Logic is
concerned
with the predicate 'true' in a special way.
Gottlob-Frege-Posthumous-Writings
We apply such a link in the sentence 'the number 2 falls under the concept prime number'; it is contained in the words 'falls under', which need to be completed in two ways-by a subject and an ac- cusative; and only because their sense is thus 'unsaturated' are they capable of serving as a link.
Only when they have been supplemented in this twofold respect do we get a complete sense, a thought.
I say that such words or phrases stand for a relation.
We now get the same
? difficulty for the relation that we were trying to avoid for the concept. For the words 'the relation of an object to the concept it falls under' designate not a relation but an object; and the three proper names 'the number 2', 'the concept prime number', 'the relation of an object to a concept it falls under', hold aloof from one another just as much as the first two do by themselves; however we put them together, we get no sentence. It is thus easy for us to see that the difficulty arising from the 'unsaturatedness' of one part of the thought can indeed be shifted, but not avoided. 'Complete' and 'un- saturated' are of course only figures of speech; but all that I wish or am able to do here is to give hints.
It may make it easier to come to un understanding if the reader com- pares my work Function und BegrifJ. For over the question what it is that is called a function in Analysis, we come up against the same obstacle; and on thorough investigation it will be found that the obstacle is essential, and founded on the nature of our language; that we cannot avoid a certain inap- propriateness of linguistic ex- pression; and that there is nothing for it but to realize this and always take it into account.
On Concept and Object 117
? ? [Comments on Sense and Meaning] 1 [1892-1895]
In an article (Uber Sinn und Bedeutung) I distinguished between sense and meaning in the first instance only for the case of proper names (or, if one prefers, singular terms). The same distinction can also be drawn for concept? words. Now it is easy to become unclear about this by confounding the division into concepts and objects with the distinction between sense and meaning, so that we run together sense and concept on the one hand and meaning and object on the other. To every concept-word or proper name, there corresponds as a rule a sense and a meaning, as I use these words. Of course in fiction words only have a sense, but in science and wherever we are concerned about truth, we are not prepared to rest content with the sense, we also attach a meaning to proper names and concept-words; and if through some oversight, say, we fail to do this, then we are making a mistake that can easily vitiate our thinking. The meaning of a proper name is the object it designates or names. A concept-word means a concept, if the word is used as is appropriate for logic. I may clarify this by drawina attention to a fact that seems to weigh heavily on the side of extensionalist as against intensionalist logicians: namely, that in any sentence we can substitute salva veritate one concept-word for another if they have the same extension, so that it is also the case that in relation to inference, and where the laws of logic are concerned, that concepts differ only in so far as their extensions are different. The fundamental logical relation is that of an object's falling under a concept: all relations between concepts can be reduced to this. If an object falls under a concept, it falls under all concepti with the same extension, and this implies what we said above. Therefore just as proper names can replace one another salva veritate, so too can concept? words, if their extension is the same. Of course the thought will alter when such replacements are made, but this is the sense of the sentence, not ita meaning. * The meaning, which is the truth-value, remains the same. For thia reason we might easily come to propose the extension of a concept as the
* Cf. my article Uber Sinn und Bedeutung.
1 These comments were not composed before 1892, the year in which the article Ober Sinn und Bedeutung appeared. They are part of a bundle of papers entitled 'Schrodersche Logik', which existed in a complete form prior to the destruction of the NachlajJ. The first part of these constituted a draft of Frege's article Kritischl Beleuchtung einiges Punkte in E. Schroder's Vorlesungen iiber die Algebra der
Logik (ed. ).
? [Comments on Sense and Meaning] 119
meaning of a concept-word; to do this, however, would be to overlook the fact that the extensions of concepts are objects and not concepts (Cf. my essay Funktion und Begrifl). Nevertheless there is a kernel of truth in this position. In order to bring it out, I need to advert to what I said in my mono- graph on Funktion und Begriff. On the view expressed there a concept is a function of one argument, whose value is always a truth-value. Here I am borrowing the term 'function' from Analysis and, whilst retaining what is essential to it, using it in a somewhat extended meaning, a procedure for which the history of Analysis itself affords a precedent. The name of a func- tion is accompanied by empty places (at least one) where the argument is to go; this is usually indicated by the letter 'x' which fills the empty places in 4uestion. But the argument is not to be counted as belonging to the function, und so the letter 'x' is not to be counted as belonging to the name of the function either. Consequently one can always speak of the name of a function as having empty places, since what fills them does not, strictly speaking, belong to them. Accordingly I call the function itself unsaturated, or in need of supplementation, because its name has first to be completed with the sign of an argument if we are to obtain a meaning that is complete in itself. I call such a meaning an object and, in this case, the value of the function for the argument that effects the supplementing or saturating. In the cases we first encounter the argument is itself an object, and it is to these that we shall mainly confine ourselves here. Now with a concept we have the special case that the value is always a truth-value. That is to say, if we complete the name of a concept with a proper name, we obtain a sentence whose sense is a thought; and this sentence has a truth value as its meaning. To acknowledge this meaning as that of the True (as the True) is to judge that the object which is taken as the argument falls under the concept. What in the case of a function is called unsaturatedness, we may, in the case of a concept, call its predicative nature. ? This comes out even in the cases in which we speak of a subject-concept ('All equilateral triangles are equiangular' means 'If anything is an equilateral triangle, then it is an equiangular triangle').
Such being the essence of a concept, there is now a great obstacle in the way of expressing ourselves correctly and making ourselves understood. If I want to speak of a concept, language, with an almost irresistible force, compels me to use an inappropriate expression which obscures-! might nlmost say falsities-the thought. One would assume, on the basis of its 1malogy with other expressions, that if I say 'the concept equilateral triangle' I am designating a concept, just as I am of course naming a planet If I say 'the planet Nepture'. But this is not the case; for we do not have anything with a predicative nature. Hence the meaning of the expression 'the
? The words 'unsaturated' and 'predicative' seem more suited to the sense than the meaning; still there must be something on the part of the meaning which corresponds to this, and I know of no better words.
Cf. Wundt's Logik.
? 120 [Comments on Sense and Meaning]
concept equilateral triangle' (if there is one in this case) is an object. We cannot avoid words like 'the concept', but where we use them we must always bear their inappropriateness in mind. ? From what we have said it follows that objects and concepts are fundamentally different and cannot stand in for one another. And the same goes for the corresponding words or signs. Proper names cannot really be used as predicates. Where they might seem to be, we find on looking more closely that the sense is such that they only form part of the predicate: concepts cannot stand in the same relations as objects. It would not be false, but impossible to think of them as doing so. Hence, the words 'relation of a subject to a predicate' designate two quite different relations, according as the subject is an object or is itself a concept. Therefore it would be best to banish the words 'subject' and 'predicate' from logic entirely, since they lead us again and again to confound two quite different relations: that of an object's falling under a concept and that of one concept being subordinated to another. The words 'all' and 'some', which go with the grammatical subject, belong in sense with the grammatical predicate, as we see if we go over to the negative (not all, nonnulli}. From this alone it immediately follows that the predicate in these cases is different from that which is asserted of an object. And in the same way the relation of equality, by which I understand complete coincidence, identity, can only be thought of as holding for objects, not concepts. If we say 'The meaning of the word "conic section" is the same as that of the concept-word "curve of the second degree"' or 'The concept conic section coincides with the concept curve o f the second degree', the words 'meaning of the concept-word "conic section"' are the name of an object, not of a concept; for their nature is not predicative, they are not unsaturated, they cannot be used with the indefinite article. The same goes for the words 'the concept conic section'. But although the relation of equality can only be thought of as holding for objects, there is an analogous relation for concepts. Since this is a relation between concepts I call it a second level relation, whereas the former relation I call a first level relation. We say that an object a is equal to an object b (in the sense of completely coinciding with it) if a falls under every concept under which b falls, and conversely. We obtain something corres- ponding to this for concepts if we switch the roles of concept and object. We could then say that the relation we had in mind above holds between the concept ([> and the concept X, if every object that falls under ([> also falls under X, and conversely. Of course in saying this we have again been unable
to avoid using the expressions 'the concept ([>','the concept X', which again obscures the real sense. So for the reader who is not frightened of the concept-script I will add the following: The unsaturatedness of a concept (of first level) is represented in the concept-script by leaving at least one empty place in its designation where the name of the object which we are saying falls under the concept is to go. This place or places always has to be filled
? I shall deal with this difficulty.
