Frege. “On Sense and Reference”, summary

by Corry Shores

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[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Gottlob Frege

“On Sense and Reference”

 

Brief Summary:

Both names (signs, terms, etc.) and whole sentences can have both reference and sense.

{a} the reference of a sign: the specific object(s) it designates

{b} the sense of a sign: its mode of presentation (the contextual elements of how it is to be grasped. The same object can be cognized using different cognitive contents)

{c} the reference of a sentence: its truth value

{d} the sense of a sentence: its thought (the cognition involved in its conceptualization, usually taking the structure of predication)

Expressions also have “ideas,” but these are the subjective associations one has when hearing a term or sentence, and they vary from person to person. So they do not factor into our analysis here of the objective components of sense and reference. The reference and sense of sentences are built compositionally from the sense and reference of the component parts (that is, of the component subclauses and terms). We might doubt that the reference of a sentence is its truth value. We can test this by trying to see if in all relevant cases, when we substitute a clause within a sentence with another clause of a different truth value, we change the overall value. If that is so, and/or if in these cases the value of the whole sentence stays the same when the substitute clause’s value is the same as the one it replaces, then we know that the reference of a sentence is its truth value. For, the component sentence structures that determine the whole sentence’s truth value have as their reference a truth value. So since the reference of the whole is composed from and determined by the reference of its parts, and since the reference of the parts are truth values which are determinative of the truth value of the whole, that means the reference of the whole is its truth value. Frege first disqualifies a number of cases of nested subordinate clauses that are not instances where the clause has an independent sense and reference and also where the clause does not have some determining effect on the truth value of the whole. Frege does find some cases that fulfill the criteria and that also confirm his thesis. One is when the common component of two clauses is designated by a proper name, another is conditional statements where the referents of the clauses are determinate, and a third is when the subsidiary clause begins with ‘although’. Thus we can say that the reference of any sentence, whether a whole sentence or a nested clause of the qualifying type, has as its reference its truth value.

Summary

Frege addresses the notion of equality and remarks that it presents difficult philosophical challenges. We wonder, is equality a relation, and if so, is it a relation between objects or between names or signs of objects? [Recall what Frege said about equality in his Begriffsschrift in section 8. A name has contents, which is what the name stands for. Equality is between names. When names are equalized, that does not mean we only need one of the names. Under a certain mode of determination, both names might have identical contents. But under a different mode of determination, they might not. For example, point A gradually comes nearer to point B until they finally coincide. When they do coincide, we can say they are equal, but when they do not coincide, they are not equal, as they have different contents (this example is explained in more depth below). In “Function and Concept”, Frege also has a notion of equality between saturated functions (that is, functions with specified arguments) that are either both true or false. Thus two statements can be said to be equal when they have the same truth value, even if they refer to different objects or different numerical values. And in “On Concept and Object,” Frege distinguished two notions of the “is”. One is the predicative is, in which something (either an object or a concept) falls under some concept, like how an argument or argument place is operated upon by the function. An example of this is “the morning star is a planet”. However, the other meaning of “is” is as an equative “is”. Here there is not something falling under a concept but rather two names are said to be equal, meaning that they share the same referent.] Frege says that in his Begriffsschrift he assumed that equality is a relation not between objects but between names or signs of objects. He then explains his reasoning for this. [The idea will be that if equality were said to hold between contents, then we would not need different names for things which have the same referents (that is, the same contents). However, there are cases where names with the same referents are not entirely reducible to each other for some reason. If we only said that there is equality in the contents, then we would only need one name. We also need to say that there is equality in the names, so that both can be established as different in some way while at the same time having contents that are no different.] He has us consider two equalities:

a = a

a = b

[In the first case, it is true by necessity. It could not be otherwise that a≠a. In fact, to think a=a is not to really think much at all. However, there is no necessity in a=b. It could have been a≠b. In fact, since they have different names, we would perhaps find their inequality to be more likely. When we think a=b, there is something more in our minds than when we think a=a. With a=a, we could have known that already without being told it. But to be told a=b is to learn something we may not have been able to deduce on our own. It has different cognitive content, you might say. a=a gives us an a priori analytic judgment, meaning that the right side of the equation does not add any information that was not already given in the left side, while the a=b gives us an a posteriori judgment in which the second side of the equation gives us information that is not necessarily deducible from the first side.]

a=a and a=b are obviously statements of differing cognitive value a=a holds 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.

(56)

[Now consider how we do not see the sun after it sets under the horizon at night. For all we know, it is a new sun each time it arises the next day. But we learn that it is the same sun. So evening sun = evening sun (that is, the sun we see one evening is the same sun we see that evening) does not give us new information in the second part of the equation, but evening sun = morning sun, in this particular context of making the discovery of identity between two previously non-identified items, does have a ‘different cognitive value’ in that it is informative. Even long after this equality is established, we might also say that there are connotational differences between morning and evening sun. We experience them differently, they mean very different things to us in our lives, so still to equate them is to match incompatible or inconsistent connotations (the situation to go to bed = the situation to wake up). Also recall from section 8 of Frege’s Begriffsschrift that when we equate two different signs, they not only stand for their content but they also stand for themselves.

Equality of content differs from conditionality and negation by relating to names, not to contents. Elsewhere, signs are mere proxies for their content, and thus any phrase they occur in just expresses a relation between their various contents; but names at once appear in propria persona so soon as they are joined p. 14] together by the symbol for equality of content; for this signifies the circumstance of two names’ having the same content. Thus, along with the introduction of a symbol for equality | of content, all symbols are necessarily given a double meaning – the same symbols stand now for their own content, now for themselves.

(Frege Begriffsschrift 10-11)

So when we say evening sun = morning sun, we are not just saying that both evening sun and morning sun share the same content. We are also saying that one name can be substituted for the other. So we are saying something about the names and not just about their contents. Frege’s point in our current text is that in these cases, we gain new knowledge. If evening sun = morning sun were simply a=a,  then we are only learning that something stands in a relation to itself. But we learn more than that. We learn about the relation between the names and not just between the content and itself, or between one name and itself. Frege’s example is not the same as this. It is rather that it is the same sun every morning. But I wanted to highlight the connotational and contextual differences of the sun as it is one night in contrast to how it is the next morning.]

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 not always a 26] 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 that the 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.

(56)

Frege continues this point. When we say a=b, we say that the signs themselves have a relation to one another, namely one of equality (whose important property is mutual substitutability). This relation is mediated through their shared content. (Perhaps we might think of this mediation as the content somehow being a middle term, as if a=c and b=c.) But, supposing that a equals b on account of them sharing the same referent, when we say a=b, we  are no longer referring to their referent but rather we are saying  that they should be taken as designating the same thing, such that we could use a or b and in either case we would be referring to one and the same object. Frege notes that this does not give us any “proper” knowledge [that is, we do not have a concept here (with something falling under a predicate) but rather something more like instructions or rules for the usage of the symbols]. However, in many cases the difference in signs is important, because the same object can have a different sense depending on its mode of presentation. [We mentioned above the circle example from Begriffsschrift section 8, but let us here examine it more closely. (The following between ellipses is taken from that summary) ...

At first sight this makes it appear as though it were here a matter of something pertaining only to expression, not to thought; as though we had no need of two symbols for the same content, and therefore no need of a symbol for equality of content either.
(Frege Begriffsschrift 11)

Frege then has us picture a geometrical situation.

There is a fixed point A, which is the endpoint of a line that spans the diameter of a circle, extending to (and past) a point B on the opposite side. We move the line from horizontal to vertical. While the line is pivoting, point B continues to mark the point of the line’s intersection with the circle’s circumference, opposite to A. As such, B revolves along the edge of the circle, moving closer and closer to A. Once the line attains a vertical position, B and A overlap.

In order to show the unreality of this appearance, I choose the following example from geometry. Let a fixed point A lie on the circumference of a circle, and let a straight line rotate around this. When this straight line forms a diameter, let us call the opposite end to A the point B corresponding to this position. Then let us go on to call the point of intersection of the straight line and the circumference, the point B corresponding to the position of the straight line at any given time; this point is given by the rule that to continuous changes in the position of the straight line there must always correspond continuous changes in the position of B.
(Frege Begriffsschrift 11)

Frege’s point with this seems to be that the name “point B” normally has its own content, which is different from that of “point A”. However, there is one instance where they have the same content, and thus the names become equalized under that circumstance. Even though they do become equalized, that does not mean we can just use one name for both of them. For, under many other circumstances their contents are not identical.

