by Corry Shores
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Logic: A Very Short Introduction
Descriptions and Existence: Did the Greeks Worship Zeus?
Very Brief Summary:
We may specify a thing by describing it in terms of its properties, thus creating a “definite description”. This thing which is itself described by predicates may also receive predication. There is a problematic principle called the Characterization Principle that is sometimes used to make arguments, thereby invalidating them in certain cases. The “CP” says that whatever is in the thing’s description is also something that can be predicated to that described thing. For example, the ontological argument says that God can only be described as having existence, thus by the CP, God does in fact exist. However, there is a rule that the predicates to non-existing described things are false, thus the ontological argument begs the question; for, it only works if you first assume that God is a real thing, but that is what was to be proven. However, the rule does not apply to fictional entities like Greek gods who on the one hand do not exist, while on the other hand, we are still able to rightly assign each of them their proper attributes.
A definite description specifies a thing satisfying certain conditions, for example, “the man who first landed on the Moon”. Descriptions can be formulated symbolically by the use of variables that are predicated. The overall formulation takes the form ιxcx. Here, the ιx means, “the object x, such that…”, and the cx gives the conditions specifying the object. In our example we could write ιx(xM & xF) to mean, “the object x such that x is a man and x first landed on the Moon”. Furthermore, we may treat the whole description as something that can take predicates, and we can use Greek letters to stand for the whole description, thus possibly making the above formulation simply μ. This abbreviation will help us examine the validity of the Characterization Principle (CP), which is used in the Ontological Argument for God. We describe God as having a variety of properties that specify God, with the final one being “exists”: ιx(xP1 & … & xPn). The CP says that a thing characterized by certain properties in fact has those properties, and thus the whole described thing is predicated by the properties given in the description. Symbolically this involves substituting all cases of x in the description with that description itself. In this formulation we would get: ιx((xP1 & … & xPn)P1 & … & (xP1 & … & xPn)Pn), which in part says that the object that is omniscient etc., and exists, is in fact omniscient, etc., and does really exist. Using the Greek letters we can render the above substitution as: γP1 & … & γPn. But there is an important rule this argument breaks, namely that any predication to a non-existing entity is false. If there is a God, then the predication that God exists is true; but if there is no God in reality, then this predication is false. This means that for the argument to work, it must assume the truth of its conclusion at the outset, and is thus invalid. Yet there are cases where this rule does not apply, for example in instances of fictional entities like Greek gods whose properties can rightly be predicated to their description even though the thing described does not exist.
Priest will discuss something called “definite descriptions” or just “descriptions,” which can be the subjects of sentences (24). But by “description” we do not here mean it in its normal broad sense, but instead it is a technical term.
Descriptions are phrases like ‘the man who first landed on the Moon’ and ‘the only man-made object on the Earth that is visible from space’. In general, descriptions have the form: the thing satisfying such and such a condition.
We will formulate descriptions following Bertrand Russell. [For this formulation, we keep in mind the idea of satisfying a condition. The man on the moon is an object. We call him x. We say, “there is an object x.” The condition that makes this man the one we have in mind is that he was the first to land on the moon. Before we get to this description that specifies this man, we need to introduce that description. So we say, “the object, x, such that…”. We write this as ιx(…). The “…” is not meant to be there. It is where the description should go. So in this case:
ιx(x is a man and x landed first on the Moon).
We will further abbreviate this. M will stand for “is a man” and F will stand for “landed first on the Moon” (p.24). We now have:
ιx(xM & xF)
Priest gives this formalization:
In general, a description is something of the form ιxcx, where cx is some condition containing occurrences of x.
