1

The structure of Frege’s thoughts

M. ZOUHAR

Institute of Philosophy, Slovak Academy of Sciences, Slovakia

Abstract: Fregean thoughts (i.e., the senses of assertoric sentences) are structured

entities because composed of simpler senses that are somehow ordered and

interconnected. The constituent senses form a unity because some of them are

„saturated‟ and some others „unsaturated‟. The present paper shows that Frege‟s

explanation of the structure of thoughts, which is based on the „saturated/unsaturated‟

distinction, is by no means sufficient because it permits what I call „wild analyses‟,

which have certain unwelcome consequences. Wild analyses are made possible because

any „unsaturated‟ sense that is a mode of presentation of a concept together with any

„saturated‟ sense forms a thought. The reason is that any concept can be applied to any

object (which is presented by a „saturated‟ sense). This stems from the fact that Frege

was willing to admit only total functions. It is also briefly suggested what should be

done to block wild analyses.

1. Introduction

According to Frege‟s post-1890 theory, a thought is a compound semantic entity that

is expressed by assertoric sentence. A thought is composed of the senses expressed by

the constituent sub-expressions of a sentence. The constituent senses are of two

fundamentally different kinds – „saturated‟ ones and „unsaturated‟ ones; a thought

emerges when the former kind of sense „saturates‟ the latter kind of sense. The resulting

thought is both compound and structured because the constituent senses can be

identified as proper parts of it.

According to my main thesis, Frege‟s explanation of the structure of thoughts is

problematic. For, if the explanation is based merely on the „saturated/unsaturated‟

distinction, there is no chance to block what I call wild analyses of thoughts. But wild

analyses have to be eliminated as possible analyses of thoughts.

To begin with, a cursory overview of Frege‟s fundamental logical and semantic

notions would be appropriate. An expression of a given language1 both expresses a

sense and designates a denotation. The sense of an expression is a mode of presentation

of its denotation (cf. Frege 1984b, 158). For every sense there is at most one denotation

of which it is a mode of presentation. There are two kinds of denotations – objects and

functions (cf. Frege 1984a, 147). A concept „is a function of one argument, whose value

is always a truth-value‟ (Frege 1979a, 119; cf. also Frege 1984a, 146). A function of

two arguments whose value is always a truth-value is a relation (cf. Frege 1984a, 154).

There are two kinds of expression – proper names and functional expressions. Proper

names designate objects and functional expressions designate functions; concept-words

designate concepts and go with functional expressions. Assertoric sentences designate

truth-values. There are two truth-values – the True and the False. Since the truth-values

are objects, assertoric sentences are proper names. The sense of an assertoric sentence is

called a thought (cf. Frege 1984b, Frege 1984f).2 Thoughts are modes of presentation of

1 For the sake of simplicity, references to language will be suppressed throughout the paper. 2 This point is somewhat simplified because it is only assertoric sentences complete in every

respect that express thoughts. According to Frege, many assertoric sentences fail to express

2

truth-values. Frege‟s theory involves simple and general criteria for distinguishing (i)

proper names from functional expressions; (ii) the senses of proper names from those of

functional expressions; and (iii) the denotations of proper names from those of

functional expressions. They are all based on the „saturated/unsaturated‟ distinction.

The paper is organized in the following way: I start with the observation that Frege‟s

thoughts are composed of simpler parts that form a certain unity rather than a mere list

of constituent senses. Frege‟s explanation of the unified nature of thoughts is presented

in Section 2. Next, I put forth the idea that this explanation permits wild analyses of

thoughts and I point out that such analyses are unacceptable (Section 3). In Section 4, I

track the possibility of wild analyses down to Frege‟s theory of functions. Finally, a

way out of these predicaments is briefly suggested (Section 5).

2. Thought and its unity

A thought expressed by a sentence is both compound and structured. It is compound

because it consists of the senses expressed by the sub-expressions of the sentence. As

Frege claims, „[a]s the proper name is part of the sentence, so its sense is part of the

thought‟ (Frege 1979d, 191). So, thoughts can be taken „as composed of simple parts‟

which „correspond to the simple parts of the sentences‟ (Frege 1984h, 390). In the case

of a sentence, we are capable to identify retrospectively its simple constituent

expressions; analogously, when we have a thought, we are capable to identify

backwards its simple constituent senses. However, being compound is not the same as

being structured; for a mere list of senses is compound as well without being structured.

What is required is that the constituent senses of a thought be mutually interconnected

in certain ways to form a unity.3 By parity of reasoning, a sentence is not just a mere list

of the constituent expressions; they form a unity because they are mutually

interconnected according to the laws of grammar.

