ch. 5)
As to truth-value, after half a century of disputes, in my view the strongest reasons favor Russell. First, it seems definitional that a proposition (a thought-content or s-thought) is the kind of thing that for intrinsic reasons given by its function of saying something must be able to have a known or at least an unknown but conceivable truth-value. Second, although one might doubt that a statement like ‘The present King of France is wise’ (Strawson) is false, just a little reflection will show that it is more reasonable to view it as false. Consider, first, examples of statements in which the singular term is empty but which have predicates that have more weight, defining ‘weight’ as the capacity of attracting our attention, either because they have a more complex semantic structure or because they are particularly relevant or curious or puzzling. Some examples:
1. I saw the present King of France strolling on the beach last week.
2. The present King of France has forbidden tourists to visit the Palace of Versailles.
3. Yesterday the present King of France was inebriated and therefore unable to perform his official duties.
4. The present King of France visited me this afternoon and we had the opportunity to discuss the EU’s inability to solve European problems.
5. The present King of France is sitting on that chair.
These statements are all perceived as intuitively false, and it seems that the reason lies in the weight of the expressions complementing the descriptions: they force us to pay attention to their complex informational content (1 to 4) or to something that should be relevant if it were not glaringly false (5).
Moreover, when we say ‘The present King of France does not exist’ this statement is obviously true; however, these statements should lack truth-value according to a Strawsonian presuppositional analysis.
Additional evidence for this point is the following statement considered by Stephen Neale:
6. The present King of France isn’t wise, because there is no present King of France.
Statement (6) seems intuitively true. But (6) could not be true if the statement ‘The present King of France is wise’ weren’t really false. If it had no truth-value, the whole statement (6) would be also devoid of truth-value.
As some have concluded (Russell 1995, ch. 18, III; Sainsbury 1979: 118; Blackburn 1984: 309-10), the reason why the statement ‘The present King of France is wise’, chosen by Strawson, appears to lack truth-value is only a pragmatic one, which can be explained as follows. First, we normally regard a statement as false if its predicative expression does not apply, while we normally assume that the singular term applies; for instance, the statement ‘Bertrand Russell was bald’ is obviously false, since this is a standard case of a predicate that does not apply to its subject. However, we are not used to considering the truth-value of singular statements when the singular term has no reference, since these statements only rarely appear in our language, for the simple reason that it is pointless to ascribe properties to something that does not exist! This is why we hesitate to say that Strawson’s statement ‘The present King of France is wise’ is false, and we tend to say instead that it is a misapprehension or devoid of sense. But strictly speaking the statement is false. Or, more weakly expressed: there is no constraint against the reasonability of our decision to generalize and treat all statements of this kind as false.
Moreover, statements that put weight on the predicative expression or on what is said complementarily to the definite description, like (1), (2), (3), (4) and (5), are seen by us as patently false. Why? Not because they belong to a different category, as some would believe. Their falsity is clear to us because of their weight: we are motivated to pay attention to their predicative or relational expressions as being clearly inapplicable, satisfying in this way our usual criterion of falsity for singular statements. However, the ultimate cause of this inapplicability is still the same: there is no object for them to be applied to in order to make the whole statement possibly true. By contrast, statements like:
7. The present King of France is a dunce.
8. The present King of France is sitting.
9. The present King of France is a human being.
do not seem to have any truth-value. Why? Because their predicates have little predicative weight. Because of this, we focus our attention on the void subject, and since we are not used to extracting falsity from a statement because its singular term lacks reference, we tend to see the whole statement as lacking truth-value and devoid of sense. However, we can say that they are all false for the same reason, namely, that we cannot ascribe these predicates to anything. For predicate ascription is our pragmatic criterion for truth attribution.
Furthermore, consider statements that in a fictional context are undoubtedly true, such as:
10. Santa Claus has a white beard.
If understood as a statement about a fictional realm, we can even say that (10) is true. But if understood as a statement about the real world, (10) is a statement like (1): it seems to have no truth-value, though it must be false. And with good reason it shows its falsity when we make a statement like ‘I trimmed Santa Claus’s white beard last Christmas’, once it assumes that Santa Claus is a man of flesh and blood belonging to our real world.
Finally, it is worth noting that we can construct verifiability rules for these statements, which also suggests their meaningfulness. One can consider ways to verify that there is no bald or wise present King of France, that there is no real Santa Claus whose beard I trimmed last Christmas, that criteria are lacking. All the given statements are directly or indirectly falsified by the absence of applicability criteria for their verifiability rules.
(ii) Negative Existential It is not so easy to give a Fregean explanation for the enigma of negative existentials. However, consider the following statement:
(1) The present King of France does not exist.