Thus the name B has an indeterminate meaning until the corresponding position of the straight line is given. We may now ask: What point corresponds to the position of the straight line in which it is perpendicular to the diameter? The answer will be: The point A. The name B thus has in this case the same content as the name A; and yet we could not antecedently use just one name, for only the answer to the question justified our doing so.
(Frege Begriffsschrift 11)

... As we see, the idea of “mode of presentation” or “mode of determination” is very important in this matter. When we have a = b in terms of them sharing the same referent, and when the mode of presentation is the same, then they have the same sense. But when the mode of presentation differs, they have different senses, even though they share the same referent. This notion of mode of presentation or mode of determination is still not entirely clear to me. With the example of the sun, it will always be presented differently in the morning than at night, as it is located in a different part of the sky, and the context within our daily rhythms is normally different in each case. That would seem to suggest that morning sun and evening sun no matter what will have different senses? I am not certain, but it seems from this and from the next geometrical example that “mode of presentation” means the way we determine the referent, and this way we determine the referent can vary depending on the way it relates to other referred givens. Let us look then at this next example. We consider a triangle with lines a, b, and c connecting vertices to the midpoints on the opposite side of the triangle.

The point of intersection of a and b,

is the same as the point of intersection of b and c.

So ‘point of intersection of a and b’  = ‘point of intersection of b and c’. They are identical in what they designate, but they differ in their mode of presentation. In the first case, the point is presented only with reference to lines a and b, and in the second only with reference to b and c. This is enough for them to have different senses. Maybe we can think of the difference then as “the point of intersection insofar as it is determined and conceptualized by means of lines a and b” and “the point of intersection insofar as it is determined and conceptualized by means of lines b and c”. In other words, there is a notable conceptual difference in the two different modes of presentation. Those differences of sense might also be understood in the following way. When we determine the point the first way, what enters our mind are lines a and b, but not line c. However, when we determine the point the second way, what enters our mind are lines b and c, but not line a. So with regard to conceptual content, there is a notable difference.]

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 be forbidden 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 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 signs corresponds 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 of intersection of b and c’) likewise indicate the mode of presentation; and hence the statement contains actual knowledge.

(56-57)

Frege then distinguishes sense from reference. The thing that the sign designates is the reference [the sun itself, the point of intersection], while the mode of presentation is the sense [the sun’s rising in the morning or its setting in the evening, or the point of intersection as determined and understood by a and b or by b and c]. Frege also here introduces his famous morning/evening star example.

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 presentation is contained. In our example, accordingly, the 27] 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.

(57)

Frege then clarifies that when he uses the terms ‘sign’ or ‘name’ he is speaking of proper name designations that refer to a definite object, and he does not mean by these terms concepts or relations. (57)

[Sense here is a little like connotation or contextual implication. We might know the reference of a proper name as being that person or thing itself, but the sense requires a little more knowledge of the language and other designations involved, since connotations and contextual implications are imbedded into these wider systems. Each a sense is only one aspect of the thing, and since the system of meanings in which sense is embedded is infinite in its complex interconnectedness, we never attain a full knowledge of all a thing’s senses.]

The sense of a proper name is grasped by everybody who is sufficiently familiar with the language or totality of designations | to which it belongs; but this serves to illuminate only a single aspect of the reference, supposing it to have one. Comprehensive knowledge of the reference would require us to be able to say immediately whether any given sense belongs to it. To such knowledge we never attain.

(57-58)

[To understand the following, we draw from a footnote. We have the sign or proper name ‘Aristotle’. It has a sense, ‘the pupil of Plato and teacher of Alexander the Great’. This sense has one object or referent, the man Aristotle himself. However, the man Aristotle can be referred to by means of different names, including perhaps ‘Aristotle’ and ‘The Philosopher’ and as well ‘Ἀριστοτέλης’ and ‘Aristotélēs’]

to the sign there corresponds a definite sense and to that in turn a definite reference, while to a given reference (an object) there does not belong only a single sign. The same sense has different expressions in different languages or even in the same language.

(58)

Another interesting thing about sense is that although every sign or expression can have a sense, not every sense will correspond to some specific referent. One example is ‘the celestial body most distant from earth’. This has sense, but [since we can never know that body or maybe since the universe is infinite] it does not have a specific referent. “In grasping a sense, one is not certainly assured of a reference” (58).

We can also refer to the signs themselves rather than to their referents, and we often use quotations for this purpose. So “Aristotle” is a word which refers to the man with that name. In that case, the referent  is the word within the quotations and not the object that would be designated without those quotations (58-59).

[Frege will now make some distinctions that are a bit tricky. So we consider reported speech. Suppose for example we say, “I did not agree with so and so when she said  ‘such and such’.” Here, we are disagreeing not with the words themselves nor with their referents but rather with the sense or thought that is being evoked. Now, when the other person said ‘such and such’ in the original context, it has a customary reference (what the words were referring to) and a customary sense (the mode by which those referents were designated and the overall “meaning” of the sentences themselves). But when we say, “I did not agree with so and so when she said  ‘such and such’,” we are not disagreeing with the words themselves or their referents but rather with their sense. Rather, we are using the quoted words indirectly. Hence the customary sense of the words is their indirect reference in indirect speech. I do not follow this section very well, so please read the quotation to follow.]

In order to speak of the sense of an expression ‘A’ one may simply use the phrase ‘the sense of the expression “A” ’. In reported speech one talks about the sense, e.g., of another person’s remarks. It is quite clear that in this way of speaking words do not have their customary reference but designate what is usually their sense. In order to have a short expression, we will say: In reported speech, words are used indirectly or have their indirect reference. We distinguish accordingly the customary from the indirect reference of a word; and its customary sense from its indirect sense. The indirect reference of a word is accordingly its customary sense. Such exceptions must always be borne in mind if the mode of connexion between sign, sense, and reference in particular cases is to be correctly under stood.

(59)

In addition to sense and reference, Frege introduces the notion of “idea,” which seems to be the associated cognitive and imaginative content bound up with the sign. The idea undergoes variance in its constitution and with regard to the feelings invested in it.

The reference and sense of a sign are to be distinguished from the associated idea. If the reference of a sign is an object perceivable by the senses, my idea of it is an internal image, arising from memories of sense impressions which I have had and acts, both internal and external, which I have performed. Such an idea is often saturated with feeling; the clarity of its separate parts varies and oscillates. The same sense is not always connected, even in the same man, with the same idea. The idea is subjective: one man’s idea is not that of another.

(59)

So we can speak of the sense of a sign (since much of it depends on custom), but for the idea of a sign, you must specify for which person, since it varies (59-60).

Frege then opposes the idea to the reference. The idea is the subjective reality of the thing while the reference is its objective reality. In between these two is the sense, which is not wholly subjective like the idea but is still not the object itself. To illustrate, Frege has us imagine a person observing the moon through a telescope. The moon itself is the reference. The real image of the moon is projected by the glass in the telescope and in that way it mediates the real thing the moon. Furthermore, that image is projected into the person’s retina. That retinal image is more like the idea, since it is the subjective experience of the moon, and the image on the telescope’s inner glass is like the sense [since it is not the moon itself but not yet the moon in its subjectivity reality. It is the moon as it would be experienced by any person looking into the telescope.]

The reference of a proper name is the object itself which we designate by its means; the idea, which we have in that case, is wholly subjective; in between lies the sense, which is indeed no longer subjective like the idea, but is yet not the object itself. The following analogy will perhaps clarify these relationships. Somebody observes the Moon through a telescope. I compare the Moon itself to the reference; it is the object of the observation, mediated by the real image projected by the object glass in the interior of the telescope, and by the retinal image of the observer. The former I compare to the sense, the latter is like the idea or experience. The optical image in the telescope is indeed one-sided and dependent upon the standpoint of observation; but it is still objective, inasmuch as it can be used by several observers. At any rate it could be arranged for several to use it simultaneously. But each one would have his own retinal image. On account of the diverse shapes of the observers’ eyes, even a geometrical congruence could hardly be achieved, and an actual coincidence would be out of the question. This analogy might be developed still further, by assuming A’s retinal image made visible to B; or A might also see his own retinal image in a mirror. In this way we might perhaps show how an idea can itself be taken as an object, but as such is not for the observer what it directly is for the person having the idea. But to pursue this would take us too far afield.

(Frege 60)

Frege now notes three levels of difference that can be found between words, between expressions, or between whole sentences. Those three levels of difference can be matters of: {1} ideas, {2} of sense but not of reference, or {3} of sense and also of reference. [I am not sure yet what is meant by levels of difference here, but it seems to be ways that one thing (idea, sense, reference) is distinguishable from another.] In the case of ideas, we must acknowledge that certain differences can hold between particular ideas for one person that do not hold for others, given the subjective nature of how the ideas associate with words and with each other differently for each person. [I might be wrong about Frege’s next point, but it seems to be that when translating a text, the translator should not employ any expressions for her ideas when they are based on the idiosyncratic ways that the words/ideas associate in her own mind. The next idea might now be about poets and not translators, or maybe it is about translating poetry. I am not sure. But suppose it is about how poets write poetry for people speaking the same language. Frege’s point seems to be that the poet will want to convey senses that cannot or preferably should not be directly stated but rather hinted at or by some other indirect means evoked in the hearer or reader. And for this to work, there way that ideas associate for each person still need to some extent to be common in order for that sense to be conveyed to all people. However, given this indirect means and the variability of how ideas associate for each person, we can never be sure if the poet succeeded at communicating the intended sense to all people. ]

We can now recognize three levels of difference between words, | expressions, or whole sentences. The difference may concern at most the ideas, or the sense but not the reference, or, finally, the reference as well. With respect to the first level, it is to be 31] noted that, on account of the uncertain connexion of ideas with words, a difference may hold for one person, which another does not find. The difference between a translation and the original text should properly not overstep the first level. To the possible differences here belong also the colouring and shading which poetic eloquence seeks to give to the sense. Such colouring and shading are not objective, and must be evoked by each hearer or reader according to the hints of the poet or the speaker. Without some affinity in human ideas art would certainly be impossible; but it can never be exactly determined how far the intentions of the poet are realized.