Since the thought is not a mere list of the constituent senses, we may ask:

[H]ow a thought comes to be constructed, and how the parts are so combined

together that the whole amounts to something more that the parts taken

separately[?] (Frege 1984h, 390)

Let us call this the problem of thought-unity. Frege‟s solution to it is ingenious. He

claims:

thoughts. He calls them sometimes quasi-sentences. A quasi-sentence „has the grammatical

form of a sentence and yet is not an expression of a thought, although it may be a part of a

sentence that does express a thought, and thus a part of a sentence proper‟ (Frege 1979d, 190).

A quasi-sentence is capable to express, at most, a mode of presentation of a concept as opposed

to that of a truth-value. In general, this holds for sentences in which „the mere wording, which

can be made permanent by writing or the gramophone, does not suffice for the expression of the

thought‟ (Frege 1984f, 357–58); glaring examples are sentences in which the present tense

indicates the time of utterance (as opposed to timelessness; cf. Frege 1984f, 358) or sentences

featuring demonstrative or indexical expressions (cf. Frege 1984f, 358). Anyway, I pretend the

sentences used as examples in the present paper do express thoughts. 3 It is easy to see that the constituent senses have to be unified somehow in order to become a

thought. According to Frege, a thought is something „to which the question “Is it true?” is in

principle applicable‟ (Frege 1979f, 253). Thus, the thoughts are bearers of truth-values.

However, a list of senses can be neither true nor false; the constituent senses have to be

interconnected in accordance with certain specific (logical) relations; being truth-evaluable can

be viewed as a product resulting from the application of these relations to the constituent senses.

3

For not all the parts of a thought can be complete; at least one must be

„unsaturated‟, or predicative; otherwise they would not hold together. For

example, the sense of the phrase „the number 2‟ does not hold together with that

of the expression „the concept prime number‟ without a link. We apply such a link

in the sentence „the number 2 falls under the concept prime number‟; it is

contained in the words „falls under‟, which need to be completed in two ways – by

a subject and an accusative; and only because their sense is thus „unsaturated‟ are

they capable of serving as a link. (Frege 1984c, 193)

Frege sorts out the constituent senses in two categories – those that are complete or

„saturated‟ and those that are incomplete or „unsaturated‟. An „unsaturated‟ sense

involves at least one empty place to be supplemented with another sense. It is

impossible for two complete senses to hold together; they form just a mere list of two

senses. But if a complete sense fills an empty place in an incomplete sense, what we get

is a unity featuring the two senses as its constituents. A thought is a unity because an

„unsaturated‟ constituent sense is completed with a „saturated‟ one.

At the level of expressions, there occurs a similar problem that may be labelled the

problem of sentence-unity. According to the traditional logic, sentences such as

„Socrates is wise‟ can be analyzed into the subject term „Socrates‟, the copula „is‟, and

the predicate term „wise‟. Now, how is it possible to distinguish the sentence „Socrates

is wise‟ from a mere list of the three expressions „Socrates‟, „is‟, and „wise‟? The

traditional logic has it that it is the copula that serves as a unifying device forming

sentences from isolated expressions. However, it remains rather unclear how it does in

fact unify the isolated expressions into a whole. The unifying force of the copula is

merely postulated without any deeper explanation.

Frege‟s analysis of sentences is radically different:

Statements in general… can be imagined to be split up into two parts; one

complete in itself, and the other in need of supplementation, or „unsaturated‟.

(Frege 1984a, 146)

Concept-words (or functional expressions in general) are „unsaturated‟ and involve

empty places ready for completions; on the other hand, proper names are complete. The

sentence „Socrates is wise‟ is analyzed into the proper name „Socrates‟ and the concept-

word „( ) is wise‟ in which the empty place is indicated by the round brackets. This

picture generalizes: Any sentence is analyzed into an „unsaturated‟ constituent having at

least one empty place and the appropriate number of („saturated‟) proper names.

The problem of sentence-unity closely resembles the one of thought-unity. The

solutions to both of them are based on the parallel „saturated/unsaturated‟ distinctions. It

remains to be explained why the empty places go with concept-words (or functional

expressions in general) rather than with proper names. Frege‟s answer can be read off

this quotation:

As a mere thing, of course, the group of letters „and‟ is no more unsaturated than

any other thing. It may be called unsaturated in respect of its employment as a

symbol meant to express a sense, for here it can have the intended sense only

when situated between two sentences: its purpose as a symbol requires completion

by a preceding and a succeeding sentence. It is really in the realm of sense that

unsaturatedness is found, and it is transferred from there to the symbol. (Frege

1984h, 393; italics mine)

4

Thus, the concept-word is incomplete because it expresses an incomplete sense. It is

senses that are „saturated‟ or „unsaturated‟ in the first line. Expressions are such only in

a derivative manner. However, this is not the end of story. There remains another open

question: Why should we suppose that it is the sense of „is wise‟ rather than that of

„Socrates‟ that is „unsaturated‟? As far as I can see, the answer lies in Frege‟s distinction

between concepts (functions) and objects; for this reason it is postponed to Section 4.

