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by Corry Shores
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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]
Summary of
Graham Priest
An Introduction to Non-Classical Logic: From If to Is
Part II:
Quantification and Identity
12.
Classical First-order Logic
12.6
Some Philosophical Issues
Brief summary:
(12.6.1) We will now examine some problems with classical first-order semantics. (12.6.2) One problem with classical first-order semantics is the following. A standard interpretation of ∃x is ‘There exists an x such that’. This means that if we have ∃xA, it tells us that something does exist which satisfies A. Furthermore, it is a logical truth in classical first-order semantics that ∃x(A ∨ ¬A). This means that within these semantics, we are forced to hold that something must exist that would satisfy either A or its negation. That furthermore implies that we have to conclude that no matter what, something must exist. But that claim does not seem like a logical truth, because we can think that it is possible that nothing exists. To deal with this problem, we cannot simply allow the domain of quantification to be empty, because that makes us unable to assign constants some denotation. Another solution that we explore later is to make the evaluation function v be a partial function, meaning that it has no value for some constants. (12.6.3) Another problem with first-order classical semantics is that Ax(a) ⊨ ∃xA is valid, meaning that anything that can be predicated must exist. But Pegasus can be predicated (for example as a mythological figure), but it does not in fact exist. (12.6.4) Even if one objects that the problem with the Pegasus example is that we wrongly think that existence is a proper predicate, we can still find true sentences where there are other sorts of predicates for objects that nonetheless denote non-existing things, like “Sherlock Holmes is a character in a work of fiction.” The denotation calls for Holmes to be in the domain, but his fictionality calls for him not to be in the domain. (12.6.5) Another problem is that on account of the validity of the substitutivity of identicals, the following could be a valid inference: a = b, ‘a’ is the first letter of the alphabet; so ‘b’ is the first letter of the alphabet. However, this argument can be easily dispelled by claiming that the quotational usage here is not proper for first-order logic. (12.6.6) This problem with the substitutivity of identicals has a more stubborn form. Suppose we have a picture of a person as a baby, and we call them a. And we can say that it is true that a is a baby. Now suppose also that we have a picture of an adult, and we call them b. And then we are informed that a and b are the same person, and thus a = b. The substitutivity of identicals would say that because a is a baby, b must be one too. But that cannot be so, because b is an adult and thus not a baby. (12.6.7) We can solve this problem by making time designations in our predications. We can say either that a-at-time-t is a baby or a is a-baby-at-time-t. In either case, even if we use the substitutivity of identicals, what we will get is something still true: b-at-time-t is a baby or b is a-baby-at-time-t. (12.6.8) But the use of temporal designations does not work in cases where we have intention states. We can have mental states where we are thinking about the novelist George Eliot. Later we can learn that George Eliot is the pen name for Mary Anne Evans. That establishes an equivalence, and on account of the substitutivity of identicals, we should be able to say that back before you leaned this equivalence, whenever you were thinking about George Eliot, you were also thinking about Many Anne Evans. But that is not really what was going on in your mind. (12.6.9) Priest ends by noting that we will return to these problems in later chapters.
[Problems with Classical First-Order Semantics]
[A Problem with Necessitating Existence]
[Another Problem: Pegasus Should Not Exist]
[Further Difficulties with the Denotation of Non-Existents]
[Problems with the Substitutivity of Identicals]
[Growth and the Problem with the Substitutivity of Identicals]
[Using Time Designations for the Growth Problem]
[The Lack of a Solution for Intentional States]
[Returning Later to These Issues]
Summary
[Problems with Classical First-Order Semantics]
[We will now examine some problems with classical first-order semantics.]
[(ditto)]
The semantics we have been considering, though orthodox, are not without their problems. In this section, we will consider some.
(275)
[A Problem with Necessitating Existence]
[One problem with classical first-order semantics is the following. A standard interpretation of ∃x is ‘There exists an x such that’. This means that if we have ∃xA, it tells us that something does exist which satisfies A. Furthermore, it is a logical truth in classical first-order semantics that ∃x(A ∨ ¬A). This means that within these semantics, we are forced to hold that something must exist that would satisfy either A or its negation. That furthermore implies that we have to conclude that no matter what, something must exist. But that claim does not seem like a logical truth, because we can think that it is possible that nothing exists. To deal with this problem, we cannot simply allow the domain of quantification to be empty, because that makes us unable to assign constants some denotation. Another solution that we explore later is to make the evaluation function v be a partial function, meaning that it has no value for some constants.]
