Gottlob Frege and the Nature of Identity

Gottlob Frege

Gottlob Frege

The following is a paper on the views of Gottlob Frege, a 19th Century German philosopher  who founded modern logic and analytic philosophy.  In two of his most famous works (Begriffsschrift and On Sense and Nominatum) he tried to determine what kind of thing an identity statement (like A=A or A=B) is.

In Begriffsschrift, Frege suggested identity is a relation between names or signs of objects.  This means that what is expressed by something like A=B is simply that the name “A” and the name “B” both refer to the same object.  This seems like a reasonable view but it is wrong.  Frege realized his error more than a decade later and returned to the topic in On Sense and Nominatum to explain why he previously thought identity was a relation between names, why this is wrong, and what to make of identity now.

The text is marginally dense but does not require prior knowledge to understand.  Do pay attention to footnotes, they serve as guideposts to keep the wayward reader on course.

I In this section I will explain why Frege correctly rejected the view that identity (an identity is something like A=A or A=B where true) is a relation between objects, the problem of viewing identity as a relation between names[1], and how he tries to resolve these problems with his theory in On Sense and Nominatum.

The problem with viewing identity as a relation between objects is A=A and A=B, if true[2], would not be different in any important respect.  Both would state a logical, a priori truth (namely that some object is identical to itself) because the names “A” and “B” would both stand for the same object and nothing more.  However, there are many cases in which A=A and A=B express genuine, non-a priori knowledge.

Imagine A represents “the sun in the sky today” and B represents “the sun in the sky yesterday”.  A=A (the sun in the sky today is the same as the sun in the sky today) is an obvious, somewhat trivial point, but A=B (the sun in the sky today is the same as the sun in the sky yesterday) represents real, scientific knowledge[3].  Thus, identity cannot simply be a relation between objects because there are identity statements that express genuine knowledge, and identity relations between objects are not capable[4] of expressing genuine knowledge.

In On Sense and Nominatum Frege puts forth the above observation as an explanation for why he previously believed identity to be, not a relation between objects, but a relation between names.  He goes on to explain that what we seem to want to express with a statement like A=B is that the sign A and the sign B both name the same thing[5].  If this were the case, then identity statements would be a relation between names.

However, there is a problem with this view as well.  If identity were a relation between names, “the sun in the sky today = the sun in the sky yesterday” would mean something like: the combination of symbols “the sun in the sky today” stands for the same object as another combination of symbols “the sun in the sky yesterday.”  In other words, this view does not account for the different thoughts expressed by “the sun in the sky today” and “the sun in the sky yesterday,” only that two different signs exist, and serve to name the same object (the sun).   Thus, like the theory of identity as a relation between objects, identity as a relation between names fails to explain how we often use identity statements to express genuine knowledge.

Frege then outlines a theory that accounts for this peculiarity in identity statements.  His theory is based on the acute observation that identities like A=B are non-trivial and express genuine knowledge when the different signs correspond to different ways in which the same object is represented[6].  Again, think back to the sun example and this seems to be intuitively correct.  “The sun in the sky today” and “the sun in the sky yesterday” are two different signs that correspond to the same object (the sun) in two importantly different ways[7], and it is equating these two different modes of correspondence that seems to be the genuine knowledge the identity statement “the sun in the sky today is the same as the sun in the sky yesterday” expresses.  From this observation Frege asserts there are two important parts of a name: a sense (the connotation or meaning) as well as a nominatum (the actual thing/object the name represents).  According to Frege, it is the sense that determines (or helps us get on to or identify) the nominatum, and the sense is an objective thing that can be shared between individuals[8].

This picture does well to address the difficulty faced by the theories of identity as objects and identity as names.  According to this view, identity statements  like A=B represent a relation between two names with different senses but the same nominatum, while statements like A=A represent a relation between the same name with the same sense and nominatum.  The difference between A=A and A=B, then, is in the latter case there exists a non-trivial distinction in the sense of the names.

