Frege On Sense and Reference
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ShaughanLavine - 22 Aug 2005
Frege distinguishes in a meaningful unit of language the sign, the sense, the idea, and the reference. The sign is just the words or letters used—he refers to the shapes of the symbols, because often what he has in mind is not ordinary language but his "Begriffschrift," which is something like today's interpreted formal logic, which he views as a perfected language. His main target after all is arithmetic.
He is working at the time non-Euclidean geometries were developed. You can Euclid's axioms for geometry and mess with the parallel postulate:
Given a line and a point not on the line there is exactly n lines through the point parallel to the original line,
where n=1. You can put in other values for n and get different geometries. Examples: n=0 gives the geometry on the surface of a sphere. n=2 gives the geometry on a saddle.
It is a waste of energy to view these geometries as completely independent: you'd like to know what follows, first, from the axioms without any parallel postulate, because those results hold for all the geometries, and you don't need to rederive them separately each time. The problem is that for that to work, you need to know in a precise sense what follows from what. The Euclidean axioms rely on diagrams, and are in fact, if you ignore the diagrams, inconsistent. There was no standard of derivation through which you could be sure you hadn't smuggled in assumptions that were not explicit. That was a lot of Frege's motivation for developing logic, and it made it possible to give a theory of "arithmetic," which actually meant most of mathematics, on an axiomatic basis, which was Frege's central accomplishment in the Grundgesetze. His formal language was then so complete that he took it as a basis to analyze all of language. (That is not chronologically correct.) If mathematics is your project, you need a liberal conception of what entities are, since you want the mathematical ones.
He declares early on (in the Grundlagen) that all uses of numbers in mathematics are reducible to ones that occur in identity sentences. He's therefore very interested in identities.
He begins with a puzzle. What are identity statements about? a=b can't be just about the signs, because such identities would only be useful for introducing previously uninterpreted signs.
What about the idea? That's a nonstarter because different people have different ideas associated with signs, and in that case a=b couldn't be used for communication.
What about the sense? That is what he advocated in the Grundlagen. Why doesn't that work? Hesperus=Phosphorus might or might not be true, so far as the meanings alone are concerned.
What about the reference? That would make a=a and a=b the same, and they aren't. There is a modern (post-Kripke) debate about whether there can be a contingently true identity statement.
Solution: We need both sense and reference.
Implicit principle: compositionality—whatever semantic properties complexes of linguistic units have will be determined from the semantic properties of the linguistic units. He says related things late in the article when he talks about eliminating ambiguity from a perfect language.
What are the linguistics components of a complex? Who knows? So, why not try sign, sense, idea, reference?
How do we figure out the sign, etc. of a linguistic complex? It is to be a function, in the mathematical sense of the sign, etc. of the linguistic units.
Kevin—Worry. Isn't this going to saddle us with a problem like that of the Kantian (and Frege knew his Kant) of getting from the senses to the world? Wittgenstein and Russell got very worried about exactly those problems for the theories they took from Frege.
Comment—This is not like Kant in that senses are
not psychological entities at all. There are two problems: how do minds get to senses, and how do senses
Top of p. 22--Complete knowledge of an object would require knowledge of all the true identity sentences about the object. Note that for mathematical purposes, knowing all the truths and knowing the object are not clearly distinct. For medium-sized dry goods, we seem to want more—Russell wanted knowledge by acquaintance. If all you are finally interested in is what follows from what, nothing more than the true identity statements matters. The same will be true if you are just trying to see which sentences are true and which false.
I think all he says it that senses are "modes of presentation," that different names can express different senses and designate different objects, and he thinks that names with the same sense designate the same object.
The linguistic components of a complex (tentatively, sign, sense, reference) are to be functions of the linguistic components of the units. At the beginning of the article, Frege looks at sign alone, reference alone, so what if we make an additional assumption: the sign of the complex is a function of the signs of the units and nothing else (obviously), the sense of the complex is a function of the senses of the units alone (he practically states this), and similarly for reference (that requires some argument). That is a greatly simplifying assumption that will give us some chance of a mathematically tractable theory. He never directly states this assumption, but his talk about "staying on the same level" and about fiction practically does.