3. Thought and wild analysis

The „saturated/unsaturated‟ distinction is crucial to presenting thoughts as structured

entities – a thought is structured because the empty place in the incomplete constituent

sense is „saturated‟ by a complete sense. Now it seems that the „saturated/unsaturated‟

distinction does not suffice to determine the unique structure of the thought. For,

a thought can be split up in many ways, so that now one thing, now another

appears as subject or predicate. The thought itself does not yet determine what is

to be regarded as the subject. (Frege 1984c, 188)4

Let us adopt, for the sake of simplicity, the following agreement: If E is an

expression, [E] designates the sense of E.5 Now, consider the following thought:

(1) [Caesar conquered Gaul]

It can be analyzed in three different ways (cf., e.g., Frege 1984a); we might get (i) the

„saturated‟ sense [Caesar] and the „unsaturated‟ sense [( ) conquered Gaul];6 (ii) the

„saturated‟ sense [Gaul] and the „unsaturated‟ sense [Caesar conquered ( )]; (iii) the

„saturated‟ senses [Caesar] and [Gaul] and the double „unsaturated‟ sense [( ) conquered

( )].7 Thus, (1) may have the following structures:

(1a) [(Caesar) conquered Gaul]

4 Despite certain vital differences between his early and mature semantic theories, Frege

adopted a similar standpoint as early as in his Begriffsschrift: „For us, the different ways in

which the same conceptual content can be considered as a function of this or that argument have

no importance so long as function and argument are completely determinate‟ (Frege 1967, 23).

So, different analyses of the same conceptual content are possible provided what we get is a

completely determinate function and argument. 5 This notation should make it plain that I am concerned here solely with thoughts and other

kinds of sense rather than with sentences and other kinds of expression. 6 The sign „( )‟ indicates empty place. Frege usually used Greek letters ζ, ξ, etc. in the same

manner. I shall sometimes write „(E)‟ to indicate that E fills the empty place. 7 Different analyses of the same thought have no bearing on the identity of the thought. In

particular, the thought (1) as analyzed in accordance with (i) and the thought (1) as analyzed in

accordance with (ii) is one and the same thought. However, this is far from being obvious.

Frege himself seemed to be of different opinions when he discussed a somewhat different

example. On the one hand, he claimed that [S] and [non-non-S] (where S is a complete sentence

expressing a thought) are different thoughts because one of them – as opposed to the other one –

involves two occurrences of [non-( )] (cf. Frege 1984g, 389). In another paper, however, he said

that [S] and [non-non-S] are the same thought (cf. Frege 1984h, 404–405) (moreover, [S] is

supposed to be the same thought as [S and S] and [S or S], etc.!). What we may see here is that

sometimes Frege took the structure of a thought as essential to its identity and sometimes he did

not (for an excellent discussion see Tichý 1988, 32).

5

(1b) [Caesar conquered (Gaul)]

(1c) [(Caesar) conquered (Gaul)]

If (1) has the structure (1a), it is about Caesar and claims that he conquered Gaul; if it

has the structure (1b), it is about Gaul and claims that it was conquered by Caesar; and

if its structure is (1c), it is about Caesar and Gaul and claims that the former conquered

the latter.

Now, what is important for us is that all these analyses obey just a single principle. It

may be called The Principle of Analysis and is based on the „saturated/unsaturated‟

distinction:

Whenever a thought is analysed, it is split up into an ‘unsaturated’ part, which

has at least one empty place, and the required number of ‘saturated’ parts that fill

up the empty place(s) in the ‘unsaturated’ sense.

Thus, whenever we analyse a thought what we have to get is a mode of presentation of a

concept or a relation and the required number of modes of presentation of objects. Any

analysis which conforms to this principle is admissible.

Frege seems to adhere also to a converse version of this principle, as witnessed by

the second sentence in the following quotation:

By analysing a singular thought we obtain components of the complete and of the

unsaturated kind, which of course cannot occur in isolation; but any component of

the one kind together with any component of the other kind will form a thought.

(Frege 1979d, 187; italics mine)

The converse version – or the Principle of Synthesis, as we may call it – runs:

Whenever an ‘unsaturated’ sense (that is a mode of presentation of a concept or a

relation), which has at least one empty place, is filled up with the required

number of ‘saturated’ senses a thought results.