[(ditto)]
It is standard to read ∃x as ‘There exists an x such that’, in which case ∃xA expresses the fact that there exists something that satisfies A. Since the domain of quantification is non-empty, ∃x(A ∨ ¬A) is a logical truth, and expresses the fact that there exists something which satisfies either A or its negation – or simply that something exists. This hardly seems to be a logical truth. It would seem entirely possible that there should be nothing. To avoid this, we could allow the domain of quantification to be empty, but we would then be unable to assign constants any denotation. Perhaps the natural remedy for this is to allow v to be a partial function (so that it may have no value for some constants). We will return to this matter in chapter 21, when we consider logics with truth value gaps.
(275)
[Another Problem: Pegasus Should Not Exist]
[Another problem with first-order classical semantics is that Ax(a) ⊨ ∃xA is valid, meaning that anything that can be predicated must exist. But Pegasus can be predicated (for example as a mythological figure), but it does not in fact exist.]
[The next idea might be the following, but I am not certain. Recall from section 12.3.2 that:
we extend the language to ensure that every member of the domain has a name. For all d ∈ D, we add a constant to the language, kd, such that v(kd) = d. The extended language is the language of ℑ, and written L(ℑ).
(Priest 265)
Now, we have the name/constant “Pegasus,” which means it would be substitutable for a variable. So it could be like the a in
Ax(a)
I am not sure what the A would be, but let me just propose for the moment that A here is something like, “is a mythological figure.” So Aa would be true. As such, there is something that satisfies A, and so ∃xA. Whenever Aa is true, then ∃xA must also be true. That means
Ax(a) ⊨ ∃xA
is generally valid. But it says that so long as Pegasus can hold for a predicate, that means Pegasus must exist. And we know that Pegasus does not exist. This is another problem.]
The fact that the denotation function is always defined also makes the following inference valid:
Ax(a) ⊨ ∃xA
Now, presumably, it is true that Pegasus does not exist. But the conclusion that there exists something that does not exist is certainly false.
(275)
[Further Difficulties with the Denotation of Non-Existents]
[Even if one objects that the problem with the Pegasus example is that we wrongly think that existence is a proper predicate, we can still find true sentences where there are other sorts of predicates for objects that nonetheless denote non-existing things, like “Sherlock Holmes is a character in a work of fiction.” The denotation calls for Holmes to be in the domain, but his fictionality calls for him not to be in the domain.]
[I may not get this next idea right, but it might be the following. In the above example, we might be thinking of the particular quantifier as an existence predicate. And maybe some might object that we cannot treat existence like other predicates, so we should not have a formulation like “Pegasus exists.” Then Priest gives examples where Sherlock Homes is assigned predicates that make true sentences. But I am not following what the problem is exactly. It seems to be that in these cases, we are denoting non-existing entities. Maybe the problem is that it both denotes something, meaning that it should be in the domain, while what is being denoting is something that is fictional and should not be in the domain. But I am guessing here.]
One might suspect that something funny is going on in this example, on the ground that existence is not a real predicate. But there seem to be other true sentences containing names that do not denote existent objects, which have nothing to do with existence, and where it is wrong to generalise existentially. Thus, consider the following:
1. Sherlock Holmes lived in Baker St.
2. Sherlock Holmes is a character in a work of fiction.
3. I am thinking about Sherlock Holmes.
In the case of the first of these, one might claim that it is not really true. What is true is that:
In the novels by Arthur Conan Doyle, Sherlock Holmes lived in Baker St.
|
But this still gives us a true sentence about Sherlock Holmes, so the problem has not been solved. In the second and third cases, not even this move seems available.