It is true that ‘the present King of France’ is a definite description that does not refer to anything. But also here the description ‘The present King of France’ has at least a conceptual sense, that is, an identifying rule that we can conceive as being applied, as we saw in the last chapter. Now, if existence is the property of effective applicability of a concept in a chosen domain, and the description ‘the present King of France’ does not apply to any object in this chosen domain, which is here the fundamental domain of real things, our conclusion is the following. The thought-content expressed by the assertive sentence (1) is true, since the predicate ‘…does not exist’ says that the conceptual sense, the identifying rule of ‘The present King of France’ isn’t satisfied, that is, it does not apply to any object in the present domain of real things as suggested, though it remains applicable in a conceivable, merely imaginary domain.
The same can be said for the denial that the referent of a proper names exist. If the sense of a proper name, as Frege indirectly suggested, is the abbreviation of clusters of definite descriptions, or, as I have in a more nuanced form defended, the abbreviation of a properly characterized disjunction of fundamental descriptions, then a similar strategy is applicable to negative existential statements with empty names, statements like ‘Vulcan does not exist’. What this sentence means is that the conceptual sense expressed by the fundamental localizing description abbreviated by the name of the small planet ‘Vulcan’ has no effective application in its proper domain, that its identifying rule isn’t satisfied by any real object, which is true (see appendix of chapter 1).
(iii) Identity Statements. The riddle of identity between descriptions can be exemplified by the most discussed sentence of analytic philosophy:
The Morning Star is the Evening Star.
For Frege this identity sentence is informative because the descriptions ‘the morning star’ and ‘the evening star’ express different senses or modes of presentation of the same object, the first as the most brilliant celestial body that appears to us at dawn, and the second as the most brilliant celestial body that appears to us in the evening… It is informative to know that these two very different modes of presentation are of the same object.
In particular concerning proper names, due to their semantic flexibility a double answer could be given depending on different contextual emphases. Suppose that we have the proper names ‘Phosphorus’ (Morning Star) and ‘Hesperus’ (Evening Star) and the sentence ‘Phosphorus is Hesperus’. There are two main ways of understanding this sentence, according to which semantic element we are emphasizing in accordance with the context:
Immediate Emphasis: In this case the modes of presentation for Phosphorus and Hesperus, their identifying rules, are emphasized, Phosphorus being understood as the last star to disappear at dawn and Hesperus as the first star to appear in the evening... The mode of presentation of Venus, which contains both and is responsible for the identity, is here made implicit, being only the resulting datum of identity that is preserved. In this case, the statement is seen as contingent a posteriori and the identity as informative because it informs us in an implicit supplementary way that these two different senses or identifying rules have the same ultimate reference, namely, the planet Venus. They refer to an apparent sub-fact and only secondarily let us infer the further grounding fact of Venus’s self-identity.
Mediated emphasis: In this case, with both names we emphasize that we mean Venus, attaching to both terms the same fundamental localizing astronomical description (say, the second planet of the solar system at least at the time of its discovery and probably later…) that forms its identification rule. Here the descriptions of Venus’ appearances to us play only the role of irrelevant auxiliary descriptions. Because of this, the sentence ‘Phosphorus is Hesperus’ is here seen as an uninformative analytic identity sentence – a necessary a priori sentence – even if it has different fringes of meaning depending on the different auxiliary descriptions related to different usual ways of presentation. In this case the assertive sentence has as its sense and intended reference the grounding fact of Venus’s self-identity expressed by the sentence ‘Venus = Venus’. In this case ‘Phosphorus = Hesperus’ means the same as ‘Venus = Venus’, even if by means of the implicit sub-fact that the morning star is the evening star, as determined by the two different senses expressed by the sentence.
Kripke’s necessary a posteriori identities between proper names seem in this case the result of a confusion of the necessity of the mediated emphasis with the contingency of the immediate emphasis.
(iv) Intentional contexts As for the enigma of intentional contexts, Frege suggests that in statements of propositional attitudes the subordinate sentence does not have its usual reference, its truth-value, but rather an indirect reference, which is its sense. Thus, in saying (1) ‘George IV believes that Scott is Scott’, the reference of the subordinate sentence ‘Scott is Scott’ isn’t either its truth-value or a corresponding fact, but simply the thought expressed by this sentence. And in saying (2) ‘George IV believes that the author of Waverley is Scott’, the subordinate sentence ‘the author of Waverley is Scott’ also refers to a thought. Since the thoughts referred to by ‘Scott is Scott’ and ‘the author of Waverley is Scott’ are different, the sentences ‘George IV believes that the author of Waverley is Scott’ and ‘George IV believes that Scott is Scott’ cannot be interchangeable salva veritate.
I do not wish to discuss here the objections of detail that could be made to Russell’s and Frege’s solutions. I want to mention only the general objection made to Fregean-kind solutions of the riddles of reference, according to which they induce us to accept some kind of Platonism of senses and thoughts, unlike Russell’s ontologically more economical solutions. Against this, the last chapter showed that we can preserve the objectivity of sense as something interpersonally accessible, if we understand senses as always being embodied semantic-cognitive rules developed as interpersonal corrigible conventions or as combinations derived from them.