Wolfgang Carl gives an apt summary of Frege‟s position that is completely

compatible with the above principles. Furthermore, he points to a possible awkwardness

which can be read off Frege‟s ideas:

This ability to consider a thought as being composed in different ways from its

constituents is limited only by the requirement that one draw a definite distinction

between concept and object, but within the limits of this distinction one is perfectly

able to form concepts for which there is no predicate in ordinary language and to take

arguments as objects that are not objects in any ordinary or metaphysical sense of the

word. (Carl 1994, 67)

The second half of Carl‟s quotation should be taken seriously. In what follows, I shall

use this observation to show that the above principles lead to difficulties for Frege‟s

theory of thought-structure. When we consider simple examples such as [Socrates is

wise] or [Caesar conquered Gaul] everything goes smoothly despite the fact that various

analyses of the same thought might be available. However, in the case of more complex

thoughts it is sometimes possible to produce also analyses deviating from conventional

ones in certain respects; I call them wild analyses. Such analyses can be offered for

6

some thoughts in which several „unsaturated‟ senses can be discerned. Typical examples

are thoughts expressed by compound sentences such as „a is Φ * b is Ψ‟, where „a is Φ‟

and „b is Ψ‟ are sentences and „*‟ is a connective.

The following example concerning the thought

(2) [Socrates thinks and Plato sings]

explains what I have in mind. Here are two analyses of the thought, one that is

conventional and another one that is wild (the arrow indicates splitting):

Conventional analysis:

1. [Socrates thinks and Plato sings] → [Socrates thinks] + [( ) and ( )] + [Plato

sings].

2. [Socrates thinks] → [Socrates] + [( ) thinks].

3. [Plato sings] → [Plato] + [( ) sings].

Wild analysis:

1. [Socrates thinks and Plato sings] → [Socrates thinks and Plato] + [( ) sings].

2. [Socrates thinks and Plato] → [Socrates thinks] + [( ) and ( )] + [Plato].

3. [Socrates thinks] → [Socrates] + [( ) thinks].

In both cases, we have the same simple constituent senses – namely [Socrates], [( )

thinks], [( ) and ( )], [Plato] and [( ) sings] – but the structure of (2) can be captured in

two different manners:

(2a) [((Socrates) thinks) and ((Plato) sings)]

(2b) [(((Socrates) thinks) and (Plato)) sings]

(2a) corresponds to the conventional analysis and (2b) to the wild one. According to the

wild analysis, the thought (2) can be derived in the following way: Complete the sense

[( ) thinks] with the sense [Socrates] – the „saturated‟ sense [Socrates thinks] results;

next, supplement the sense [( ) and ( )], which is double „unsaturated‟, with two

„saturated‟ senses [Socrates thinks] and [Plato] – the „saturated‟ sense [Socrates thinks

and Plato] arises; finally, supplement the sense [( ) sings] with the sense [Socrates

thinks and Plato] – the „saturated‟ sense [Socrates thinks and Plato sings] results. What

is unusual in this derivation is that the sense [( ) and ( )], which is supposed to be

„saturated‟ with two thoughts under normal circumstances, is filled with different kinds

of sense. However, this move is in complete agreement with the Principle of Synthesis.

Both analyses are permitted because they follow the „saturated/unsaturated‟

distinction. They obey Frege‟s dictum quoted above that „a thought can be split up in

many ways, so that now one thing, now another appears as subject or predicate‟. This

dictum is preserved in any of the following partitions of (2) into subject and predicate

((2c) and (2f) are simplified versions of (2a) and (2b), respectively):8

8 It can be seen, moreover, that the following partition is not possible:

(2g) [(Socrates) thinks and Plato sings]

The reason is that [( ) thinks and Plato sings] cannot be generated from [( ) thinks], [( ) and ( )],

[Plato] and [( ) sings]. In particular, [( ) thinks] (qua an „unsaturated‟ sense) cannot fill an empty

place in [( ) and ( )] which can be „saturated‟ only with complete senses. In other words,

[Socrates] is not a proper subject of (2) and [( ) thinks and Plato sings] is not a proper predicate

of it.