? ? [Comments on Sense and Meaning] 121
in some way or other. Besides being filled by a proper name it can also be filled by a sign which only indicates an object. We can see from this that the sign of equality, or one analogous to it, can never be flanked by the designation of a concept alone, but in addition to the concept an object must also be designated or indicated as well. Even if we only indicate concepts schematically by a function-letter, we must see to it that we give expression to their unsaturatedness by an accompanying empty place as in C/>( ) and X( ). In other words, we may only use the letters (cl>, X}, which are meant to indicate or designate concepts, as function-letters, i. e. in such a way that they are accompanied by a place for the argument (the space between the following brackets). This being so, we may not write cp = X, because here the letters cp and X do not occur as function-letters. But nor may we write C/>( ) = X( }, because the argument-places have to be filled. But when they are filled, it is not the functions (concepts) themselves that are put equal to one another: in addition to the function-letter there will be something else on either side of the equality sign, something not belonging to the function.
These letters cannot be replaced by letters that are not used as function- letters: there must always be an argument-place to receive the 'd. The idea might occur to one simply to write cp = X. This may seem all right so long as we are indicating concepts schematically, but a mode of designation that is really adequate must provide for all cases. Let us take an example which I have already used in my paper on Funktion und Begriff.
For every argument the function x 2 = 1 has the same (truth-} value as the function (x + 1}2 = 2(x + 1) i. e. every object falling under the concept less by 1 than a number whose square is equal to its double falls under the concept square root of1 and conversely. If we expressed this thought in the
way that we gave above, 1 we should have
(al=1)~ ((a+1}2=2(a+1))
What we have here is that second level relation which corresponds to, but should not be confused with, equality (complete coincidence) between objects. If we write it-0-~(a2 = 1) = ((a + 1)2 = 2(a + 1}}, we have expressed what is essentially the same thought, construed as an equation between values of functions that holds generally. We have here the same second level relation; we have in addition the sign of equality, but this does not suffice on its own to designate this relation: it has to be used in combination with the sign for generality: in the first line we have a general statement but not an equation. In e(e2 = 1) = il((a + 1)2 = 2(a + 1}} we do have an equation, but not between concepts (which is impossible} but between objects, namely extensions of concepts.
Now we have seen that the relation of equality between objects cannot be
1 It may be that the notation used in the following formula, which Frege has not explained above, was introduced in the lost first part of the manuscript (see footnote lop. II8)(ed. ).
? ? 122 [Comments on Sense and Meaning]
conceived as holding between concepts too, but that there is a correspond- ing relation for concepts. It follows that the word 'the same' that is used to designate the former relation between objects cannot properly be used to designate the latter relation as well. If we try to use it to do this, the only recourse we really have is to say 'the concept ([> is the same as the concept X' and in saying this we have of course named a relation between objects,? where what is intended is a relation between concepts. We have the same case if we say 'the meaning of the concept-word A is the same as that of the concept word B'. Indeed we should really outlaw the expression 'the meaning of the concept-word A', because the definite article before 'meaning' points to an object and belies the predicative nature of a concept. It would be better to confine ourselves to saying 'what the concept word A means', for this at any rate is to be used predicatively: 'Jesus is, what the concept word "man" means' in the sense of 'Jesus is a man'.
Now if we bear all this in mind, we shall be well able to assert 'what two concept-words mean is the same if and only if the extensions of the corresponding concepts coincide' without being led astray by the improper use of the word 'the same'. And with this statement we have, I believe, made an important concession to the extensionalist logicians. They are right when they show by their preference for the extension, as against the intension, of a concept that they regard the meaning and not the sense of words as the essential thing for logic. The intensionalist logicians are only too happy not to go beyond the sense; for what they call the intension, if it is not an idea, is nothing other than the sense. They forget that logic is not concerned with how thoughts, regardless of truth-value, follow from thoughts, that the step from thought to truth-value-more generally, the step from sense to meaning-has to be taken. They forget that the laws of logic are first and foremost laws in the realm of meanings and only relate indirectly to sense. If it is a question of the truth of something-and truth is the goal of logic-we also have to inquire after meanings; we have to throw aside proper names that do not designate or name an object, though they may have a sense; we have to throw aside concept-words that do not have a meaning. These are not such as, say, contain a contradiction-for there is nothing at all wrong in a concept's being empty-but such as have vague boundaries. It must be determinate for every object whether it falls under a concept or not; a concept word which does not meet this requirement on its meaning is
meaningless. E. g. the word 'wiJ). v' (Homer, Odyssey X, 305) belongs to this class, although it is true that certain characteristic marks are supplied. For this reason the context cited need not lack a sense, any more than other contexts in which the name 'Nausicaa', which probably does not mean or name anything, occurs. But it behaves as if it names a girl, and it is thus assured of a sense. And for fiction the sense is enough. The tkought, though it is devoid of meaning, of truth-value, is enough, but not for science.
? These objects have the names 'the concept ([>'and 'the concept X'.
? ? ? [Comments on Sense and Meaning] 123
In my Grundlagen and the paper Uber formale Theorien der Arithmetik I showed that for certain proofs it is far from being a matter of indifference whether a combination of signs--e. g. F l - h a s a meaning* or not, that, on the contrary, the whole cogency of the proof stands or falls with this. The meaning is thus shown at every point to be the essential thing for science. Therefore even if we concede to the intensionalist logicians that it is the concept as opposed to the extension that is the fundamental thing, this does not mean that it is to be taken as the sense of a concept-word: it is its meaning, and the extensionalist logicians come closer to the truth in so far us they are presenting-in the extension-a meaning as the essential thing. Though this meaning is certainly not the concept itself, it is still very closely connected with it.
1
Husserl takes Schr6der to task for the unclarity in his discussion of the words 'unsinnig' [without sense], 'einsinnig' [having one sense], and 'mehrsinnig' [having more than one sense], 'undeutig' [without meaning], 'eindeutig' [having one meaning], 'mehrdeutig' [having more than one meaning] (pp. 48 ff. and 69),2 and unclarity indeed there is, but even the distinctions Husserl draws are inadequate. It was hardly to be expected that Schroder's use of the particles 'sinnig' and 'deutig' would not differ from my own; still less can I take issue with him over this, since when his work uppeared nothing had been published by me in this connection. For him this distinction is connected with that between common names and proper names, and the unclarity springs from a faulty conception of the distinction between concept and object. According to him there is nothing amiss with common names that are mehrdeutig, they are this when more than. one
? It is true that I had not then settled upon my present use of the words 'sense' and 'meaning', so that sometimes I said 'sense' where I should now say 'meaning'.
1 In what follows Frege is referring to the review of Schroder's Vorlesungen iiber die Algebra der Logik (Exakte Logik) I (Leipzig 1890), which Husserl had written for the Gottingischen Gelehrten Anzeigen (pp. 243-278, April 1891) (ed. ).
2 In the place referred to by Frege Schroder fixes on the adjectives ending in 'deutig' as terms for the sizes of extensions of concepts. Schroder speaks generally of names and calls proper names 'eindeutig', common names like 'my hand' 'zweideutig' [having two meanings], common names in general 'mehrdeutig' or 'vieldeutig' [having many meanings]. The corresponding formations with 'sinnig' are employed by Schroder to distinguish terms whose use is precisely fixed {'einsinnig' or 'univocal'), from terms with multiple meanings ('doppelsinnig' ! having a double sense], 'mehrsinnig' or 'equivocal') and from formations without sense ('unsinnig'; 'round square' in Schroder's example). With Husserl Frege chiefly criticizes Schroder for calling a name like 'round square' 'undeutig' when for this label to apply the? name is surely already presupposed as being significant as Much, so that it cannot at the same time be designated as 'unsinnlg' (ed. ).
