On Sense and Reference

A little bit of Frege by someone who probably doesn’t understand.

a = a holds a priori. Kant would say it’s “analytic”.

Using * to denote “the referent of”, expressions of the kind *a = *b, rather than being self-evident, instead contain extension of knowledge. We’ve gained the insight that a and b refer to the same thing: as a silly example, the learning that “the ratio of a circle’s circumference to its diameter” refers to the same thing as “the infinite sum 4 (1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots )“.

We can have *a = *b without it being the case that a = b. The signs are not the same. In what ways do they differ?

Even if the referents are the same, the sense is also important.


  • a referent potentially has many signs.
  • (we hope) a sign in a given context has at most referent.
  • It may have none – consider the sign “the celestial body most distant from the earth”.

sign has a sense, refers to a referent

In natural language normally when we use words (without quotes) we intend to speak of their referent. If we want to refer to the sign itself we can quote the words. We can also call out the sense explicitly. When using natural language we quite often presuppose the existence of a referent.

We can attempt to make some definitions a tad more concrete.

  • a sense/thought is the way in which a sign refers to an object.
  • a proper name/sign/word/expression expresses/has a sense and refers to/references/designates its referent.
  • a referent is an object of some kind.
  • a sentence is a special kind of expression which has a truth-value as its referent
  • .

  • a truth-value is something which is either true or false, with no other possibilities.

Now, we postulate to the following invariant: the truth-value of a sentence should remain unchanged when we replace a part of the sentence with an expression having the same referent. The truth value of the sentence is in the space of referents and isn’t effected by the sense.
Performing a substitution may well change the sense.

Subtleties here include:

  • Due to the quirks of natural language, parts of a sentence might be implicitly quoted – the referent of the quoted expression is the sense of the unquoted expression.
  • Frege enumerates many different grammatical constructs with edge-cases.
  • A ‘part’ of a sentence may not be a contiguous string of words, so one has to be careful when performing replacements.

Apparently Russell blows a lot of this stuff up, but I’m not there yet.

On Sense and Reference