On the Theory of Identity

by: Alfred Tarski

Logical concepts not belonging to sentential calculus; the concept of identity

Sentential calculus, to which the preceding chapter was devoted, forms only one of the parts of logic. It constitutes undoubtedly the most fundamental part, - at least inasmuch as one necessarily uses its terms and laws in defining other terms, and in formulating and demonstrating logical laws that do not belong to this calculus. Sentential calculus by itself, however, does not form an adequate basis for the foundations of other sciences and, in particular, not for the foundations of mathematics; various concepts from other branches of logic are constantly encountered in mathematical definitions, theorems, and proofs. Some of them will be discussed in the present [ and in the following two ] chapter[s].

Among the logical concepts not belonging to sentential calculus, the concept of IDENTITY, or, of EQUALITY, is perhaps the one which has the greatest importance. It occurs in phrases such as:

x is identical with y,
x is the same as y,
x equals y.

We ascribe the same meaning to all three phrases; for the sake of brevity, they will be replaced by the symbolic expression:

x = y

Moreover, instead of writing: x is not identical with y or x is different from y we employ the formula: x ≠ y

The general laws governing the above expressions constitute a part of logic, which we shall call the THEORY OF IDENTITY.

Fundamental laws of the theory of identity

Among logical laws which involve the concept of identity, the most fundamental is the following: I. x = y if, and only if, x has every property which y has, and y has every property which x has.

We could also say more simply:

x = y if, and only if, x and y have every property in common.

Other and perhaps more transparent, though less correct, formulations of the same law are known, for instance (1):

x = y if, and only if, everything that may be said about one of the objects x or y may also be said about the other.

Law I was first stated by Leibniz (+) (although in somewhat different terms) and hence may be called LEIBNIZ'S LAW. It has the form of an equivalence, and enables us to replace the formula:

x = y,

which is the left side of the equivalence, by its right side, that is, by an expression no longer containing the symbol of identity. In view of its form, this law may be considered as the definition of the symbol "=", and it was so considered by Leibniz himself. (Of course, it would make sense to regard Leibniz's law as a definition only if the meaning of the symbol seemed to us less evident than the meaning of the expressions on the corresponding right side, such as "x has every property which y has"; [ cf. Section 11.] ) As a consequence of Leibniz's law we have the following RULE OF REPLACEMENT OF EQUALS, which is of great practical importance: if, in a certain context, a formula having the form of an equation, e.g.: x = y²

has been assumed or proved, then one is allowed to replace in any formula or sentence occurring in this context the left side of the equation by its right side, "x" by "y²" in this case, and the right side by the left side. It is moreover understood that, should "x" occur at several places in a formula, it may be left unchanged at some places, and at others replaced by y² ; we have therefore an essential difference between the rule now under consideration and the rule of substitution discussed [ in Section 15 ], which does not permit such a partial replacements of one symbol by another. (2)

From Leibniz's law we can derive a number of other laws which belong to the theory of identity, and which are frequently applied in various considerations and especially in mathematical proofs. The most important of these will be listed here, together with sketches of their proofs; we shall then see by way of concrete examples that there is no essential difference between reasonings in the field of logic and those in mathematics.

II. Every object is equal to itself: x = x.

Proof.(3) Use the rule of substitution, and substitute in Leibniz's law "x" for "y" ; we obtain:

x = x if, and only if, x has every property which x
has, and x has every property which x has.

We can, of course, simplify this sentence by omitting its last part "and x has ..." (this is a consequence of one of the laws of tautology which were stated [ in Section 13 ]). The sentence assumes then the following form:

x = x if, and only if, x has every property which x has.

Obviously, the right side of this equivalence is always satisfied (for, according to the law of identity [ in Section 12 ], if x has a certain property, then it has this property). Hence the left side of the equivalence must also be satisfied; in other words, we always have:

x = x,

which was to be proved. III. If x = y, then y = x.

Proof. By substituting in Leibniz's law "x" for "y" and "y" for "x", we obtain: y = x if, and only if, y has every property which x has, and x has every property which y has.

