gottlob frege's analysis of language - class notes by bernard weiss

8/9/2019 Gottlob Frege's analysis of language - class notes by Bernard Weiss

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The Fregean Background

(i) Freges Analysis of Language

At its simplest, our task is to analyse sentences such as Fred is red; Gertrude is

loud; Everest is high. How do we explain the fact that these sentences mean whatthey do on the basis of what the component words mean?

Frege, a mathematician interested in the foundations of mathematics, comes at

this problem from a distinctly mathematical perspective. So lets set aside for the

moment the question about language and take a simple mathematical function, say,

the function of adding 2. Heres a table of the results of applying this function to a

series of numbers:

x x+2

1 3

2 4

3 5

5 7

OK this is all easy enough. We obviously understand the function and know what the

result of applying the function to any number is. Freges question concerns the nature

of the function and how we are to represent it. It is tempting to represent the function,

as we did in the table, by saying it is the function x+2. This isnt entirely wrong but is

apt to be misunderstood. If x simply designates an indeterminate/imprecise number

then so to does x+2; it doesnt represent the function itself. In order to get at the

function we have to see what is common between the numbers in the right handcolumn. That is we want to express the pattern we can see in:

x x+2

1 1+2

2 2+2

3 3+2

5 5+2

That is we want to see the numbers on the right as arising from adding two to thenumbers on the left. The function is something which takes each of the numbers on

the right to the appropriate number on the left. So it is something which gives the

numbers on the right when we replace the x in x+2 by the numbers on the left. So

the expression for the function should be thought of as something essentially

incomplete since it yields expressions for certain numbers when other numbers are

placed in a specified position. So the expression for the function should be something

which makes this incompleteness explicit, viz.,

( )+2

Importantly, Frege thinks that since we must represent the function to ourselves in thisway we must think of the function itself as being essentially incomplete.


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So, to sum this up, a function is an incomplete entity which takes numbers (certain

objects) to other numbers (other objects). The expression for a function should make

its incompleteness explicit; we need, that is, to indicate the functions argument place

or places.

Examples of functions:

( ).3+5; ( ).( ); ( )2-67; ( ); ( )+ { }

(Here the different styles of brackets indicate that these gaps may be completed by

using different numbers.)

In general whenever we have a (complex) expression for a number we can extract

some of the symbols for numbers and arrive at a function.

E.g., we may have 5.3+52 from which we might extract the symbol 5 to arrive at

the following expression for a function: ( ).3+( )2. Or we might only extract onesymbol 5 to arrive at: 5.3+( )2. We might also extract the symbol 5 and 3 to

arrive at: ( ).{ }+ ( )2.

As long as remember how to interpret the x, i.e., as indicating a gap, not as

indicating an indeterminate number, then we can replace the brackets by xs and ys.

Note that the function 2.(x+1)-2.x, whose value is always 2 no matter what we

substitute for x must still be sharply distinguished from the number 2 itself: the one is

a function the other a number, that is, a kind of object.

Consider this somewhat different case. Take the arithmetical expression 2+3=5, if

we extract the symbol 3 then what we should have left according to Frege is an

expression for a function. And obviously we can substitute names for other numbers

in the place occupied by 3 to get:

x 2+x=5

1 2+1=5

2 2+2=5

3 2+3=5

5 2+5=5

But the expressions on the right are complete, so, according to Frege, they should be

names. But what do they name? Clearly they differ from expressions such as 2+3 in

that they can be true or false (all but the third is false). But Frege treats them like

these other expressions: he simply says that they name Truth and Falsity. So the

function 2+x=5 is a function which takes numbers to truth-values. Any function

whose values are only the two truth-values is said by Frege to be a concept.

Natural Language


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Lets try to apply this framework to natural language, as Frege did. Take our sentence

Fred is red if we take the name Fred from this sentence were left with x is red.

So this according to Frege is an expression for a function, specifically, a concept: it is

a concept which takes objects, including Fred to truth-values.