Descriptions can be subjects to which predicates are assigned. We will use U to mean “was born in the USA.” This means we would write, “the man who first landed on the Moon was born in the USA” as:
ιx(xM & xF)U
Now we would like to make this look like a normal predication with just two symbols. We will abbreviate ιx(xM & xF) as μ. We now have
Instead of our “was born in the USA” predicate U, we will return to the Moon examples. Thus we would write “The first man to land on the Moon is a man and he landed first on the moon” as
μM & μF
Recall what we said about quantifiers in the last section. In those cases, the truth of the formulations had something to do with the quantities of the things described. So for “All people are happy” to be true, that happiness must be shared by every person. However, descriptions are names and not quantifiers. So,
The man who first landed on the Moon was born in the USA”, μU, is true just if the particular person referred to by the phrase μ has the property expressed by U.
Now consider proper names, like “Annika” or “the Big Bang” (26). These merely designate the thing. They do not contribute additional information about what they refer to. But definite descriptions do carry this extra info. “Thus, for example, ‘the man who first landed on the Moon’ carries the information that the object referred to has the property of being a man and being first on the Moon” (26). This is actually not so trivial. Descriptions prove useful in certain mathematical and philosophical arguments. Priest will use the example of the Ontological Argument for the existence of God, which can take the following simple form:
God is the being with all the perfections.
But existence is a perfection.
So God possesses existence.
In other words, God exists. Perfections include omniscience, omnipotence, moral perfection… “[i]n general, the perfections are all those properties that it is a jolly good thing to have” (26). Now we wonder, why does the second premise say that existence is a perfection? The answer is a bit complicated and involves Plato’s philosophy. Priest says we can work around this issue by 1) making “a list of properties like omniscience, omnipotence, etc., include existence in the list, and simply let ‘perfection’ mean any property on the list” [as we will see, there is a problem with the structure of the argument, and it would not matter really how we justify this premise], and 2) we will “take ‘God’ to be synonymous with a certain description, namely, ‘the being which has all the perfections (i.e., those properties on the list)’” (p.27). Now in that light, consider the first premise: “God is the being with all the perfections”. This is the same as our second stipulation. So we can eliminate it for now. The second premise also: “Existence is a perfection” is the same as the first stipulation, and so we can eliminate it too. The argument now has one line:
The object which is omniscient, omnipotent, morally perfect, … and exists, exists.
We will make this more apparent by abbreviating. We will write God’s list of properties as P1, P2, …, Pn. The last one on the list, Pn, is the property of existence. Now let us use our definite description notation. We will predicate all these properties to the description for God:
ιx(xP1 & … & xPn)
We will abbreviate this whole description as γ. [In the following, it seems we regard the full sentence that we are abbreviating to be: “The object which is omniscient, omnipotent, morally perfect, … and exists, is omniscient, omnipotent, morally perfect, … exists.”] The one-line conclusion then becomes
γP1 & … & γPn (from which γPn follows).
Priest explains that we are dealing here with an instance of something called the Characterization Principle. It can be stated as “a thing has those properties by which it is characterized” or “the thing satisfying such and such a condition, satisfies that very condition” (27). We will abbreviate the Characterization Principle as CP. Above we had another instance of it, namely, “The first man to land on the Moon is a man and he first landed on the Moon”, or μM & μF.
In general, we obtain a case of the CP if we take some description, ιxcx, and substitute it for every occurrence of x in the condition cx.
[So take for example again “The first man to land on the Moon is a man and he landed first on the Moon”. We begin simply with the description
ιx(xM & xF),
that is, “the object x who is a man and who first landed on the Moon.” Here, the condition cx is (xM & xF). We see there are two occurrences of x. So now we substitute (xM & xF) into the x’s in the parentheses of
ιx(xM & xF).
We then get
ιx((xM & xF)M & (xM & xF)F)).
In other words, we get, “the first man to land on the Moon is a man and the first man to land on the moon is the first to have landed on the moon”.]
[So the formulation of CP again is “the thing satisfying such and such a condition, satisfies that very condition.” This seems tautological. Thus,] CP looks true by definition. But actually it is false, since it implies many things that are no doubt untrue (27).