Frege’s picture as presented above (in a very basic form) is not without its own problems, some of which I will address later in this paper.  However, the question at hand is how well does this view address the problems of identity as elucidated by the theories of identity as a relation between names and identity as a relation between objects, and what should he say about identity as a whole?  I believe Frege’s picture successfully addresses the previous problems of identity and serves to advance the conversation.  However, as previously noted, this new theory gives rise to new, complex difficulties that must be confronted.

II Consider the sentence “Joe is beautiful.”  Prior to reading this sentence someone who speaks the English language will already have an understanding[9] of the meaning (or sense) of “is” and “beautiful” as well as the conjunction “is beautiful.”  However, the thought that is given by the sentence “Joe is beautiful” is incomplete without also an understanding of “Joe.”  Without this, the sentence would translate in to a thought like “____ is beautiful,” where “____” represents a blank spot or gap in the understanding.  This is another point that is fairly easy to get on to and subject to little resistance.  The fact that “Joe is beautiful” is meaningfully different from “___ is beautiful” seems absolutely correct.  Thus, the question become: how (or with what) does the name Joe fill this gap in cognition?

Bertrand Russell believed the gap in cognition was filled by the nominatum of the name (the thing itself).  There are many problems with this view, I will outline three.  First, the objection Frege posed[10] goes something like the following.  “The nominatum can’t be the gap filler because it cannot be the case that I have this physical thing in my mind.  Thoughts are non-physical, while the nominatum is a material thing that exists in the physical world, so how can this physical thing come to exist in my thought?  It clearly cannot.  Furthermore, if the nominatum were the gap filler, I would have to have every part of the object in my cognition.  Certainly this is not the case.  Consider something like the Pacific Ocean.  It would be absurd to suggest that I have a thought about every drop of water in the Pacific Ocean when I use or perceive the name “Pacific Ocean.”  However, despite this truth I do (successfully) use names like “Pacific Ocean,” so Russell is incorrect in asserting that the nominatum itself is the gap filler.”

A second problem with Russell’s view is, the aforementioned objection aside, Russell’s picture still does not solve the problem Frege observed with identity as a relation between objects or names.  Namely, if A=A and A=B are true, then how can A=B present different cognitive content than A=A[11][12]?  For, if the cognitive gap filler of a name were simply the thing itself, there would be no meaningful difference between A=A and A=B.  As I have established[13], we know this is a flawed view so Russell’s theory cannot be accurate.

A third reason why a name’s nominatum cannot be the cognitive gap filler is there are some names that we use perfectly well that have no nominatum.  Consider names from mythology or literature.  Take Beowulf for example.  There is no medieval Geatish hero who slayed the evil Grendel and was buried in a burrow.  However, even though the sign ‘Beowulf’ does not name any real person (and thus has no nominatum), “Beowulf was a brave warrior” is a complete and coherent expression.  This is not possible according to Russell’s view.  According to Russell, the nominatum fills the cognitive gap created by the name “Beowulf.”  But because Beowulf has no nominatum, the gap remains empty and, according to Russell, “Beowulf was a brave warrior” and “___ was a brave warrior” are analogous, incomplete expressions.  Again, this seems quite clearly not to be the case.

III Earlier I signaled an important clarification regarding what Frege uses the term ‘name’ to represent[14].  I will now discuss the importance of this distinction with respect to his theory of meaning, explain why it is problematic to group his two kinds of names into the same category, and address an additional difficulty for his theory (indexicals).

Frege’s theory serves quite well to explain the function of definite descriptions.  Definite descriptions are generally combinations of properties, properties that we get on to through real world experience.  Consider the definite description ‘the tallest blonde woman in this room.’  Through visual perception, formal education and observing others, I understand the meanings (or senses) of properties like tallest, blonde, and woman; the concept room; the function of is.  I also know how linguistic rules govern the conjunction of all seven words in this instance and serve to communicate one singular meaning (sense) that uniquely picks out one object (nominatum), which in this case is a tall, blonde woman.  The sense of the definite description ‘the tallest blonde woman in this room’ determines the nominatum (the singular object that is uniquely identified by this, specific, definite description).  With respect to names that are definite descriptions, Frege’s theory of sense and nominatum seems to be exactly right.