(60-61)

Frege now announces that he will no longer discuss ideas and experiences. His purpose for discussing them up to this point was merely “to ensure that the idea aroused in the hearer by a word shall not be confused with its sense or its reference” (61). [So let us be clear about the difference between the idea and the sense and reference for words. The reference is the object that is being referred to. The sense is the overall manner of presenting a reference, which carries with it certain connotations, implications, or shades of meaning, depending on the conceptual contents employed when presenting that reference. The idea is a complex of subjective, arbitrary, associative images, feelings, and cognitions that are conjured up in an individual upon recognizing a sign or grasping a sense. Frege has shown ideas to play an important role in communication. But he also saw them to be subjective and arbitrary, and as well, their role in effective communication depends on there actually being a lack of subjective variance in the ideas that are conjured in each person. So instead we will focus on sense and reference, which are not (as) variable and which are given objectively in expressions.]

Frege now makes a terminological distinction between expression and standing for or designating, with regard to proper names, words, signs, sign combinations, and expressions. The sign is said to express its sense but stand for or designate its reference.

A proper name (word, sign, sign combination, expression) expresses its sense, stands for or designates its reference. By means of a sign we express its sense and designate its reference.

(61)

[Frege then addresses an objection that an idealist or skeptic might make. I think that objection is that Frege is speaking of an objective reference, but we cannot be sure that there is such a thing, or perhaps we even believe that indeed there is no such thing, and instead we can only speak of the ideas we have. Frege’s reply seems to be that our sentences will not have their sense if we do not assume them to have references. He gives the example of the sentence “the Moon is smaller than the Earth”. If in this sentence, “the Moon” only concerned the idea of the Moon, then we are missing the sense of the sentence. (I am not exactly sure why, but I guess he is saying that we are not claiming in this sentence that the idea of the Moon is somehow smaller than the idea of the Earth, but rather the Moon itself is smaller than the Earth itself. Furthermore) if the speaker wanted us to have in mind the idea of the Moon rather than the Moon itself, she would have literally stated, “my idea of the Moon.” I might have this wrong, but Frege’s next point seems to be that the speaker intending for their words to refer to some object is enough to justify there being such a reference, so long as that reference does exist. (We might here wonder about non-existing things like unicorns. Perhaps in these cases Frege would say that it has no reference but still has a sense and an idea. So when we speak of unicorns, we are discussing ideas and not things. I am just guessing.)]

Idealists or sceptics will perhaps long since have objected: ‘You talk, without further ado, of the Moon as an object; but how do you know that the name ‘the Moon’ has any reference? How do you know that anything whatsoever has a reference ?’ I reply that when we say ‘the Moon,’ we do not intend to speak of our idea of the Moon, nor are we satisfied with the sense alone, but we presuppose a reference. To assume that in the sentence ‘The Moon is smaller than the Earth’ the idea of the Moon is in question, would be flatly to misunderstand the sense. If this is what the speaker wanted, he would use the phrase ‘my idea of the Moon.’ Now we can of course be mistaken in the presupposition, and such mistakes have indeed occurred. But the question whether the presupposition is perhaps always mistaken 32] need not be answered here; in order to justify mention of the reference of a sign it is enough, at first, to point out our intention | in speaking or thinking. (We must then add the reservation: provided such reference exists.)

(61-62)

So we have explained what the sense and reference is of a name. We now wonder what it would be for a whole declarative sentence. Frege will use the term “thought”, and he defines it in a footnote. A thought is not like an idea. The idea is subjective content. But the thought is the objective content of thinking, and by objective he means that it is the “common property of several thinkers”. [Perhaps we might think of when we define something, and the definition will have conceptual content that every person will put into the conception of that idea.] Frege says that a sentence contains a thought, and he wonders of the thought should be understood as the sense or as the reference. He then has us think that a sentence has a reference, but he does not explain yet what it means for a sentence to have a reference. We just suppose that it has a reference of some kind. [Let us use the sentence, ‘the Morning Star is a planet with a shorter period of revolution than the Earth’. We are not specifying what its reference is yet.]  Now suppose we change one word in the sentence with another word that has the same reference. [In our example, ‘the Morning Evening Star is a planet with a shorter period of revolution than the Earth’.] Frege says that this will not change the reference of the sentence. [He does not state specifically why this is so in this case. The reasoning seems to be that if nothing changes in the syntax and reference of the words in a sentence, then nothing can be said to have changed with the sentence’s own reference. So it seems a sentence’s reference depends somehow on the references of the terms. For, were one of the terms changed such that the reference of that term now changes, then supposedly the reference of the whole sentence could change.] However, even if we change a word with a synonym, the thought of the sentence changes. Now, since the sentence’s reference can be the same while the thought changes, that means the thought cannot be understood as being the reference. It instead must be considered as the sense [I suppose because there is no third option. It is still a bit vague what the “thought” is however.] Frege now wonders what we can say about the sentence’s reference, if anything can be said about it, and also if it even has a reference. [We might think perhaps of a term that has sense but no reference, like “unicorn”.] Frege notes that just like with sentence parts, there are cases where a whole sentence has sense but no reference. An example of such sentences would be one that has terms which have sense but no reference. He says that “Odysseus was set ashore at Ithaca while sound asleep” has sense, but since it is doubtful that “Odysseus” has a reference, we might further doubt that the whole sentence has a reference; “for it is of the reference of the name that the predicate is affirmed or denied.” [The reasoning here seems to be that the reference of the sentence has something to do with whether or not the predicate is affirmed or denied. Since in “Odysseus was set ashore at Ithaca while sound asleep” we cannot assign a reference, that means we cannot affirm or deny that he was “set ashore at Ithica while sound asleep”, and thus we cannot say what its reference is.] Frege’s next point is the following. So we just said that because “Odysseus” has no reference, then we cannot affirm or deny that the predication holds. [This seems to be saying that we cannot assign a truth value to the sentence.] But suppose someone did think that the sentence were either true or false. The only way such a person would be able to make that determination is if they gave “Odysseus” a reference and were thereby able to say that the predication held or did not. For, “Whoever does not admit the name has reference can neither apply nor withhold the predicate” (62). Frege’s next point is the following. Let us suppose we are not concerned with the truth value of the sentence. We in that case need not be concerned with the reference’s of the sentence’s parts, and instead we can just be concerned with the thought and thus with just the sense. Furthermore, were we only concerned with the thought, we would not need to be concerned with the reference of any part of the sentence, because “only the sense, not the reference, of the part is relevant to the sense of the whole sentence” (63). [I am not entirely sure, but the idea here seems to be that if we only are concerned with the sense of the sentence, that we need only think about the mode of presentation of the terms. Odysseus for example is presented as a figure of uncertain historical reality but no doubt a rich fictional and mythological life. He is presented as a being who seems to have all the features of a normal person, like needing to sleep, wanting to go home when far away from it, and so on.] Frege’s next point is that since we are in fact concerned with the reference of a part of the sentence, that means we also expect there to be a reference for the whole sentence. For, “The thought loses value for us as soon as we recognize that the reference of one of its parts is missing. We are therefore justified in not being satisfied with the sense of a sentence, and in inquiring also as to its reference” (63). When we are concerned with the references of the sentence’s parts, it is because we are concerned with the truth value of the sentence. [Of course at this point we might object that in fiction writing we are not concerned with the reference of the terms.] However, sometimes we are not concerned with the truth value of the sentence, as when enjoying an epic poem. Instead of its truth value, we are interested in its aesthetic features and in the “sense of the sentences and the images and feelings thereby aroused” (63). In that case, we are not interested in whether or not “Odysseus” has a reference. Rather, it is only when we strive for truth that our concerns move from sense to reference. Thus, “the reference of a sentence may always be sought, whenever the reference of its components is involved; and [...] this is the case when and only when we are inquiring after the truth value” (63).