One problem with its implications is that on the basis of the CP we can “deduce the existence of all kinds of things that do not really exist” (28). We can for example on its basis conclude that there is a greatest integer, when of course there is not [there can always be one greater by adding 1.] So we first consider the non-negative integers, going from 0, 1, 2, 3, and on and on. Now we select a condition, which will be our cx in ιxcx. We then make cx be “x is the greatest integer and x exists.” [Let us formulate this so it is similar to the above, where we had the substitution:
ιx((xM & xF)M & (xM & xF)F)),
which stood for: “the first man to land on the Moon is a man and the first man to land on the moon is the first to have landed on the moon”. So let us in this case make ιxcx be:
ιx(xG & xE),
meaning “the object x that is the greatest integer and it exists”. Now we substitute (xG & xE) for each case of x in the conditions. This gives us
ιx((xG & xE)G & (xG & xE)E).
So here perhaps we have “the greatest integer, which does exist, is the greatest integer, and it exists.” It is not entirely clear to me why the CP was needed for the absurdity. Before applying it, we already said that it exists. Perhaps the idea is the following. In the first case, “exists” is not predicated to x as much as it is just a description said to apply to it, putting aside whether x is a real thing or not. So we could for example formulate legitimate descriptions of things that could not possibly exist, like a square circle: x is a circle and x has sides like a square. We could even add that it exists. This is perhaps merely just ascribing properties, putting aside whether the described thing can actually take such predicates as “exists” or “is a possible object”. But with the CP, perhaps what we are doing is adding predication to the description. We are saying that the thing which is circular, square, and existing is a real thing. Here is perhaps where the problems arise, since we are moving away from combining properties to saying something about the thing which is affirmable or deniable. But while still just describing it, we are not yet interested in the affirmability or deniability of these combinations of properties. This is not how Priest describes it, however. Perhaps for him, in both cases there is predication that is affirmable or deniable, or that can be true or false. He will say that the second level of predication can only be true if the predications in the description refer to a real thing.]
Let cx be the condition ‘x is the greatest integer & x exists’. Let δ be ιxcx. The the CP gives us ‘δ is the greatest integer, and δ exists’.
There are even more absurdities that the CP leads to. [Let us also put the next example into our longer form just to make the mechanics visible. So again, we begin with the formulation ιxcx. We will make the condition “x married the Pope,” and let us here write that xP. So we have ιx(xP). Now we substitute xP in for each case of x in the condition. This gives us now ιx(xP)P, which might be read something like, “the one who is married to the Pope is married to the Pope”. Again, it would seem that there is enough for the absurdity without the repetition of the qualification that this individual is married to the Pope. Once would seem to be enough. So I again wonder if the issue is that description is not on the level of truth and falsity, but the second instance of the predication is, since it is perhaps no longer part of the description but rather is a predication to a described object. Or, in light of what Priest later says, perhaps description is not a matter of truth or falsity, but rather one of referring or not-referring. And predications to non-referring descriptions will in most cases be false.]
The absurdities do not end there. Consider some unmarried person, say the Pope. We can prove that he is married. Let cx be the condition ‘x married the Pope’. Let δ be the description ιxcx. The CP gives us ‘δ married the Pope’. So someone married the Pope, i.e., the Pope is married.
Priest then addresses what is going on that makes these instances problematic. His answer seems to be that in those absurd instances, the description does not refer to anything. [Perhaps this then invalidates the predication. Since there is no person who married the Pope, it does not matter which predications we give to it, since there is nothing to which those predicates may apply.]
What is to be said about all this? A fairly standard modern answer goes as follows. Consider the description ιxcx. If there is a unique object that satisfies the condition cx in some situation, then the description refers to it. Otherwise, it refers to nothing: it is an ‘empty name’. Thus, there is a unique x, such that x is a man and x landed first on the moon, Armstrong. So ‘the x such that x is a man and x landed first on the moon’ refers to Armstrong. Similarly, there is a unique least integer, namely 0; hence, the description ‘the object which is the least integer’ denotes 0. But since there is no greatest integer, ‘the object which is the greatest integer’ fails to refer to anything. Similarly, the description ‘the city in Australia which has more than a million people’ also fails to refer. Not, this time, because there are no such cities, but because there are several of them.