Indexicals like the first person pronoun ‘I,’ are more difficult for Frege’s picture to account for.  For example, when I say “I am hungry” what is expressed is “Joe Cure is hungry.”  However, when my mother says “I am hungry” what is expressed is “Mary Cure is hungry.”  In each case ‘I’ has a different nominatum.  In the first instance Joe Cure[15] is the nominatum, in the second Mary Cure is the nominatum.  However, in both cases both my mother and I understand what is said, which suggests the meaning (or sense) of ‘I’ remains the same while the nominatum changes.  This is a problem for Frege because he believes that sense determines nominatum, and in this case it seems this isn’t true because the nominatum changes but the sense does not.  If sense determined the nominatum in the phrase “I am hungry” that is spoken by me, when my mother says the same exact phrase “I am hungry” ‘I’ would have to refer to me, not her.  As I have already established, this does not happen.

One way Frege’s theory may be able to survive this apparent inconsistency is to admit that in the case of idexicals, senses are not objective and sharable.  In certain respects, this seems to be a reasonable concession and one that can still be compatible with Frege’s theory as a whole.  Indexicals are different kinds of words than given names or definite descriptions, and it seems reasonable to think that different rules would apply to them.

Consider the consequences of believing senses of indexicals to be subjective and non-sharable.  For example, when I say “I am beautiful” the person I am speaking to and I (myself) would have to have two different thoughts about the meaning or sense of the word “I” in that sentence.  In some respects this is correct, for I certainly have a different idea of who or what I am than anyone else does.

But then how could communication be possible[16]?  One explanation may be that communication is possible because the two senses of “I” are sufficiently similar for the two people to both get on to the nominatum in question (the physical being Joe Cure).   It is kind of like as if someone were to say “I went to Sea World and saw a huge black and white fish do tricks.”  This person’s understanding of the definite description “huge black and white fish that does tricks at Sea World” is different than my understanding of the same description because I know the animal he/she is speaking about is actually a mammal, not a fish[17].  However, the definite description is close enough to get us both on to the same nominatum although our understandings of the definite desctiption are different.  This is to say, indexicals are problematic for Frege’s general picture but it may be that a reasonable (and justified) adjustment can be made that allows Frege’s theory to succeed.

Given names pose the most difficult problem to Frege’s picture.  He takes the position that given names are similar to properties like tall or blonde.  We get on to the sense of “tall” through experiences and understand the sense of “tall” to be some kind of group of properties or descriptions.  We then use this sense to get on to nominata of “tall” and “blonde.”  The problem is while there are properties of “blonde” that determine what it is to be blonde, and these properties help to make up the sense, it doesn’t seem like there are certain x,y,z properties that determine what it is to be a given name like Joe Cure.  Actually, the opposite seems to be true.  When I was created my parents decided they would call this thing “Joe Cure” regardless of what properties it (I) came to have.  I could have been blonde, brunette, tall, short, etc. (read: have any sense) and still have been called Joe Cure.  Being male wasn’t even a necessary characteristic of having the name “Joe Cure.”

Recall from section II, Frege establishes given names (and names as a whole) contain cognitive information.  His task then is to explain what this cognitive content is.  Frege claims the cognitive content expressed by given names is the sense of the name (which is a group of descriptions or properties of the thing named).  What I have shown in the previous paragraph places this position in serious doubt because it does not seem like this idea of the sense of a given name is essential in any way to determining its nominatum, even to the extent where the nominatum could be prior to the sense.  We know that one function of a given name is get us on to the actual thing named, and because the sense (as understood by Frege[18]) clearly does not do this, it seems like the cognitive content expressed by a name cannot be its sense.

IV Kaplan suggests a way to resolve Frege’s problem with given names in which the spirit of the general picture may be preserved: the cognitive content expressed by (and the sense of) a given name is simply the name itself[19].  One strength of this position is the sense remains objective and sharable.  This explains how we are able to communicate using given names.  The sense of the name “Joe Cure” for me is the same as the sense of the name “Joe Cure” for you, so when either one of us uses the name we both can immediately pick out its nominatum.  It seems like language does work this way so that is a strength of the theory.