? ? ? 124 [Comments on Sense and Meaning]
object falls under the corresponding concept. * On this view it would be possible for a common name to be undeutig too, like 'the round square', without its being defective. Schroder, however, calls it unsinnig as well and is thus untrue to his own way of speaking; for according to this the 'round square' would have to be called einsinnig, and Husserl was right when he called it a univocal common name; for 'univocal' and 'equivocal' correspond to Schroder's 'einsinnig' and 'mehrsinnii. Husserl says (p. 250) 'Obviously he confuses two quite different questions here, namely (1) whether a name has a Bedeutung (a 'Sinn'); and (2) whether there does or does not exist an object corresponding to the name'. This distinction is inadequate. The word 'common name' leads to the mistaken assumption that a common name is related to objects in essentially the same way as is a proper name, the difference being only that the latter names just one thing whilst the former is usually applicable to more than one. But this is false, and that is why I prefer 'concept-word' to 'common name'. A proper name must at least have a sense (as I use the word); otherwise it would be an empty sequence of sounds and it would be wrong to call it a name. But if it is to have a use in science we must require that it have a meaning too, that it designates or names an object. Thus it is via a sense, and only via a sense that a proper name is related to an object. 1 _
A concept-word must have a sense too and if it is to have a use in science, a meaning; but this consists neither of one object nor of a plurality of objects: it is a concept. Now in the case of a concept it can of course again be asked whether one object falls under it, or more than one or none. But
this relates directly to the concept and nothing else. So a concept-word can be absolutely impeccable, logically speaking, without there being an object to which it is related through its sense and meaning (the concept itself). As we see, this relation to an object is more indirect and inessential, so that there seems little point in dividing concept-words up according as no object falls under the corresponding concepts or one object or more than one.
* If, as Husserl says in the first footnote to p. 252, a distributive name is one 'whose Bedeutung is such that it designates any one of a plurality of things', then a concept-word (common name) is at any rate not a distributive name.
1 Since Schroder and Husserl did not distinguish, in the way Frege did, between the Sinn and Bedeutung of an expression, we have thought it best in this paragraph to preserve the actual German where these terms or (more commonly) their cognates with 'sinnig' and 'deutig' occur in quotation from these authors, or where Frege himself uses the latter in alluding to their views. We have given what help we could to the reader by providing renderings in square brackets; he should only remember not to attribute to the words 'sense' and 'meaning', as they occur in these renderings, the significance they have in the main body of the text, where they are of course used to render Frege's 'Sinn' and 'Bedeutung' (trans. ).
? [Comments on Sense and Meaning] 125
Logic must demand not only of proper names but of concept-words as well that the step from the word to the sense and from the sense to the meaning be determinate beyond any doubt. Otherwise we should not be entitled to speak of a meaning at all. Of course this holds for all signs and combinations of signs with the same function as proper names or concept- words.
? ? [128]
[129]
[129] [130] [131]
[131f. ]
Logic 1 [1897]
The word 'true' specifies the goal.
Logic is concerned with the predicate 'true' in a special way. The word 'true' characterizes logic. True cannot be defined; we cannot say: an idea is true if it agrees with reality.
True primitive and simple. This feature of our predicate is to be brought out by comparing it with others. Predicating it is always included in predicating anything whatever. To locate the domain of application of the predicate 'true'. Not applicable to what is material. It is most frequently ascribed to sentences-but only to assertoric sentences. Not, however, to sentences as series of sounds.
Translation.
We do not need to consider mock assertions in logic.
The sense of a sentence is called a thought. The predicate 'true' applies to thoughts. Does it also apply to ideas? Even where an idea is called true, it is really a thought to which this predicate is ascribed. A thought is not an idea and is not composed of ideas. Thoughts and ideas are fundamentally different. By associating ideas we never arrive at anything that could be true.
The proper means of expression for a thought is a sentence. On the other hand, a sentence is hardly an appropriate vehicle for conveying ideas. By contrast, pictures and musical compositions are unsuited
for expressing thoughts. The predicate 'true' compared with 'beautiful'. The latter admits of degree, not the former. What is beautiful is beautiful only for him who experiences it as such. There is no disputing tastes. What is true is true in itself; nothing is beautiful in itself. The assumption of a normal human being always underlies objective judgements of beauty. However, what is normal?
The objective sense of 'beautiful' can only be based on the subjective sense. Nor is it of any use to replace the assumption of a normal human being by that of an ideal one. A work of art is a structure of ideas within us. Each of us has his own. Aesthetic judgements don't contradict one another. Anyone who asserts that it -is only our recognizing a thing as true that makes it so, would, by so doing, contradict the content of his own assertion. In reality he could assert
[132f. ]
1 The date mentioned on p. 147 and the quotation on p. J58 make it probable that this essay was composed in 1897 (ed. ). '
? Logic 127
nothing. Every opinion would then be unjustified; there would be no science. Properly speaking, there would be nothing that is true. The independence of being recognized by us is integral to the sense of the word 'true'. Nor do thoughts have to be thought by us in order to be true. Laws of nature are discovered (not invented).
l133f. ) Thoughts are independent of our thinking. A thought does not belong specially to the person who thinks it, as an idea does to the person who has it: whoever thinks it encounters it in the same way, as the same thought. Otherwise two people would never attach the same thought to the same sentence. A contradiction between the assertions of different people would be impossible. A dispute about the truth of something would be futile. There would be no common ground to fight on. As each man makes a judgement about his poem, if the
judgement is an aesthetic one, so each man would make a judgement about his thought, if the thought were related to a sentence as the auditory ideas of the spoken sounds are related to the sound waves. If a thought were something mental, then its truth could only consist in a relation to something external, and that this relation obtained would be a thought into the truth of which we could inquire.
l134f. ) Treadmill. A thought is something impersonal. Writing on a wall. Objection: what of a sentence like 'I am cold'? The spoken word often needs to be supplemented. The word 'I' does not always designate the same person. A sentence containing 'I' can be cast into a more appropriate form. Interjections are different. The words 'now' and 'here' analogous to 'I'. Identity of the speaker essential where a subjective judgement of taste is concerned.
[135) Objection:myuseoftheword'thought'isoutoftheordinary. 1136] Footnote. Dedekind's way of using it agrees with mine.
[137) Thoughts are not generated by, but grasped by, thinking.
l137f. ) A thought not something spatial, material. Only in a special sense is it something actual.
[138) False thoughts are also independent of the speaker. The predicate 'true' is predicated in predicating anything. In an assertoric sentence the expression of a thought and the recognition of its truth usually go hand in hand. This does not have to be so. An assertoric sentence does not always contain an assertion. Grasping a thought usually precedes the recognition of truth. Judging, asserting.
ll39f. ] A sentence is also meant to have an effect on the imagination and feelings. It is able to do this because it consists of heard sounds. Onomatopoeia. Words also have an effect on the imagination through the sense they have. But ideas and sense must not be confused. A word by itself does not determine an idea. Different ideas answer to the same word. Words furnish hints to the imagination.
t140] The means available to the poet. 'Dog' and 'cur' can be substituted
? ? 128
Logic
[141 j
for one another without altering the thought. What distinguishes them is of the nature of an interjection. Criterion. To distinguish thoughts that are expressed from those that are merely evoked in us. A sad tone of voice, 'ab', 'unfortunately'.
Changes in language give rise to borderline cases. 1
Introduction
The predicate true, thoughts, consequences for the treatment of logic
When entering upon the study of a science, we need to have some idea, if only a provisional one, of its nature. We want to have in sight a goal to strive towards; we want some point to aim at that will guide our steps in the right direction. The word 'true' can be used to indicate such a goal for logic, just as can 'good' for ethics and 'beautiful' for aesthetics. Of course all the sciences have truth as their goal, but logic is concerned with the predicate 'true' in a quite special way, namely in a way analogous to that in which physics has to do with the predicates 'heavy' and 'warm' or chemistry with the predicates 'acid' and 'alkaline'. There is, however, the difference that these sciences have to take into account other properties besides these we have mentioned, and that there is no one property by which their nature is so completely characterized as logic is by the word 'true'.
