I am going to speak today about general propositions and existence. The two subjects really belong together; they are the same topic, although it might not have seemed so at the first glance.
Bertrand Russell begins this task by first clarifying his subject matter, namely those propositions which evoke indefiniteness in their subject, such as those characterized by words like, ‘all’, ‘some’, ‘a’, ‘any’, etc. Of these, they may be further divided into general propositions and existence propositions. These two classes of propositions serve as each others negation; as the name suggests, existence propositions assert existence whereas general propositions deny existence. This is the case even in what appear to be positive assertions, such as “All Greeks are men”. What this is actually asserting is not that there are men who are Greeks, but rather that it is not the case that there are some Greeks that are not men. Notice that that in order to expresses the meaning of the general proposition, a negated existence proposition is used, and not merely a negative general proposition. The reason for this is that the proposition, “No Greeks are men” does not contradict “All Greeks are men” if in fact there are no Greeks, but that it does contradict the claim that there are some Greeks that are not men even if there are no Greeks.
What then is established in general propositions if not the existence? It is the veracity of the propositional function that is demonstrated. And a propositional function is “any expression containing an undetermined constituent, or several undetermined constituents, and becoming a proposition as soon as the undetermined constituents are determined.” The distinction between propositional function and proposition is illustrated be the algebraic formula (x+y)(x-y)=x2+y2, a propositional function, and (2+1)(2-1)=5, a proposition. Since propositional functions do not have content until the variables have been determined, what can be said of them is what modality they have, namely necessary, possible, or impossible. Propositions do not have this characteristic, as once the variable has been determined the relationship can only be true or false.
Of these three modes, it is the possible that Russell finds to intersect with out concept of existence. In fact, he states that possible propositional functions are “the fundamental meaning of ‘existence’”. It is this relationship, between possibility and existence that demonstrates existence to be a property not of individual things, but rather of propositional functions. Russell points out that to believe that existence applied to propositional functions is somehow carried over to the individuals which satisfy it is to commit an error similar to claiming that if men are numerous and Socrates is man, that therefore Socrates is numerous. The attributes of the general are not carried over to the particular. Which is not to say that the individual things do not exist, since that would be the same error expressed differently. Both the assertion and its denial are nonsense for the individual. This distinction is clearly made in such common phrases as ‘I met a man on my way home’. We know from this statement that men exist, and can do so while remaining ignorant of any individual being picked out.
What does this mean regarding propositional functions? For they are nothing but empty ciphers waiting to be filled with the identities they identify. It is these things that populate the world, namely propositions, both atomic and general, and facts, again both particular and general. It seems to a small challenge to accept particular facts, but what of these general facts? Well, it is demonstrable that we cannot arrive at general propositions with particulars alone, for even if we were to succeed in complete induction, we would still lack the necessary proposition that we have in fact succeeded in our effort at complete induction and have excluded no pertinent particular fact. To thus make any sense of general propositions requires that we postulate some general fact to accompany the particulars being evaluated. Which leads us back to propositional functions as the products of our analysis of the structure of these general propositions and facts.
The purest form of propositions and propositional functions is found in logic, where only variables remain within the schema of the proposition. Here, only the form of the proposition remains. This is the subject of logic, the validity of the form. It is this form that persists through the various propositions that can be obtained through the substitution of alternate variables. Which is not to say that logic contains all forms. For there are some, such as “there is at least one thing in the world”, for which it picks out nothing, and so has only variables, and yet which is determined not from its structure, but rather from empirical investigation. Thus, while it is a characteristic of logical propositions that they have only variables, this is not the only attribute that defines them. There must also be a certain tautological flavor to them, so that the truth of them may be known a priori, from the nature of the proposition itself.
Given all of this, it now becomes clear that what occurs with our description of individuals does in no way involve us in affirming the existence of that individual, and doing so involves precisely the same mistake as was noted earlier with the misappropriations of a predicate that properly belongs to the propositional function for the individual so delineated by the function. What then is going on with statements that select a constituent that is not in the world? It would be nonsense to assert or deny anything when there is in fact no subject. What we must conclude is that something else is under analysis in such statements, and it is about this that we are able to assert or deny existence. Russell contends that what we are really talking about is not an individual at all, but rather a description, a propositional function, for which we find that there is no value for the variable being described. The confusion arises from the misleading nature of the grammar that makes us think something is being named, when in fact nothing of the sort is actually going on.