3. Reviewing Fregean assumptions
Who is right? Russell or Frege? As I noted at the outset, my hypothesis is that it is not a matter of choosing between two views. The fact that we have achieved no consensus regarding the right theory reinforces the suspicion that both theories have some truth. This is why I suppose that each of them has insightful content mixed with very implausible metaphysical assumptions, and that these implausible metaphysical assumptions are what make them appear irreconcilable. Thus, in the course of this appendix I will reconstruct these theories by eschewing their metaphysical assumptions and filling the resulting gaps with more plausible ideas.
Let’s start with Frege. We have already seen that we can eliminate the anachronistic ontological realism of senses if we replace it with any psychological instantiation of a semantic-cognitive rule qualitatively identical to the one with which we are associating the term. Repeating what has already been proposed in our reading of Ernst Tugendhat in the introductory chapter, it is perfectly plausible to identify what Frege called the senses of singular sentences in terms of semantic rules, so that:
(i) The sense of a nominative expression (the mode of presentation of the object) is the same as the identifying rule (Identifikationsregel) of a singular term, whose immediate criteria of application are adequate configurations of identifying s-properties (tropes) of the object.
(ii) The sense of a predicative expression (as its conceptual content) is the same as its ascription rule (Verwendungsregel), whose first criteria of application are s-properties dependently associated with the object.
(iii) The sense of an assertive sentence (its s-thought or thought-content) is the same as its verifiability rule (Verifikationsregel), whose immediate criteria of application are its possible truthmakers, which as we have seen can often be better identified (differing from Frege) with the sub-fact referred to by the sentence, which remits to a grounding fact. (See Tugendhat 1976: 262; Tugendhat & Wolf 1983, ch. 13)
A second point is to reject some of Frege’s odd ideas concerning reference, like those of an unsaturated concept as the reference of a predicate and of truth-value as the reference of a sentence, as I argued in the last chapter. It is much more plausible to see the concept in a natural way as the sense of a predicative expression – a conventionally grounded rule – and the reference of a sentence not as a truth-value, but simply as a fact.
A further thing we did in the last chapter was to paraphrase the Fregean concept of existence. For Frege existence was the property of a concept of being satisfied by at least one object. Fur us existence is the property of a conceptual sense – of a semantic-cognitive rule – of applying effectively (and not merely putatively) to at least one referent belonging to a chosen domain during some period of time (the period in which the object is said to exist). Thus, to know that a referent exists is to know that its conceptual rule is effectively and continuously applicable in its proper domain during the time during which the object can be said to exist. Moreover, as we have seen, this does not deprive existence of objectivity, because if the effective applicability of a conceptual rule is a property of the rule, it is also a second order property of the referent, which is that of having its conceptual rule effectively applicable to it. This is what allows us to envisage an object as really existing in the outside world.
This result can be conceded for each of the rules (senses) already supposed by Tugendhat: (i) The existence of an object (made up of a certain relatively independent compresent bundles of s-properties) is the same as the effective applicability of the identifying rule for the singular term that names it. (ii) The existence of an s-property – differing from the object to which it is attached by its relative dependence – is the same as the effective applicability of the ascription rules of its predicative expression to an s-property dependent of the object. (iii) By symmetry with cases (i) and (ii), the existence of a fact (minimally an arrangement of an independent bundle of compresent s-properties and a dependent s-property) is the same as the effective applicability of the verifiability rule constitutive of the s-thoughts to the verifier (or truthmaker) of this fact. Since the verifiability rule is the real Fregean thought, then the existence of the fact is also the same as the effective applicability of the thought-content expressed by the assertive sentence. Existence here is also called ‘truth’ in the derived sense of the reality of a fact.
Finally, even in the context of Fregean theory, I want to treat sentences without a reference as ultimately false and not as simply devoid of truth-value, as Frege has suggested. After all, the reason Frege believed that sentences with components that lack reference are devoid of truth-value lies in his insistence on the indefensible idea that the reference of a sentence should be its truth-value. However, at this point we are already certain that the reference of a sentence is a fact. Therefore, the absence of such a fact (verified by the emptiness of the singular term) just leads us to the falsity of the whole sentence, as we have shown in our discussion of the Fregean solution to the question of the reference of non-existents. This heavily corrected version of Frege’s view is already close to the position of Russell, who regarded sentences with empty attributive definite descriptions as false.
4. Reviewing Russellian Assumptions
Now it is time to review the assumptions of Russell’s theory of descriptions. A first step is to rule out (i): the thesis according to which definite descriptions and even our usual names (which for him were clusters of descriptions) are not to be viewed as referential terms, but rather as incomplete symbols.