Like ethics, logic can also be called a normative science. How must I think in order to reach the goal, truth? We expect logic to give us the answer to this question, but we do not demand of it that it should go into what is peculiar to each branch of knowledge and its subject-matter. On the. contrary, the task we assign logic is only that of saying what holds with the utmost generality for all thinking, whatever its subject-matter. We must assume that the rules for our thinking and for our holding something to be true are prescribed by the laws of truth. The former are given along with the latter. Consequently we can also say: logic is the science of the most general laws of truth. The reader may find that he can form no very precise impression from this description of what is meant. The author's inadequacy and the awkwardness of language are probably to blame for this. But it il only a question of giving a rough indication of the goal of logic. What is still lacking in the account will have to be made good as we go on.
Now it would be futile to employ a definition in order to make it clearer what is to be understood by 'true'. If, for example, we wished to say 'an idea is true if it agrees with reality' nothing would have been achieved, since in order to apply this definition we should have to decide whether some idea or other did agree with reality. Thus we should have to presuppose the very thing that is being defined. The same would hold of any definition of the form 'A is true if and only if it has such-and-such properties or stands in
1 Frege only took the table of contents as far as p. 141 (ed. ).
? Logic 129
such-and-such a relation to such-and-such a thing'. In each case in hand it would always come back to the question whether it is true that A has such- and-such properties, or stands in such-and-such a relation to such-and-such a thing. Truth is obviously something so primitive and simple that it is not possible to reduce it to anything still simpler. Consequently we have no alternative but to bring out the peculiarity of our predicate by comparing it with others. What, in the first place, distinguishes it from all other predicates is that predicating it is always included in predicating anything whatever.
If I assert that the sum of 2 and 3 is 5, then I thereby assert that it is true that 2 and 3 make 5. So I assert that it is true that my idea of Cologne Cathedral agrees with reality, ifl assert that it agrees with reality. Therefore it is really by using the form of an assertoric sentence that we assert truth, and to do this we do not need the word 'true'. Indeed we can say that even where we use the form of expression 'it is true that . . . 'the essential thing is really the assertoric form of the sentence.
We now ask: what can the predicate 'true' be applied to? The issue here is to delimit the range of application of the word. Whatever else may be the case, the word cannot be applied to anything that is material. If there is any doubt about this, it could arise only for works of art. But if we speak of truth in connection with these, then we are surely using the word with a different meaning from the one that is meant here. In any case it is only as a work of art that a thing is called true. If a thing had come into existence through the
blind play of natural forces, our predicate would be clearly inappropriate. Por the same reason we are excluding from consideration the use that is made by, say, an art critic when he calls feelings and experiences true.
No one would deny that our predicate is, for the most part, ascribed to sentences. We are not, however, concerned with sentences expressing wishes, questions, requests and commands, but only with assertoric sentences, sentences that is to say, in which we communicate facts and propound mathematical laws or laws of nature.
Further, it is clear that we do not, properly speaking, ascribe truth to the series of sounds which constitute a sentence, but to its sense; for, on the one hand, the truth of a sentence is preserved when it is correctly translated into another language, and, on the other hand, it is at least conceivable that the same series of sounds should have a true sense in one language and a false sense in another.
We are here including under the word 'sentence' the main clause of a sentence and clauses that are subordinate to it.
In the cases which alone concern logic the sense of an assertoric sentence is either true or false, and then we have what we call a thought proper. But there remains a third case of which at least some mention must be made here.
The sentence 'Scylla has six heads' is not true, but the sentence 'Scylla does not have six heads' is not true either; for it to be true the proper name 'Scylla' would have to designate something. Perhaps we think that the name
? ? 130 Logic
'Scylla' does designate something, namely an idea. In that case the first question to ask is 'Whose idea? ' We often speak as if one and the same idea occurred to different men, but that is false, at least if the word 'idea' is used in the psychological sense: each man has his own idea. But then an idea does not have heads, and so one cannot cut heads off an idea either. The name 'Scylla' does not therefore designate an idea. Names that fail to fulfil the usual role of a proper name, which is to name something, may be called mock proper names. Although the tale of Willian Tell is a legend and not history and the name 'William Tell' is a mock proper name, we cannot deny it a sense. But the sense of the sentence 'William Tell shot an apple off his son's head' is no more true than is that of the sentence 'William Tell did not shoot an apple off his son's head'. I do not say, however, that this sense is false either, but I characterize it as fictitious. This may elucidate the sense in which I am using the word 'false', which is as little susceptible of a definition proper as is the word 'true'.
If the idealist theory of knowledge is correct then all the sciences would belong to the realm of fiction. Indeed one might try to reinterpret all sentences in such a way that they were about ideas. By doing this, however, their sense would be completely changed and we should obtain quite a different science; this new science would be a branch of psychology.
Instead of speaking of 'fiction', we could speak of 'mock thoughts'. Thus if the sense of an assertoric sentence is not true, it is either false or fictitious, and it will generally be the latter if it contains a mock proper name. * The writer, in common with, for example, the painter, has his eye on appearances. Assertions in fiction are not to be taken seriously: they are only mock assertions. Even the thoughts are not to be taken seriously as in the sciences: they are only mock thoughts. If Schiller's Don Car/os were to be regarded as a piece of history, then to a large extent the drama would be false. But a work of fiction is not meant to be taken seriously in this way ati all: it's all play. Even the proper names in the drama, though they: correspond to names of historical personages, are mock proper names; they are not meant to be taken seriously in the work. We have a similar thing in the case of an historical painting. As a work of art it simply does not claim to give a visual representation of things that actually happened. A picture that was intended to portray some significant moment in history with photographic accuracy would not be a work of art in the higher sense of the word, but would be comparable rather to an anatomical drawing in a scientific work.
The logician does not have to bother with mock thoughts, just as 1 physicist, who sets out to investigate thunder, will not pay any attention to stage-thunder. When we speak of thoughts in what follows we mean thoughts proper, thoughts that are either true or false.
*We have an exception where a mock proper name occurs within a clause in indirect speech.
? ? Logic 131
The sense of an assertoric sentence I call a thought. Examples of thoughts are laws of nature, mathematical laws, historical facts: all these find expression in assertoric sentences. I can now be more precise and say: The predicate 'true' applies to thoughts.
Of course we speak of true ideas1 as well. By an idea we understand a picture that is called up by the imagination: unlike a perception it does not consist of present impressions, but of the reactivated traces of past impressions or actions. Like any other picture, an idea is not true in itself, hut only in relation to something to which it is meant to correspond. If it is said that a picture is meant to represent Cologne Cathedral, fair enough; it can then be asked whether this intention is realized; if there is no reference to an intention to depict something, there can be no question of the truth of a picture. It can be seen from this that the predicate true is not really conferred on the idea itself, but on the thought that the idea depicts a certain object. And this thought is not an idea, nor is it made up of ideas in any way. Thoughts are fundamentally different from ideas (in the psychological sense). The idea of a red rose is something different from the thought that this rose is red. Associate ideas or run them together as we may, we shall still finish up with an idea and never with something that could be true. This difference also comes out in the modes we have of communicating. The proper means of expression for a thought is a sentence. But a sentence is hardly an appropriate vehicle for conveying an idea. I have only to remind you how inadequate any description is by comparison with a pictorial representation. Things are not so bad where it is a matter of representing sounds, since we have the resources of onomatopoeia; but onomatopoeia has nothing whatever to do with the expression of thoughts, and whilst in translation the play of sounds is easily lost, the thought must be preserved if we are to speak of a translation at all. Conversely, pictures and musical
compositions without accompanying words are hardly suited for expressing thoughts. It is true that we may associate all kinds of thoughts with some work of art or other but there is no necessary connection between the two, and we are not surprised when someone else associates different thoughts with it.
