by Corry Shores
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[Bracketed commentary and boldface are my own, unless otherwise noted (for example, there are some symbols that are boldface in the original). Please forgive my typos, as proofreading is incomplete. I am not trained in logic, so at times my summaries may be unhelpful or misleading. Please consult and trust the original text, which is absolutely wonderful.]
Logic: A Very Short Introduction
Identity and Change. Is Anything Ever the Same?
Over time, something’s properties might change. But it might either keep its identity or it might take on another one altogether. This presents a difficulty for philosophy and logic, especially since identity is a foundational concept in our thinking. We first distinguish objects and their properties, and we note that the properties may be variable while the objects remain constant. The ‘is’ of predication (x is red, or Rx) is different from the ‘is’ of identity (x is y, or x=y). However, Leibniz’s Law [of indiscernibles] uses properties to define identity. If two things share the same properties, then they are identical, and vice versa. This is a useful law in most applications, as for example when we use it for substituting terms in algebra. There are some other instances that at first seem to cast doubts on the applicability of the law, but these cases can be shown in the end to be mistaken for other reasons. However, there is one case that presents a big problem for the Law. We assume that identical things always were and always will be identical. When an amoeba A splits into amoebae B and C, then A has transformed into two other things in the sense of it having taken on new guises. This means that before the split, B and C were identical to A and thus were identical to each other. However, after the split they are non-identical. This contradicts the assumption that things that are identical always are so.
We look now at “what is to be said about the identity of objects that change through time” (63).
The problem is that things might change but remain the same thing, or they might change into something completely different. It is not obvious then what to say on the topic of identity and temporal variation in things.
Here is an example. We all think that objects can survive through change. For example, when I paint a cupboard, although its colour may change, it is still the same cupboard. Or when you change your hair style, or if you are unfortunate enough to lose a limb, you are still you. But how can anything survive change? After all, when you change your hairstyle, the person that results is different, not the same at all. And if the person is different, it is a different person; so the old you has gone out of existence. In exactly the same way, it may be argued, no object persists through any change whatsoever. For any change means that the old object goes out of existence, and is replaced by a quite different object.
Logicians see this problem arising from an ambiguity between the object and its properties.
Arguments like this appear at various places in the history of philosophy, but it would be generally agreed by logicians, now, that they are mistaken, and rest on a simple ambiguity. We must distinguish between an object and its properties. When we say that you, with a different hairstyle, are different, we are saying that you have different | properties. It does not follow that you are literally a different person, in the way that I am a different person from you.
Priest says this confusion arises because we use the verb “to be” for both designating identity as well as predicating properties to objects. Symbolically, however, it is easy to distinguish these two sorts of relation. The identity relation uses the equals symbol, =, while predication places upper and lower-case letters directly beside one another, like Hm [see chapter 3].
One reason why one may fail to distinguish between being a certain object and having certain properties is that, in English, the verb ‘to be’ and its various grammatical forms – ‘is’, ‘am’, and so on – can be used to express both of these things. (And the same goes for similar words in other languages.) If we say ‘The table is red’, ‘Your hair is now short’, and similar things, we are attributing a property to an object. But if someone says ‘I am Graham Priest’, ‘The person who won the race is the same person who won it last year’, and so on, then they are identifying an object in a certain way. That is, they are stating its identity.
Logicians call the first use of ‘is’ the ‘is’ of predication; they call the second use of ‘is’ the ‘is’ of identity. And because these have somewhat different properties, they write them in different ways. The ‘is’ of predication we have already met in Chapter 3. ‘John is red’ is typically written in the form jR. (Actually, as I noted in Chapter 3, it is more common to write this the other way round, as Rj.) The ‘is’ of identity is written with =, familiar from school mathematics. Thus, ‘John is the person who won the race’ is written: j=w. (The name w is a description here; but this is of no significance in the present matter.) Sentences like this are called identities.
[This next part is interesting and perhaps controversial. Identity is a relation held between something and itself. The idea that something can have a self-relation is presented as something non-problematic, as with the example of someone seeing themselves in the mirror. I am not entirely convinced yet that a relation, especially identity, can hold for something regarded as having singularity in an absolute sense. There seems to be a tension in the concept of identity, that in order to think of something having self-identity, we need in some sense to conceive it, if ever so slightly, as being apart from itself so that it may be related to itself. (Even in the mirror example of seeing oneself, we are seeing our reflection). When we take into account the flow of time, we might see the self-relation of identity as being a matter of two temporal instantiations of a thing as belonging to one identical thing that endured through time. But what does it mean to say that something within one instant is self-identical? How is that relation of self to self to be conceived, if not by multiplying that self in one way or another in our conceptualization of it? I wonder for example when we say something is identical to itself if we first conceive it hypothetically as distinct from itself, and then negate that distinction, and then say that it and itself (its two potentially distinct instantiations) are not really two different things after all. Even when we symbolically represent identity, we use two symbolic instantiations of the thing in question (x=x). Another issue is the question of self-identity and external differentiation. When we say something has an identity, do we also thereby mean that it is not reducible to other things that do not share in its identity, in other words, that it is unique? If so, then a self-relation of identity is thereby an external relation of non-identity with other things. Then to conceive something’s identity involves not just the conception of one thing but necessarily a conception of other things, potential or real, which would not identify with the self-relating thing. Let me return to Priest.]