This means that the CP is not problematic in those cases when the ιxcx, that is, the unique object satisfying cx, actually exists. In those cases, the CP holds. (28d)
[This next point gets very interesting I think. If the described object does not exist, this makes all predications to it false. Why this is so might be interesting to discuss. For, could the predications not be neither true nor false, or be things to which affirmation or denial cannot rightly apply? To say it is false might carry with it metaphysical assumptions, namely, that nothing true can be said of things that do not exist. I wonder what to do with the notion that ‘the Pope’s wife does not exist”? Would the predication, ‘is non-existent’ be a false predication too? Perhaps the idea is that for non-existing things, all ‘predications’ would be part of the description. So maybe we can describe a non-existing thing all we want using predicates, but we cannot predicate the description itself. I am not sure. Here is what Priest writes:]
But what if there is no unique object satisfying cx? If n is a name and P is a predicate, the sentence nP is true just if there is an object that n refers | to, and it has the property expressed by P. Hence, if n denotes no object, nP must be false. Thus, if there is no unique thing having the property P, (if, for example, P is ‘is a winged horse’) (ιx xP)P is false. As is to be expected, under these conditions, the CP may fail.
Priest now asks how all this applies to the Ontological Argument? [The basic idea here will be that on the basis of the description, we do not know whether or not the described thing exists. The argument only works if we assume that there really is the thing described. But God’s existence was to be proved, and thus it cannot be assumed.] Recall that we described God with a series of predications, among which is that God exists: ιx(xP1 & … & xPn). We then made γ stand for the description, and we applied the CP to get γP1 & … & γPn. This in effect predicated existence to God. But, before we move to these predications of the description, we first need to establish whether or not the described thing exists. For otherwise the predications will be false.
So γ refers to this thing, and γP1 & … & γPn is true. If there is not, then γ refers to nothing; so each conjunct of γP1 & & . . . & γPn is false; as, therefore, is the whole conjunction. In other words, the instance of the CP used in the argument is true enough if God exists; but it is false if God does not exist. So if one is arguing for the existence of God, one cannot simply invoke this instance of the CP: that would just be assuming what one is supposed to be proving. Philosophers say that such an argument begs the question; that is, begs to be granted exactly what is in question. And an argument that begs the question clearly does not work.
Priest ends by noting a problem with this rule that no true predications can be given to non-existing entities. [It seems similar to what we said above in brackets. In this case, we] consider a mythological figure, Zeus. [The insight seems to be that there are non-existing things that rightly have certain predicates.] Zeus’ description could be “the most powerful of the ancient Greek gods,” and his predicates could be “lived on Mount Olympus,” “was worshipped by the Greeks,” and so on (29). [These predicates are true, but the description refers to a non-existing entity. It seems here that instead of saying that on the basis of its non-existence the predicates are false, we instead say that the predicates are true despite its non-existence.] So if it is right that no Greek gods existed,
then the description ‘the most powerful of the ancient Greek gods’ does not refer to anything. But in that case, there are true subject/predicate sentences in which the subject term fails to refer to anything, such as ‘The most powerful | of the ancient Greek gods was worshipped by the Greeks’. To put it tendentiously, there are truths about non-existent objects, after all.
[I wonder then how this would apply to the God example. Could it be said that God is a fictional entity to which it rightly can be said that God exists? Is there another notion of truth at work here, perhaps something that could be called, fictional truth?]
[The following section is entirely quotation.]
Main Idea of the Chapter
● ιxcx is true in a situation just if, in that situation, there is a unique object, α, satisfying cx and αP.
(quoted from Priest, 30, boldface his)
Priest, Graham. Logic: A Very Short Introduction. Oxford: Oxford University, 2000.