In order to shed a clearer light on the peculiarity of the predicate true, let us compare it with the predicate beautiful.
? difficulty for the relation that we were trying to avoid for the concept. For the words 'the relation of an object to the concept it falls under' designate not a relation but an object; and the three proper names 'the number 2', 'the concept prime number', 'the relation of an object to a concept it falls under', hold aloof from one another just as much as the first two do by themselves; however we put them together, we get no sentence. It is thus easy for us to see that the difficulty arising from the 'unsaturatedness' of one part of the thought can indeed be shifted, but not avoided. 'Complete' and 'un- saturated' are of course only figures of speech; but all that I wish or am able to do here is to give hints.
It may make it easier to come to un understanding if the reader com- pares my work Function und BegrifJ. For over the question what it is that is called a function in Analysis, we come up against the same obstacle; and on thorough investigation it will be found that the obstacle is essential, and founded on the nature of our language; that we cannot avoid a certain inap- propriateness of linguistic ex- pression; and that there is nothing for it but to realize this and always take it into account.
On Concept and Object 117
? ? [Comments on Sense and Meaning] 1 [1892-1895]
In an article (Uber Sinn und Bedeutung) I distinguished between sense and meaning in the first instance only for the case of proper names (or, if one prefers, singular terms). The same distinction can also be drawn for concept? words. Now it is easy to become unclear about this by confounding the division into concepts and objects with the distinction between sense and meaning, so that we run together sense and concept on the one hand and meaning and object on the other. To every concept-word or proper name, there corresponds as a rule a sense and a meaning, as I use these words. Of course in fiction words only have a sense, but in science and wherever we are concerned about truth, we are not prepared to rest content with the sense, we also attach a meaning to proper names and concept-words; and if through some oversight, say, we fail to do this, then we are making a mistake that can easily vitiate our thinking. The meaning of a proper name is the object it designates or names. A concept-word means a concept, if the word is used as is appropriate for logic. I may clarify this by drawina attention to a fact that seems to weigh heavily on the side of extensionalist as against intensionalist logicians: namely, that in any sentence we can substitute salva veritate one concept-word for another if they have the same extension, so that it is also the case that in relation to inference, and where the laws of logic are concerned, that concepts differ only in so far as their extensions are different. The fundamental logical relation is that of an object's falling under a concept: all relations between concepts can be reduced to this. If an object falls under a concept, it falls under all concepti with the same extension, and this implies what we said above. Therefore just as proper names can replace one another salva veritate, so too can concept? words, if their extension is the same. Of course the thought will alter when such replacements are made, but this is the sense of the sentence, not ita meaning. * The meaning, which is the truth-value, remains the same. For thia reason we might easily come to propose the extension of a concept as the
* Cf. my article Uber Sinn und Bedeutung.
1 These comments were not composed before 1892, the year in which the article Ober Sinn und Bedeutung appeared. They are part of a bundle of papers entitled 'Schrodersche Logik', which existed in a complete form prior to the destruction of the NachlajJ. The first part of these constituted a draft of Frege's article Kritischl Beleuchtung einiges Punkte in E. Schroder's Vorlesungen iiber die Algebra der
Logik (ed. ).
? [Comments on Sense and Meaning] 119
meaning of a concept-word; to do this, however, would be to overlook the fact that the extensions of concepts are objects and not concepts (Cf. my essay Funktion und Begrifl). Nevertheless there is a kernel of truth in this position. In order to bring it out, I need to advert to what I said in my mono- graph on Funktion und Begriff. On the view expressed there a concept is a function of one argument, whose value is always a truth-value. Here I am borrowing the term 'function' from Analysis and, whilst retaining what is essential to it, using it in a somewhat extended meaning, a procedure for which the history of Analysis itself affords a precedent. The name of a func- tion is accompanied by empty places (at least one) where the argument is to go; this is usually indicated by the letter 'x' which fills the empty places in 4uestion. But the argument is not to be counted as belonging to the function, und so the letter 'x' is not to be counted as belonging to the name of the function either. Consequently one can always speak of the name of a function as having empty places, since what fills them does not, strictly speaking, belong to them. Accordingly I call the function itself unsaturated, or in need of supplementation, because its name has first to be completed with the sign of an argument if we are to obtain a meaning that is complete in itself. I call such a meaning an object and, in this case, the value of the function for the argument that effects the supplementing or saturating. In the cases we first encounter the argument is itself an object, and it is to these that we shall mainly confine ourselves here. Now with a concept we have the special case that the value is always a truth-value. That is to say, if we complete the name of a concept with a proper name, we obtain a sentence whose sense is a thought; and this sentence has a truth value as its meaning. To acknowledge this meaning as that of the True (as the True) is to judge that the object which is taken as the argument falls under the concept. What in the case of a function is called unsaturatedness, we may, in the case of a concept, call its predicative nature. ? This comes out even in the cases in which we speak of a subject-concept ('All equilateral triangles are equiangular' means 'If anything is an equilateral triangle, then it is an equiangular triangle').
Such being the essence of a concept, there is now a great obstacle in the way of expressing ourselves correctly and making ourselves understood. If I want to speak of a concept, language, with an almost irresistible force, compels me to use an inappropriate expression which obscures-! might nlmost say falsities-the thought. One would assume, on the basis of its 1malogy with other expressions, that if I say 'the concept equilateral triangle' I am designating a concept, just as I am of course naming a planet If I say 'the planet Nepture'. But this is not the case; for we do not have anything with a predicative nature. Hence the meaning of the expression 'the
? The words 'unsaturated' and 'predicative' seem more suited to the sense than the meaning; still there must be something on the part of the meaning which corresponds to this, and I know of no better words.
Cf. Wundt's Logik.
? 120 [Comments on Sense and Meaning]
concept equilateral triangle' (if there is one in this case) is an object. We cannot avoid words like 'the concept', but where we use them we must always bear their inappropriateness in mind. ? From what we have said it follows that objects and concepts are fundamentally different and cannot stand in for one another. And the same goes for the corresponding words or signs. Proper names cannot really be used as predicates. Where they might seem to be, we find on looking more closely that the sense is such that they only form part of the predicate: concepts cannot stand in the same relations as objects. It would not be false, but impossible to think of them as doing so. Hence, the words 'relation of a subject to a predicate' designate two quite different relations, according as the subject is an object or is itself a concept. Therefore it would be best to banish the words 'subject' and 'predicate' from logic entirely, since they lead us again and again to confound two quite different relations: that of an object's falling under a concept and that of one concept being subordinated to another. The words 'all' and 'some', which go with the grammatical subject, belong in sense with the grammatical predicate, as we see if we go over to the negative (not all, nonnulli}. From this alone it immediately follows that the predicate in these cases is different from that which is asserted of an object. And in the same way the relation of equality, by which I understand complete coincidence, identity, can only be thought of as holding for objects, not concepts. If we say 'The meaning of the word "conic section" is the same as that of the concept-word "curve of the second degree"' or 'The concept conic section coincides with the concept curve o f the second degree', the words 'meaning of the concept-word "conic section"' are the name of an object, not of a concept; for their nature is not predicative, they are not unsaturated, they cannot be used with the indefinite article. The same goes for the words 'the concept conic section'. But although the relation of equality can only be thought of as holding for objects, there is an analogous relation for concepts. Since this is a relation between concepts I call it a second level relation, whereas the former relation I call a first level relation. We say that an object a is equal to an object b (in the sense of completely coinciding with it) if a falls under every concept under which b falls, and conversely. We obtain something corres- ponding to this for concepts if we switch the roles of concept and object. We could then say that the relation we had in mind above holds between the concept ([> and the concept X, if every object that falls under ([> also falls under X, and conversely. Of course in saying this we have again been unable
to avoid using the expressions 'the concept ([>','the concept X', which again obscures the real sense. So for the reader who is not frightened of the concept-script I will add the following: The unsaturatedness of a concept (of first level) is represented in the concept-script by leaving at least one empty place in its designation where the name of the object which we are saying falls under the concept is to go. This place or places always has to be filled
? I shall deal with this difficulty.
