(in A.W. Moore, ed., Meaning and Reference, OUP 1993)
An extract from Chapter XVI of Russell’s Introduction to Mathematical Philosophy (London: Allen & Unwin, 1919).
In this excerpt, Russell presents a semantic theory (a theory of meaning) for a particular type of expression: descriptions. Russell delimits definite descriptions from indefinite ones, saying that the former take the form ‘the so-and-so’ while the latter take the form ‘a so-and-so’. Russell attempts to establish two fundamental points: one, descriptions do not have meaning in isolation, and two, the propositions in which descriptions occur include a quantifier phrase plus a propositional function rather than singular terms.
Russell establishes the first point separately for both sorts of description. In the case of indefinite descriptions, Russell argues that to define an indefinite description (and thus assign it meaning) would require specifying a definite object it describes. Since such descriptions are necessarily ambiguous, they do not describe any definite object and as such cannot be assigned meaning in isolation (e.g., there is no definite object described by the indefinite description ‘a man’, making it impossible to define the term and thus impossible to assign it meaning). (49, ¶2)
Russell then argues that definite descriptions do not have meaning in isolation either, since their doing so would require that they be singular terms. Since definite descriptions are not singular terms, they do not have meaning in isolation (Russell here uses the term ‘name’ for a singular term, defining it as a symbol with parts that are not symbols and which has as its meaning its referent). (50, ¶4)
Russell thinks definite descriptions aren’t singular terms because substituting a singular term for a definite description in a proposition – even when the definite description is describing the referent of the singular term – always results in the expression of a different proposition. (52, ¶3) Russell gives the example of substituting ‘Scott’ (a singular term) for ‘the author of Waverly’ (a definite description) in the proposition ‘Scott is the author of Waverly’. The result is the proposition ‘Scott is Scott’, which is clearly a different proposition than when the definite description is included. If definite descriptions were singular terms, then swapping them out for singular terms whose referents they describe would not change the proposition expressed; since such change occurs, it follows that such descriptions are not singular terms, and hence not meaningful in isolation.
Russell thinks this first part of his theory is a virtue because it is supposed to explain how discourse about non-existent entities is possible: such discourse is possible because, given that descriptions aren’t meaningful in isolation, there aren’t non-existent entities (e.g., ‘a unicorn’) as constituents of propositions; rather, descriptions which do not describe anything are constituents of propositions. On the contrary, if terms denoting non-existent entities were singular terms rather than descriptions, then the entities denoted would have to be constituents of propositions and thus would have to exist in some sense. (48, ¶2)
Since descriptions in isolation have no meaning, in order to provide a semantic theory for these expressions Russell must analyze the propositions in which they occur. Russell does so for either sort of description. In both cases, the proposition expressed is said to include a quantifier phrase with a propositional function as a part*.
Russell early in the excerpt provides an example, indicating that ‘I met a man’ should be translated as “The propositional function ‘I met x and x is human’ is sometimes true.” (47,¶1) It is obvious that this proposition includes a propositional function. But it also includes a quantifier phrase because it can be translated into ‘∃x[I met x and x is human]’, which means the same thing since both indicate that at least one proposition resulting from the propositional function ‘I met x and x is human’ is true.
Russell thinks the same of the propositions in which definite descriptions occur, albeit with an added proviso: such propositions must uniquely denote an object, i.e., there can only be a single object denoted. We cannot speak of ‘the inhabitant of London’, since there is more than one person inhabiting London. (52, ¶3) Thus, the proposition ‘I met the author of Waverly’ will have the same form as ‘I met a man’, except that the former will also have in its translation an element indicating that there is only one such author, viz., “The propositional function ‘I met x and x wrote Waverly and only x wrote Waverly’ is sometimes true”, which in turn can be translated into ‘∃x∀y[I met x and (y wrote Waverly↔y=x)]’. The latter format is to be preferred because it succinctly captures what Russell takes to be the logical form of the propositions in which descriptions occur.
A subsidiary issue Russell discusses in the latter portion of the excerpt is the status of proper names. To Russell, the fact that one can question the existence of a so-and-so, i.e., of a description, and not a name (it would be meaningless to question the existence of a name since the term wouldn’t have meaning if it didn’t refer, i.e., if the object named didn’t exist) indicates that what we take to be proper names oftentimes are properly construed as descriptions, since we do in fact legitimately question the existence of that which is named. (54, ¶2) It is worth noting, however, the Russell does not indicate that all uses of what we take to be proper names are in fact descriptions: early in the excerpt Russell differentiates the propositions expressed by ‘I met John’ and ‘I met a man.’ (47, ¶1) If all proper names really functioned as descriptions, then these two sentences ought to have propositions of the same form, viz, propositions with a quantifier phrase and propositional function, since descriptions would occur in both. But Russell denies this, indicating that ‘I met John’ “names an actual person.” (ibid.) Therefore, Russell seems to think that at least some uses of proper names succeed in expressing a proposition with a named object as constituent.
*Unfortunately, it isn’t readily abundant what Russell means by ‘propositional function’, despite its centrality to his work here. On a provisional basis I took it to mean a basic function that maps a proposition onto objects in a domain, e.g., ‘I met x and x is human’ maps onto objects O(1) … O(n) such that the propositions ‘I met O1 and O1 is human’ and so forth result. Then ‘I met a man’ is true iff at least one of the propositions resulting from the propositional function is true.