This Russellian thesis flies in the face of our most fundamental ordinary language intuitions. For what could better exemplify a referential expression than a proper name or even a definite description? One could even say that our usual names, definite descriptions and indexicals are the templates that allow us to define singular referential terms as those whose function is to select precisely one object, indicating which it is among all other objects of a certain domain. To try to change this is to distort natural language in a way that spreads confusion. Thus, without denying that definite descriptions are incomplete symbols, I will maintain that definite descriptions are typical referential terms.
Russell’s intention with his logical atomism and semantic referentialism was to eschew the supposed referential and semantic role of definite descriptions with the ultimate goal of replacing ordinary language referential expressions with what he called logical proper names, the only truly referential expressions. However, as we have already seen earlier in this book (ch. 3, sec. 3), this doctrine is hopeless, and his semantic referentialism indefensible (Tugendhat 1976: 437; Kripke 2013, ch. 1).
Once we reject Russell’s atomist doctrine of logical proper names, there is no reason to deny that usual names and definite descriptions are referential terms. Even when definite descriptions are analyzed in the form of a conjunction of quantified predicative expressions, as Russell does, together they can continue to do the same referential work of a singular term, since it is assumed that they are able to pick out a single object and distinguish it from all other objects of a given domain. After all, this is all that is required for an expression to be a singular term.
We must also reject a second assumption made by Russell, namely, his mysterious suggestion that (ii) definite descriptions do not have any meaning in themselves. Within the semantic referentialism of Russell’s logic atomism this assumption makes sense: since for him descriptions aren’t referential expressions and reference is the source of meaning, then it is justified to say that they aren’t intrinsically meaningful. But even if you complete them by constructing meaningful statements like ‘The man who wrote “On Denoting” was a philosopher’, it seems impossible to explain why the addition of a new predicate produces a meaningful statement. Assumption (ii) only reaffirms the incoherence of Russell’s semantic referentialism. One cannot reasonably doubt that definite descriptions have meanings in themselves or that they are referential expressions.
Now, once we reject Russell’s semantic referentialism, admitting that we usually make our references by means of semantic-cognitive rules, one thing is clear. The Russellian requirement of applying a predicate to a single object with such-and-such characteristics already constructs something at least close to an identifying rule with a complete sense allowing us to refer to something unique.
5. Building a bridge between both views
Once in possession of a depurated understanding of Frege’s and Russell’s analysis – one that deprives them of their implausible speculative wrappers – we are ready to take the final step. We need to use the semantic-cognitive rules constitutive of senses, together with the concept of existence as the effective application of these rules, in order to build a ‘Tugendhatian bridge’ allowing us to travel from Fregean solutions of riddles of reference to Russellian ones and vice versa. In this way, I will demonstrate that their answers to puzzles of reference are in essence inter-translatable and therefore reconcilable. Here is how this can be done:
(i) Reference to non-existents. As we have seen, the most reasonable answer to the Fregean problem of how to give meanings to statements referring to non-existent objects is that we can at least conceive how we can supplement the dependent (unsaturated) sense of a predicative expression with the independent (saturated) sense of a singular term, thus constituting the complete content of a thought. This is what allows us to think of the present King of France as bald or wise without having to admit his actual existence.
A better understanding emerges when we translate Fregean senses in terms of semantic-cognitive rules. In this case – following Tugendhat – we normally say that the true ascription rule of the predicate always applies to its usual reference as a consequence of the application of the identifying rule. Returning to an example considered in the introduction: Seeing the Earth from outside the earth’s atmosphere for the first time, Russian cosmonaut Yuri Gagarin remarked: ‘The Earth is blue’. But in order to formulate this thought, he first had to identify something outside his space capsule, an object, the planet Earth. Only by means of this identification could he apply the predicate ‘…is blue’ to the s-property of the object he had visually located. We see that the rule for the application of the predicate ‘…is blue’ needs to be first, say, driven by the selective application of the identifying rule in order to find the object called ‘earth’, only then being able to be applied in the identification of the singularized property (trope) of this object of being blue.
As already noted in the introduction, one could object that perhaps Gagarin first saw the blue out there and then identified it as covering the earth, inverting the order of application. However, in this case, his first thought should be ‘Outside there… is blue’ or ‘Something out there… is blue’, first applying an indexical identifying rule and then an ascription rule. That is, the supposed exception confirms the rule: there must first be at least an identifying rule for a spatio-temporal location. Surely, after this spatio-temporal identification Gagarin could invert the order: he could apply to the blue thing a rule identifying it as being the earth and then reapply the ascription rule for the predicate ‘…is blue’. Anyway, the predicate’s ascription rule must always assume the application of some identifying rule, so that it can be decided whether the reference satisfies it or not. On the other hand, if the statements were ‘The earth is red’ or ‘Out there is red’, they would be false because the object/place located by means of the identifying rule would not have the s-property satisfying the ascription rule of the predicate ‘…is red’.
