Signification and assertion

Every departmental science has a subject, and its literature talks about or refers to that subject. Physics talks about heavy bodies and momentum and energy, chemistry talks about compounds, biology talks about flora and fauna etc. What does semantics, the science of meaning, talk about?

And there is the problem. Sometimes we cannot refer to what we signify.






Frege recognised this problem in 1892, in his essay ‘On Concept and Object’. A sentence consists of words, each of which has a signification or sense. What the whole sentence signifies is thus a compound of the senses corresponding to the words. (See e.g. his undated letter to Jourdain, in Frege’s Philosophical and Mathematical Correspondence, ed Gabriel and Hermes, 1980). The possibility of understanding a sentence we have never heard before depends on this property. What the sentence signifies is something new and perhaps previously unknown to us, but the signification of the words of which it is composed must be known, otherwise we would be incapable of understanding the sentence. For example ‘Socrates is a man’ is composed of the expressions ‘Socrates’ and ‘is a man’, both of which we know and understand.

The problem that Frege grapples with in ‘On Concept and Object’ is that while we can talk about what ‘Socrates’ signifies, namely Socrates himself, we can’t talk about what ‘is a man’ signifies. Or suppose we can. Let’s refer to it by the expression ‘The signification of “is a man”’. Will that do? No, because that expression is what Frege calls an Object term, an expression that refers to an object like Socrates. Thus the sentence ‘The signification of “is a man” is an Object’ is true. But it cannot signify an object, otherwise the sentence ‘Socrates is a man’ would be composed of two terms for objects. But two such terms cannot compose a sentence, any more than ‘Socrates Plato’ can. The sentence would be a mere list of words. Frege says, enigmatically ‘the concept horse is not a concept’, and attributes it to ‘an awkwardness of language’.

There is a similar problem regarding what I call signs of assertion. Consider

It is false that Socrates is a horse

I have not asserted that Socrates is a horse. On the contrary, I have denied it. Yet the four words ‘Socrates is a horse’ occur inside the eight word sentence ‘It is false that Socrates is a horse’. Perhaps we can explain this as follows. The eight word sentence can be split into ‘It is false’ and ‘that Socrates is a horse’. The latter is what Frege calls an object term. It refers to something a mad person might assert as true, the very thing I stand in the relation of denying to. So the meaning of the eight word sentence is changed by putting ‘It is false’ in front, and so if the meaning of the whole sentence is a composite of the meaning of ‘it is false’ and ‘that Socrates is a horse’, the composite is what ‘It is false that Socrates is a horse’ signifies. But of course that can’t be so, for the very fact that we could signify that Socrates was not a horse, would require that Socrates not being a horse was a fact. Worse, ‘It is true that Socrates is a horse’ would signify Socrates being a horse, so would require the existence of Socrates being a horse. Both those contradictory facts would have to exist in order for the contradictory sentences to be significant. Impossible!

Frege alludes to this problem in a much later essay (‘Negation’) published in 1918. He distinguishes between a question (my example is ‘is Socrates a horse’) from the thought corresponding to an answer like ‘yes’ or ‘no’. For if the sense of the question contained the sense of ‘yes’ or ‘no’, then the question would contain its own answer. The question would express a thought ‘whose being consists in its being true’.

Grasping the sense [of the question] would at the same time be an act of judging, and the utterance of the interrogative sentence would at the same time be an assertion, and so an answer to the question. But in an interrogative sentence neither the truth nor the falsity of the sense may be asserted.

Fair enough, but Frege does not see this as a challenge to his compositional semantics. Consider ‘Is Socrates a horse? No’. The first part signifies the question. If adding the sign ‘No’ completes the sense, then what is signified by the whole thing, namely question plus answer, must indeed be something whose being consists in being true, which Frege apparently denies.

In summary, if the signification of the whole is made up of the signification of the parts, then we should be able to refer to the signification of the whole, if semantics is to be a proper science. But we can’t, otherwise the subject of our science would include items like Socrates not being a horse, as well as Socrates being a horse. Which is impossible. Therefore semantics is not a science, at least not a proper science.

Number, concepts and existence

The Maverick Philosopher is agonising about number and existence in this post. It would be simpler if we returned to the original text of Frege which started all this (Die Grundlagen der Arithmetik 1884. Page numbers are to the original edition).

Frege claims with a concept the question is always whether anything, and if so what, falls under it. With a proper name such questions make no sense. (§51, p. 64). He also claims that when you add the definite article to a concept word, it ceases to function as a concept word, although it still so functions with the indefinite article, or in the plural (ibid).