? ? [Comments on Sense and Meaning] 121
in some way or other. Besides being filled by a proper name it can also be filled by a sign which only indicates an object. We can see from this that the sign of equality, or one analogous to it, can never be flanked by the designation of a concept alone, but in addition to the concept an object must also be designated or indicated as well. Even if we only indicate concepts schematically by a function-letter, we must see to it that we give expression to their unsaturatedness by an accompanying empty place as in C/>( ) and X( ). In other words, we may only use the letters (cl>, X}, which are meant to indicate or designate concepts, as function-letters, i. e. in such a way that they are accompanied by a place for the argument (the space between the following brackets). This being so, we may not write cp = X, because here the letters cp and X do not occur as function-letters. But nor may we write C/>( ) = X( }, because the argument-places have to be filled. But when they are filled, it is not the functions (concepts) themselves that are put equal to one another: in addition to the function-letter there will be something else on either side of the equality sign, something not belonging to the function.
These letters cannot be replaced by letters that are not used as function- letters: there must always be an argument-place to receive the 'd. The idea might occur to one simply to write cp = X. This may seem all right so long as we are indicating concepts schematically, but a mode of designation that is really adequate must provide for all cases. Let us take an example which I have already used in my paper on Funktion und Begriff.
For every argument the function x 2 = 1 has the same (truth-} value as the function (x + 1}2 = 2(x + 1) i. e. every object falling under the concept less by 1 than a number whose square is equal to its double falls under the concept square root of1 and conversely. If we expressed this thought in the
way that we gave above, 1 we should have
(al=1)~ ((a+1}2=2(a+1))
What we have here is that second level relation which corresponds to, but should not be confused with, equality (complete coincidence) between objects. If we write it-0-~(a2 = 1) = ((a + 1)2 = 2(a + 1}}, we have expressed what is essentially the same thought, construed as an equation between values of functions that holds generally. We have here the same second level relation; we have in addition the sign of equality, but this does not suffice on its own to designate this relation: it has to be used in combination with the sign for generality: in the first line we have a general statement but not an equation. In e(e2 = 1) = il((a + 1)2 = 2(a + 1}} we do have an equation, but not between concepts (which is impossible} but between objects, namely extensions of concepts.
Now we have seen that the relation of equality between objects cannot be
1 It may be that the notation used in the following formula, which Frege has not explained above, was introduced in the lost first part of the manuscript (see footnote lop. II8)(ed. ).
? ? 122 [Comments on Sense and Meaning]
conceived as holding between concepts too, but that there is a correspond- ing relation for concepts. It follows that the word 'the same' that is used to designate the former relation between objects cannot properly be used to designate the latter relation as well. If we try to use it to do this, the only recourse we really have is to say 'the concept ([> is the same as the concept X' and in saying this we have of course named a relation between objects,? where what is intended is a relation between concepts. We have the same case if we say 'the meaning of the concept-word A is the same as that of the concept word B'. Indeed we should really outlaw the expression 'the meaning of the concept-word A', because the definite article before 'meaning' points to an object and belies the predicative nature of a concept. It would be better to confine ourselves to saying 'what the concept word A means', for this at any rate is to be used predicatively: 'Jesus is, what the concept word "man" means' in the sense of 'Jesus is a man'.
Now if we bear all this in mind, we shall be well able to assert 'what two concept-words mean is the same if and only if the extensions of the corresponding concepts coincide' without being led astray by the improper use of the word 'the same'. And with this statement we have, I believe, made an important concession to the extensionalist logicians. They are right when they show by their preference for the extension, as against the intension, of a concept that they regard the meaning and not the sense of words as the essential thing for logic. The intensionalist logicians are only too happy not to go beyond the sense; for what they call the intension, if it is not an idea, is nothing other than the sense. They forget that logic is not concerned with how thoughts, regardless of truth-value, follow from thoughts, that the step from thought to truth-value-more generally, the step from sense to meaning-has to be taken. They forget that the laws of logic are first and foremost laws in the realm of meanings and only relate indirectly to sense. If it is a question of the truth of something-and truth is the goal of logic-we also have to inquire after meanings; we have to throw aside proper names that do not designate or name an object, though they may have a sense; we have to throw aside concept-words that do not have a meaning. These are not such as, say, contain a contradiction-for there is nothing at all wrong in a concept's being empty-but such as have vague boundaries. It must be determinate for every object whether it falls under a concept or not; a concept word which does not meet this requirement on its meaning is
meaningless. E. g. the word 'wiJ). v' (Homer, Odyssey X, 305) belongs to this class, although it is true that certain characteristic marks are supplied. For this reason the context cited need not lack a sense, any more than other contexts in which the name 'Nausicaa', which probably does not mean or name anything, occurs. But it behaves as if it names a girl, and it is thus assured of a sense. And for fiction the sense is enough. The tkought, though it is devoid of meaning, of truth-value, is enough, but not for science.
? These objects have the names 'the concept ([>'and 'the concept X'.
? ? ? [Comments on Sense and Meaning] 123
In my Grundlagen and the paper Uber formale Theorien der Arithmetik I showed that for certain proofs it is far from being a matter of indifference whether a combination of signs--e. g. F l - h a s a meaning* or not, that, on the contrary, the whole cogency of the proof stands or falls with this. The meaning is thus shown at every point to be the essential thing for science. Therefore even if we concede to the intensionalist logicians that it is the concept as opposed to the extension that is the fundamental thing, this does not mean that it is to be taken as the sense of a concept-word: it is its meaning, and the extensionalist logicians come closer to the truth in so far us they are presenting-in the extension-a meaning as the essential thing. Though this meaning is certainly not the concept itself, it is still very closely connected with it.
1
Husserl takes Schr6der to task for the unclarity in his discussion of the words 'unsinnig' [without sense], 'einsinnig' [having one sense], and 'mehrsinnig' [having more than one sense], 'undeutig' [without meaning], 'eindeutig' [having one meaning], 'mehrdeutig' [having more than one meaning] (pp. 48 ff. and 69),2 and unclarity indeed there is, but even the distinctions Husserl draws are inadequate. It was hardly to be expected that Schroder's use of the particles 'sinnig' and 'deutig' would not differ from my own; still less can I take issue with him over this, since when his work uppeared nothing had been published by me in this connection. For him this distinction is connected with that between common names and proper names, and the unclarity springs from a faulty conception of the distinction between concept and object. According to him there is nothing amiss with common names that are mehrdeutig, they are this when more than. one
? It is true that I had not then settled upon my present use of the words 'sense' and 'meaning', so that sometimes I said 'sense' where I should now say 'meaning'.
1 In what follows Frege is referring to the review of Schroder's Vorlesungen iiber die Algebra der Logik (Exakte Logik) I (Leipzig 1890), which Husserl had written for the Gottingischen Gelehrten Anzeigen (pp. 243-278, April 1891) (ed. ).
2 In the place referred to by Frege Schroder fixes on the adjectives ending in 'deutig' as terms for the sizes of extensions of concepts. Schroder speaks generally of names and calls proper names 'eindeutig', common names like 'my hand' 'zweideutig' [having two meanings], common names in general 'mehrdeutig' or 'vieldeutig' [having many meanings]. The corresponding formations with 'sinnig' are employed by Schroder to distinguish terms whose use is precisely fixed {'einsinnig' or 'univocal'), from terms with multiple meanings ('doppelsinnig' ! having a double sense], 'mehrsinnig' or 'equivocal') and from formations without sense ('unsinnig'; 'round square' in Schroder's example). With Husserl Frege chiefly criticizes Schroder for calling a name like 'round square' 'undeutig' when for this label to apply the? name is surely already presupposed as being significant as Much, so that it cannot at the same time be designated as 'unsinnlg' (ed. ).