So far we have considered the sense and reference only of such expressions, words, or signs as we have called proper names. We now inquire concerning the sense and reference for an entire declarative sentence. Such a sentence contains a thought.* Is this thought, now, to be regarded as its sense or its reference? Let us assume for the time being that the sentence has reference. If we now replace one word of the sentence by another having the same reference, but a different sense, this can have no bearing upon the reference of the sentence. Yet we can see that in such a case the thought changes; since, e.g., the thought in the sentence ‘The morning star is a body illuminated by the Sun’ differs from that in the sentence ‘The evening star is a body illuminated by the Sun.’ Anybody who did not know that the evening star is the morning star might hold the one thought to be true, the other false. The thought, accordingly, cannot be the reference of the sentence, but must rather be considered as the sense. What is the position now with regard to the reference? Have we a right even to inquire about it? Is it possible that a sentence as a whole has only a sense, but no reference? At any rate, one might expect that such sentences occur, just as there are parts of sentences having sense but no reference. And sentences which contain proper names without reference will be of this kind. The sentence ‘Odysseus was set ashore at Ithaca while sound asleep’ obviously has a sense. But since it is doubtful whether the name ‘Odysseus,’ occurring therein, has reference, it is also doubtful whether the whole sentence has one. Yet it is certain, nevertheless, that anyone who seriously took the sentence to be true or false would ascribe to the name ‘Odysseus’ a reference, not merely 33] a sense; for it is of the reference of the name that the predicate is affirmed or denied. Whoever does not admit the name has reference can neither apply nor withhold the predicate. But in that case it would be superfluous to advance to the reference of the name; one could be satisfied with the sense, if one wanted to go no further than the thought. If it were a question only of the sense of the sentence, the thought, it would be | unnecessary to bother with the reference of a part of the sentence; only the sense, not the reference, of the part is relevant to the sense of the whole sentence. The thought remains the same whether ‘Odysseus’ has reference or not. The fact that we concern ourselves at all about the reference of a part of the sentence indicates that we generally recognize and expect a reference for the sentence itself. The thought loses value for us as soon as we recognize that the reference of one of its parts is missing. We are therefore justified in not being satisfied with the sense of a sentence, and in inquiring also as to its reference. But now why do we want every proper name to have not only a sense, but also a reference? Why is the thought not enough for us? Because, and to the extent that, we are concerned with its truth value. This is not always the case. In hearing an epic poem, for instance, apart from the euphony of the language we are interested only in the sense of the sentences and the images and feelings thereby aroused. The question of truth would cause us to abandon aesthetic delight for an attitude of scientific investigation. Hence it is a matter of no concern to us whether the name ‘Odysseus,’ for instance, has reference, so long as we accept the poem as a work of art. It is the striving for truth that drives us always to advance from the sense to the reference. We have seen that the reference of a sentence may always be sought, whenever the reference of its components is involved; and that this is the case when and only when we are inquiring after the truth value.

* By a thought I understand not the subjective performance of thinking but its objective content, which is capable of being the common property of several thinkers.

(62-63)

Frege then concludes that we have no choice but to consider the truth value of a sentence as being its reference. [His reasoning again seems to be that the sense of a sentence is built from the sense of its parts, and likewise, the reference of a sentence is built from the reference of its parts. So, since when we are concerned with the reference of the terms we are concerned with determining the truth value of the whole sentence, we therefore understand the reference of the sentence to be its truth value. We might at this point think that this is not necessarily the only way to conceive of the reference of a sentence, even given what Frege is saying about our interest in the reference of its parts being an interest in the sentence’s truth value. Could we not think of the reference of the sentence as a state of affairs or an event in the world? Could not the reference of “the cat is on the mat” be the cat’s being on the mat? Suppose we are concerned with the reference of “cat” and “mat” and I suppose also of “being on”. That concern is not incompatible with us saying that the reference of the whole sentence is the cat’s being on the mat. Frege’s point seems to be the following. Even if we are thinking of the reference as the cat’s being on the mat, the fact that we want to know what the cat is and what the mat is and maybe even what this sort of “being on” is that we want to know if the described situation is a real state of affairs. And perhaps also there is here the idea that there is no such thing as a false state of affairs. So insofar as we are interested in the reference of the terms, we are interested in a  real state of affairs, and insofar as we are interested in a real state of affairs, we are interested in truth. Frege will then clarify that what he means by the truth value is the “circumstance that it is true or false”. Perhaps this is something like the truth conditions, but I am not sure. The circumstance that the sentence is true is called the True, and the circumstance that the sentence is false is called the False. These distinctions so far correspond to how models are made. Recall a model we modified in Agler’s Symbolic Logic section 6.4.2.

D = {Alfred, Bill, Corinne}
I (a) = {Alfred}
I (b) = {Bill}
I (c) = {Corinne}
I (Lxy) = {<Bill, Corinne>, <Corinne, Alfred>}
I (Lca) = {<Corinne, Alfred>}
I (Lcb) = {<Corinne, Bill>}
v(Lca) = T
v(Lcb) = F

Here we are only dealing with reference. As we can see, the reference of names like ‘a’ and ‘b’ are objects, Alfred and Bill. We can say that they are in the extension of the names. We also have unsaturated sentences, like Lxy. Here the extension is a set of ordered couples. But what is the reference or extension of a saturated sentence? It is the values true or false. Frege then says, “Every declarative sentence concerned with the reference of its words is therefore to be regarded as a proper name” but I do not understand what he means by a sentence being concerned with the reference of its words. I would need an example I think. And furthermore, I do not understand what it would mean for a sentence to be understood as a proper name, unless we are mentioning it in another sentence.]

34] We are therefore driven into accepting the truth value of a sentence as constituting its reference. By the truth value of a sentence I understand the circumstance that it is true or false. There are no further truth values. For brevity I call the one the True, the other the False. Every declarative sentence concerned with the reference of its words is therefore to be regarded as a proper name, and its reference, if it has one, is either the True or the False. These two objects are recognized, if only implicitly, by everybody who judges something to be true – and so even by a sceptic. The designation of the truth values as objects may | appear to be an arbitrary fancy or perhaps a mere play upon words, from which no profound consequences could be drawn. What! mean by an object can be more exactly discussed only in connexion with concept and relation. I will reserve this for another article. But so much should already be clear, that in every judgment, no matter how trivial, the step from the level of thoughts to the level of reference (the objective) has already been taken.

(Frege 63-64)

Frege’s next point is that we cannot think of the relation between the thought to the True as being like the relation between the subject and predicate rather than as the relation between sense and reference. [Frege’s reasoning seems to be the following. Were the relation to be one of subject and predicate, that would involve us predicating the truth or falsity to the thought. And were this so, that means a new concept would be born from that predication. For, a predication involves a concept, and so the predication of true or false to a thought would itself produce a concept. But in fact, by judging a sentence as true or false does not add any new conceptual content to what was already there. Therefore, the relation of the thought to its truth value is not like predication.]

One might be tempted to regard the relation of the thought to the True not as that of sense to reference, but rather as that of subject to predicate. One can, indeed, say: ‘The thought, that 5 is a prime number, is true.’ But closer examination shows that nothing more has been said than in the simple sentence ‘5 is a prime number.’ The truth claim arises in each case from the form of the declarative sentence, and when the latter lacks its usual force, e.g., in the mouth of an actor upon the stage, even the sentence ‘The thought that 5 is a prime number is true’ contains only a thought, and indeed the same thought as the simple ‘5 is a prime number.’ It follows that the relation of the thought to the True may not be compared with that of subject to predicate. 35] Subject and predicate (understood in the logical sense) are indeed elements of thought; they stand on the same level for knowledge. By combining subject and predicate, one reaches only a thought, never passes from sense to reference, never from a thought to its truth value. One moves at the same level but never advances from one level to the next. A truth value cannot be a part of a thought, any more than, say, the Sun can, for it is not a sense but an object.

(64)

[Frege then appeals to what seems to be something like Leibniz’s law (identity of indiscernibles or indiscernibility of identicals, see Nolt Logics section 6.3) in order to emphasize that the truth value of the sentence, that is, its reference, will not change if we substitute any of its terms with synonyms. But I am not certain.]

If our supposition that the reference of a sentence is its truth value is correct, the latter must remain unchanged when a part of the sentence is replaced by an expression having the same reference. And this is in fact the case. Leibniz gives the definition: ‘Eadem sunt, quae sibi mutuo substitui possunt, salva veritate.’ What else but the truth value could be found, that belongs quite generally to every sentence if the reference of its components is relevant, and remains unchanged by substitutions of the kind in question?

(Frege 64)

Now, what follows from this is that all true sentences have the same thing, the True, as their reference, and similarly for all false ones, they have the same reference, the False. So we cannot just be interested in a sentence’s reference [because our interest in sentences cannot possibly be limited just to their truth value]. But also, if we are just concerned with its sense, then this will yield no knowledge [I am not sure why, but I suppose because knowledge should only concern true things. Learning a lot of things about Odysseus the fictional character, then, would seem not to constitute knowledge.] Thus we need to be concerned both with the sense and reference of sentences (65). Frege’s next point is that when we make a judgment, we are advancing from a thought to a truth value [possibly because a judgment regards something as being the case, like how the judgment-stroke functions in Frege’s Begriffsschrift, see section 2.] [I do not understand what Frege means when he says judgments are distinctions of parts within truth values. What are the parts and what are the distinctions? I will quote since I cannot summarize this part adequately.]

