Russell on the Structure of Propositions
Russell's semantic views before 1905, the internal contradictoriness of which led further to the theory of descriptions and at the same time to the acceptance of two-level semantics, become clear during the comparison of them with the theory of Frege. This comparison, the basis of which is constituted not only by works of Frege, but also by his correspondence with Russell in the period of 1902 - 1904, is undertaken first of all by Russell himself in Appendix A to The Principles of Mathematics (1903). The confrontation of views of both authors is a matter of great interest because they have similar reasons for the study of semantic problems. The examination of principles of mathematics as well as the ideas of logicism raise a question about logical structure of mathematical and logical propositions and about relation between linguistic signs and logical forms. In spite of the similarity of problems and of ideas suggested for their decision there is a considerable difference between approaches of both authors. Russell's approach to a notable extent is determined by his ideas about ontological structure of the world that is cognized as well as by accepted by him epistemology, whose construction is an additional independent source of Russell's admission of one definite typology of meanings.
One of the essential differences between Frege's and Russell's logical theories is in Russell's opinion the solution of the question whether a concept can be a logical subject of proposition or not.
Frege holds that concept (Begriff) is unable to perform this function, i.e. it is impossible to predicate something of it in a sentence. Russell has a contrary opinion. But the comparison of the notions "concept" of Russell and "Begriff" of Frege itself seems to be not fully correct. The grammatical interpretation of Begriff and concept by them is nearly identical - neither Frege's nor Russell's concepts can be expressed in the language in proper names. Their presence is indicated with the help of words "all", "any", "some", "no one". In spite of this Frege's and Russell's concepts are entities of different types.
Frege believes concepts to belong to the field of references (Bedeutung). Concept is "the reference of a grammatical predicate" (3). In his letter to Husserl (24.05.1891) Frege suggests a scheme representing the semantic place of concepts (4).
Frege defines concept also as a function, the value of which is always a truth-value (2). Even when a word that stands for a concept has in a sentence the place of logical subject, the structure of a thought that is stated in this sentence is not identical with the structure of subject-predicate proposition, in which the predicate is assigned to the subject. It has structure of a relation between two predicates. Frege says that what is mentioned in such a sentence is not that an object falls under a concept, but that one concept falls into another one of a higher level.
Russell regards concepts as constituents of propositions.
It should be noted that Russell uses the word "proposition" at various periods in various meanings. This meaning can often be defined only proceeding from the context. But at the period under review Russell considers proposition not as a linguistic formation, but as something for what a linguistic sign stands, and such a sign is a declarative sentence. In The Principles of Mathematics, the correspondence with Frege and some manuscripts and papers (1903 - 1904) he defines main features of propositions.
1. They are initially true or false.
2. They do not exist, but they have being in a sense that they are objects of thinking.
3. They are objects of thinking independently of the fact that somebody thinks about them at some definite moment or not.
4. These formations are complexes. Their complexity is cognizable, it is initial and independent of the cognizing subject.
5. These complexes contain terms as their constituents. By the term Russell understands all that may be object of thinking and that can be enumerated. There are two kinds of terms or objects: some of them have some place in space or time, the other - not.
All these features with the exception of the last correspond to the characteristic that is peculiar to Frege's thought (Gedanke), i.e. to the sense of a sentence. Russell's proposition differs from Frege's sense only in supposition that it can contain objects from the field of space and time. In Frege's opinion thought can contain only senses of words that stand for such objects. It must be noted that Russell does not accept the distinction between sense (Sinn) and reference (Bedeutung) made by Frege. The comparison of his terminology and terminology of Frege that is undertaken in their correspondence shows that by proposition Russell means the sense of the sentence, but considers it as the only object corresponding to this sentence. So he translates the term "meaning" always used by him into German as "Bedeutung" (4) and remarks that he does not accept the idea of truth-value as the reference of the sentence. The reference of the sentence is proposition that can be true or false.