Are there functions which arent concepts in natural language? Indeed there are.Consider this way of talking of South Africa: The homeland of Nelson Mandela.

Here we have an expression for a function, namely, The homeland of x which takes

people (among other things) to countries. The example can obviously be multiplied.

Lets think now about logical vocabulary. Take, for example, the conjunction The

weather is foul and I am tired of getting soaked. We can extract the two sentences to

get an expression for a concept: x and y. The role of this concept is to take truth-

values to truth-values. We might sum this up as follows:

x y x and y

True True True

True False False

False True False

False False False

Here we can ignore which particular sentences we replace x and y by since all those

sentences simply function as names for one or other of the two truth values. So we

can capture the meaning of x and y completely by the above table which exhausts

the possible combinations of truth-values. Conjunction is thus an example of a truth

function and we arrive at whats now commonly called the truth functional definitionof the sentential connectives (x and y; x or y; not x; and if x then y).

Finally lets consider the sentence All men are mortal. Our aim is to work out what

sort of expression all is. Lets start though with men and mortal These have the

same meaning here as in Socrates is a man and Socrates is mortal respectively. (If

they didnt have the same meaning then the syllogismAll men are mortal. Socrates

is a man. Therefore, Socrates is mortal.would commit the fallacy of equivocation,

which it clearly does not.) That is we have two functions x is a man and x is

mortal. So the form of the sentence is better represented as For all x, if x is a man

then x is mortal. And here we apply the function For all x to the complex function

if x is a man then x is mortal; that is, we have a function of functions.

So, For all x, Fx says of the concept F that it is satisfied by every object. There is

an x such that Fx says of the concept F that it has at least one instance.

What weve been given is an account of how the semantic value of a complex

expression is determined by the semantic values of its parts. The semantic value of an

expression is an item in the world with which it is correlated, which it refers to or

denotes. So the semantic value of a name is an object, the semantic value of a

functional expression is a function and the semantic value of a sentence is a truth-

value.


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We need to say a little more about functions. A function, as weve noted is something

which takes objects to other objects. So we can think of a function such as x+3 as

pairing each number with another. The list of pairs so determined is called the value

range or extension of the function. So for x+3 wed have: (1,4); (2,5); (3,6) etc.

Frege treats functions extensionally, which is to say that he identifies functions which

have the same extension. This is because he believes that the semantic value of acompound expression is completely determined by the semantic values of its

components. So, provided two expressions have the same extension, interchange of

one with the other wont affect the semantic value of the whole.

And now we can see how wed give an account of how the semantic-value of

complex expression is determined. That is, if we had specifications of what the names

in our language refer to and of the extensions of the functional expressions then wed

be able to work out the truth-values of elementary sentences.

Example of a Primitive Semantic Theory

Axioms

(i) The denotation of Freda is Freda

(ii) The denotation of Mary is Mary

(iii) The denotation of Lucy is Lucy

(I) The extension of The mother of x is (Freda, Mary); (Mary, Lucy)

(II) The extension of x is a lawyer is (Freda, False); (Mary, True); (Lucy, True)

(C) A name formed from a name and a function denotes the second object in the

ordered pair whose first member is the object denoted by the name, in the extension of

the function.

Examples:

(a) The mother of Mary

1. The mother of Mary denotes the second object in the ordered pair whose first

member is the denotation of Mary, in the extension of The mother of x (C)

2. The mother of Mary denotes the second object in the ordered pair whose first

member is Mary, in the extension of The mother of x (ii)

3. The mother of Mary denotes Lucy (I)

We could use the same method to work out the denotation of, for example, The

mother of the mother of Freda

(b) The mother of Freda is a lawyer

1. The mother of Freda is a lawyer denotes the second object in the ordered pair

whose first member is the object denoted by The mother of Freda in the

extension of x is a lawyer (C)


whose first member is (the second object in the ordered pair whose first object

is the denotation of Freda in the extension of The mother of Freda) in the

extension of x is a lawyer (C)