Judgments can be regarded as advances from a thought to a truth value. Naturally this cannot be a definition. Judgment is something quite peculiar and incomparable. One might also say that judgments are distinctions of parts within truth values. Such distinction occurs by a return to the thought. To every sense belonging to a truth value there would correspond its own manner of analysis. However, I have here used the word ‘part’ in a special sense. I have in fact transferred the relation between the parts and the whole of the sentence to its reference, by calling the reference of a word part of the reference of the sentence, if the 36] word itself is a part of the sentence. This way of speaking can certainly be attacked, because the whole reference and one part of it do not suffice to determine the remainder, and because the word ‘part’ is already used in another sense of bodies. A special term would need to be invented.

(65)

Frege will now further test his claim that the truth value of sentences are their references. [We said that within a sentence, were you to change a term with a synonym, that is, with a another term sharing exactly the same reference, then the reference of the sentence will not change. Now, consider a sentence that contains another sentence.] If sentences’ references are their truth values, then we should be able to change a nested sentence with any other one whatsoever that shares the same truth value, and the whole sentence should still keep its own truth value. Exceptions to this would be when the sentence in question is quoted, because then the words do not keep their normal reference [recall from above the ideas of customary sense and so on for indirect speech.] For, “In direct quotation, a sentence designates another sentence, and in indirect quotation a thought” (65).

So since we are not talking about directly quoted or paraphrased sentences being nested into larger sentences, we instead need to look at cases where the nested sentence is structurally a direct part of the whole sentence, as with subordinate sentences or clauses (65d). We said that the reference of a quoted sentence is its sense. We ask now if the reference of the subordinate sentence is its truth value. Since grammarians classify subordinate clauses in accordance with parts of speech, like noun clauses, adjective clauses, and so on, we might then think that the reference of a subordinate clause is not a truth value like for full sentences but rather is more like the reference for nouns, adjectives, and so on, which in that case would be just a part of a thought. Frege will explore this matter by grouping clauses not by grammatical categories but rather by logical ones. And he will begin by looking for cases where the sense of a subordinate clause is not an independent thought (66).

[Frege’s first category seems like indirect speech. In this case, it seems that we use a clause to state the sense of some other sentence or to state some thought. The basic conclusion that we seem to draw from this is that these cases are disqualified from being possible examples, because they do not have a truth-value. But we need them to have a truth value if we want to substitute the clause for another with the same truth value.]

37] The case of an abstract noun clause, introduced by ‘that,’ includes the case of indirect quotation, in which we have seen the words to have their indirect reference coinciding with what is customarily their sense. In this case, then, the subordinate clause has for its reference a thought, not a truth value; as sense not a thought, but the sense of the words ‘the thought, that . . . ,’ which is only a part of the thought in the entire complex sentence. This happens after ‘say,’ ‘hear,’ ‘be of the opinion,’ ‘be convinced,’ ‘conclude,’ and similar words. There is a different, and indeed somewhat complicated, situation after words like ‘perceive,’ ‘know,’ ‘fancy,’ which are to be considered later.

(66)

[What Frege does next is he shows why in cases like these the reference of the clause is its thought and not its truth value. For, were the reference its truth value, then that truth value should potentially affect the truth value (the reference) of the whole sentence. But since it does not, that means its reference is its sense and not its truth value. He demonstrates this with an example that shows it does not matter whether or not the subordinate clause is deemed true or false. The sentence is the same in both cases. ‘Copernicus believed that the planetary orbits are circles’ and ‘Copernicus believed that the apparent motion of the sun is produced by the real motion of the Earth.’ (I think we need to regard ‘the planetary orbits are circles’ as being false, as they are actually elliptical, but I am not sure.) So here we can change the subclause to one with a different truth value, and that does not affect the sentence’s truth value, which is to be determined by whether or not Copernicus believed the sense given in the subordinate clause.]

That in the cases of the first kind the reference of the subordinate clause is in fact the thought can also be recognized by seeing that it is indifferent to the truth of the whole whether the subordinate clause is true or false. Let us compare, for instance, the two sentences ‘Copernicus believed that the planetary orbits are circles’ and ‘Copernicus believed that the apparent motion of the sun is produced by the real motion of the Earth.’ One subordinate clause can be substituted for the other without harm to the truth. The main clause and the subordinate clause together have as their sense only a single thought, and the truth of the whole includes neither the truth nor the untruth of the subordinate clause. | In such cases it is not permissible to replace one expression in the subordinate clause by another having the same customary reference, but only by one having the same indirect reference, i.e. the same customary sense. If somebody were to conclude: The reference of a sentence is not its truth value, for in that case it could always be replaced by another sentence of the same truth value; he would prove too much; one might just as well claim that the reference of ‘morning star’ is not Venus, since one may not always say ‘Venus’ in place of ‘morning star.’ One has the right to conclude only that the reference of a sentence is not always its truth value, and that ‘morning star’ does not always 38] stand for the planet Venus, viz. when the word has its indirect reference. An exception of such a kind occurs in the subordinate clause just considered which has a thought as its reference.

(66-67)

This situation [where the subordinate clause’s truth-value is irrelevant to the truth value of the whole sentence] also holds for cases of ‘It seems that...’ and also for expressions beginning with ‘to be pleased,’ ‘to regret,’ ‘to approve,’ ‘to blame,’ ‘to hope,’ ‘to fear.’ [I think the idea is the sentence would begin with ‘that’ and be preceded by one of these verb phrases, like “I hope that it is not raining right now.” That hope is such whether or not it is actually raining.] Frege gives this nice example:

If, toward the end of the battle of Waterloo, Wellington was glad that the Prussians were coming, the basis for his joy was a conviction. Had he been deceived, he would have been no less pleased so long as his illusion lasted; and before he became so convinced he could not-have been pleased that the Prussians were coming – even though in fact they might have been already approaching.

(67)

[In the next paragraph, it seems Frege is saying that there is a variation on this structure of ‘that’ clauses where you first state the grounds for some other belief stated in the sentence. He seems to be saying that the same things hold in these situations, namely, that the truth or falsity of the subordinate clause does not affect the truth value of the whole sentence; however, it would matter to its sense were we to change some part of the subordinate clause with a synonym having a different sense.]

Just as a conviction or a belief is the ground of a feeling, it can, as in inference, also be the ground of a conviction. In the sentence: ‘Columbus inferred from the roundness of the Earth that he could reach India by travelling towards the west,’ we have as the reference of the parts two thoughts, that the Earth is round, and that Columbus by travelling to the west could reach India. All that is relevant here is that Columbus was convinced of both, and that the one conviction was a ground for the other. Whether the Earth is really round and Columbus could really reach India by travelling west, as he thought, is immaterial to the truth of our sentence; but it is not immaterial whether we replace ‘the Earth’ by ‘the planet which is accompanied by a | moon whose diameter is greater than the fourth part of its own.’ Here also we have the indirect reference of the words.

(67-68)

 

Frege then says that adverbial final clauses beginning with ‘in order that’ function the same way. [I am not exactly sure I know what these clauses are or why they function the same way, but Frege explains that:] “for obviously the purpose is a thought; therefore: indirect reference for the words, subjunctive mood” (68).

 

[Frege’s next point seems to be that clauses which are directives do not have truth values (references) but they do have sense, which might be what they are requesting or demanding.]

A subordinate clause with ‘that’ after ‘command,’ ‘ask,’ ‘forbid,’ would appear in direct speech as an imperative. Such a clause has no reference but only a sense. A command, a request, are indeed not thoughts, yet they stand on the same level as thoughts. Hence in subordinate clauses depending upon 39] ‘command,’ ‘ask,’ etc., words have their indirect reference. The reference of such a clause is therefore not a truth value but a command, a request, and so forth.

(68)

Frege than says the situation is similar for “the dependent question in phrases such as ‘doubt whether,’ and ‘not to know what.’ [I am not sure what such constructions are like.] He says that the words here have their indirect reference [that is, their sense]. [I also do not understand what he means by the next construction, which are dependent clauses expressing questions, beginning with who, what, where, how, by what means, and so on. I do not understand grammar well enough to know how a dependent clause can ask a question. I would think that the independent clause would need to. But with what is said later about dependent and independent clauses, I wonder if in this text they have the opposite meanings than what I am used to them having.]

The case is similar for the dependent question in phrases such as ‘doubt whether,’ ‘not to know what.’ It is easy to see that here also the words are to be taken to have their indirect reference. Dependent clauses expressing questions and beginning with ‘who,’ ‘what,’ ‘where, ‘when,’ ‘how,’ ‘by what means,’ etc., seem at times to approximate very closely to adverbial clauses in which words have their customary references. These cases are distinguished linguistically [in German] by the mood of the verb. With the subjunctive, we have a dependent question and indirect reference of the words, so that a proper name cannot in general be replaced by another name of the same object.

(68)

Frege then sums up what we have found with the cases so far. We saw that because the words of the subordinate clauses had an indirect reference [a sense], that means the whole subordinate clause had an indirect reference rather than a truth value, and that indirect reference is a thought, command, request, question, etc. [But I do not follow the next point. Let me quote:] “The subordinate clause could be regarded as a noun, indeed one could say: as a proper name of that thought, that command, etc., which it represented in the context of the sentence structure” (68).