Just that very Russell's supposition, according to which objects taking some place in space or time are constituents of some of the propositions, leads him to the rejection of Frege's thesis about identity of all true (and all false) propositions. He considers the replacement of one of the constituents of a sentence by the other one with the same reference but another sense as impossible. It is impossible by such a substitution to keep the identity of propositions, i.e. meanings of the both sentences - the initial one and the result of replacement. In his opinion this conclusion is confirmed by the antinomy of propositions described in The Principles (§ 500).
For his part, Frege thinks that such a situation, when object itself is a constituent of such an intensional formation as thought, is inadmissible. Indeed, it seems to be strange that in the meaning of the sentence "The Mont Blanc is more than 4000 meters high" the Mont Blanc itself occurs as one of its constituents (4). However this idea conforms to other Russell's conceptions. This thesis depends on his theory of understanding. It is necessary to be acquainted with the object that is designated by a linguistic sign in order to understand the latter. Otherwise, without acquaintance with Mont Blanc it is impossible to understand the corresponding proposition.
Russell holds that the analysis of propositions into their constituents is not the only possible. He speaks about the possibility of another kind of analysis already in The Principles. The basis of this one is the notion of propositional function. During such an analysis one determines on the one hand one or more terms that can be replaced by the other, and on the other hand a constant form. When proposition is analyzed in such a way - into one or more arguments and a function - the last one becomes the basis of classification of propositions. Then one can consider each proposition as a particular case of some type of propositions. The basis of each such type is a propositional function. It is evident that this kind of analysis is similar to Frege's one. Russell himself considers his propositional functions to be analogous to concepts of Frege. The differences are following.
1. The value of propositional function for one definite argument is not a truth-value, but a proposition itself.
2. Unlike the concept, no independent linguistic sign corresponds to propositonal function. The basis of this thesis is Russell's idea, according to which the structure of language corresponds somehow to the structure of objective world. Russell considers the significance of linguistic expressions to be precondition of their use (8, 13). This significance cannot be subjective. For Russell all subjective is psychological and as a result of it incognizable. In his opinion, the starting point of cognition is experience. In experience a subjective idea (presentation) and an objective reference form a whole. Such a whole is not however knowledge. Knowledge appears when the thinking breaks up this whole into parts, relates its subjective part to the field of psychical and converts its objective part into the content of knowledge that can be reported to other subject (11). Only the possibility to be reported to other subject as well as to be obtained by this subject independently makes knowledge objective and trustworthy. Being an instrument of communication language has as its purpose the transmission of objective meanings. Russell thinks also that on the basis of grammatical structure of language one can suppose, what kinds of objects exist in objective world. In his opinion, each linguistic expression is a name in the sense that it designates something in objective world. There are distinguished three kinds of designation.
Some words (these are mainly proper names) indicate particular objects. This kind of designation may be represented in following scheme.
Other expressions stand for concepts.
Some concepts are predicates which are meanings of words and at the same time themselves are symbols. However, this symbolism has quite other nature than the symbolism of linguistic formations, since these symbols unlike words are not created by cognizing subject. They symbolize objects that are their exemplifications. Russell calls the relation between concepts and their particular exemplifications denotation. This relation is objective or, as Russell also says, logical.
Another part of concepts is formed by meanings of corresponding linguistic expressions that mean objects of this kind. These are relations. As distinct from predicates they have no particular exemplifications and are identical with themselves in each case of their occurrence, in spite of the fact that they can occur between different particular objects, or a particular object and a predicate, or different predicates.
The relation of designation takes place also between sentences and propositions. Therefore it seems possible to apply the notion of denotation also to sentences. The idea of such a possibility considerably depends on the variety of grammatical potentialities of language. Thus, when we assert some sentence, for example "Caesar died", it is possible to give other grammatical formulation of this sentence in the expression "The death of Caesar". These two expressions seem to stand for one and the same object. Russell names the object designated by the expression "the death of Caesar" "propositional concept". He thinks its name to express what is asserted in the corresponding sentence. If sentence is considered as a denoting expression, denoting can be part of the nature of proposition. In that case a sentence expresses the proposition which is its meaning and indicates the fact (or complex) that is none other than propositional concept.