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whose first member is (the second object in the ordered pair whose first object

is Freda in the extension of The mother of Freda) in the extension of x is a

lawyer (i)


whose first member is Mary, in the extension of x is a lawyer (I)5. The mother of Freda is a lawyer denotes the True. (II)

The Referential View of the Meanings of Names

Kilimanjaro means the highest mountain in Africa

is grammatical, and, just possibly, an ordinary sentence of English. In this case the

analysis would be something like the following:

Kilimanjaro means the highest mountain in Africa iff Kilimanjaro refers tothe highest mountain in Africa.1

What we shall learn in this section is that this analysis faces severe problems. These

problems are sufficient to cast doubt on the original sentences as being an acceptable

starting point. These lessons are due to Frege. We shall then go on to look at Russells

alternative account which serves to show that the logico-syntactic category of singular

terms is highly moot. Although we shall be looking at these questions from the point

of view of giving an analysis of meaning, the discussion takes us to the heart of

philosophical attempts to understand names and definite descriptions. This is scarcely

surprising since the general point that we are pushing is that fixing on the starting

point of analysis already requires considerable philosophical insight into how

language works.

Freges Argument for the Notion of Sense

Freges argument2 focuses on identity statements. But this is primarily a heuristic

device: he could have made the same point by focusing on elementary sentences in

general. However its worth following through his reasoning about identity statements

simply because, in doing so, we restrict our attention to the way names function.

We often assert statements of identity. For instance: The morning star is the

evening star; Everest is Gaurisanker; Clark Kent is Superman; Portia isBalthazar. The crucial thing to note is that these statements often carry useful

information. If you remember the plot ofThe Merchant of Venice youll know that

Bassanio would not have got himself in a pickle with Portia had he known that Portia

and Balthazar are identical. Similarly Lois Lane would have acted quite differently

had she known that Clark Kent is Superman: that piece of knowledge would have

been informative to her. So statements of identity can be informative. No account of

language and, in particular, of the functioning of names can be acceptable unless it

makes sense of this fact.

1 Note: we should not confuse these two claims. (i) The meaning of e is m iff e refers to m; and (ii) the

meaning of e is given by the fact that e refers to m. (i) is the claim in the text; (ii) is a claim of quite adifferent sort that we shall only come to discuss in chapter six.2 See inter alia Frege (1980: 56-78)


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Frege asks himself the question: what does the relation of identity hold

between? It would seem that on the analysis of names with which we have so far been

provided we can give only two answers. The relation either holds between the objects

named by the names or it holds between the names themselves. Lets consider each

suggestion in turn. Suppose that a and b are names and that a=b is true. If the

relation holds between the objects denoted by the names then a=b cannot differ froma=a. For, since a=b is true, a and b name the same object. But clearly a=b

and a=a differ because on many occasions (such as those above) statements of the

form a=b are informative whereas those of the form a=a are uninformative since

theyre known a priori to be true.

Should we then take the identity statement to assert a relation between names?

Frege says not since this can never be informative. Why so? Well what a sign means

is a matter of speakers conferring meaning on the sign. So, if identity asserts a relation

between names, then either a=b is false since a and b are different names or it

registers a stipulation (which were entirely entitled to make) that the signs a and b

are to be treated as the same name. On neither reading can a=b be treated as an

informative statement.The point seems to be this. The character of a particular language is arbitrary

in the sense that theres no intrinsic connection between the signs we use and what

those signs refer to: whatever connections actually exist can be imagined to be quite

other without supposing any difference in extralinguistic reality. But we cant simply

suppose that the link of a name to its reference is purely arbitrary: that there is no

saying how the use or the meaning of the name establishes the link. Why not?

Because then there couldnt be informative identity statements, these would just be

the laying down of arbitrary conventions. What we need to be able to see is that there

is both an element of convention or arbitrariness and an element that is non-

conventional or non-arbitrary in the relation of a name to its referent. That is, we

need to see the link as mediated: the name is conventionally or arbitrarily related

to something and that thing is non-conventionally, non-arbitrarily related to the

referent. Frege calls this thing the sense of the name. How does this help with our

problem?