[I also am not certain of the next point, but he might be saying that we will look now at clauses that are similar to the ones before only now the clause does not have a thought as its sense, but it also does not have a truth value as its reference. This is difficult to follow. But his point will be that in certain clauses, the subject has no sense of its own, and thus the dependent clause is not a complete thought, which means it has no truth value. Thus it has neither a thought nor a truth value. His example is: “Whoever discovered the elliptic form of the planetary orbits died in misery.” Here “whoever discovered...orbits” has the subject “whoever,” but it has no sense.  The clause, however, has as its reference Kepler. Frege then addresses an objection. Frege is saying that the clause “Whoever discovered the elliptic form of the planetary orbits” does not contain a thought, because part of it does not contain a sense. The objection seems to be that even with that part being undetermined, there is still a thought in the clause, namely, that there was someone who first discovered the elliptic form of the planetary orbits. I am not sure, but Frege might be saying that the objector’s reasoning here would be that since the whole sentence has a truth value, we cannot say that some part of it does not. I again am not following, but it seems also Frege is saying that the objector is claiming that any sentence that has sense also is a sentence whose parts have definite reference. And maybe his point is that we are misled into thinking this, because for “Whoever discovered the elliptic form of the planetary orbits” depends on the sentence “There was someone who discovered the elliptic form of the planetary orbits” being true, and then for some reason we think that “Whoever discovered the elliptic form of the planetary orbits” implies that it has some reference. I will quote this section, as I cannot explain it. What we might be able to take away from this section is that formal languages, like the ones invented in logic, do not have such problems of ambiguity, but we have to recognize and accommodate for them when they appear in natural language (and even in mathematics). But as much as possible, in the sciences at least, we should avoid terms that do not designate an object.]

We now come to other subordinate clauses, in which the words do have their customary reference without however a thought occurring as sense and a truth value as reference. How this is possible is best made clear by examples.

Whoever discovered the elliptic form of the planetary orbits died in misery.

If the sense of the subordinate clause were here a thought, it would have to be possible to express it also in a separate sentence. | But this does not work, because the grammatical subject ‘whoever’ has no independent sense and only mediates the relation with the consequent clause ‘died in misery.’ For this reason the sense of the subordinate clause is not a complete thought, and its reference is Kepler, not a truth value. One might object that the sense of the whole does contain a thought as part, viz. that there was somebody who first discovered the elliptic form of the planetary orbits; for whoever takes the whole to be true cannot 40] deny this part. This is undoubtedly so; but only because otherwise the dependent clause ‘whoever discovered the elliptic form of the planetary orbits’ would have no reference. If anything is asserted there is always an obvious presupposition that the simple or compound proper names used have reference. If one therefore asserts ‘Kepler died in misery,’ there is a presupposition that the name ‘Kepler’ designates something; but it does not follow that the sense of the sentence ‘Kepler died in misery’ contains the thought that the name ‘Kepler’ designates something. If this were the case the negation would have to run not

Kepler did not die in misery

but

Kepler did not die in misery, or the name ‘Kepler’ has no reference.

That the name ‘Kepler’ designates something is just as much a presupposition for the assertion Kepler died in misery as for the contrary assertion. Now languages have the fault of containing expressions which fail to designate an object (although their grammatical form seems to qualify them for that purpose) because the truth of some sentence is a prerequisite. Thus it depends on the truth of the sentence:

There was someone who discovered the elliptic form of the planetary orbits

whether the subordinate clause

Whoever discovered the elliptic form of the planetary orbits

really designates an object or only seems to do so while having in fact no reference. And thus it may appear as if our subordinate clause contained as a part of its sense the thought that there was | somebody who discovered the elliptic form of the planetary orbits. If this were right the negation would run:

Either whoever discovered the elliptic form of the planetary orbits did not die in misery or there was nobody who discovered the elliptic form of the planetary orbits.

 

41] This arises from an imperfection of language, from which even the symbolic language of mathematical analysis is not altogether free; even there combinations of symbols can occur that seem to stand for something but have (at least so far) no reference, e.g. divergent infinite series. This can be avoided, e.g., by means of the special stipulation that divergent infinite series shall stand for the number o. A logically perfect language (Begriffsschrift) should satisfy the conditions, that every expression grammatically well constructed as a proper name out of signs already introduced shall in fact designate an object, and that no new sign shall be introduced as a proper name without being secured a reference. The logic books contain warnings against logical mistakes arising from the ambiguity of expressions. I regard as no less pertinent a warning against apparent proper names having no reference. The history of mathematics supplies errors which have arisen in this way. This lends itself to demagogic abuse as easily as ambiguity-perhaps more easily. ‘The will of the people’ can serve as an example; for it is easy to establish that there is at any rate no generally accepted reference for this expression. It is therefore by no means unimportant to eliminate the source of these mistakes, at least in science, once and for all. Then such objections as the one discussed above would become impossible, because it could never depend upon the truth of a thought whether a proper name had a reference.

(68-70)

Logically similar to these noun clauses are adverbial and adjective clauses (70).

Adjective clauses help to construct compound proper names, but by themselves they cannot accomplish this (70).

Adjective clauses do not have a thought as sense or a truth value as reference. At best, their sense is only partial, being just part of a thought (71).

Since places, instants, and durations are, logically speaking, objects, that means expressions designating them are proper nouns. We can use adverbial clauses to construct such designations (71).

Frege turns now to conditional clauses. [I do not follow the terminology so well. I would think that the conditional clause is the “if...” clause of an “if...then...” sentence, and also that it is the dependent clause. However, from the wording in the paragraph, I am not so sure. Just to be clear, as far as I knew, the independent clause is the one that cannot stand on its own. But that does not seem to be how the term has been used in this text. His main idea seems to be that the conditional terms “if” and “then” refer one clause to the other, and only together do they form a connected whole and single thought. I will quote this part as it is a bit difficult.]

In conditional clauses, also, there may usually be recognized to | 43] occur an indefinite indicator, having a similar correlate in the dependent clause. (We have already seen this occur in noun, adjective, and adverbial clauses.) In so far as each indicator refers to the other, both clauses together form a connected whole, which as a rule expresses only a single thought. In the sentence

If a number is less than 1 and greater than o, its square is less than 1 and greater than o

the component in question is ‘a number’ in the conditional clause and ‘its’ in the dependent clause. It is by means of this very indefiniteness that the sense acquires the generality expected of a law. It is this which is responsible for the fact that the antecedent clause alone has no complete thought as its sense and in combination with the consequent clause expresses one and only one thought, whose parts are no longer thoughts. It is, in general, incorrect to say that in the hypothetical judgment two judgments are put in reciprocal relationship. If this or something similar is said, the word ‘judgment’ is used in the same sense as I have connected with the word ‘thought,’ so that I would use the formulation: ‘A hypothetical thought establishes a reciprocal relationship between two thoughts.’ This could be true only if an indefinite indicator is absent; but in such a case there would also be no generality.

(72)

Frege then says that when both parts of a conditional sentence refers to some instant of time, often the present tense is used. For example, ‘When the Sun is in the tropic of Cancer, the longest day in the northern hemisphere occurs.’ [His next point seems to reinforce the idea that the clauses in conditional sentences do not have sense on their own but only in their combination. For, in this case of present tense, were we to extract one clause, it would imply the real present tense. Let me quote.]

If an instant of time is to be indefinitely indicated in both conditional and dependent clauses, this is often achieved merely by using the present tense of the verb, which in such a case however does not indicate the temporal present. This grammatical form is then the indefinite indicator in the main and 44] subordinate clauses. An example of this is: ‘When the Sun is in the tropic of Cancer, the longest day in the northern hemisphere occurs.’ Here, also, it is impossible to express the sense of the subordinate clause in a full sentence, because this sense is not a complete thought. If we say: ‘The Sun is in the tropic of Cancer,’ this would refer to our present time and thereby change the sense. Just as little is the sense of the main clause a thought; only the whole, composed of main and subordinate clauses, has such a sense. It may be added that several common components in the antecedent and consequent clauses may be indefinitely indicated.

(72)

 

He then says that noun clauses beginning with ‘who’ or ‘what’ and adverbial clauses beginning with ‘where,’ ‘when,’ ‘wherever,’ ‘whenever’ often have the sense of conditional clauses, as we see for example in ‘who touches pitch, defiles himself’ (73).

 

He then notes that adjective clauses can also take the place of conditional clauses. [Previously we had the sentence, ‘If a number is less than 1 and greater than o, its square is less than 1 and greater than o’.] For example we can change the conditional sentence from before to ‘The square of a number which is less than 1 and greater than o is less than 1and greater than o’ (73).