Let us consider now the case of empty singular terms, the alleged reference to non-existents, as found in the sentence ‘Vulcan is red’. As we know, ‘Vulcan’ is the name of a small planet that astronomers once believed should exist between the Sun and Mercury in order to explain variations in the perihelion of the later. According to the calculations of the astronomer Le Verrier, this small planet would have been located approximately 21 million kilometers from the Sun… This is the Fregean sense of this name; the mode of presentation of its reference, the identifying rule. However, since we now know that the planet Vulcan does not exist, the name’s reference is empty and its identifying rule inapplicable. As a result, the effective application of the ascription rule of the predicate ‘…is red’ is also impossible. As the identifying rule of the singular term doesn’t apply to any expected object, an application of the predicative rule cannot be made, remaining non-satisfied by any actually given s-property. Thus the predicate cannot be applied, making the sentence false (pace Frege and Strawson).
As already noted above, we do not need complex metaphysical theories to explain what happens in this case. The right explanation appeals to our capacity for imagination. We are at least in some measure able to conceive what it would be like to apply both rules in combination, even if we cannot find a way to apply them to the real world. To use a Wittgensteinian expression, we are able to conceive the reference of a statement like ‘Vulcan is red’ as a possible state of affairs (Wittgenstein 1984a, 3.02). It is only to the extent that we are able to conceive the possibility of applying both combined rules in the constitution of a verifiability rule that we can understand the cognitive meaning of the statement. By doing this we realize that the proper name is empty and that this thought-content (cognitive meaning) has no effective application to a real fact in the world. This is why the statement ‘The present King of France is bald’ is already able to express a complete sense, a thought-content. We are capable of conceiving the two rules used in combination in order to form the verifiability rule, the thought-content, the sense of the statement, imaginatively applicable in our minds to a possible fact or state of affairs, but without effective application in its proper domain as a real fact in the world; and this makes this rule-thought-content false.
To the question of how it is possible to assign wisdom to something that does not exist, the answer is now clear: we are capable, at least in some measure, of conceiving the application of semantic-cognitive rules and their combinations, and by doing this we give meaning to the terms and the statement as a whole. We are able to make a fictive predication, even if only to a limited degree, without the proper assertive or judicative force.
Now, in the light of this reconstruction it is easier to make the theory of sense agree with the theory of descriptions. We can paraphrase the description ‘the present King of France’ in a Russellian way as:
1. At least one x and at most one x, such that x is a present King of France.
And we can say that what is expressed here is a different formulation of the Fregean sense, of the same identifying rule for the present King of France, which is seen as having two components:
(i) the condition of uniqueness,
(ii) the ascription rule for the predicative expression ‘…is a present King of France ’.
Together (i) and (ii) form a kind of rule of identification because they give us the possibility of distinguishing at least one and at most one object by means of the criterial properties derived from the predicate, such as the presence of a hereditary head of state governing France today.
The non-existence of the present King of France corresponds to the lack of effective applicability of the identifying rule roughly expressed by the conjunction of (i) and (ii) and, therefore, to the lack of reference. As for the predicate ‘x is wise’, its ascription rule also does not apply, since no one has the property of being the present King of France to whom the rule could apply. But this predicate also expresses an ascription rule as a conceivable Fregean sense. Pulling the threads together, with the statement (2) ‘There is at least one x and at most one x, such that x is a present King of France, and x is bald’ we do nothing more than attempt to apply the same verifiability rule expressed by the statement (3) ‘The present King of France is bald’. That is, we realize that the identifying rule cannot find a bearer and that consequently the ascription rule is also inapplicable, the same being the case with their combination, namely, the verifiability rule. In this way analyzing the case of reference of non-existents, we are already able to see how we can exchange a ‘Fregean’ explanation for a ‘Russellian’ explanation and vice versa.
(ii) Negative Existentials. In the last chapter (despite Frege’s view) we identified the concept with the sense of a predicative expression. This also means that to say ‘The present King of France does not exist’ becomes the same as saying that the sense of ‘the present King of France’ does not determine its reference.
How would we express this by using semantic-cognitive rules in place of the sense? Well, we would say again that the sense or meaning expressed by a singular term like ‘the present King of France’ consists in the identifying rule of this definite description in its only conceivable application. We know this because we know we can at least imagine to some extent how we would apply this definite description. But we cannot gain any awareness of the effective applicability of this rule, that is, we cannot say that the object that should be referred to by this definite description exists, since we know that this rule cannot be effectively applied in its proper domain.
Finally, we come to the corresponding ‘Russellian’ analysis. A description like ‘the present King of France’ is here transformed into
1. At least one x and at most one x is such that x is a present King of France.
Here again, what we have is one identifying rule for a particular object, which is composed of two sub-rules:
(i) the condition of unity,
(ii) the rule of application of the predicate ‘...is a present King of France’.