Another strength of holding that the sense of a given name is the name itself is that the name is cognitively accessible only to those with language.  This is an important point.  Without language an individual could come into contact with me and have some cognitive content associated with who I am, but not with the name “Joe Cure.”  If the sense of a given name is the name itself, then it is not possible for someone to get on to the sense of Joe Cure without language, which seems correct.  Additionally, this picture avoids all three problems encountered by Russell’s theory and better addresses those faced by Frege’s official account of given names (which essentially is an unsuccessful attempt to reduce given names to abbreviations of definite descriptions).

However, it is still unclear whether in this picture the sense of a given name (the name itself) actually determines the nominatum, or if it is even possible (in any picture) for the sense of a given name to determine its nominatum.  The way in which it could be said that the sense determines the nominatum in Kaplan’s theory is something like the name Kelly Cure was created when my sister was born and it was said that this barely sentient infant would be named Kelly Cure.  A consequence is one would have to maintain that Kelly Cure in this case is a different name from any other given name Kelly Cure that has ever and will ever exist.  In one sense this seems right because this given name Kelly Cure refers to one singular object that no other given name Kelly Cure will ever refer to.  But, in another sense it seems odd to say this name Kelly Cure is different from another name Kelly Cure.

What I want to say about Kaplan’s suggestion is I believe it is the best possible picture that is compatible with Frege’s general theory.  If it does fail, it is because it adheres to Frege’s (possibly mistaken) belief that the sense of a given name determines its nominatum[20].  In this case Kaplan’s theory would be similar to the best account of the physical world in which the earth is flat, or of a universe that revolves around the earth.  There are worthwhile points to be taken from the theory, but if it is shown to be based on the fundamentally flawed understanding of how given names are determined it will ultimately be incorrect.


[1] Frege uses “name” or “sign” to mean both definite descriptions that pick out definite objects like “the tallest man in the state of California,” as well as given or proper names like Barak Obama.  I will use “name” in the same way where a distinction between the two kinds is not important for my purposes and be more specific where necessary.

[2] Wherever “A=A and A=B” is referenced in this paper I will assume they are both true identity statements unless expressly stated otherwise

[3] I will refer back to this case as the sun example

[4] Because, again, an identity statement between objects simply states one object is identical to itself, regardless of what symbol(s) is used to represent the object

[5] On Sense And Nominatum p. 199

[6] On Sense And Nominatum p. 199

[7] Namely, the time at which the object was perceived

[8] Frege thinks the sense must be objective and sharable because if it were not, communication like ‘please pass the salt’ would seem impossible.  For my sense of ‘salt’ would be different than yours and possibly then refer to a different nominatum.

[9] I use thought, cognition, and [the] understanding to mean roughly the same thing

[10] Russell’s picture challenged Frege’s theory of names because Frege held there was both a sense and nominatum for each name, while Russell tried to show there was only a nominatum.

[11] Things like the sun example definitively show there are cases where A=A and A=B are true identity statements relating to the same object, but express different cognitive information or knowledge.

[12] Recall Frege’s solution is that names are comprised of two parts: a sense and a nominatum, so for A=A both sides of the statement have the same sense and nominatum but for A=B each side represents the same nominatum but different senses.

[13] (e.g. the sun example)

[14] Frege uses “name” or “sign” to mean both definite descriptions that pick out definite objects like ‘the tallest man in the state of California’, as well as given or proper names like ‘Barak Obama’ (from p.1, footnote 2)

[15] The physical person, not the name

[16] In other words, if two people have different understandings of the word “I,” how is it that the word “I” still succeeds in picking out the same thing/person for both individuals?

[17] Our understandings of the phrase differ in that I know parts of it to be false, while the speaker does not.

[18] Again, the sense of a given name is simply a group of definite descriptions or properties of the thing named

[19] Or the cognition attached to/associated with the name (but it must be an objective, not subjective cognition)

[20] I explained why one might think this to be incorrect in the second to last paragraph of section III

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