by Corry Shores
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[Bracketed commentary and boldface (unless otherwise indicated) are my own.]
Logic: A Very Short Introduction
Names and Quantifiers: Is Nothing Something?
When we speak of things, we might refer to some specific thing by name, like if we say, “Marcus came to the party”. In this case, what we are saying refers just to this one named person or thing. Or we might speak broadly and universally of all of a group of things, like if we said, “everyone came to the party”. In this case, what we say of the people or things applies to all of them. Or, we might refer to some thing, but without designating it specifically with a name, like when we say, “Someone came to the party”. Here we are saying something about a person or thing, but we are not specifying which one. When we want to speak of some thing or another, as in, “someone is happy,” we could use the existential quantifier and formulate this as, ∃x xH, meaning, there is some x such that x is happy. Or if we wanted to say, “Everyone is happy,” we could write ∀x xH, meaning, for all x, x is happy. Note that from just one quantified sentence an inference can be drawn. For example, if all people are happy, then there is some person who is happy. By using quantification, we can settle debates in mathematics and philosophy.
Previously we examined inferences with phrases like “or” and “it is not the case that” (17). These words are joined to “whole sentences to make other whole sentences” (17). But there are inferences that work in ways different from these. Consider this one:
Marcus gave me a book.
Someone game me a book.
[Previously we determined validity not on the basis of what was within each sentence but rather on the basis of rules of the combinations, additive modifications, and inferences of whole sentences. Here, however we are not combining or modifying sentences.] “Neither the premiss nor the conclusion has a part which is itself a whole sentence. If this inference is valid, it is so because of what is going on within whole sentences” (17).
The simplest whole sentences have a subject and predicate. Priest has us consider these examples:
1) Marcus saw the elephant.
2) Annika fell asleep.
3) Someone hit me.
4) Nobody came to the party.
The subject tells us what the sentence is about, and the predicate tells us what is said about the subject (17-19). We now wonder, what makes these sentences true? They would be true if the subject really does have the property ascribed to it by the predicate. “Take the second example. It is true if the object referred to by the subject ‘Annika’ has the property expressed by the predicate, that is, fell asleep” (19).
But, consider again sentence 3: “Someone hit me”. What is its subject? Perhaps it is the person who hit you. But what if the speaker is lying, and thus no one hit you. Sentence four, “Nobody came to the party” is even less certain about who the subject would be, since “’nobody’ does not refer to a person – or to anything else” (19). While ‘Marcus’ and ‘Annika’ are proper names that refer to some specific person, ‘nobody’, ‘somebody’, and ‘everyone’ are quantifiers [since they refer to some quantity of subjects] (19).
We now look at the standard modern explanation for how quantifiers work. To do this, we will use notation to simplify the matter. A subject will be represented with a lower case letter, and the predicate with an upper case one. We combine the lower and upper to signify the predication of the subject. That predication is true if the subject has that predicated property.
A situation comes furnished with a stock of objects. In our case, the relevant objects are all people. All the names which occur in our reasoning about this situation refer to one of the objects in this collection. Thus, if we write m for 'Marcus', m refers | to one of these objects. And if we write H for 'is happy', then the sentence mH is true in the situation just if the object referred to by m has the property expressed by H. (For perverse reasons of their own, logicians usually reverse the order, and write Hm, instead of mH. This is just a matter of convention.)
Now, when we use the quantifier “someone”, it is like a sort of variable. We mean that there is some object in the collection which has that predicated property, and we use ∃x, the particular quantifier, to represent this object.
Now consider the sentence ‘Someone is happy’. This is true in the situation just if there is some object or other, in the collection of objects, that is happy – that is, some object in the collection, call it x, is such that x is happy. Let us write ‘Some object, x, is such that’ as ∃x. Then we may write the sentence as: ‘∃x x is happy’; or remembering that we are writing ‘is happy’ as H, as: ∃x xH. Logicians sometimes call ∃x a particular quantifier.
The universal quantifier, then, would be when we speak of every object, as in everyone, and we write it ∀x.
What about ‘Everyone is happy’? This is true in a situation if every object in the relevant collection is happy. That is, every object, x, in the collection is such that x is happy. If we write ‘Every object, x, is such that’ as ∀x, then we can write this as ∀x xH. Logicians usually call ∀x a universal quantifier.
For ‘Nobody is happy’, that is, for cases where there is no object, x, in the relevant collection, such that x is happy, we merely write:
rather than make a new symbol, “For to say that no one is happy is to say it is not the case that somebody is happy” (20).
Names and quantifiers work very differently. The fact that we write ‘Marcus is happy’ and ‘Someone is happy’ in these two different ways:
tells us that “not all grammatical subjects are equal” (20d). Now, recall the original inference:
Marcus gave me a book.
Someone game me a book.
Which we may write:
We see now why it is valid. If at least one person, Marcus, gave me a book, that means someone game me a book. Now let us look at an inferences from sentences with “nobody.” Previously Priest quoted these famous lines from Lewis Carroll’s Through the Looking Glass:
‘Just look along the road, and tell me if you can see . . . [the Messenger]’
’I see nobody on the road.’ said Alice.