? ? ? 124 [Comments on Sense and Meaning]
object falls under the corresponding concept. * On this view it would be possible for a common name to be undeutig too, like 'the round square', without its being defective. Schroder, however, calls it unsinnig as well and is thus untrue to his own way of speaking; for according to this the 'round square' would have to be called einsinnig, and Husserl was right when he called it a univocal common name; for 'univocal' and 'equivocal' correspond to Schroder's 'einsinnig' and 'mehrsinnii. Husserl says (p. 250) 'Obviously he confuses two quite different questions here, namely (1) whether a name has a Bedeutung (a 'Sinn'); and (2) whether there does or does not exist an object corresponding to the name'. This distinction is inadequate. The word 'common name' leads to the mistaken assumption that a common name is related to objects in essentially the same way as is a proper name, the difference being only that the latter names just one thing whilst the former is usually applicable to more than one. But this is false, and that is why I prefer 'concept-word' to 'common name'. A proper name must at least have a sense (as I use the word); otherwise it would be an empty sequence of sounds and it would be wrong to call it a name. But if it is to have a use in science we must require that it have a meaning too, that it designates or names an object. Thus it is via a sense, and only via a sense that a proper name is related to an object. 1 _
A concept-word must have a sense too and if it is to have a use in science, a meaning; but this consists neither of one object nor of a plurality of objects: it is a concept. Now in the case of a concept it can of course again be asked whether one object falls under it, or more than one or none. But
this relates directly to the concept and nothing else. So a concept-word can be absolutely impeccable, logically speaking, without there being an object to which it is related through its sense and meaning (the concept itself). As we see, this relation to an object is more indirect and inessential, so that there seems little point in dividing concept-words up according as no object falls under the corresponding concepts or one object or more than one.
* If, as Husserl says in the first footnote to p. 252, a distributive name is one 'whose Bedeutung is such that it designates any one of a plurality of things', then a concept-word (common name) is at any rate not a distributive name.
1 Since Schroder and Husserl did not distinguish, in the way Frege did, between the Sinn and Bedeutung of an expression, we have thought it best in this paragraph to preserve the actual German where these terms or (more commonly) their cognates with 'sinnig' and 'deutig' occur in quotation from these authors, or where Frege himself uses the latter in alluding to their views. We have given what help we could to the reader by providing renderings in square brackets; he should only remember not to attribute to the words 'sense' and 'meaning', as they occur in these renderings, the significance they have in the main body of the text, where they are of course used to render Frege's 'Sinn' and 'Bedeutung' (trans. ).
? [Comments on Sense and Meaning] 125
Logic must demand not only of proper names but of concept-words as well that the step from the word to the sense and from the sense to the meaning be determinate beyond any doubt. Otherwise we should not be entitled to speak of a meaning at all. Of course this holds for all signs and combinations of signs with the same function as proper names or concept- words.
? ? [128]
[129]
[129] [130] [131]
[131f. ]
Logic 1 [1897]
The word 'true' specifies the goal.
Logic is concerned with the predicate 'true' in a special way. The word 'true' characterizes logic. True cannot be defined; we cannot say: an idea is true if it agrees with reality.
True primitive and simple. This feature of our predicate is to be brought out by comparing it with others. Predicating it is always included in predicating anything whatever. To locate the domain of application of the predicate 'true'. Not applicable to what is material. It is most frequently ascribed to sentences-but only to assertoric sentences. Not, however, to sentences as series of sounds.
Translation.
We do not need to consider mock assertions in logic.
The sense of a sentence is called a thought. The predicate 'true' applies to thoughts. Does it also apply to ideas? Even where an idea is called true, it is really a thought to which this predicate is ascribed. A thought is not an idea and is not composed of ideas. Thoughts and ideas are fundamentally different. By associating ideas we never arrive at anything that could be true.
The proper means of expression for a thought is a sentence. On the other hand, a sentence is hardly an appropriate vehicle for conveying ideas. By contrast, pictures and musical compositions are unsuited
for expressing thoughts. The predicate 'true' compared with 'beautiful'. The latter admits of degree, not the former. What is beautiful is beautiful only for him who experiences it as such. There is no disputing tastes. What is true is true in itself; nothing is beautiful in itself. The assumption of a normal human being always underlies objective judgements of beauty. However, what is normal?
The objective sense of 'beautiful' can only be based on the subjective sense. Nor is it of any use to replace the assumption of a normal human being by that of an ideal one. A work of art is a structure of ideas within us. Each of us has his own. Aesthetic judgements don't contradict one another. Anyone who asserts that it -is only our recognizing a thing as true that makes it so, would, by so doing, contradict the content of his own assertion. In reality he could assert
[132f. ]
1 The date mentioned on p. 147 and the quotation on p. J58 make it probable that this essay was composed in 1897 (ed. ). '
? Logic 127
nothing. Every opinion would then be unjustified; there would be no science. Properly speaking, there would be nothing that is true. The independence of being recognized by us is integral to the sense of the word 'true'. Nor do thoughts have to be thought by us in order to be true. Laws of nature are discovered (not invented).
l133f. ) Thoughts are independent of our thinking. A thought does not belong specially to the person who thinks it, as an idea does to the person who has it: whoever thinks it encounters it in the same way, as the same thought. Otherwise two people would never attach the same thought to the same sentence. A contradiction between the assertions of different people would be impossible. A dispute about the truth of something would be futile. There would be no common ground to fight on. As each man makes a judgement about his poem, if the
judgement is an aesthetic one, so each man would make a judgement about his thought, if the thought were related to a sentence as the auditory ideas of the spoken sounds are related to the sound waves. If a thought were something mental, then its truth could only consist in a relation to something external, and that this relation obtained would be a thought into the truth of which we could inquire.
l134f. ) Treadmill. A thought is something impersonal. Writing on a wall. Objection: what of a sentence like 'I am cold'? The spoken word often needs to be supplemented. The word 'I' does not always designate the same person. A sentence containing 'I' can be cast into a more appropriate form. Interjections are different. The words 'now' and 'here' analogous to 'I'. Identity of the speaker essential where a subjective judgement of taste is concerned.
[135) Objection:myuseoftheword'thought'isoutoftheordinary. 1136] Footnote. Dedekind's way of using it agrees with mine.
[137) Thoughts are not generated by, but grasped by, thinking.
l137f. ) A thought not something spatial, material. Only in a special sense is it something actual.
[138) False thoughts are also independent of the speaker. The predicate 'true' is predicated in predicating anything. In an assertoric sentence the expression of a thought and the recognition of its truth usually go hand in hand. This does not have to be so. An assertoric sentence does not always contain an assertion. Grasping a thought usually precedes the recognition of truth. Judging, asserting.
ll39f. ] A sentence is also meant to have an effect on the imagination and feelings. It is able to do this because it consists of heard sounds. Onomatopoeia. Words also have an effect on the imagination through the sense they have. But ideas and sense must not be confused. A word by itself does not determine an idea. Different ideas answer to the same word. Words furnish hints to the imagination.
t140] The means available to the poet. 'Dog' and 'cur' can be substituted
? ? 128
Logic
[141 j
for one another without altering the thought. What distinguishes them is of the nature of an interjection. Criterion. To distinguish thoughts that are expressed from those that are merely evoked in us. A sad tone of voice, 'ab', 'unfortunately'.
Changes in language give rise to borderline cases. 1
Introduction
The predicate true, thoughts, consequences for the treatment of logic
When entering upon the study of a science, we need to have some idea, if only a provisional one, of its nature. We want to have in sight a goal to strive towards; we want some point to aim at that will guide our steps in the right direction. The word 'true' can be used to indicate such a goal for logic, just as can 'good' for ethics and 'beautiful' for aesthetics. Of course all the sciences have truth as their goal, but logic is concerned with the predicate 'true' in a quite special way, namely in a way analogous to that in which physics has to do with the predicates 'heavy' and 'warm' or chemistry with the predicates 'acid' and 'alkaline'. There is, however, the difference that these sciences have to take into account other properties besides these we have mentioned, and that there is no one property by which their nature is so completely characterized as logic is by the word 'true'.
