On Sense and Reference
Equality* gives rise to challenging questions which are not altogether easy to answer. Is it a relation? A relation between objects, or between names or signs of objects? In my Begriffsschrift1 I assumed the latter. The reasons which seem to favour this are the following: a a and a b are obviously statements of differing cognitive value; a aholds a priori and, according to Kant, is to be labelled analytic, while statements of the form a b often contain very valuable extensions of our knowledge and cannot always be established a priori. The discovery that the rising sun is not new every morning, but always the same, was one of the most fertile astronomical discoveries. Even to-day the identification of a small planet or a comet is notalways a matter of course. Now if we were to regard equality as a relation between that which the names `a' and `b' designate, it would seem that a b could not differ from a a (i.e. provided a b is true). A relation would thereby be expressed of a thing to itself, and indeed one in which each thing stands to itself but to no other thing. What is intended to be said by a b seems to be thatthe signs or names `a' and `b' designate the same thing, so that those signs themselves would be under discussion; a relation between them would be asserted. But this relation would hold between the names or signs only in so far as they named or designated something. It would be mediated by the connexion of each of the two signs with the same designated thing. But this is arbitrary. Nobody can beforbidden to use any arbitrarily producible event or object as a sign for something. In that case the sentence a b would no longer refer to the subject matter, but only to its mode of designation; we would express no proper knowledge by its
* I use this word in the sense of identity, and understand `a b' to have the sense of `a is the same as b' or `a and b coincide.'
On Sense andReference
means. But in many cases this is just what we want to do. If the sign `a' is distinguished from the sign `b' only as object (here, by means of its shape), not as sign (i.e. not by the manner in which it designates something), the cognitive value of a a becomes essentially equal to that of a b, provided a b is true. A difference can arise only if the difference between the signscorresponds to a difference in the mode of presentation of that which is designated. Let a, b, c be the lines connecting the vertices of a triangle with the midpoints of the opposite sides. The point of intersection of a and b is then the same as the point of intersection of b and c. So we have different designations for the same point, and these names (`point of intersection of a and b,' `point ofintersection of b and c') likewise indicate the mode of presentation; and hence the statement contains actual knowledge. It is natural, now, to think of there being connected with a sign (name, combination of words, letter), besides that to which the sign refers, which may be called the reference of the sign, also what I should like to call the sense of the sign, wherein the mode of presentationis contained. In our example, accordingly, the reference of the expressions `the point of intersection of a and b' and `the point of intersection of b and c' would be the same, but not their senses. The reference of `evening star' would be the same as that of `morning star,' but not the sense. It is clear from the context that by `sign' and `name' I have here understood any designationrepresenting a proper name, which thus has as its reference a definite object (this word taken in the widest range), but not a concept or a relation, which shall be discussed further in another article.2 The designation of a single object can also consist of several words or other signs. For brevity, let every such designation be called a proper name. The sense of a proper name is grasped by everybody who...