7

(2c) [(Socrates thinks) and (Plato sings)]

(2d) [(Socrates thinks) and Plato sings]

(2e) [Socrates thinks and (Plato sings)]

(2f) [(Socrates thinks and Plato) sings]

Now let us turn to some consequences. For the sake of argument, suppose it is true

that Socrates thinks as well as that Plato sings; thus, (2) is true. It can be shown the two

analyses lead to widely different results. When we apply the concepts presented by [( )

thinks] and [( ) sings] to Socrates and Plato, respectively, we get the True in both cases

and when we apply the relation presented by [( ) and ( )] to the True twice over, the

result is again the True. The conventional analysis conforms to our assumption. The

wild analysis, however, gives another result. When we apply the concept presented by [(

) thinks] to Socrates, the True results; next, when we apply the relation presented by [( )

and ( )] to the True and Plato as its arguments, its value is the False (see Section 4 for

explanation); finally, when we apply the concept presented by [( ) sings] to the False,

the resulting value is again the False. So, the wild analysis leads to different results than

the conventional one.

This is unacceptable for Frege‟s theory of senses. For he claims that for any sense

there is at most one denotation; consequently, if the thoughts T1 and T2 are modes of

presentation of different truth-values, they have to be different thoughts. When we

assume the wild analysis concerns the very same thought as the conventional one, we

offend against this essential feature of Frege‟s theory. So, in spite of the fact that (2f) (or

(2b)) involves the mode of presentation of a completely determinate concept and that of

a completely determinate object, „saturating‟ the former with the latter should not

produce (2). This is at odds with the Principle of Analysis which admits that a thought

can be split up at will provided the „saturated/unsaturated‟ distinction is preserved.

Now, the above reason for rejecting (2f) (or (2b)) seems to be just a specific feature

of this kind of example. By way of comparison, consider the thought

(3) [1 is even and Plato sings]

We have:

Conventional analysis:

1. [1 is even and Plato sings] → [1 is even] + [( ) and ( )] + [Plato sings].

2. [1 is even] → [1] + [( ) is even].

3. [Plato sings] → [Plato] + [( ) sings].

Wild analysis:

1. [1 is even and Plato sings] → [1 is even and Plato] + [( ) sings].

2. [1 is even and Plato] → [1 is even] + [( ) and ( )] + [Plato].

3. [1 is even] → [1] + [( ) is even].

The structure of (3) is parallel to that of (2); so, we have either (3a) or (3b):

(3a) [((1) is even) and ((Plato) sings)]

(3b) [(((1) is even) and (Plato)) sings]

8

It is easy to see that the thought (3) has to be necessarily false. Both analyses

conform to this fact. Firstly, consider the conventional analysis: Applying the concept

presented by [( ) is even] to 1 gives us the False and applying the concept presented by

[( ) sings] to Plato gives us the True; finally, applying the relation presented by [( ) and (

)] to the False and the True results in the False. The same outcome is achieved when the

wild analysis is followed: the value of the concept presented by [( ) is even] for the

argument 1 is the False and the value of the relation presented by [( ) and ( )] for the

arguments the False and Plato is the False as well; finally the value of the concept

presented by [( ) sings] for the False is again the False. So, the reason for rejecting (2b)

(and (2f)) as a proper analysis of (2) need not be generalized for (3b).

Both thoughts (2) and (3) are perfectly on a par; they can be taken as substitution

instances of [x is φ and y is ψ]. Yet, the latter can be analyzed both conventionally and

wildly without doing any harm to some other postulates of Frege‟s theory while the

former can be analyzed either merely conventionally or merely wildly. Wild analyses

are such because they may lead to unpredictable consequences and portray different

thoughts (albeit very similar to each other in certain respects) as requiring different

treatments.

4. Total functions and wild analysis again

Since it is the Principle of Analysis what makes wild analyses possible, it might be

instructive to explore the motivations underlying it. In so doing we have to return to

Frege‟s ontology of objects and concepts (functions). For Frege‟s ontology justifies why

the function presented by [( ) and ( )], despite being a truth function, is capable to take

as its arguments both truth-values and non-truth-values.

First of all, it is important to notice that

[t]he peculiarity of functional signs, which we here called „unsaturatedness‟,

naturally has something answering to it in the functions themselves. They too may

be called „unsaturated‟ (Frege 1984e, 292)

and that

the argument does not belong with a function, but goes together with the function

to make up a complete whole; for a function by itself must be called incomplete,

in need of supplementation, or „unsaturated‟ (Frege 1984a, 140).

On the other hand, since an „object is anything that is not a function‟ (Frege 1984a,

147), objects are complete or „saturated‟. Furthermore,

[w]hat in the case of a function is called unsaturatedness, we may, in the case of

a concept, call its predicative nature (Frege 1979a, 119).

The predicative nature of concepts is just a special case of their „unsaturatedness‟

(Frege 1984c, 187n).9 To say that concepts can be applied, or predicated, to objects is

tantamount to saying that objects fall, or can be subsumed, under concepts (Frege

1989d, 193; Frege 1979d, 183). So, to say that concepts are „unsaturated‟ or that they

9 Although Frege points out that „[t]he words “unsaturated” and “predicative” seem more suited

to the sense than the [denotation]‟, he readily adds that „still there must be something on the part

of the [denotation] which corresponds to this, and I know of no better words‟ (Frege 1979a,

119n).