Russell supposes this scheme of relations between the sentence and the object designated by it to hold also on following grounds. Understanding of a sentence does not demand assertion which distinguishes proposition from propositional concept, but requires at least its assumption (10). At the same time it is not propositional concept, but proposition that is asserted in a sentence. All that makes the role of propositional concept and its relation with proposition doubtful and vague. However, whatever status can be received by propositions depending on the treatment of their relations with linguistic expressions, they are subject to analysis, one of the forms of which is the apparatus of propositional functions. Any propositional function that is designated by an arbitrarily chosen sign represents some type of propositions, and for this reason is not itself an objective constituent of the world, but a way of consideration of its constituents.
3. The third distinction between Frege's concepts and Russell's propositional functions was pointed out by Bochenski (1). He thinks that Russell proceeds not from the mathematical notion of function as Frege, but from the analysis of statement by Aristotle. Russell himself underlines in one of his manuscripts (1904) that the mathematical form of function is not a form that is fundamental also in the field of philosophical investigation. In such a fundamental form all constituents of the proposition, the type of which is determined by a propositional function, must be enumerated. The mode of combination of these constituents must be also pointed out (9).
Before 1905 the analysis of propositions by the means of propositional functions is not accepted as a fundamental one. The reasons of this unacceptance are following.
1. Any propositional function is a denoting complex that denotes indefinitely until the value of a variable or variables contained in it is determined. As soon as the value of a variable is determined, a linguistic expression which formerly had no definite meaning receives it. In contrast to propositional function proposition itself being a value of the former denotes nothing. Propositions are units of meaning which constitute subject of cognition and content of knowledge. Russell considers proposition not as a pure logical notion in the spirit of extensional logic that sets aside any content and takes into account only truth-values of sentences.
2. From Russell's point of view, unity of any proposition is formed by some relation. On that ground he considers the expression of the structure of propositions by the means of relations to be more fundamental than by the means of propositional functions, although the last mode seems to him to be simpler.
This preference is connected to a large extent with the theory of external relations, the main ideas of which were formulated in 1899 - 1900. Earlier Russell believed relations to be deduced from properties. But already at the very beginning of the work on the problems of philosophy of mathematics further transformed into the program of logicism it turned out, that many mathematical propositions and axioms suppose plurality and diversity of logical subjects (5) and also the existence between them of asymmetrical relations which are not reducible to identity or difference of predicates of these subjects (7). This theoretical fact was one of the main sources of Russell's interest in the logical structure of propositions. Investigations undertaken in that direction showed that the truth of the necessary mathematical propositions and mathematical axioms in particular does not depend on real existence of plurality of subjects, but is a priori. Such propositions are not liable to empirical verification or falsification - by their consideration the question is rather their possibility or impossibility. Russell in general considers necessary propositions to be demonstrable only in the sense, that they can be reduced to other necessary propositions whose truth is evident. The necessity of propositions is defined in early Russell's works even as perceptible (6).
Asymmetrical relations are also regarded by Russell as perceptible - their consideration implies the reference to experience (5). Being perceived these relations have properties, that are self-evident and accordingly do not permit of substantiation by the means of logical reasoning. The study of these self-evident properties will serve in 1911 as a ground for the proof of existence of particulars (12).
The notion of particular is essential for Russell's semantics. Russell acknowledges the real existence of plurality of particulars - of particular things that are complexes of properties and relations and of particular exemplifications of properties. Russell holds that the "acquaintance" with particular, i.e. the immediate knowledge of it obtained by the means of perception is the starting point of cognition. Another means of receiving of knowledge, to wit reflection has as its material what is perceived through the senses, namely particular. A statement about particular is possible only as a result of activity of thinking that fastens the object of perception in memory and makes it the subject of abstraction. As a result of this process the particular is as a rule eliminated. However, the knowledge of truth, i.e. the knowledge of propositions, which Russell considers to be more genuine than the knowledge of things, always implies the knowledge of particulars.