‘I only wish I had such eyes,’ the King remarked in a fretful tone. ‘To be able to see Nobody! And at that distance tool! Why, it's as much as I can do to see real people, by this light!’
(qtd. in Priest 19)
Here, Alice sees nobody, and from that fact the king infers that she sees somebody, namely, “Nobody.” So let us write the predicate “is seen by Alice” as A to formulate this inference:
This of course is invalid. There is a “relevant domain” [presumably, the things that are visible to Alice in that situation]. And we are saying there is no object in that relevant domain. Obviously then, it is not true that there is some object in that domain (21).
Priest notes that quantifiers are important also in very serious debates in mathematics and philosophy. Priest proceeds to show how they are useful in a certain debate regarding the existence of God. We begin by assuming that there is a reason or explanation for everything. “people don’t get ill for no reason; cars don’t break down without a fault” (21). So everything has a cause. But then we ask, what is the cause of everything? [The use of ‘everything’ might seem ambiguous here, since perhaps we are thinking about a long chain of causal relations and we want to know what the first one is, or perhaps it means something like, ‘the reason why all things are here in the first place and are acting causally upon one another’. But this ambiguity, as we will see, is what is at issue here.] The cause of everything cannot be a physical thing like a person for example. Could it be something like the Big Bang of cosmology? No, because even this would have a cause. It must be something metaphysical, and “God is the obvious candidate” (21).
The above is the idea behind an argument for the existence of God called the ‘cosmological argument’. As we will see, it is based on a logical fallacy that we can uncover using quantification. “Everything has a cause” ambiguously means two things: 1) that for each event, there is yet another event which caused the first one we mentioned, “that is, for every x, there is a y, such that x is caused by y,” or 2) that there is one single thing or event that causes each and every other one, “that is, there is some y such that for every x, x is caused by y” (21-22). So we have sort of relation, which is one thing being caused by another thing. We will call this relation C. And we will then write ‘x is caused by y’ as xCy. We can then write formulate the two above meanings using quantification, which will make plain their very different meanings.
1. ∀x ∃y xCy
2. ∃y ∀x xCy
[I will venture a rewording. The first one is saying that for all x’s, there is a y such that x is caused by y. In other words, consider one x or another. Every one you consider will have a cause, y. However, we do not know if there is just one y for all of them, or if there is a different y for each x, or perhaps if some x’s share a common y. The second one says that there is a y such that for all x’s, each one is caused by y. In other words, there are many x’s, and they all are caused by the same y. So it seems what is important is the order of the quantification. For, the question could be, why is it just by switching the order of the quantifications does the meaning change? I am guessing here, but perhaps the first one sets a context for the following quantifications. When we begin with a universal, like in the first case, the focus is on all members of this domain. Then when we follow that with the particular quantification, it is in reference to all those members, and so there can perhaps be many of these particular things. However, when we begin with the particular quantification, we are now talking about this one thing. Then when we follow with the universal, all these many things are now understood in terms of the first singular one.]
Priest then observes that the two formulations we made are not logically equivalent, since from the second one we can infer the first; however, from the first we cannot infer the second. If, as the second one says, there is one thing that causes everything else, then we also know, like the first says, that everything has some cause. However, from the fact that everything has a cause does not mean that we can infer that there is one cause for everything. For, there could be a different cause for each effect.
The Cosmological Argument, then, operates incorrectly on the basis of the ambiguity in the formulation.
From this example we can see why we must be clear about our quantifiers. Also, we see that for the most part “something” and “nothing” “do not stand for objects, but function in a completely different way” (22). But Priest then notes that in certain cases they can stand for some certain thing. We consider two claims: 1) the cosmos goes back infinitely into the past, and thus has no beginning, or 2) the cosmos came into existence at some particular time. Now we pose a formulation, “the cosmos came out of nothing.” Which of the two does it apply to? It does not apply to the first one, since here the cosmos was always there, and thus, it did not come out of nothing. Rather, it applies to the second one. For, if the cosmos comes about at a particular time, then presumably there was nothing before it, and thus it comes out of nothing. Now, let us formulate “the cosmos came out of nothing” using quantifiers. We will write ‘x came into existence out of y’' as xEy, and we will call the cosmos c. What do we get?
1) ¬∃x cEx
So the formulation applies both to “the cosmos came out of nothing” and “the cosmos came to be at some particular time.” This is what we wanted. We would also hope that it does not apply to the first case. Does it? In the first case, where there is no beginning to the cosmos, there also is nothing coming before it. So, ‘to come out of nothing’ cannot simply mean that there was not some thing that came before everything else. Rather,
When we say that in the second cosmology the cosmos came into existence out of nothing, we mean that it came into being from nothingness. So nothing can be a thing. The White King was not so foolish after all.
[The following is entirely quotation.]
Main Ideas of the Chapter
● The sentence nP is true in a situation if the object referred to by n has the property expressed by P in that situation.
● ∃x xP is true in a situation just if some object in the situation, x, is such that xP.
● ∀x xP is true in a situation just if every object in the situation, x, is such that xP.
Priest, Graham. Logic: A Very Short Introduction. Oxford: Oxford University, 2000.