9

involve empty places to be filled with arguments is just a metaphorical way of saying

that they can be applied to something.

Now it should be observed that all functions are total for Frege. Consequently,

concerning concepts, it holds that

for any argument they shall have a truth-value as their value; that it shall be

determinate, for any object, whether it falls under the concept or not. In other

words: as regards concepts we have a requirement of sharp delimitation. (Frege

1984a, 148; italics mine; cf. also Frege 1984d, 245)

This fact is crucial for Frege and it can be illustrated by various passages. For example:

The sentence „ > 2‟ [where „‟ designates the sun] is false, because the sun is

not a number, and only numbers can be greater than 2. Accordingly the sentence

„( > 2) (2 > 2)‟ would be true, regardless of whether its right hand side were

true or false – and it ought to be one or the other. (Frege 1984d, 245)

Frege is not bound to say that „ > 2‟ is an ill-formed sentence; for the relation

designated by „>‟ is total and, thus, defined for any couple of arguments whatsoever.

Similarly, when he introduced the so-called horizontal stroke, i.e. the sign „─ x‟ (where

x is a variable), he presented it as a sign for a total function: its value is the True if its

argument is the True; otherwise – i.e. when its argument is either the False or something

that is not a truth-value at all – its value is the False (Frege 1984a, 149). As a result,

both „─ 1 + 3 = 5‟ and „─ 4‟ designate the False, because „1 + 3 = 5‟ designates the

False while „4‟ designates a non-truth-value. In a similar vein, Frege introduced another

total function designated by „┬ x‟ such that its value

is the False just for those argument for which the value of ─ x is the True, and,

conversely, is the True for the arguments for which the value of ─ x is the False.

(Frege 1984a, 150)

Since the value of ─ x is the True just for the argument the True, the value of ┬ x is the

False provided it takes as its argument the True; and since the value of ─ x is the False

if it takes as its argument either the False or a non-truth-value, so the value of ┬ x is the

True if it is applied either to the False or to a non-truth-value. In particular, both „┬ 1 +

3 = 5‟ and „┬ 4‟ designate the True.10

Finally, Frege defines the implicative function „if

y, then x‟ such that its value

10 I should stress that Frege‟s discussion concerning the function ┬ x is rather murky. For if we

take the definition quoted in the main text, it seems that ┬ x is a total function that is defined for

all objects whatsoever, be they truth-values or not: the function ┬ x is allowed there to take as

its arguments everything that can be an argument of the function ─ x. However, a few lines

below it is written: „I conceive of ┬ x as a function with the argument ─ x: (┬ x) = (┬(─ x))

where I imagine the two horizontal strokes to be fused together. But we also have: (─ (┬ x)) =

(┬ x), since the value of ┬ x is always a truth-value‟ (Frege 1984a, 150). This suggests that ┬ x

may take as its arguments exclusively truth-values because the value of ─ x is a truth-value for

whatever argument it takes. So, it seems that ┬ x is a partial function.

Now, the question is how to handle this apparent contradiction. I think we should take the

official definition of ┬ x as what is superior to all other formulations concerning ┬ x. Anyway,

we are obliged to explain away the seeming contrariwise formulations. The quotation given

above continues: „I thus regard the bits of the stroke in “┬ x” to the right and to the left of the

10

is to be the False if we take the True as the y-argument and at the same time take

some object that is not the True as the x-argument. (Frege 1984a, 154)

Thus, if the x-argument is the True or the y-argument is either the False or a non-truth-

value, the value of the implicative function is the True.

As far as I can see, there are at least two important reasons why Frege is willing to

permit only total functions.11

Firstly, if it would be possible that there be at least one

argument for which a concept does not give any value, the concept would not have

sharp boundaries and could not be well defined according to Frege‟s standards. No

uncertainty is permitted and for any object whatsoever it has to be clear whether it falls

under the concept in question or does not.12

Secondly, if the concept designated by „F(

)‟ would not give any value for the argument designated by „a‟, the compound

expression „F(a)‟ would designate nothing at all and could not appear in scientific

discourse. This would be problematic even for expressions such as „3 † 0‟ which occur

in mathematical discourse; for, the sentences such as „3 † 0 = 1‟ could not be taken to

say something false while the sentences such as „3 † 0 ≠ 1‟ could not be taken to say

something true.13

stroke of negation as horizontals‟ (Frege 1984a, 150). This implies that the two horizontal

strokes form a compound sign with the stroke of negation; this compound sign can be taken as a

unit that can be prefixed to a sign for an argument of the function. The sign of its argument need

not involve the horizontal stroke because the stroke is already a part of the compound sign.