Added later: The sense is independent of whether or not there is a reference: "The thought remains the same whether 'Odysseus' has reference or not." (24)
Now, think like a mathematician: If you have a complex term, if you replace some parts of it with parts of equal value, you get something with the same value as the original. Thus, given his assumption, if you replace a word in a sentence by another with the same sense, the resulting sentence will have the same sense as the original. Similarly if you replace a word in a sentence by another with the same reference, the new sentence will have the same reference as the original. These principles yield a test we can use to determine what the sense and reference of a sentence must be.
So, what happens when you replace, in a=b, a with another proper name with the same sense? Nothing interesting: the resulting sentence expresses the same "thought" it did before, it has the same cognitive significance. Example:
Hesperus=Phosphorus
The Morning Star=Phosphorus.
What about when you replace a name with another with same reference? Example:
a=b
a=a.
What stays the same? Certainly not the thought. What remains the same is whether or not the sentence is true. Thus, the reference of a sentence is whether or not it is true. That isn't grammatically a reference, so he converts it into whether or not the sentence refers to "The True." What is "The True"? Well, we know what expressions refer to it, we know how it affects the truth values of sentences, what more could we ask?
Frege does not assume that there is any intuitive, prior notion of the reference of a sentence. He takes that to be a term of theory to be supplied as is appropriate for the theory.
Afterthoughts
Kevin said he thought that there is an intuitive notion of the reference of a sentence: the state of affairs described by the sentence. I asked for an example of two sentences that describe the same state of affairs but that have different senses. Annie offered the following, which I accepted, but should not have:
My cat is on the rug.
The cat who jumped out the window yesterday is on the rug.
These do not represent the same state of affairs because the second, but not the first, requires that windows exist. That is not so important, because, it seems to me, that the state of affairs that Hesperus is Phosphorus and the state of affairs that Hesperus is Hesperus could reasonably be taken to be the same.
Here is why states of affairs won't do: It is a requirement of the postulated functional relationship that a sentence have a reference if and only if every word in the sentence does. For example, "The fact that we concern ourselves at all about the reference of a part of the sentence indicates that we generally recognize and expect a reference for the sentence itself" and "the reference of a sentence may always be sought, whenever the reference of its components is involved" (24). But then states of affairs won't do, because a false sentence all of whose parts refer (Hesperus=the Red Planet) has a reference but does not describe a state of affairs. Possible states of affairs won't do either: now false sentences all of whose parts refer have reference (the possible state of affairs that Hesperus=the Red Planet) but sentences with nonreferring parts also have reference, contrary to the requirements ("The famous cocaine-addicted detective who wears deerstalker hats lives on Baker Street" describes the possible state of affairs that the famous cocaine-addicted detective who wears deerstalker hats lives on Baker Street.) Of course, one can criticize the requirements, and many have, but it does seem to me to be correct that it follows from them that the reference is the truth value.
Here is what I wanted to discuss but didn't get to. This material is important in part because it is what Russell picks up on or doesn't pick up on but needed.
Use vs mention: 22 "If words are used in the ordinary way, what one intends to speak of is their reference. It can also happen, however, that one wishes to talk about the words themselves ... We then have signs of signs. In writing, the words are in this case enclosed in quotation marks."
Indirect reference: 22 "In order to speak of the sense of an expression 'A' one may simply use the phrase 'the sense of the expression "A"'. In reported speech one talks about the sense, e.g., of another person's remarks. ... in this way of speaking words do not have their customary reference but designate what is usually their sense. ... we will say: In reported speech, words are used
indirectly or have their
indirect reference. We distinguish ... the
customary from the indirect reference of a word; and its
customary sense from its
indirect sense."
Definite descriptions (Russell's term—Frege's is "compound proper names"): Note the discussion of the definite descriptions "whoever discovered the elliptic form of the planetary orbits," "the square root of 4 which is smaller than 0," and "the negative square root of 4" on 27–28 and also notes 14 and 15. Frege says that the assertion of (the truth of) a sentence in which a definite description occurs in the customary sense presupposes that the definite description refers, but that the thought that the definite description refers is not a part of the thought expressed; that the existence of definite descriptions that do not refer is an imperfection of language to be avoided by means of special stipulation; and that a sentence in which a definite description that fails to refer occurs in the customary sense does not have a reference, but that if the sentence is interpreted so that the thought that the definite description refers is a part of the thought expressed by the sentence, rather than a presupposition, then the sentence will have a thought expressed in it that is false, beside a part that is without reference.
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ShaughanLavine - 30 Aug 2005