Now, to say ‘The present King of France does not exist’ is to say ‘It is not the case that there is at least one x and at most one x such that x is a present King of France’. This is the same as to say that the identifying rule roughly composed of conditions (i) and (ii) is not effectively applicable. What is the difference between this rule and the Fregean sense of the description? The answer is again that the ‘Russellian’ analysis only decomposes the identifying rule of ‘The present king of France’ into two rules: a unity rule and a rule of application for the predicate. Saying that the present king of France does not exist is to say that the ascription rule of the predicate ‘…is a present King of France’ does not effectively apply in its proper domain, in this case because it does not fulfill the existential condition implicit in (i). Once more, the ‘Russellian’ and ‘Fregean’ analyses of negative existentials seem to be two different ways to say approximately the same.
(iii) Identity. Consider now identity sentences like ‘the Morning Star is the Evening Star’. How can this sentence be informative, if the two descriptions refer to the same object? Frege’s reply is that despite the fact that these descriptions refer to the same object, they express different modes of presentation of this object and that because of this they are informative.
Paraphrasing the concept of meaning in terms of a semantic-cognitive rule, what Fregean semantics suggests is that the sentence above is informative because it tells us that we identify the same object using two different identifying rules. These identifying rules are respectively a rule for the last star to disappear at dawn and the rule for the first star to appear in the evening. These rules call for different criterial settings. That they in the end refer to the same object is, in the context considered by Frege, a further piece of information, a further identifying rule for the planet Venus. We disregard them in the present identification, which makes the thought-content contingent a posteriori (case of immediate emphasis in our discussion in section 2 above).
In Russellian terms, letting M abbreviate the predicate ‘…is a morning star’ and the E abbreviate the predicate ‘…is an evening star’, the identity sentence can be symbolized as:
(1) Ǝx [(Mx & Ex) & (y) (My → y = x) & (z) (Ez → z = x)].
In other words:
(2) There is at least one x and at most one x that is the Morning Star and this same x is the Evening Star.
In this case, what we are doing with the identity sentence is (i) making a conjunction of two different ascription rules of predicates and adding to it the condition (ii) that they both apply to one and the same object. Thus, the ‘Russellian’ analysis only assures us that the identifying rule constituted by ‘Ǝx [Mx & (y) (My → y = x)]’ applies to the same object that the identifying rule constituted by ‘Ǝx [Ex & (z) (Ez → z = x)]’ applies to, since by transitivity if y = x and x = z, then y = z. But this is like saying that we have two different identifying rules, two different Fregean modes of presentation further known as being of the same object. Again, the two analyses turn out to be interchangeable.
(iv) Intentional Contexts. Finally, consider expressions of propositional attitudes such as:
(1) George IV believes that Scott is Scott.
(2) George IV believes that the author of Waverley is Scott.
Why doesn’t the truth of (1) guarantee the truth of (2), if both subordinate clauses are identity sentences about the same person?
As we have noted, for Frege the answer is that in such cases a subordinate sentence does not have its usual reference, which for him is its truth-value. Subordinate sentences refer, he thinks, to the thoughts expressed by them, and the thoughts expressed by them in (1) and (2) are different. Hence, the truth-value of the sentence that expresses a propositional attitude cannot be derived from the truth-value of the subordinate sentence, making inter-substitution salve veritate impossible.
Since we reject Frege’s artificial idea that the normal reference of a sentence should be its truth-value, we must first reformulate his solution. An isolated statement such as ‘The author of Waverley is Scott’ has for us as its immediate reference the aspectually given sub-fact represented by the identification of the modes of presentation of the singular terms flanking the identity relation. This sub-fact can be represented by the statement ‘The author of Waverley = Scott’, while the mediated reference, the grounding fact, can be represented by the statement ‘Scott = Scott’ (The underscore ‘_’ signals that I am speaking about facts). As already explained, these facts are complex arrangements of s-properties.
Now, what fact is represented in the case of propositional attitudes? First, we can preserve Frege’s idea that in utterances of propositional attitudes the reference of the subordinate sentence is its sense, for us a thought-content or s-thought – a mental fact. But there is more to the matter. This mental fact is part of the whole fact represented by a propositional attitude, which has the form aAp, in which a abbreviates the person who has the attitude, p abbreviates the thought referred to by the subordinate sentence, and A abbreviates the attitudinal verb, which can be one of belief, knowledge, desire, etc. The reference of ‘Henry IV believes that the author of Waverley is Scott’ is not any fact in the world. It is a fact consisting in the psychological belief of the real Henry IV that the author of Waverley is Scott. In other words, a propositional attitude conventionally refers to a partly mental fact: the (mental) attitude of a (partly non-mental) speaker (a person) concerning a certain (mental) thought-content (s-thought) that we can symbolize as aAp. Here p refers to a thought-content (dispositional or not) in the mind of person a, such that it no longer refers to any fact in the external world that could possibly match p, making it true. Here if aAp represents the partly mental fact that aAp, then the statement is true, otherwise it is false; and while a as a person should be a bundle of compresent (physical and mental) s-properties in the world, Ap is a mental relational s-property appropriately linking person a with a factual arrangement of mental s-properties. In other words, in statements of propositional attitudes, what matters is a certain relationship between the contents of the main clause (usually expressing a dispositional mood or mental act of the speaker) and the thought-content expressed by the subordinate clause. And this must be, so that the truth of a sentence of propositional attitude depends only on the fact of this attitudinal relationship to p really being in the mind of person a, independently of the truth or falsity of the thought expressed by p concerning a fact in the real word. We now see more clearly the reason why the thought expressed by the subordinate clause cannot be replaced salva veritate in (1) and (2): The mental dispositions or acts of a concern different factual thought contents expressed by different subordinate clauses. Finally, it is worth noting that the person who judges these propositional attitudes is a third person or even the first person in an introspective mood, and there is no distinction between the senses and the facts reported, once they are the same.