 

Frege then notes that the situation is special when the common component of both clauses is some proper name. [I think his point is that in other cases, we did not regard each clause as having its own sense and reference. But in this case both clauses are like self-standing assertions, and the whole sentence is like a conjunction, where if one part is false, the whole thing is false. I am not exactly sure why that is so. But this is potentially one instance that meets the criteria for a subclause that has an independent sense and reference and that plays determining role in the overall sentence’s truth value. Let me quote.]

The situation is quite different if the common component of the two clauses is designated by a proper name. In the sentence:

Napoleon, who recognized the danger to his right flank, himself led his guards against the enemy position two thoughts are expressed:

1 . Napoleon recognized the danger to his right flank

2. Napoleon himself led his guards against the enemy position.

When and where this happened is to be fixed only by the context, but is nevertheless to be taken as definitely determined thereby. If the entire sentence is uttered as an assertion, we thereby simultaneously assert both component sentences. If one of the parts is false, the whole is false. Here we have the case that the subordinate clause by itself has a complete thought as sense (if we complete it by indication of place and time). The reference of the subordinate clause is accordingly a truth value. We can therefore expect that it may be replaced, without harm to the truth value of the whole, by a sentence having the same truth 45] value. This is indeed the case; but it is to be noticed that for purely grammatical reasons, its subject must be ‘Napoleon,’ for only then can it be brought into the form of an adjective clause belonging to ‘Napoleon.’ But if the demand that it be expressed in this form be waived, and the connexion be shown by ‘and,’ this restriction disappears.

(73)

 

Frege then says that subsidiary clauses that begin with ‘although’ also in themselves express complete thoughts, and they can be replaced with others of the same truth value. However,  were we to do so, “the light in which the clause is placed by the | conjunction might then easily appear unsuitable, as if a song with  a sad subject were to be sung in a lively fashion. (73-74).

 

If a conditional clause has a proper name instead of an indefinite indicator, then the truth value of the whole depends on that of the component parts [with that truth value being determined by the truth functionality of the material conditional. Perhaps I misunderstand, so let me quote. The idea here might instead be that these cases do not interest us, because although in certain instances of conditionals we can change one part for another with the same truth value, this is only because the truth value of that part will not contribute to the truth value of the whole anyway, as in the case where we deny the antecedent or affirm the consequent. I am not sure, so consult the quotation below.]

In the last cases the truth of the whole included the truth of the component clauses. The case is different if a conditional clause expresses a complete thought by containing, in place of an indefinite indicator, a proper name or something which is to be regarded as equivalent. In the sentence

If the Sun has already risen, the sky is very cloudy

the time is the present, that is to say, definite. And the place is also to be thought of as definite. Here it can be said that a relation between the truth values of conditional and dependent clauses has been asserted, viz. such that the case does not occur in which the antecedent stands for the True and the consequent for the False. Accordingly, our sentence is true if the Sun has not yet risen, whether the sky is very cloudy or not, and also if the Sun has risen and the sky is very cloudy. Since only truth values are here in question, each component clause can be replaced by another of the same truth value without changing the truth value of the whole. To be sure, the light in which the subject then appears would usually be unsuitable; the thought 46] might easily seem distorted; but this has nothing to do with its truth value. One must always take care not to clash with the subsidiary thoughts, which are however not explicitly expressed and therefore should not be reckoned in the sense. Hence, also, no account need be taken of their truth values.

(74)

 
Frege now reviews what we have learned so far (74).
 

[First recall the reason we have been examining subordinate clauses. Frege’s claim was that the truth value of sentences is their reference. Given his assumption of compositionality (in the sense of a sentence’s full sense or reference is built up somehow from that of its parts), were we to substitute some part of a sentence for some other part, with them both having the same reference, then the overall full sentence should not see a change in its truth value. Frege then began looking for a sentence with a subclause that fulfills two main qualifications: {a} the subclause has a sense and also a truth value, and {b} the subclause’s truth value contributes compositionally to the truth value of the whole sentence. Frege first eliminates a number of relevant cases that do not fulfill these qualifications, and found just a couple so far that do. He first excluded sentences given in indirect speech as by quotation or paraphrase. We excluded them on the basis that they do not qualify as sentences that may have truth values as their reference, and thus they would not be parts of a sentence whose truth value could affect the overall sentence’s truth value. Frege then further explained why this is so. Were the nested sentence directly quoted, then the reference is the sentence itself (indicated in quotation marks) and not its truth value. Or if it is indirectly mentioned (as by paraphrase for example), it designates the thought of that sentence. So, we have eliminated cases where the internal sentence is structurally distinct from that of the main sentence. That leaves us with clauses that are structurally a part of the larger sentence they are found in. Of these we have distinguished many types. The first are clauses beginning with ‘that’ and which state some thought, like “the thought, that ...”. This applies also to expressions like “hear that...”, “believe that,” “conclude that...”. Again, in these cases the reference of the clause is not a truth value and thus it will have no bearing on the truth value of the overall sentence. The situation is the same for clauses beginning with {a} “it seems that...”, “to be pleased that...”, “to hope that...” and so on, {b} “in order that”, {c} “command that...,” “ask that...”, “forbid that...”, and {d} “doubt whether,” and a question word followed with a question clause. So in these cases the subclauses have a sense (a thought) but not a reference (no truth value). Frege next examined subclauses that have neither a sense nor a truth value, for example, “Whoever discovered the elliptic form of the planetary orbits died in misery.” Here the subject has no reference, which means it neither completes a thought (and thus does not have a sense) nor does it allow for a truth value (as we do not know whether or not the predicate holds, as the subject of the predication is not specified). Frege then mentions adverbial and adjective clauses, which likewise have neither a complete thought nor a truth value. And then he examined conditional clauses. Frege does find some cases that fulfill the criteria. One is when the common component of two clauses is designated by a proper name, another is when in conditional clauses the referents are determinate, and a third is when the subsidiary clause begins with ‘although’. (Note: I am not entirely sure if I have all of this right, so please consult the text to be sure.) Now let us look at how Frege summarizes these parts. He says that a subordinate clause usually only expresses a partial thought, either because of indirect reference or because the subordinate clause depends on its relation to the main clause for it to have sense. But, he notes, we saw cases where the subclause expresses a complete thought and can be substituted by another with the same truth value without changing the truth value of the whole. Let me quote:]

The simple cases have now been discussed. Let us review what we have learned.

 

The subordinate clause usually has for its sense not a thought, but only a part of one, and consequently no truth value as reference. The reason for this is either that the words in the subordinate clause have indirect reference, so that the reference, not the sense, of the subordinate clause is a thought; or else that, on account of the presence of an indefinite indicator, the subordinate clause is incomplete and expresses a thought only when combined with the main clause. It may happen, however, that the sense of the subsidiary clause is a complete thought, in | which case it can be replaced by another of the same truth value without harm to the truth of the whole-provided there are no grammatical obstacles.

(74-75)

Frege then says that there can be many cases which do not fit neatly into his categories above. The reason for this can be that the sentence will have parts, but as well it will have another sense that is not reducible to the parts and is not stated directly in the whole either. He gives the example of

Napoleon, who recognized the danger to his right flank, himself led his guards against the enemy position.
(75)

He says that it on the one hand expresses the two thoughts shown. [I think those thought are: {1} Napoleon recognized the danger to his right flank, and {2} Napoleon led his guards against the enemy position.] But it also can have a third thought, namely, {3} “that the knowledge of the danger was the reason why he led the guards against the enemy position” (75). Frege will now explain how we can test to see if this third subsidiary thought should be included as part of the whole sentence’s sense. [So we are saying that the reason Napoleon led his guards against the enemy position was because he recognized the danger of his right flank. Now, if Napoleon already made his decision prior to learning about the danger, then the danger would not be the cause of his decision.] To make this test, we ask, would the sentence be false if Napoleon made his decision before recognizing the danger? [If the sentence’s value does not change by negating this subsidiary thought, then the sense of this subsidiary sentence  has no influence on the overall sense of sentence. I am not sure why, but the idea might be that the sense depends on the reference.] “If our sentence could be true in spite of this, the subsidiary thought should not be understood as part of the sense” (75). Frege says that probably it indeed would have no effect on the sentence’s truth value. [I am not sure I follow his reasoning, but it seems to be the following. He says suppose instead that the falsity of the subsidiary clause changes the truth value of the whole. This will create problems when we try substituting the parts with others of equal truth value, because that substitution will thereby alter the subsidiary clause’s sense. For instance, suppose we make the following substitution in our example sentence: “Napoleon, who was already more than 45 years old, himself led his guards against the enemy position.” Frege says that this somehow changes the first thought. I am not sure, but is he saying it changes the thought “Napoleon himself led his guards against the enemy position”? In that case I do not know how that changes. Or does he just mean that it changes in the obvious way of substitution? At any rate, he says it also changes the subsidiary meaning. I am not exactly sure how, but perhaps the idea is that it implies something like, “It is because of his experience as a general (as he was already more than 45 years old) that he led his guards against the enemy position.” I am not sure. Frege’s main point will be that in such cases of there being an important third sense, we cannot exchange internal clauses of the same truth value. This is because it also can influence the truth value of the third sentence as a consequence of changing its sense.]