Russell fears that something similar is also occurring with those who wish to assert that definite descriptions are in some sense naming something. First, he thinks it is peculiar because these phrases involve more than simply knowing that this is something the individual is called, and this additional information is applicable even if no one references him with the phrase. This is the difference between a legitimate name, which is just a sort of symbolic demonstrative, and a definite description, even truncated ones like Pegasus or Romulus. An easy rule to distinguish the two is that if the phrase under question is either a tautology or false, then it is a name; all else are descriptions that must be further analyzed. Thus, we know that statements like, “Scott is the author of Waverly” involve something more than names since it assert nothing about the name ‘Scott’. In this case, the name is merely a token demonstrative similar to ‘this’ or ‘that’.
What do we make of identity then? Clearly, the name ‘Scott’ and the definite description ‘author of Waverly’ are not referring to two different individuals, and yet it is not a tautology. Something more is occurring than the ‘S’=’W’ direct identification that we see with mere names. In fact, the confusion is precisely this, that the definite description is mistaken for a constituent of the proposition in the same way that a name would, when a proper analysis would have broken the description up into its various elements. This is why propositions like “Unicorns do not exist” can in fact be not only meaningful, but true, for the word ‘Unicorn’ is not in fact a name, picking something out, but rather an abbreviated description, something that would not be possible if we were required to think that the word served to mark some constituent of the proposition.
But what does this tell us about things we do assert to exist? First, that these statements are propositional functions entailing possibility and so must have the lower limit of the variable be no less than one in order to be true. The second property is that the upper limit of the variable be no greater than one. Hence, our statement about the author of Waverly being Scott becomes something like “if ‘x’ is ‘Scott’, then ‘x’ is the one and only thing which wrote Waverly”, and then asserting that there is a possible value of ‘x’ which can make the proposition true. Thus we realize that such questions about existence regarding the author of Waverly are in fact questions about the modal nature of the propositional function understood in the phrase, and not some individual so marked out. This is also why such questions can never arise regarding an actual name, since a name does indeed mark something out, and so puts an end to the question before it even arises. So when disputing the claim “The present king of France is bald” there is no need to even take up the question of baldness, since “The present king of France” is not a constituent of the phrase, but is rather a propositional phrase, of which it must be ascertained if it is possible or not along with any considerations of baldness. A denial of either constitutes a rejection of the claim.
What seems to be demonstrated in this analysis is that it is a mistaken notion to think existence is somehow a property that can be applied to individuals. What this ends up doing is leaving us with the question of what to do with individuals that lack this property, when in fact there is no such thing and cannot be since existence equivalent to, to possible propositional functions, and so pertains only to classes which in fact have members. It is then nonsense to even ask about non-existent things. As Russell puts it:
So the individuals that there are in the world do not exist, or rather it is nonsense to say that they exist and nonsense to say that they do not exist. It is not a thing you can say when you have named them, but only when you have described them. When you say ‘Homer exists’, you mean ‘Homer’ is a description which applies to something.
The mistake is in assuming that the surface grammar that these statements share with naming statements is somehow indicative of their logical structure as well, and this is not the case. It is only with the proper identification, and delineation, of these various propositions that one can begin to offer an adequate analysis of the facts and the world.
That was a beast of a summary to write, both because of its length and due to Russell’s rambling style of writing. It is interesting that so much of my agitation caused at the beginning of Russell’s article, and my solution to that agitation, should be exactly what Russell himself proposes as the solution in the conclusion of his paper. Which ought to teach me to wait until the end of a paper before panicking about something that I think is wrong or doesn’t make sense.
First, I would like to thank for posting this. Second, I have a couple of questions, because I do not entirely understand some of the concepts. For example, since my name is ‘Jose’, would it still be incorrect for me to say: “Jose exists”? If so, what would I have to say? Because even if I said: “the person born on June 26, 1982,” it would still like it does not pick out one individual, but many individuals – all that were born that day. Can this type of thing be avoided?