Now, to paraphrase thought-contents as verifiability rules of sentences, we can say that the verifiability rules of the subordinate sentences of (1) and (2) are different, applying only to the mental fact of p, without committing ourselves to the effective applicability of these rules to what p could refer to in the real world. So, considering the sense of the proper name Scott as an identifying rule, we can in many cases paraphrase (1) as:
(1’) George IV believes that the identifying rule (i) (sense (i)) that he has for ‘Scott’ applies to the same object as the identifying rule (i) (sense (i)) that he has for ‘Scott’.
This belief is true even if George IV knows nothing about Scott. We can paraphrase (2) as:
(2’) George IV believes that the identifying rule (i) (sense (i)) that he has for ‘Scott’ applies to the same object as the different identifying rule (ii) (sense (ii)) that he has for ‘the author of Waverley’.
The obvious argument drawn from this is the following:
1. In (1’) and (2’), the senses represented by the identifying rules considered by George IV are different.
2. Therefore, the two thought-contents are different.
3. The truth-value of the propositional attitude statements depends only on the (essentially) mental fact that a thought-content p is the object of the attitude A of person a.
4. These (essentially) mental facts are different in each case.
5. Conclusion: We do not need to preserve the same truth-value in statements (1) and (2).
The subordinate clauses cannot replace one another salva veritate because the factual thoughts-as-references expressed by them are different.
Finally, consider the Russellian paraphrases. Statement (1) can be formulated as ‘George IV believes that Ǝx [(Axw & (y) (Ayw → y = x)) & (s = s)]’, or simply as:
(1’’) George IV believes that there is at least one x and at most one x, such that x is Scott and that x is Scott.
And statement (2) can be (in a secondary occurrence) formulated as ‘George IV believes that Ǝx [(Axw & (y) (Ayw → y = x)) & (x = s)]’ or, more naturally:
(2’’) George IV believes that there is at least one x and at most one x, such that x is the author of Waverley and x is Scott.
Now, as the subordinate clauses expressing George IV’s beliefs (i) ‘there is precisely one x that is Scott’ and (ii) ‘there is precisely one x that is the author of Waverley’ are different, ‘Scott is Scott’ cannot mean the same mental fact as ‘Scott is the author of Waverley’.
A point to note is that Russellian analysis only better clarifies one aspect of our version of Fregean analysis. After all, we can present the Fregean analysis in (2’), for example, as:
(2’’’) George IV believes that there is at least one x and at most one x, such that the rule of identification (i) for Scott (sense (i)), as well as that the rule of identification (ii) for the description ‘the author of Waverley’ (sense (ii)) effectively applies to x.
But (2’) and (2’’’) do not differ significantly. After all, suppose we say based on Russell that George IV believes the rule of identification (i) that he knows for the name ‘Scott’ and that the ascription rule (ii) that he knows for the predicate ‘…is the author of Waverley’ effectively apply to precisely one object. This amounts to nearly the same thing as to say, based on Frege, that George IV believes that the identifying rule (i) (the sense (i)) he knows for the singular term ‘Scott’ has the same referent as the rule of identification (ii) (the sense (ii)) of the definite description ‘the author of Waverley’. Now it is clear: in the case of propositional attitudes, the two analyses are also inter-translatable.
Summarizing, we can analyze the referential function of definite descriptions in at least three ways: (a) in terms of abstract entities, as did Frege when speaking of senses, (b) in terms of semantic-cognitive criterial rules inspired by approaches like that of Tugendhat, and (c) using resources from predicative logic, as Russell did in his theory of descriptions. These are only complementary ways of saying approximately the same.