This is part of a section of the Grundlagen where he develops the thesis that number is a property of concepts, not of things. Thus if I say (§46, p. 59) that ‘the King’s carriage is drawn by four horses’, I am ascribing the number 4 to the concept horse that draws the King’s carriage. The number is not a property of the horses, either individually or collectively, but of a concept.

From these two claims, namely that number is a property of concept words, and that proper names are not concept words, it seems to follow that ‘Socrates exists’ makes no sense. If ‘Socrates’ is not a concept word, then it seems no concept corresponds to it, but existence means that some concept is instantiated, so ‘Socrates exists’cannot express existence. This is the difficulty that Bill is grappling with.

But why, from the fact that ‘Socrates’ is not a concept word, does it follow that there is no corresponding concept? Frege has already told us that a concept word ceases to be such when we attach the definite article to it. So while ‘teacher of Plato’ signifies a concept, ‘the teacher of Plato does not. Why can’t the definite noun phrase ‘Socrates’ be the same, except that the definiteness is built into the proper name, rather than a syntactical compound of definite article and concept word. Why can’t ‘Socrates’ be semantically compound? So that it embeds a concept like person identical with Socrates, which with the definite article appended gives us ‘Socrates’?

As I argued in one of the comments, the following three concepts all have a number

C1: {any man at all}
C2: {any man besides Socrates}
C3: {satisfies C1 but not C2}

If the number corresponding to C1 is n, then the number of C2 is n-1. And the number of C3 is of course 1, and if C3 is satisfied, then Socrates exists. Simple.

Frege on compositionality

A proposition has a sense, and the sense of the proposition has a corresponding thought.  This is is how we communicate.  But we don't attach a simple sign to each thought, otherwise we could only communicate as many thoughts as there were simple signs.  So a proposition is composed of simple signs, each of which has a sense.  We can put these simple signs together in as many ways as we like, so we can communicate new thoughts.

I do not believe that we can dispense with the sense of a name in logic; for a proposition must have a sense if it is to be useful. But a proposition consists of parts which must somehow contribute to the expression of the sense of the proposition: so they themselves must somehow have a sense. Take the proposition 'Etna is higher than Vesuvius'. This contains the name 'Etna', which occurs also in other propositions, e.g., in the proposition 'Etna is in Sicily'. The possibility of our understanding propositions which we have never heard before rests evidently on this, that we construct the sense of a proposition out of parts that correspond to the words. If we find the same word in two propositions, e.g., 'Etna', then we also recognize something common to the corresponding thoughts, something corresponding to this word. Without this, language in the proper sense would be impossible. We could indeed adopt the convention that certain signs were to express certain thoughts, like railway signals ('The track is clear'); but in this way we would always be restricted to a very narrow area, and we could not form a completely new proposition, one which would be understood by another person even though no special convention had been adopted beforehand for this case.

Subsumption and subordination

Maverick dedicates his post here to me, saying that I like “sophisms and scholastic arcana”. Yes, and Fregean arcana too, Bill, heh heh. His post is about the odd-looking inference

Man is a species; Socrates is a man; ergo, Socrates is a species.

Note that the scholastics would have resolved this by treating the conclusion as a reduplicative proposition: Socrates insofar as he is a man, is a species, but never mind that. Maverick goes on to discuss the ‘modern’ Fregean treatment of propositions like ‘man is a species’. According to Frege, a universal proposition like ‘every man is an animal’ has a fundamentally different form from ‘Socrates is an animal’. In the former, one concept is subordinated to another: the concept ‘man’ is said to be subordinate to the concept ‘animal’. In the latter, the individual Socratres is subsumed under the concept ‘animal’.

Fregegives some very bad arguments for this in his famous essay ‘On Concept and Object’, but I won’t discuss those now. For the moment, here are two arguments against his view.

1. Argument from obviousness. It is obvious that ‘every man is animal’ does not say that one concept is subordinate to another, for the simple reason that it does not say anything about concepts at all. What it says is that every man is an animal. Thus, Socrates is an animal, Plato is an animal. It is talking about every man, not about some concept.