An examination of all the subordinate clauses which one may encounter will soon provide some which do not fit well into these categories. The reason, so far as I can see, is that these subordinate clauses have no such simple sense. Almost always, it seems, we connect with the main thoughts expressed by us subsidiary thoughts which, although not expressed, are associated with our words, in accordance with psychological laws, by the hearer. And since the subsidiary thought appears to be connected with our words of its own accord, almost like the main thought itself, we want it also to be expressed. The sense of the sentence is thereby enriched, and it may well happen that we have more simple thoughts than clauses. In many cases the sentence must be understood in this way, in others it may be doubtful whether the subsidiary thought belongs to the sense of the sentence or only 47] accompanies it. One might perhaps find that the sentence

Napoleon, who recognized the danger to his right flank, himself led his guards against the enemy position

expresses not only the two thoughts shown above, but also the thought that the knowledge of the danger was the reason why he led the guards against the enemy position. One may in fact doubt whether this thought is merely slightly suggested or really expressed. Let the question be considered whether our sentence be false if Napoleon’s decision had already been made before he recognized the danger. If our sentence could be true in spite of this, the subsidiary thought should not be understood as part of the sense. One would probably decide in favour of this. The alternative would make for a quite complicated situation: We would have more simple thoughts than clauses. If the sentence

Napoleon recognized the danger to his right flank

were now to be replaced by another having the same truth value, e.g.

Napoleon was already more than 45 years old

not only would our first thought be changed, but also our third | one. Hence the truth value of the latter might change – viz. if his age was not the reason for the decision to lead the guards against the enemy. This shows why clauses of equal truth value cannot always be substituted for one another in such cases. The clause expresses more through its connexion with another than it does in isolation.

(75-76)

 

Frege will now look at cases where a clause “expresses more through its connexion with another than it does in isolation.” He gives this example:

Bebel fancies that the return of Alsace-Lorraine would appease France’s desire for revenge.

(76)

Frege says it expresses two thoughts.

(1) Bebel believes that the return of Alsace-Lorraine would appease France’s desire for revenge

(2) the return of Alsace-Lorraine would not appease France’s desire for revenge.

(Frege 76)

Frege points out that the subordinate clause [(believes) that the return of Alsace-Lorraine would appease France’s desire for revenge] has indirect reference. [This means that it does not have a truth value as its reference but rather its sense is its reference. What Bebel is believing is the sense of that clause.] However, the subordinate clause in the second case has its customary reference [which is the True.] “This shows that the subordinate clause in our original complex sentence is to be taken twice over, with different reference, standing once for a thought, once for a truth value” (76). [Recall that we are looking at cases where a clause “expresses more through its connexion with another than it does in isolation.”] So we see that the truth value is not the whole reference of the subordinate clause [because it also has as its indirect reference, that is, its sense. This means that were we to substitute it for another sentence, we are losing that sense.] Since the truth value is not the whole reference of the subordinate clause, we cannot simply replace the latter by another of equal truth value. Similar considerations apply to expressions such as ‘know,’ ‘discover,’ ‘it is known that’” (76).

 

Frege then gives other examples where “By means of a subordinate causal clause and the associated main clause we express several thoughts, which however do not correspond separately to the original clauses” (76). [The example he gives seems like an enthymeme.] Consider the sentence, “Because ice is less dense than water, it floats on water”. This has these three thoughts:

(1) Ice is less dense than water;

(2) If anything is less dense than water, it floats on water;

(3) Ice floats on water.

(76)

Frege notes that the third thought is contained in the first two. So it does not need to be explicitly stated. But suppose instead we just have senses 1 and 3 or just 2 and 3? That will not give us the sense of the whole sentence. Thus, “because ice is less dense than water” is somehow a part of the first and second thoughts. So if we substituted the subsidiary clause with another of equal truth value, it “would alter our second thought and thereby might well alter its truth value.” [I am not sure what a good example would be. Is Frege saying that if we substitute for “because it is less dense than water”, that will change the meaning of “If anything is less dense than water, it floats on water”? For example, if we said, “Because is colder than water, it floats on water,” would that then mean we have as sense 2, “If anything is colder than water, it floats on water”, which is false, even though ice is colder than water is true?]

Frege then says we have a similar situation with:

If iron were less dense than water, it would float on water.

(77)

For, it has the same structural feature where the subsidiary clause “expresses one thought and a part of the other” (77).

This situation can also be found with the sentence:

After Schleswig-Holstein was separated from Denmark, Prussia and Austria quarrelled.

(77)

[Frege says that the subordinate clause (“After Schleswig-Holstein was separated from Denmark”) somehow expresses both the thoughts that Schleswig-Holstein was once separated from Denmark and Prussia and Austria quarrelled. But I am not sure how. Is it because it was a causal relation? Let me quote so you can see for yourself.]

If we interpret the sentence already considered

After Schleswig-Holstein was separated from Denmark, Prussia and Austria quarrelled

in such a way that it expresses the thought that Schleswig-Holstein was once separated from Denmark, we have first this thought, and secondly the thought that at a time, more closely determined by the subordinate clause, Prussia and Austria quarrelled. Here also the subordinate clause expresses not only one thought but also a part of another. Therefore it may not in general be replaced by another of the same truth value.

(77)

Although Frege cannot examine every possible situation in a natural language, he at least has given us some basic reasons “why a subordinate clause may not always be replaced by another of equal truth value without harm to the truth of the whole sentence structure” (77). He then lists the reasons for this:

(1) when the subordinate clause does not stand for a truth value, inasmuch as it expresses only a part of a thought;

(2) when the subordinate clause does stand for a truth value but is not restricted to so doing, inasmuch as its sense includes one thought and part of another.

(Frege 77)

Frege then says that the first case [where the subordinate clause does not stand for a truth value, inasmuch as it expresses only a part of a thought] happens in the following two situations:

(a) in indirect reference of words

(b) if a part of the sentence is only an indefinite indicator instead of a proper name.

(Frege 77)

In the second case [where the subordinate clause does stand for a truth value but is not restricted to so doing, inasmuch as its sense includes one thought and part of another], “the subsidiary clause may have to be taken twice over, viz. once in its customary reference, and the other time in indirect reference; or the sense of a part of the subordinate clause may likewise be a component of another thought, which, taken together with the thought directly expressed by the subordinate clause, makes up the sense of the whole sentence” (78).

[So again, Frege’s main point in all this is that the reference of a sentence is its truth value. One test for this is to see if by changing a component of a sentence for another of the same truth value will change the overall sentence’s truth value. If it does change the value in all qualifying cases, then his theory is correct that the sentences’ reference is its truth value. He showed some valid cases where the substitution did have a compositional affect on the overall sentence value but still did not change it. He also found many cases of subordinate clauses which should not factor into our assessment, because for example they do not have a proper truth value all on their own. In certain such cases, they do have a value, but their sense is so tied up with other parts of the sentence that the substitution will have unpredictable effects on those other parts and thus on the whole. But these cases do not count, because their value is not independent of other parts of the sentence.]

It follows with sufficient probability from the foregoing that the cases where a subordinate clause is not replaceable by another of the same value cannot be brought in disproof of our view 50] that a truth value is the reference of a sentence having a thought as its sense.

(78)

[Now having established that the reference of a sentence is its truth-value,] Frege will return to the points we began with. [It seems for his first point of a=a and a=b regards a and b as sentences, but I am not sure. His point might be that, yes, since two sentences have the same reference, that is, the same truth value, one can equal the other, but they probably each express a different thought and thus has a distinct cognitive value, and this is just as important as its truth value. And furthermore, a judgment made with regard to a sentence is also different from the sentence’s sense and reference. But I might be wrong about that last point.]

Let us return to our starting point. When we found ‘a=a’ and ‘a=b’ to have different cognitive values, the explanation is that for the purpose of knowledge, the sense of the sentence, viz., the thought expressed by it, is no less relevant than its reference, i.e. its truth value. If now a=b, then indeed the reference of ‘b’ is the same as that of ‘a,’ and hence the truth value of ‘a=b’ is the same as that of ‘a= a.’ In spite of this, the sense of ‘b’ may differ from that of ‘ a’, and thereby the thought expressed in ‘a=b’ differs from that of a= a.’ In that case the two sentences do not have the same cognitive value. If we understand by ‘judgment’ the advance from the thought to its truth value, as in the above paper, we can also say that the judgments are different.

(78)

Frege, Gottlob. “On Sense and Reference”. Transl. P.T. Geach. In Translations from the Philosophical Writings of Gottlob Frege. Eds. P.T. Geach and Max Black. Oxford: Basil Blackwell, 1960, second edition (1952 first edition), pp.56-78.

Or if otherwise noted:
Frege, Gottlob. “Begriffsschrift (Chapter 1)”. Transl. P.T. Geach. In Translations from the Philosophical Writings of Gottlob Frege. Eds. P.T. Geach and Max Black. Oxford: Basil Blackwell, 1960, second edition (1952 first edition).