What properties does identity have? First, it is a relation. A relation is something that relates two objects. For example, seeing is a relation. If we say 'john sees Mary' we are stating a relation between them. The objects related by a relation do not necessarily have to be different. If we say John sees himself’ (maybe in a mirror), we are stating a relation that John bears to John. Now, identity is a very special relation. It is a relation that every object bears to itself and to nothing else.
Priest then observes that propositions of identity are not limited to useless affirmations that the thing in question is itself, or that names and objects are assigned to each other. We can obtain new information from statements of identity, as in the proposition, “John is the person who won the race”.
Priest says that what is most important about identity are the inferences that they can be part of. He gives this example:
John is the person who won the race.
The person who won the race got a prize.
So John got a prize.
This can be formulated symbolically as:
j = w wP
(p.65) The reasoning is that if something is identical to something else, then they share the same properties. This is Leibniz’s Law [of Indiscernibles]. As we saw above, we may apply Leibniz’s Law by first stating the identity between two things, then saying one term has certain properties, and finally concluding the other term has those properties as well.
This inference is valid in virtue of the fact that, for any objects, x and y, if x=y, then x has any properties that y has, and vice versa. One and the same object either has the property in question, or it doesn’t. This is usually called Leibniz’s Law, after Leibniz, whom we met in Chapter 6. In an application of Leibniz’s law, one premiss is an identity statement, say m=n; the second premiss is a sentence containing one of the names that flanks the identity sign, say m; and the conclusion is obtained by substituting n for m in this.
Priest explains that Leibniz’s Law is quite important. And for the most part, it is applied without creating any problems. He gives the example of substitution in algebra.
For example, high school algebra assures us that (x+y)(x-y)=x2-y2. So if you are solving a problem, and establish that, x2-y2=3, you can apply Leibniz’s Law to infer that (x+y)(x-y)=3·
But in fact, there are many problematic applications. He has us consider this inference, for example:
John is the person who won the race.
Mary knows that the person who won the race got a prize.
So Mary knows that John got a prize.
Here the premises are true, but it is quite possible that the conclusion is false, which calls into question the validity of the substitution. However, it is a misapplication of the Law. [I might be misunderstanding this next part. Suppose we use these abbreviations from above: j=w, Mary knows that wP, thus Mary knows that jP. Priest’s point is that ‘Mary knows that x got a prize’ is not a property of x but rather a property of Mary. I am not sure how to work that into the formulation. I guess the basic idea is that j=w, wP, thus jP is a valid inference and a correct application of the Law. But when we deal with people’s knowledge of such facts, we are not dealing with properties of the things being known and thus those properties cannot be said to be shared necessarily by the substitutable terms. Perhaps if I knew how to formulate “knowing that,” I could see better how it is mistakenly treated as a property of the known thing in this example.]
Yet it is clear that the premisses could well be true without the conclusion being true: Mary might not know that John is the person who won the race. Is this a violation of Leibniz’s Law? Not | necessarily. The law says that if x=y then any property of x is a property of y. Now, does the condition ‘Mary knows that x got a prize’ express a property of x? Not really: it would seem, rather, to express a property of Mary. If Mary were suddenly to go out of existence, this would not change x at all! (The logic of phrases such as ‘knows that’ is still very much sub judice in logic.)
In the next example, we have two things connecting and forming one larger thing. We might misapply Leibniz’s Law to say that the properties of the one part are held by the other part. But this would merely to be lacking precision in our language.
Another sort of problem is as follows. Here is a road: it is a tarmac road; call it t. And here is a road; it is a dirt road; call it d. The two roads, though, are the same road, t=d. It is just that the tarmac runs out towards the end of the road. So Leibniz’s Law tells us that t is a dirt road, and d is a tarmac road – which they are not. What has gone wrong here? We cannot say that being dirt or tarmac are not really properties of the road. They certainly are. What has gone wrong (arguably) is this: we are not being precise enough in our specification of properties. The relevant it properties are being tarmac at such and such a point, and being dirt at a such and such a point. Since t and d are the same road, they have both properties, and we do not have a violation of Leibniz’s Law.
So the above two cases seem to call into question the validity Leibniz Law in certain inferences, and Priest showed in each case that there was really no problem with the Law itself but rather with how it is thought to be applied. The next example is much more problematic. We first recall the tense operators from chapter 8. Recall G: ‘it is always going to be the case that’. We first note that something has the property of being identical to itself. Thus: x=x. We also note that something which is self-identical is such despite the passage of time. [I am not sure that part is so obvious, since things it seems in many senses differ from themselves as time goes on. Perhaps the idea is that if something comes to differ from itself so much that it no longer is identical with its prior instantiation, then it would not really be the same thing after all and instead has a new self-consistent identity. Yet I also wonder if things can change from one identity to another, but do so very gradually and thus also without us being able to say that at every moment it was only one or the other identity.] This means that the identity will hold for all times in the future. Then, we will consider an inference where we use Leibniz’s Law to substitute terms, using this temporal formulation.