What we wanted to be able to explain is that we can learn something when we

are told a=b. Well a is arbitrarily related to the sense of a and the sense of a is

non-arbitrarily related to the referent of a. Likewise, b is arbitrarily related to the

sense of b and the sense of b is non-arbitrarily related to the referent of b. What

we learn when we are told that a=b is that the sense of a and the sense of b are

related to the same object. And, since neither of these relations, is a matter of arbitrary

this is a genuine piece of knowledge: conceivably, things might have been different.The point can be made in terms of what speakers understand when they grasp

the meaning of an expression. If speakers grasped the reference of an expression then,

for those regions of language which they understood, their understanding would

suffice for them to arrive at a verdict on the truth-value of an identity statement. For,

in such a case, theyd know the referent of each name as part of their understanding of

it. They would simply need to reflect on this understanding in order to appreciate the

truth or falsity of an identity statement. Clearly this misconstrues the nature of our

understanding of language. So, we might say that in grasping the meaning of an

expression speakers do not grasp as much as the reference of an expression.

Conversely, it is clear that speakers do not grasp as little as the reference of an

expression. For, if they did, then co-referring terms (terms which refer to the samething) would be synonymousindeed since Frege treats sentences as names for truth-


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values all true sentences would be synonymous with one another, as would all false

sentences. In this case we couldnt distinguish the sentences a=a and a=b in terms

of meaning.

The conclusion is that the reference of a term is not an ingredient of what

speakers understand when they understand the terms meaning. In short, reference is

not part of meaning. What do speakers grasp when they understand a term? Amongstother things they grasp its sense.

Now if we are to explain the informativeness of a=b by saying that it informs

us that the sense of a and the sense of b share a reference. Then it had better be the

case that a and b have different senses. So it is possible for the same referent to

have many senses.

We are able now to explain or, better, to allow for the possibility of

meaningful terms which dont have a reference. If we insist on thinking of the

meaning of a term as what it refers to then terms which fail to refer will fail to have a

meaning. But, once we introduce the notion of sense, it may be possible for a term to

have a sense, and thus to be meaningful, yet fail to have a reference.

Let us sum up so far what weve discovered about sense. (i) The sense of aterm is part of what speakers understand when they understand the terms meaning.

(ii) The sense is non-conventionally, non-arbitrarily related to the terms reference.

(iii) In fact the sense of the term is that ingredient of a terms meaning which

determines its reference. For Frege, sense determines reference. (iv) The relation of

sense to reference is many-one, that is, many senses may share the same reference. (v)

In addition, a sense may fail to have a reference.

We should note also that Frege thinks that these arguments can be extended to

cover all terms of the language: all expressions have a sense. The sense of a sentence

is a thought.

So, according to Freges argument our analysis of the meaning of a name is

wrong. The reasons for this relate to the information content conveyed by sentences

which include names. And this is brought out by consideration of routine uses of such

sentences. So the starting point of analysis (Kilimanjaro means the highest mountain

in Africa) itself seems threatened; such sentences, even if they are sometimes used,

cannot be thought to capture the essentials of the way names mean since they are

blind to the names possession of a sense. Just to be clear, we are not saying that

names have senses. Frege says this. But we want a starting point of analysis that

doesnt beg the question against Frege, whether or not hes is right.

Sense and Statements ofBelief

Consider the following argument:

Premise 1: King George doubts that Scott is the author ofWaverley.

Premise 2: Scott is the author ofWaverley.Conclusion: Therefore King George doubts that Scott is Scott.

Clearly something has gone very awry here. The first premise attributes an

understandable doubt to King George who may simply be not very well informed

about Scotts literary productions. The second premise states a truth. But the

conclusion attributes to King George a doubt about the law of identity and clearly it is


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unlikely that even he was so confused: such a conclusion is surely not warranted by

the premises3.