Consequently, the function designated by the compound sign may take as its arguments any x; it

is not required that it be applied only to something designated as „─ x‟ (for any x). The identity

„(┬ x) = (┬(─ x))‟ can be taken to mean that when we apply the function ┬ x to a particular

argument, α, what we get is the same as when we apply it to the argument ─ α. It need not be

taken to mean that whenever the function is applied to something, it is, in fact, applied only to

truth-values (designated by „─ x‟).

In fact, there are at least two different kinds of evidence for the claim that Frege permitted

the function ┬ x to take as its arguments also non-truth-values. Firstly, anything that can be

substituted for x in „─ x‟ can be substituted also for x in „┬ x‟. The letter x is the same variable

in both „┬ x‟ and „─ x‟. Secondly, Frege suggests to read „┬ 2‟ as „2 is not the True‟ (Frege

1984a, 150); i.e., the function designated by „( ) is not the True‟ (which is just the function ┬ x)

takes here as its argument the number 2 (which is a non-truth-value). If the function ┬ x was

allowed to take just truth-values as its arguments, „2 is not the True‟ should be properly

reformulated as „(─ 2) is not the True‟ and taken as saying that the False is not the True

(because „─ 2‟ designates the False). So, Frege probably did not take ┬ x as a partial function. 11 Pavel Tichý presents a thoroughgoing analysis of Frege‟s horror vacui – as he describes his

repudiation of partial functions – in Tichý 1988, 31. 12 This fact is highlighted in numerous places in Frege‟s writings; cf. Frege 1979c, 15; Frege

1979d, 195–196; Frege 1979e, 229, 241, 243–244; Frege 1984a, 148. Frege adopted it also in

his pre-1890 works (cf. Frege 1950, 87). The paramount motive underlying his views on logic

was that proofs in logic and mathematics should be conducted in the most rigorous manner

possible (cf. Frege 1967, 5ff.). The call for sharp definitions of concepts was an essential part of

this enterprise. 13

Frege claims that „the proper name formed from this function-name and any proper name

whatever in the argument-place must always have a [denotation], provided only that this last

proper name [designates] something. For the proper name thus formed out of our function-name

and this proper name is a part of the proper name of the True‟ (Frege 1979c, 155). If an

expression designates nothing, it belongs to fictional discourse and cannot be used in scientific

discourse; cf. Frege 1979a, 118; Frege 1979b, 130; Frege 1979d, 194; Frege 1984b, 163.

11

Now we are ready to justify the idea that wild analyses of thoughts are not excluded

by Frege‟s overall theory. It is easy to see that Frege‟s claim quoted above that „any

component of the [“unsaturated”] kind together with any component of the [“saturated”]

kind will form a thought‟ is made possible because of his insistence that all functions he

is willing to recognize are total. „Unsaturated‟ senses are such because they are modes

of presentations of concepts (functions) and „saturated‟ senses are such because they are

modes of presentations of objects. And as we are allowed to apply any concept to any

object without any restrictions, so we have to be allowed to supplement any

„unsaturated‟ sense with any „saturated‟ sense without any restrictions. For this reason it

is possible to analyse the thought [1 is even and Plato sings] wildly, i.e. to split it up into

[1 is even and Plato] and [( ) sings]. For, the sense [1 is even and Plato] can be correctly

generated from the senses [1 is even], [( ) and ( )] and [Plato], respectively. The double

„unsaturated‟ sense [( ) and ( )] can be „saturated‟ with any couple of (complete) senses

because the function presented by [( ) and ( )] has to give some value for any couple of

arguments (objects); thus, it can be „saturated‟ with a thought (e.g., the sense [1 is

even]) as well as with something that is not a thought (e.g., the sense [Plato]).

5. Conclusions

Let us recapitulate. There are two fundamentally different kinds of senses, „saturated‟

ones and „unsaturated‟ ones. A sense is „saturated‟ because it is a mode of presentation

of an object that is itself „saturated‟; and a sense is „unsaturated‟ because it is a mode of

presentation of a concept (or, more generally, a function) that is itself „unsaturated‟. If

we „saturate‟ the latter kind of sense with a former kind of sense, what we get is a

thought that is compound (because the constituent senses are proper parts of the

thought) and structured (because the constituent senses form a unity). There is no other

restriction that is to be met in combining senses into thoughts. As a result, it is possible

to analyse certain kinds of thought wildly, i.e., to split them up into rather unexpected

constituent senses.