(275-276)
[Problems with the Substitutivity of Identicals]
[Another problem is that on account of the validity of the substitutivity of identicals, the following could be a valid inference: a = b, ‘a’ is the first letter of the alphabet; so ‘b’ is the first letter of the alphabet. However, this argument can be easily dispelled by claiming that the quotational usage here is not proper for first-order logic.]
[(ditto) (I did not grasp the problem here, so please see the quotation. Maybe the issue is that when you put the quotes around the letters, you thereby may be designating some other object. In other words, you can say that the constant a equals the constant b, but if you put quotes around ‘a’ and predicate it as the first letter of the alphabet, then we are no longer dealing with the same denotation in the domain as the constant a. For, the name of the alphabetic letter A denotes an object distinct from that of B. So they cannot be one and the same thing in the domain; thus they cannot share the same denotation as the a and b in a = b. I am guessing.)]
The semantics of first-order logic also validates the general law of the substitutivity of identicals (see 12.9.2):
a = b, Ax(a) ⊨ Ax(b)
(I will also abbreviate this general form as SI.) There are a number of apparent counter-examples to this, such as the following:
a = b, ‘a’ is the first letter of the alphabet; so ‘b’ is the first letter of the alphabet.
The standard response to this is to say that the context
‘. . .’ is the first letter of the alphabet
and similar quotational contexts, are not predicates in the sense of first-order logic. That is, the claim that ‘a’ is the first letter of the alphabet is not about a at all. “ ‘a’ ” simply refers to the letter ‘a’; the referent of ‘a’ itself is irrelevant.
(276)
[Growth and the Problem with the Substitutivity of Identicals]
[This problem with the substitutivity of identicals has a more stubborn form. Suppose we have a picture of a person as a baby, and we call them a. And we can say that it is true that a is a baby. Now suppose also that we have a picture of an adult, and we call them b. And then we are informed that a and b are the same person, and thus a = b. The substitutivity of identicals would say that because a is a baby, b must be one too. But that cannot be so, because b is an adult and thus not a baby.]
[(ditto)]
Other examples are not so easily defused. Thus, suppose that I show you a picture of a baby. Let us call the person involved a. I then show you a picture of an adult. Let us call the person involved b. Suppose that, as a matter of fact, a and b are the same person (at different stages of her life). Then a = b and a is a baby; but it is not true that b is a baby.
(276)
[Using Time Designations for the Growth Problem]
[We can solve this problem by making time designations in our predications. We can say either that a-at-time-t is a baby or a is a-baby-at-time-t. In either case, even if we use the substitutivity of identicals, what we will get is something still true: b-at-time-t is a baby or b is a-baby-at-time-t.]
[(ditto)]
It is natural to try to solve this problem by bringing time into the matter explicitly. There are two obvious ways this can be done, depending on whether we understand the sentence ‘a is a baby’ as:
a-at-time-t is a baby
or as
a is a-baby-at-time-t
(where t is the time when the photograph was taken). Some deep metaphysical issues hang on this difference, but these need not concern us here. In either case SI can now be admitted: b-at-time-t is a baby, and b is a-baby-at-time-t.
(276)
[The Lack of a Solution for Intentional States]
[But the use of temporal designations does not work in cases where we have intention states. We can have mental states where we are thinking about the novelist George Eliot. Later we can learn that George Eliot is the pen name for Mary Anne Evans. That establishes an equivalence, and on account of the substitutivity of identicals, we should be able to say that back before you leaned this equivalence, whenever you were thinking about George Eliot, you were also thinking about Many Anne Evans. But that is not really what was going on in your mind.]
[(ditto)]
There are cases where even this move is not available, however. Substitution into intentional contexts (that is, contexts containing predicates for certain kinds of mental states) causes problems of the following kind. The real name of the novelist George Eliot was ‘Mary Anne Evans’. For many years I knew that George Eliot was a novelist; I had no idea that Mary Anne Evans was a novelist. And I knew that George Eliot was George Eliot; I had no idea that George Eliot was Mary Anne Evans. And from time to time I thought about George Eliot, but I was not thinking about Mary Anne Evans.
(277)
[Returning Later to These Issues]
[Priest ends by noting that we will return to these problems in later chapters.]
[(ditto)]
We will return to a number of these problems in subsequent chapters.
(277)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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