Now it is hard to explain what goes wrong here. What, in effect, premise 2

tells us is that Scott and the author of Waverley have the same reference. And in

that case we should be able to substitute one term for the other without changing the

truth-value of any sentence in which they occur. This is a consequence of Fregescompositional view of semantic value. The semantic value of a name is its referent;

that of a sentence is its truth-value. According to compositionality, the semantic value

of a complex expression is determined by the semantic values of its components.

What seems to be our problem is that applying this principle in the context of

statements about someones beliefs, doubts, knowledge etc., leads from plausible

truths to evident untruths and absurdities.

How should we resolve the situation? We could either say that in the context

of statements of belief etc. the principle of the compositionality of semantic values no

longer holds. Or we could say that the semantic values of expressions in the context of

statements of belief etc. change so that terms which shared a semantic value, that is,

which referred to the same thing, may no longer do so. But what could the terms nowrefer to? Freges answer is that the terms shift their reference from what is their usual

reference to what is usually their sense. So in the context of the sentence, King

George doubts that Scott is the author of Waverley, the name Scott does not refer to

Scott, it refers to the sense of Scott. Similarly the author of Waverley no longer

refers to the author of Waverley (i.e., to Scott) it refers to the sense of the author of

Waverley. Since the sense of Scott and the sense of the author of Waverley are

different we can no longer be assured that when we substitute Scott for the author

of Waverley the truth-value of the containing sentence remains unchanged. Note that,

as a consequence of this account, senses are in the realm of reference.

The Objectivity of Sense

Different speakers can grasp the same sense. Indeed grasping the same sense is a

condition for communication. When we communicate one speaker uses a sentence to

express a certain thought; to do so she must grasp that thought. She successfully

communicates only if her audience takes the sentence which she utters to express just

that thought. In order to do so the audience must grasp that thought. Thus

communication requires that speakers share the senses of their terms.

The sense of an expression must thus be sharply distinguished from any set of

subjective impressions associated with it. Ideas, impressions feelings are all things

that are had by a certain subject: one can always ask whose idea, whose feeling etc.Something which can be had in this way cant be shared or, perhaps better, cant be

known to be shared. The most that we can determine is qualitative differences

between different peoples feelings, sensations or ideas. We cannot determine that

they are qualitatively identical since for that, the sensations would have to be

compared by a single subject. And thats clearly impossible since no subject can have

anothers sensations or impressions. So sensations, feelings and ideas cannot be

known to be shared. Thus, since communication requires that we share and can know

that we share senses, senses must be sharply distinguished from ideas. That is, senses

are not part of the subjective realm and, for Frege, this means that they are objective.

(Another option might be to say that they belong to the inter-subjective realm but

Frege doesnt consider this as a possible challenge to his dichotomy of subjective and3 See Russell (1956: 39-55).


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objective.) Moreover for Frege to say that senses are objective is to reify them: senses

are objective items in the world. He thus distinguishes three realms of existents. There

is the realm of the actual, which includes ordinary middle-sized objects etc., then

there is the subjective realm consisting of ideas and sense impressions and the like

and finally there is a third realm consisting of senses and abstract objects such as

numbers and the truth-values.

Frege, G. Function and Concept in his Collected Papers and in his Philosophical

Writings

For exposition see the chapter by Grayling and Weiss in Philosophy: Further through

the Subject (Anthony Grayling ed.) and also chapter 1 of Alexander Millers

Philosophy of Language Freges Distinction between Sense and Reference4

4 Terminological Note:

There is an unfortunate lack of agreement amongst commentators about how to

translate Freges German term bedeutung. It is variously translated as: denotation,

nominatum, meaning and reference. Ill use reference and denotation pretty

much interchangeably as translations. meaning wont be used in this way at all. And

nominatum wont be used at all.

gottlob frege's analysis of language - class notes by bernard weiss

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