Now, what should be done to block wild analyses? First of all, notice that all

functions are total for Frege because he took it that to give some value for any object

whatsoever was the same as to have sharp boundaries. Thus, the totality of concepts

(functions) is essential to them to be well defined. If the two things were separated the

above problems need not arise. It means that what we need is to preserve the concepts

as having sharp boundaries while allowing them to be partial. This would be possible

provided Frege‟s theory of functions be supplemented with a version of the theory of

types. Given this supplementation, the concepts are allowed to give no value for certain

kinds of argument. If all objects were assorted into types, the concepts might be defined

only for certain types, i.e., for certain subsets of the set of all objects. The concepts

would have sharp boundaries in that sense that for every item in a given subset it gives

some value or other; on the other hand, they would remain undefined for the members

of the other subsets. This would not mean that the concepts are not well defined but that

it is inadmissible to apply them to the members of the other subsets.

Given this suggestion, it is easy to see that the above problems would not arise.

Consider again the thought [1 is even and Plato sings]. If the function presented by [( )

and ( )] was allowed to take only truth-values as its arguments, it would be forbidden to

fill the empty places in [( ) and ( )] with senses that are not modes of presentation of

truth-values. The double „unsaturated‟ sense [( ) and ( )] could be „saturated‟ only with

thoughts. Since [Plato] is not one, it could not be used as a completion of [( ) and ( )].

Consequently, there would be no way to build up [1 is even and Plato]. At the same

12

time, it would be forbidden to split up the thought [1 is even and Plato sings] into [( )

sings] and [1 is even and Plato]. The wild analysis is thus blocked.

Acknowledgements

I am indebted to Pavel Cmorej and three anonymous referees of History and Philosophy

of Logic for their valuable suggestions.

References

Carl, W. 1994. Frege’s Theory of Sense and Reference: Its Origins and Scope,

Cambridge: Cambridge University Press.

Frege, G. 1950. The Foundations of Arithmetic, Oxford: Basil Blackwell.

Frege, G. 1967. Begriffsschrift, a formula language, modeled upon that of arithmetic,

for pure thought, in J. van Heijenoort, ed., From Frege to Gödel, Cambridge (Mass.):

Harvard University Press, 5–82.

Frege, G. 1979a. Comments on sense and meaning, in G. Frege, Posthumous Writings,

Oxford: Basil Blackwell, 118–25.

Frege, G. 1979b. Logic, in G. Frege, Posthumous Writings, Oxford: Basil Blackwell,

126–51.

Frege, G. 1979c. The argument for my stricter canons of definitions, in G. Frege,

Posthumous Writings, Oxford: Basil Blackwell, 152–56.

Frege, G. 1979d. Introduction to logic, in G. Frege, Posthumous Writings, Oxford: Basil

Blackwell, 185–96.

Frege, G. 1979e. Logic in mathematics, in G. Frege, Posthumous Writings, Oxford:

Basil Blackwell, 203–50.

Frege, G. 1979f. Notes for Ludwig Darmstaedter, in G. Frege, Posthumous Writings,

Oxford: Basil Blackwell, 253–57.

Frege, G. 1984a. Function and concept, in G. Frege, Collected Papers on Mathematics,

Logic, and Philosophy. Oxford: Basil Blackwell, 137–56.

Frege, G. 1984b. On sense and meaning, in G. Frege, Collected Papers on Mathematics,

Logic, and Philosophy. Oxford: Basil Blackwell, 157–77.

Frege, G. 1984c. On concept and object, in G. Frege, Collected Papers on Mathematics,

Logic, and Philosophy. Oxford: Basil Blackwell, 182–94.

Frege, G. 1984d. On Mr. Peano‟s conceptual notation and my own, in G. Frege,

Collected Papers on Mathematics, Logic, and Philosophy. Oxford: Basil Blackwell,

234–48.

Frege, G. 1984e. What is a function? in G. Frege, Collected Papers on Mathematics,

Logic, and Philosophy. Oxford: Basil Blackwell, 285–92.

Frege, G. 1984f. Thoughts, in G. Frege, Collected Papers on Mathematics, Logic, and

Philosophy. Oxford: Basil Blackwell, 351–72.

Frege, G. 1984g. Negation, in G. Frege, Collected Papers on Mathematics, Logic, and

Philosophy. Oxford: Basil Blackwell, 373–89.

Frege, G. 1984h. Compound thoughts, in G. Frege, Collected Papers on Mathematics,

Logic, and Philosophy. Oxford: Basil Blackwell, 390–406.

Tichý, P. 1988. Foundations of Frege’s Logic, Berlin: Walter de Gruyter.