Let us compare this sentence with the original formulation of Leibniz's law. We have here two equivalences, such that their right sides are conjunctions which differ only in the order of their members. Hence the right sides are equivalent (cf. the commutative law for logical multiplication [ in Section 13 ]), and therefore the left sides, that is, the formulas:

x = y and y = x

must also be equivalent. A fortiori we may assert that the second of these formulas follows from the first, and our law is established. IV. If x = y and y = z, then x = z.

Proof. By hypothesis, the two formulas:

(1) x = y
and
(2) y = z

are assumed valid.(4) According to Leibniz's law, it follows from formula (2) that every property of y applies also to z. Hence we may replace the variable "y" by "z" in formula (1) (in fact, one could justify this step simply by referring to the rule of replacement of equals), and we obtain the desired conclusion: x = z.
V. If x = z and y = z, then x = y; in other words, two objects which are equal to the same object are equal to each other.

This law can be proved by an argument analogous to the preceding (it can also be deduced from Laws III and IV, without using Leibniz's law). Laws II, III, and IV axe called the LAWS OF REFLEXIVITY, SYMMETRY, and TRANSITIVITY, respectively, for the relation of identity.

Identity of objects and identity of their designations; the use of quotation marks

Although the meaning of such expressions as: x = y and x ≠ y

appears to be evident, these expressions are sometimes misunderstood. It seems obvious, for instance, that the formula: 3 = 2 + 1

is a true assertion, and yet some people remain skeptical as to its truth. In their opinion, this formula seems to state that the symbols "3" and "2 + 1" are identical, which is obviously false since these symbols have entirely different shapes, and therefore, it is not true that everything which may be said about one of these symbols may also be said about the other (we might add, the first symbol is a single sign, while the second is not).

In order to dispel any lingering doubts of this kind, one should make clear a general and very important principle, which must be retained if a language is to be usefully employed. According to this principle, whenever we wish to state a sentence about a certain object, we have to use in this sentence not the object itself, but its name or designation.
The application of this principle does not give rise to any problems as long as the object in question is riot a word or a symbol or, more generally, an expression of a language. Let us imagine, for example, that we have a small blue stone in front of us, and that we write the following sentence:

this stone is blue.

To no one, presumably, would it occur to replace in this sentence the words "this stone" (which together constitute the designation of the object) by the object itself, that is to say, to blot or cut these words out and to place in their stead the stone. For, in doing so, we would arrive at a whole consisting partly of a stone and partly of words, and therefore at something which would not be a linguistic expression, and all the more, would not be a true sentence.
This principle, however, is frequently violated if the object which is talked about happens to be a word or a symbol. And yet the application of the principle is indispensable also in this case; for otherwise we would arrive at a whole which, though being a linguistic expression, would fail to convey the thought intended by us, and might even be a meaningless aggregate of words. Let us consider, for example, the following two words: good, Mary.

Clearly, the first consists of four letters, and the second is a proper name. But let us suppose that we would express these thoughts, which are quite correct, in the following manner: (1) good consists of four letters;
(2) Mary is a proper name;

then, in talking about words, we would be using the words themselves and not their names. And if we examine expressions (1) and (2) more closely, we must admit that the first is not a sentence at all, since the subject of a sentence can only be a noun and not an adjective; the second might be considered a meaningful sentence, but, at any rate, would be a false one since no woman is a proper name.
In order to clarify these situations, we have to realize that when the words "good" and "Mary" occur in such contexts as those of (1) and (2), then their meanings differ from the usual ones, and that here these words function as their own names. In generalizing this viewpoint, we should have to admit that any word may, at times, function as its own name; to rise the terminology of medieval logic, we may say that in such a case the word is used in SUPPOSITIO MATERIALIS, as opposed to its use in SUPPOSITIO FORMALIS, that is, in its ordinary meaning.(5) As a consequence, every word of common or scientific language could take on two or more different meanings, and one does not have to look far for examples of situations where serious doubts might arise as to which meaning was intended. We do not wish to reconcile ourselves with such ambiguities, and therefore we will make it a rule that every expression should differ (at least in writing) from its name.