As I have noted, the initial impression of strangeness of the proposed view comes from the acceptance of the metaphysical assumptions that permeate what Frege and Russell wrote on the issue. Against Russell’s own belief, his paraphrases of definite descriptions are nothing more than expressions of semantic rules. These paraphrases make it possible to express the referential function of definite descriptions in their attributive use by means of quantified predicative expressions used in a domain that grants them singularizing application. In this reading, they are reformulations of senses or modes of presentation which cannot be more than semantic-cognitive criterial rules. Assuming that these last rules only exist in their applications, either in imaginative psychological rehearsals or in real cognitive instantiations, the compatibility of the so-understood theory of descriptions with our cognitivist approach is clear.
 This answer has a well-known flaw: it does not preclude the possibility that statement (2) is true in the case of existing more than one present king of France.
 Saul Kripke has denied this, suggesting that Russell and Frege appealed to a simplified model of descriptivism with only one definite description. But one need only read with attention chapter 5 of Russell’s The Problems of Philosophy (1912) and Frege’s remarks (1882, 1918) to see that both were well aware that proper names abbreviate complex sets of descriptions.
 In his book on logic, Strawson suggested that statements without reference like ‘The present king of France is wise’ have no truth-value, because in order to have truth-value such statements must assume the truth of the presupposed statement ‘The present king of France exists’. (Strawson 1952: 185)
 See Stephen Neale, who in my view settled the case in favor of Russell’s analysis, in his book Descriptions (1990: 26-28).
 Defending his necessary a posteriori, Kripke could offer his counterexample of the dislodged Venus (Kripke 1980: 57-58): Suppose that Venus had long before been struck by a comet, shifting its orbit so that it occupied a very different position in the sky. Venus would be neither the second planet nor the brightest... However, we could still identify this planet as Venus precisely because it is seen, by reason of the application of its identifying rule, as having at the time of its discovery and soon afterwards occupied the position of the second planet in our solar system. In Venus’ case, the identification rule is restricted to the sufficient satisfaction of this fundamental localizing description of the planet. (See appendix to chapter 2)
 Certainly, all these three cases can be formulated using Russellian devices in which referential terms are transformed into predicative expressions. Thus, consider the existence of what is predicated in the statement ‘Marsupials exist’: symbolizing ‘…is a marsupial’ as M, we have ‘(Ǝx) (Mx)’. Consider now the definite description in the statement ‘The Morning Star exists’: symbolizing the predicate ‘… is a morning star’ as M, we have ‘Ǝx [Mx & (y) (My → y = x)]’. For the proper name in the statement ‘Socrates exists’, abbreviating the descriptive content that the name contains with the predicate ‘socratizes’ and symbolizing this last predicate as ‘S’, we have (Ǝx) [Sx & (y) (Sy → y = x)]. Finally, consider the fact asserted by ‘Socrates is wise’: symbolizing ‘…socratizes’ by S and ‘…is wise’ by W, we have (Ǝx) [Sx & (y) (Sy → y = x) & Wx]. Although replacements like these cannot meet some Kripkean objections, I effectively answer them with my own version of the cluster theory of proper names summarized in the appendix of Chapter 1 in this book.
 As Ernst Tugendhat noted, to refer to one object is not only to coordinate the name with it, but ‘to distinguish it from all the others belonging to a certain domain.’ (Tugendhat & Wolf 1983: 153).
 It is true that definite descriptions can refer to different individuals in different possible worlds, unlike proper names. However, as we saw in the appendix of chapter 1, definite descriptions are accidental or flaccid designators only when we associate them semantically with proper names, otherwise they tend to become rigid. This demonstrates that there is nothing metaphysical or mystical about the semantics of ordinary proper names, contrary to what Saul Kripke and others seem to have believed.
 He wrote, ‘I advocate that a denoting phrase is essentially part of a statement, and does not, like most single words, have any significance on its own account’ (Russell 1994: 51).
 I will leave aside all the complexities related to ‘non-Russellians’ definite descriptions like ‘the round table in this room’ (indexical use), ‘the man drinking a martini over there’ (referential use) or ‘the White Anglo-Saxon Protestant’ (general use). They divert us from our intended point, creating unnecessary distractions.
 I use ‘conceive’ and ‘imagine’ as equivalent verbs thought with different emphasis. Not all imagination is imagistic. We can speak, for example, of ‘mathematical imagination’.
 I understand here a person in P. F. Strawson’s manner as an object of both (physical) p-predicates and (mental) m-predicates (1959, I, ch. 3).
 Since the reference is determined by the sense, for Frege there must be a second indirect sense here determining the indirect sense of the subordinate clause. But no one was able to point to this hidden indirect-indirect sense, which is apt to produce a regress. We circumvent this by holding that the whole attitude ‘aAp’ is first that of a third person (or by the same person in a reflexive mood) concerning the fact that ‘aAp’. For example: ‘[I am sure that] Anna believes that Goya painted The third of May, 1808’. Here the fact that Anna believes that Goya painted the Third of May, 1808, must have an external mode of presentation for me. This could be because she visited the Prado Museum with me yesterday, which determines the reference or fact, in a case where I use a p-thought to refer to Anna’s belief in her thought-content.