2. Frege’s position requires taking on the absurd idea of ‘object dependence’, i.e. that the meaning of a proper name depends on the existence of some object referred to. I discuss this at length here, with reference to another less well-known essay of Frege's*. Briefly, if we allow that a proper name N can be meaningful and empty, there must be some relation which holds between the name and its referent when its referent exists, and which fails to hold when there is no referent. But then a proper name is not essentially different from a common name like ‘man’. We can say ‘there are no men’ if nothing falls under the meaning of ‘man’, and we can say ‘there is no Socrates’ if nothing falls under the meaning of ‘Socrates’. Frege correctly rules this out as inconsistent with the concept-object distinction. In summary: the distinction between ‘subordination’ and ‘subsumption’ implies and is implied by the object-concept distinction. And the object-concept distinction implies and is implied by the position that the meaning of a proper name is object-dependent.

*Though I note that Maverick mentions this essay, at second-hand, in his book here.

Ockham's Fregean moment

I am working on chapter 67 of Ockham's master work, where I find something that reminds me of Frege.  Ockham is talking about 'material supposition'.  This is a mode of supposition where a word stands for itself.  It's rather like when a word is enclosed in quotation marks (which the medievals didn't have), except the whole point of quotation marks is that we create a new word to stand for the unquoted word, and so the word precisely doesn't stand for itself.

Ockham claims that there are situations when a word signifies something that it does not materially supposit for. His example is the sentence "animal is predicated of man", in Latin: animal praedicatur de homine.  Note the word 'homine', which is the ablative case of the word 'homo'.  This is important, for Ockham's sentence is true because 'a man is an animal' is true, which in Latin is homo est animal.  Note the nominative case of 'homo'.  So when we say "animal is predicated of man", we mean that animal is predicated of homo, not homine.  But we can only say that using the word homine, at least in Latin.  (Or perhaps the same is really true in English, except the ablative of the word 'man' is the same as the nominative?)  Thus the quoted homine signifies itself, i.e. homine, but supposits for its nominative homo. (Hope that makes sense).

That seems trivial.  But it seems remarkably close to Frege's famous puzzle about the concept horse.  Frege held that simple propositions like 'Red Rum is a horse' are composed of Object and Concept.  The object is signified by the object word "Red Rum" and is thus Red Rum himself.  The concept is signified by the predicate " - is a horse".  Let's call that the concept Horse.  But now the puzzle.  The sentence "the concept Horse is a concept" is also a simple sentence, where the term "the concept Horse" occurs in subject position.  So it signifies an Object, according to Frege.  And so is not a Concept.   Thus, the concept Horse is not a concept.  Frege admits this is odd.

It must indeed be recognised that here we are confronted by an awkwardness of language, which I admit cannot be avoided, if we say that the concept horse is not a concept, whereas, e.g., the city of Berlin is a city, and the volcano Vesuvius is a volcano.

This seems remarkably similar to Ockham's puzzle about homine, although it would take some work to tease out the underlying basis of it, if any.

A thought whose being consists in its being true?

The question being discussed here about truthmakers has some affinity with a question discussed by Frege in his late essay “Negation”. He compares grasp of a thought with understanding of a question. And, just as we can understand a question without understanding the correct answer, so he argues that there cannot be a thought ‘whose being consists in its being true’

The content of a question is that as to which we must decide. Consequently truth cannot be counted as going along with the content of the question. When I raise the question whether the Sun is bigger than the Moon, I am seeing the sense of the interrogative sentence 'Is the Sun bigger than the Moon?' Now if this sense were a thought whose being consisted in its being true, then I should at the same time see that this sense was true. Grasping the sense would at the same time be an act of judging; and the utterance of the interrogative sentence would at the same time be an assertion, and so an answer to the question. But in an interrogative sentence neither the truth nor the falsity of the sense may be asserted. Hence an interrogative sentence has not as its sense something whose being consists in its being true.
[...]
And since the sense of an interrogative sentence is always also inherent in the assertoric sentence that gives an answer to the question, this separation must be carried out for assertoric sentences too. It is a matter of what we take the word 'thought' to mean. In any case, we need a short term for what can be the sense of an interrogative sentence. I call this a thought. If we use language this way, not all thoughts are true. The being of a thought thus does not consist in its being true. (Negation, from Translations from Philosophical Writings of Gottlob Frege, ed. Peter Geach, p.125).

I wonder if a ‘truthmaker’ as understood by the advocates of truthmaking is the same sort of thing as Frege’s marvelous but impossible thought. Something that if we perceived it for what it was, would simultaneously communicate to us the truth of what it includes.