The problem arises as to how we can go about forming names of words and expressions. There are various devices that can be used. The simplest of them is based on the convention, that one forms a name of an expression by placing the latter between quotation marks. On the basis of this convention, the thoughts tentatively expressed in (1) and (2) can now be, stated correctly and without ambiguity in the following way:

(I) "good" consists of four letters;
(11) "Mary" is a proper name.

In the light of these remarks, all possible doubts as to the meaning and the truth of such formulas as: 3 = 2 + 1

are dispelled. This formula contains symbols designating certain numbers, but it does not contain the names of any such symbols. Therefore this formula states something about numbers and not about symbols designating numbers; the number 3 is obviously equal to the number 2 + 1, so that the formula is a true assertion. We may, admittedly, replace this formula by a sentence which expresses the same idea but is about symbols, namely, by a sentence which asserts that the symbols "3" and "2 + 1" designate the same number. But this by no means implies that the symbols themselves are identical; for it is well known that a given object - and in particular, a given number can have many different designations. The symbols "3" and "2 + 1" are, no doubt, different, and this observation can also be expressed by a new formula:

"3" ≠ "2 + 1"

which, of course, does not contradict in any way the formula previously stated.(6)

Equality in arithmetic and in geometry, and its relationship to logical identity

In this book we consider the notion of equality among numbers always as a special case of the general concept of logical identity. (7) One should add, however, that there have been (and perhaps still are) mathematicians who - as opposed to the standpoint adopted here - did not identify the symbol "=" occurring in arithmetic with the symbol of logical identity; they did not consider equal numbers to be necessarily identical, and therefore looked upon the notion of equality among numbers as a specifically arithmetical concept. In connection with this outlook, those mathematicians rejected Leibniz's law in its general form, and instead, recognized various of its consequences, such as those which relate to numbers; the consequences in question are then of a less general character and are therefore regarded as laws which are specifically mathematical. Among these consequences there are Laws II to V of Section 17, as well as assertions to the effect that, whenever x = y and x satisfies some formula which is constructed out of arithmetical symbols only, then y satisfies the same formula; the following law is an example: if x = y and x < z then y < z.

In our opinion, this point of view can claim no particular theoretical advantages, while in practice, it entails considerable complications in the presentation of arithmetic. For one rejects here the general rule which allows us - on the assumption that an equation holds - to replace everywhere the left side of the equation by its right side; however, such a replacement is indispensable in various arguments, and so it becomes necessary in each particular case to give a special proof that this replacement is permissible. To illustrate this kind of situation by an example, let us consider a system of equations in two variables, for instance: x = y²,
x² + y² = 2x - 3y + 18.

In order to solve this system of equations by means of the so-called method of substitution, one has to form a new system of equations, which is obtained by laying the first equation unchanged and by replacing in the second equation "x" by "y²" throughout. And the question arises whether this transformation is permissible, that is, whether the new system is equivalent to the old. The answer is undoubtedly in the affirmative, no matter what conception of the notion of equality among numbers is adopted. But if the symbol "=" is understood to designate logical identity, and if Leibniz's law is assumed, the answer is obvious; the assumption: x = y²

permits us to replace "x" everywhere by "y²" , and vice versa. Otherwise it would first be necessary to give reasons for the affirmative answer, and although such a justification would not involve any essential difficulties, it could at any rate be rather long and tedious (such proofs can be found in some of the less recent elementary textbooks).

As to the notion of equality in geometry, traditionally there has been a point of view of an entirely different kind. One sometimes says that two geometrical figures, such as two line segments. or two angles, or two polygons, are equal, in the sense of "congruent", and one does not, intend to assert their identity by such a statement. Rather, he, or she only wishes to state that the two figures have the same size and shape, in other words - to use a figurative if not quite correct mode of expression -, that each figure would exactly cover the other if it were placed on top of the other. For example, a triangle is capable of having two sides, or even three, which are equal in this sense - we shall say in the present discussion that the sides are then "geometrically equal" - and yet these sides are certainly not identical. On the other hand, there are also cases in which it is not a question of the geometrical equality of two figures, but of their logical identity; in an isosceles triangle, for instance, the altitude upon the base and the median to the base are not only geometrically equal, but they are simply the same line segment. Therefore, in order to avoid any confusion, it would be recommendable to avoid consistently the term "equality" in all those cases where it is not a question of logical identity, and instead, to speak of geometrically equal figures as congruent figures, replacing at the same time - as it is often done anyhow - the symbol "=" by a different one, such as "≡".

