I use the following diagram when teaching students Fregean semantics.
According to Frege, linguistic items, such as terms, predicates, and sentences, each have both a referent and a sense. The referent of a simple subject-predicate sentence is a truth value, and is determined by the referent of the subject and the referent of a predicate. The referent of the subject term is an object, and the referent of the predicate is a function, which takes an object as input and yields a truth value as output. The sense of a simple subject-predicate sentence is a proposition (or, in Fregean terms, a thought), and is determined by the sense of the subject term and the sense of the predicate. The nature of the sense of a subject term can be conceived of as a description associated with the subject term. Like its referent, the sense of a predicate may also be conceived as a function, which takes the sense of a singular term as input and yields a proposition as output.
I faced a challenge when TAing for Introduction to the Philosophy of Language, when we were covering Fregean semantics. The course is lower-division, and so most students in the class were rather new to philosophy. Yet the course covers some relatively sophisticated and technical topics. Many students have difficulty understanding the usual textual and oral presentations of Frege's semantics, as they are rather dense (like the paragraph immediately preceding this one). Technical notions like that of a function often present extra hurdles to students who are not mathematically inclined. They find no solace when the instructor explains, "a function f is just a relation such that if x = y then f(x) = f(y), and a relation is just a set of ordered pairs". Most students are more comfortable understanding a function as a dynamic object, i.e., as a machine. Each machine contains an input slot and an output slot. When something is inserted into the input slot of a machine, it ejects something from its output slot.
Keeping this in mind, I presented the Fregean treatment of the sentence 'Obama is a democrat' on the chalkboard as I have on the previous page. d is a function--the referent of the predicate 'is a democrat'. When d takes Obama as its argument, it yields the truth value true. e is also a function--the sense of the predicate. When e takes the sense of 'Obama' as its argument, it yields the proposition that Obama is a democrat. The picture makes clear Frege's distinction between the levels of sense and the reference, as well as how the various notions on each level relate to one another.
This way of presenting Fregean semantics also makes it clear why, according to Fregean semantics, sentences like 'Santa Clause is jolly', while meaningful, are not false, but rather have no truth value.
Because 'Santa Clause' doesn't have a referent, there is no object to put into the input slot of the referent of 'is jolly' (the function j). As a result, the machine does not have any "raw material" from which to make an output, and so it does not yield a truth value. Because, however, 'Santa Clause' does have a sense, there is an object to put into the input slot of the sense of 'is jolly' (the function k). As a result, this machine does have some "raw material", and so does yield a proposition as output. Thus, this sentence, while neither true nor false according to Frege's account, does express a proposition, and therefore is meaningful. The students found this very helpful. It was satisfying to see the students explain a new example in the written exams. Many people's reliance on the pictorial interpretation of Frege was clear due to the wording of their explanations on exams, and was often completely obvious due to diagrams drawn in the margins.
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