Reference and Bedeutung

William Vallicella takes me to task here for saying 'referential' when 'directly referential' was meant.  This is a good point, as the English word 'reference' has a variety of meanings, even in philosophical usage.  Nonetheless, it gets its philosophical meaning from the translation of Frege's work in German "Über Sinn und Bedeutung" as "On Sense and Reference". The German word Bedeutung means meaning or significance.

Frege thought that the semantic power of an expression consists in its being associated with an extra-linguistic item. Even a whole sentence has a Bedeutung or reference, namely one or the other of the two Platonic objects 'The True' or 'The False'. Thus the Bedeutung or Reference or Signification of a complete sentence consists in its being True or False.

According to him, the semantics of the sentence is compositional. Each significant expression has a semantic value which affects the truth-value of the sentence in which it occurs. The semantic power of a a concept expression such as '- is wise' is to signify a Concept, a Platonic extra-linguistic entity which maps all and only wise objects to the value 'True'. The semantic value of a singular term is also an extra-linguistic item, an Object such as Socrates. Thus the proper name 'Socrates' has the role of introducing an object to the concept-expression '- is wise', thereby determining the Truth Value of the sentence 'Socrates is wise'.

If a proper name (eg. 'Bilbo') fails to be associated with an extra-linguistic item, it fails to introduce an Object to the concept-expression with which it is combined. As a consequence, the resulting sentence will (according to Frege) lack a Truth-value. It will signify neither the True nor the False and thus, since the signification of the sentence consists in its being True or False, will lack a signification. Thus Frege writes

A sentence can be true or untrue only if it is an expression for a thought. The sentence "Leo Sachse is a man" is the expression of a thought only if 'Leo Sachse' designates something. And so too the sentence "this table is round" is the expression of a thought only if the words 'this table' are not empty sounds but designate something specific for me" (Posthumous Writings, p.174).

And again:

Names that fail to fulfil the usual role of a proper name, which is to name something, may be called mock proper names ... Instead of speaking about 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.  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 ...  The logician does not have to bother with mock thoughts, just as a physicist, who sets out to investigate thunder, will not pay any attention to stage-thunder. When we speak of thoughts in what folows we mean thoughts proper, thoughts that are either true or false. (Posthumous Writings, p. 130, my emphasis).

In summary.  The philosophical word 'reference' derives its meaning as a translation of Frege's word Bedeutung, meaning signification.  The signification of a complete sentence consists in its signifying the True or the False.  Thus, on his account, a fictional name, which fails to introduce any Object to the Concept-expression to which it is adjoined in a complete sentence, and thus cannot contribute to the Truth value of the sentence, cannot have a signification either.

Can a relation have non-existent relata?

Can a relation have non-existent relata, asks Bill Vallicella. Surely not. For we have to ask what kind of relation it is – one-termed, two-termed, three-termed, or whatever. Let’s suppose it is two-termed, i.e. it has two relata. If one of those were destroyed, i.e. made non-existent, it would only have one relatum. If both destroyed, none. Frege said it well: that the number zero represents the denial of existence. If the number of Fs is zero, then there are no Fs: no Fs exist.

Thus if a relation has two relata, it has two existing relata.

Frege's Point

What Geach calls 'Frege's point' is the claim that the same thought or proposition may occur now asserted, now unasserted. Frege expresses this very clearly in a short essay*, posthumously published, probably written in 1915, where he argues that asserting a sentence is a matter of the assertoric force with which the sentence is uttered, and that assertion is not the function of the word 'true'.

His argument is as follows. Uttering a sentence 'sea water is salt' merely expresses a thought. Nothing is meant to be asserted (behauptet werden solle). This becomes clear when turn the sentence into a that-clause: 'that sea-water is salty'. The that-clause does not assert anything. Or we could have the sentence spoken by an actor on a stage, where the actor does not speak with 'assertoric force' (or at least only seems to).

We can make this even clearer, he says, by adding the words 'it is true' to the expression 'that seawater is salty'. This forms a sentence that we can also turn into a that-clause: 'that it is true that sea-water is salty'. Thus the sense of the word 'true' does not make any essential contribution to the thought. If I assert 'it is true that sea-water is salty', I assert the same thing as if I assert 'sea-water is salty'. Thus the assertion is not to be found in the word 'true', but in the assertoric force with which the sentence is uttered.

Readers of this blog will recognise this as the thesis that William Vallicella has been defending from his website in Arizona. I shall discuss Frege's argument in my next post.

* "My Basic Logical Insights" (Posthumous Writings, transl. P. Long and R. White, 251-2. This is a translation of Nachgelassene Schriften 271-2).