(1) Throughout this book the word "object" may refer not only to concrete, physical objects, but also to abstractions like numbers and points. Such abstractions (and more sophisticated ones) of course occur constantly in mathematics, and are also referred to as entities. - The reader may have observed that properties of objects played a secondary role in the preceding chapter. However, objects and their properties will be central for the discussions from now on.

(2) The reader no doubt remembers the rule of replacement of equals from his or her first lessons on the solution of equations, and appreciates its significance. There are reasons, however, for not including it among the basic rules of inference when discussing complete proofs [( cf. Section 15 )]. We note that in the previous editions of Introduction to Logic this rule was introduced in an informal manner, without a special emphasis, and was hardly mentioned in subsequent arguments. Leibniz's law was cited instead. In the present edition the law is cited as in the previous editions, and the rule is referred to on several occasions as well.

(3) We recall again the notion of a complete proof. (cf. the preceding footnote [ and Section 15 ].) Often, however, demonstrations of logic and mathematics are presented without strict adherence to rules such as were described, and this proof is an example. Such proofs are sometimes called INFORMAL.

(4) We should like to comment on the two words, "valid" and "true". The term "true sentence" occurs throughout [ Chapters I and II ]. This term and its opposite, "false sentence", have been the subject of many philosophical debates, but they are used in this book in a straightforward manner: with reference to sentences of elementary mathematical theories or to simple examples from everyday language. The term "valid", on the other hand, is broader in scope (and it has not given rise to such controversies). It is often used in mathematics in a similar way as "true". Thus, "valid" may mean "true" when referring to a special context (in particular, when referring to chosen models and interpretations [, cf. Chapters VI and IX ]. In fact, situations sometimes arise when the two words are used interchangeably. There is, moreover, a certain preference for using "valid" when one deals with sentential functions, as in the above proof. Then "valid" means about the same as "satisfied under every substitution of variables". Furthermore, sometimes "valid" means the same as "proved" or "provable". - Other uses of this word are familiar, and one says, for instance, "a valid argument".

(5) The reader may find it instructive to compare the preceding discussion with that of Section 9. Both discussions concern expressions, but the expressions are of different kinds. In essence, those of one kind denote objects, while those of the other refer to assertions, respectively. - With regard to the latter, i.e. to sentences and sentential functions, we say that we USE a sentence when making a direct assertion, while in an indirect formulation. as in loc. cit., we MENTION the given sentence.

(6) The convention concerning the use of quotation marks has been adhered to in this book pretty consistently. We deviate from it only in special cases, by way of a concession to traditional usage. For instance, we state formulas and sentences without quotation marks, if they are displayed in a separate line or if they occur in the formulation of mathematical or logical theorems; and we do not put quotation marks about expressions which are preceded by such phrases as "is called", "is known as", and so on. But other precautionary measures are taken in these cases, the expression in question is often preceded by a colon, and usually it is printed in a different kind of print (small capitals or italics). It should moreover be observed that, in everyday language, quotation marks are used also under certain other circumstances; and examples of such usage can be found in this book, too.

(7) Several footnotes of this chapter have dealt with the usage of words. We should like to consider now "concept", and specifically "concept of identity". In general, one can think of a concept as a kind of an inclusive idea, a synthesis resulting from several ingredients. Accordingly, if a broad interpretation is envisaged, then the concept of identity could encompass the material of the whole present chapter. However, the word 'concept' is sometimes used in a more restrictive way, and one might then think of the concept of identity as simply the term "=", when this term is defined (or characterized) by Leibniz's law. And speaking of words, instead of saying 'the term "="', one could just as well say 'the (logical) constant "="', or even, 'the symbol "="'. - The usage of "concept" in this book is primarily of the latter, more restrictive kind.