Like ethics, logic can also be called a normative science. How must I think in order to reach the goal, truth? We expect logic to give us the answer to this question, but we do not demand of it that it should go into what is peculiar to each branch of knowledge and its subject-matter. On the. contrary, the task we assign logic is only that of saying what holds with the utmost generality for all thinking, whatever its subject-matter. We must assume that the rules for our thinking and for our holding something to be true are prescribed by the laws of truth. The former are given along with the latter. Consequently we can also say: logic is the science of the most general laws of truth. The reader may find that he can form no very precise impression from this description of what is meant. The author's inadequacy and the awkwardness of language are probably to blame for this. But it il only a question of giving a rough indication of the goal of logic. What is still lacking in the account will have to be made good as we go on.
Now it would be futile to employ a definition in order to make it clearer what is to be understood by 'true'. If, for example, we wished to say 'an idea is true if it agrees with reality' nothing would have been achieved, since in order to apply this definition we should have to decide whether some idea or other did agree with reality. Thus we should have to presuppose the very thing that is being defined. The same would hold of any definition of the form 'A is true if and only if it has such-and-such properties or stands in
1 Frege only took the table of contents as far as p. 141 (ed. ).
? Logic 129
such-and-such a relation to such-and-such a thing'. In each case in hand it would always come back to the question whether it is true that A has such- and-such properties, or stands in such-and-such a relation to such-and-such a thing. Truth is obviously something so primitive and simple that it is not possible to reduce it to anything still simpler. Consequently we have no alternative but to bring out the peculiarity of our predicate by comparing it with others. What, in the first place, distinguishes it from all other predicates is that predicating it is always included in predicating anything whatever.
If I assert that the sum of 2 and 3 is 5, then I thereby assert that it is true that 2 and 3 make 5. So I assert that it is true that my idea of Cologne Cathedral agrees with reality, ifl assert that it agrees with reality. Therefore it is really by using the form of an assertoric sentence that we assert truth, and to do this we do not need the word 'true'. Indeed we can say that even where we use the form of expression 'it is true that . . . 'the essential thing is really the assertoric form of the sentence.
We now ask: what can the predicate 'true' be applied to? The issue here is to delimit the range of application of the word. Whatever else may be the case, the word cannot be applied to anything that is material. If there is any doubt about this, it could arise only for works of art. But if we speak of truth in connection with these, then we are surely using the word with a different meaning from the one that is meant here. In any case it is only as a work of art that a thing is called true. If a thing had come into existence through the
blind play of natural forces, our predicate would be clearly inappropriate. Por the same reason we are excluding from consideration the use that is made by, say, an art critic when he calls feelings and experiences true.
No one would deny that our predicate is, for the most part, ascribed to sentences. We are not, however, concerned with sentences expressing wishes, questions, requests and commands, but only with assertoric sentences, sentences that is to say, in which we communicate facts and propound mathematical laws or laws of nature.
Further, it is clear that we do not, properly speaking, ascribe truth to the series of sounds which constitute a sentence, but to its sense; for, on the one hand, the truth of a sentence is preserved when it is correctly translated into another language, and, on the other hand, it is at least conceivable that the same series of sounds should have a true sense in one language and a false sense in another.
We are here including under the word 'sentence' the main clause of a sentence and clauses that are subordinate to it.
In the cases which alone concern logic the sense of an assertoric sentence is either true or false, and then we have what we call a thought proper. But there remains a third case of which at least some mention must be made here.
The sentence 'Scylla has six heads' is not true, but the sentence 'Scylla does not have six heads' is not true either; for it to be true the proper name 'Scylla' would have to designate something. Perhaps we think that the name
? ? 130 Logic
'Scylla' does designate something, namely an idea. In that case the first question to ask is 'Whose idea? ' We often speak as if one and the same idea occurred to different men, but that is false, at least if the word 'idea' is used in the psychological sense: each man has his own idea. But then an idea does not have heads, and so one cannot cut heads off an idea either. The name 'Scylla' does not therefore designate an idea. Names that fail to fulfil the usual role of a proper name, which is to name something, may be called mock proper names. Although the tale of Willian Tell is a legend and not history and the name 'William Tell' is a mock proper name, we cannot deny it a sense. But the sense of the sentence 'William Tell shot an apple off his son's head' is no more true than is that of the sentence 'William Tell did not shoot an apple off his son's head'. I do not say, however, that this sense is false either, but I characterize it as fictitious. This may elucidate the sense in which I am using the word 'false', which is as little susceptible of a definition proper as is the word 'true'.
If the idealist theory of knowledge is correct then all the sciences would belong to the realm of fiction. Indeed one might try to reinterpret all sentences in such a way that they were about ideas. By doing this, however, their sense would be completely changed and we should obtain quite a different science; this new science would be a branch of psychology.
Instead of speaking of 'fiction', we could speak of 'mock thoughts'. Thus if the sense of an assertoric sentence is not true, it is either false or fictitious, and it will generally be the latter if it contains a mock proper name. * The writer, in common with, for example, the painter, has his eye on appearances. Assertions in fiction are not to be taken seriously: they are only mock assertions. Even the thoughts are not to be taken seriously as in the sciences: they are only mock thoughts. If Schiller's Don Car/os were to be regarded as a piece of history, then to a large extent the drama would be false. But a work of fiction is not meant to be taken seriously in this way ati all: it's all play. Even the proper names in the drama, though they: correspond to names of historical personages, are mock proper names; they are not meant to be taken seriously in the work. We have a similar thing in the case of an historical painting. As a work of art it simply does not claim to give a visual representation of things that actually happened. A picture that was intended to portray some significant moment in history with photographic accuracy would not be a work of art in the higher sense of the word, but would be comparable rather to an anatomical drawing in a scientific work.
The logician does not have to bother with mock thoughts, just as 1 physicist, who sets out to investigate thunder, will not pay any attention to stage-thunder. When we speak of thoughts in what follows we mean thoughts proper, thoughts that are either true or false.
*We have an exception where a mock proper name occurs within a clause in indirect speech.
? ? Logic 131
The sense of an assertoric sentence I call a thought. Examples of thoughts are laws of nature, mathematical laws, historical facts: all these find expression in assertoric sentences. I can now be more precise and say: The predicate 'true' applies to thoughts.
Of course we speak of true ideas1 as well. By an idea we understand a picture that is called up by the imagination: unlike a perception it does not consist of present impressions, but of the reactivated traces of past impressions or actions. Like any other picture, an idea is not true in itself, hut only in relation to something to which it is meant to correspond. If it is said that a picture is meant to represent Cologne Cathedral, fair enough; it can then be asked whether this intention is realized; if there is no reference to an intention to depict something, there can be no question of the truth of a picture. It can be seen from this that the predicate true is not really conferred on the idea itself, but on the thought that the idea depicts a certain object. And this thought is not an idea, nor is it made up of ideas in any way. Thoughts are fundamentally different from ideas (in the psychological sense). The idea of a red rose is something different from the thought that this rose is red. Associate ideas or run them together as we may, we shall still finish up with an idea and never with something that could be true. This difference also comes out in the modes we have of communicating. The proper means of expression for a thought is a sentence. But a sentence is hardly an appropriate vehicle for conveying an idea. I have only to remind you how inadequate any description is by comparison with a pictorial representation. Things are not so bad where it is a matter of representing sounds, since we have the resources of onomatopoeia; but onomatopoeia has nothing whatever to do with the expression of thoughts, and whilst in translation the play of sounds is easily lost, the thought must be preserved if we are to speak of a translation at all. Conversely, pictures and musical
compositions without accompanying words are hardly suited for expressing thoughts. It is true that we may associate all kinds of thoughts with some work of art or other but there is no necessary connection between the two, and we are not surprised when someone else associates different thoughts with it.
In order to shed a clearer light on the peculiarity of the predicate true, let us compare it with the predicate beautiful.