Let x be anything you like, a tree, a person; and consider the statement x=x. This says that x has the property of being identical to x – which is obviously true: it’s part of the very meaning of identity. And this is so, regardless of time. It is true now, true at all times future, and true at all times past. In particular, then , Gx=x is true. Now, here is an instance of Leibniz’s Law:
(p.67, boldface on G, as in all cases, is not mine, but is the convention)
Priest then makes a point in parentheses. It seems that we might have expected the conclusion to be Gy=y. But he gives an example which assures us that we do not need to substitute both instances. “Just consider: ‘John is the person who won the race; John sees John; so John sees the person who won the race’ ” (68). Priest that clarifies what this inference [the prior one] is saying.
What the inference shows is that if x is identical to y, and x has the property of being identical to x at all future times, so does y. And since the second premiss is true, as we have just noted, it follows that if two things are identical, they will always be identical.
But there are many obvious counter examples to this claim. [The instance he gives is a little bit confusing for me. Let me quote it first.]
For example, consider an amoeba. Amoebas are single-celled water creatures that multiply by fission: an amoeba will split down the middle to become two amoebas. Now, take some amoeba, A, that divides to become two amoebas, B and C. Before the split, both B and C were A. So before the split, B=C. After the split, though, B and C are distinct amoebas, ¬B=C. So if two things are the same, it does not necessarily follow that they are always going to be the same.
[What is confusing for me is how that last sentence applies to the example: “if two things [B and C] are the same, it does not necessarily follow that they are always going to be the same.” What I do not understand is how they are understood to exist before they were created through the splitting. Priest deals with this objection later. It seems the idea is that A is identical to both B and C. A has merely taken on new properties. This furthermore means that B and C do not originate. They were still there as A back before the split. ]
According to Priest, we cannot easily escape this problem, because if something is self-identical, it must be so for all future times (68).
One response is to say that before the split, B was just a part of A, and thus B was never identical to A. However, since A is a single-celled amoeba, it cannot have more cells within it. Thus B cannot be a part of A [and therefore we are still left to conclude that B was identical to A before the split] (68).
[Priest’s next consideration responds to my above confusion.] We might also say that B an C did not exist before the split. Rather, they only came into existence when the split happened. And, “If they did not exist before the split, then they were not A before the split. So it’s not the case that B=C before the split” (68). [I might misunderstand, but I think the insight he gives next is the following. B and C had to exist before the split, because they are both A under a new guise. Since A existed before the split, so too did its guised manifestations B and C that were only explicitly obvious after the split. In other words, Priest seems to be saying that A transforms into two things, B and C. This furthermore implies that B and C are still identical to A after the split, even though they are not identical to each other. I will quote below, as I might have that wrong. Putting aside the interpretation, we will need to conclude from this that we cannot object that we are being imprecise with our language. B and C really did exist before the split, and they really were identical to A, and thus they really were identical to each other before the split but non-identical after the split.]
More radically, one might suggest that B and C did not really exist before the split, that they came into existence then. If they did not exist before the split, then they were not A before the split. So it’s not the case that B=C before the split. But that seems wrong too. B is not a new amoeba; it is simply A, though some of its properties have changed. If this is not clear, just imagine that C were to die at the split. In this case, we would have no hesitation in saying that B is A. (It would just be like a | snake shedding its skin.) Now, the identity of something can’t be affected by whether there are other things around. So A is B. Likewise, A is C.
[I might misunderstand the final point. I think the idea is the following. We might say that when A becomes B and C, then it has taken on new properties. We might also then say that it is a different thing, and thus after the split, A is not B or C. Furthermore, that might suggest that we cannot, as we seem to have done above, consider B and C to be identical to A before the split. And therefore we need not think of B and C ever at one time being identical. I am not entirely sure what the problem is with this alternate explanation. Perhaps it is the following. Suppose we take that line of reasoning. This means that we seem to be assuming that any change of property changes the things itself. But we can think of many examples of something taking only relatively small alterations without losing its identity. Thus we cannot say that B and C are non-identical to A just because they have different properties than A. Thus furthermore, we cannot dissolve the problem by trying to disidentify B and C with A.]
Of course, one might insist that just because A takes on new properties, it is, strictly speaking, a new object; not merely an old object with new properties. So B is not really A. Likewise C. But now we are back with the problem with which we started the chapter.
[The following is quotation.]
Main Ideas of the Chapter
● m = n is true just if the names m and n refer to the same object.
● If two objects are the same, any property of one is a property of the other (Leibniz’s Law).
(quoted from Priest, 69, boldface his)
Priest, Graham. Logic: A Very Short Introduction. Oxford: Oxford University, 2000.