The admission of the real existence of particulars means the admission of a definite relation of predication between subject and predicate of proposition (12). In such propositions about perceptible as, for instance, "This is red" the predicate cannot be considered as a predicate in the true sense, i.e. that, the nature of what is constituted by the ability to be predicated of some subject, that is to say to be its property. If some particular red patch had the predicate red, this predicate would have realization in space or time and therefore, where there was one patch of red color, there would be already two patches, and so on. If the predicate red cannot be predicated of a particular red patch, i.e. it cannot be said that the proposition "This is red" has the subject-predicate structure, the only possible way to represent the fact, that the perceptible patch indicated by the word "this" is red, is to determine the relation of redness to the place that is taken by this patch. The sentence "This is red" can be reformulated into the sentence "Redness exists here". But in the proposition designated by the last sentence the predicate occurs not as a predicate, but as a term. As a matter of fact a predicate can be predicated only of a particular existing thing, but the sentences designating propositions which involve things permit of similar reformulation. The propositions including no particulars presuppose the consideration of predicate as a term and are as a rule relations of terms-predicates. Thus the idea of the use of predicates as terms can be one of the reasons of Russell's thesis, that predicates can be logical subjects: predicates as predicates can be predicated only of existents, and logic includes no propositions about existents.
Russell observes that the relation of predication, the existence of which is one of the proofs of inadequacy of the theory of internal relations, is really unconsciously presupposed by it - the reducing of relations between subjects to identity or diversity of their predicates and hence to the comparison of these last presupposes the consideration of predicates as terms.
Russell does not accept the reducing of relations to predicates also because of the reason, that a direction or, as Russell says, sense is not peculiar to predicates. By the sense he understands the order of connection of constituents of a proposition which provides the unity of the last, the unity not reducible to the simple sum of its constituents (4). Russell observes that "in every proposition some relation is asserted as regards the terms of the proposition. The classification of relations is, therefore, the classification of the types of proposition" (7). Relations are as ultimate constituents of objective complex entities (propositions) as other terms, and they depend upon the terms that are correlated by them no more than these terms depend upon relations.
1. Bochenski, I.M., Formale Logik, Verlag Karl Alber, Freiburg, München, 1956.
2. Frege, G., "Funktion und Begriff", Gottlob Frege. Funktion, Begriff, Bedeutung. Fünf Logische Studien, Vandenhoeck & Ruprecht, Göttingen, 1969.
3. Frege, G., "Über Begriff und Gegenstand", Gottlob Frege. Funktion, Begriff, Bedeutung.
4. Gottlob Frege. Wissenschaftlicher Briefwechsel, Felix Meiner, Hamburg, 1976.
5. Russell, B., "An Analysis of Mathematical Reasoning Being an Inquiry into the Subject-Matter, the Fundamental Conceptions, and the Necessary Postulates of Mathematics", The Collected Papers of Bertrand Russell, vol. 2, Unwin Hyman, London, Boston, Sydney, Wellington, 1990.
6. Russell, B., "Are Euclid's Axioms Empirical?", The Collected Papers of Bertrand Russell, vol. 2.
7. Russell, B., "The Classification of Relations", The Collected Papers of Bertrand Russell, vol. 2.
8. Russell, B., A Critical Exposition of the Philosophy of Leibniz, George Allen & Unwin LTD, London, 1951.
9. Russell, B., "Fundamental Notions", The Collected Papers of Bertrand Russell, vol. 4, Routledge, London and New York, 1994.
10. Russell, B., "Meinong's Theory of Complexes and Assumptions", Mind, vol. 13, N 49 - 52, 1904.
11. Russell, B., "On the Distinction between the Psychological and Metaphysical Point of View", The Collected Papers of Bertrand Russell, vol. 1, George Allen & Unwin, London, Boston, Sydney, 1985.
12. Russell, B., "On the Relations of Universals and Particulars", Logic and Knowledge: Essays, 1901 - 1950, George Allen & Unwin LTD, London, 1956.
13. Russell, B., The Principles of Mathematics, University Press, Cambridge, 1903.