Frege's theory of sense and denotation
1. Motivating puzzles
1.1. Identity Statements
"Mark Twain" is just the pen-name of Samuel Clemens. So "Mark Twain" and "Samuel Clemens" both pick out the same individual in the world. Then why do the sentences:
Mark Twain wrote Huckleberry Finn
and
Samuel Clemens wrote Huckleberry Finn
seem to have different congnitive significance?
Or, consider the sentences:
- 12 + 3 = 15
- Samuel Clemens is Mark Twain
- The morning star is the evening star
These sentences of the form 'a=b' are true iff a and b are identical (aside: this is a T-scheme sentence). But these sentences seem to have a different meaning than:
- 15=15
- Samuel Clemens is Samuel Clemens
- The morning star is the morning star
The sentences in the latter group can be seen to be true by inspection. But the sentences in the former group require some knowledge about the world.
So, when we talk about meaning, we need to talk about something additional besides, "the conditions under which a sentence is true".
1.2. Belief attribution
1.2.1. Preliminaries
A propositional attitude is the relationship between an individual and a psychological state regarding a proposition \(p\). When we make a statement about this state, we are making a propositional attitude report. For example,
- John believes that \(p\)
- Mary thinks that \(p\)
The Principle of Identity Substitution says that if a sentence \(S(m)\) only differs from \(S(n)\) in that the name \(m\) is replaced with \(n\), and \(m=n\), then \(S(m) \Rightarrow S(n)\).
1.2.2. Problem
However, there are situations when the principle of identity subsitution does not hold. For example, consider the sentence:
John believes that Mark Twain wrote Huckleberry Finn
This does not entail the sentence:
John believes that Samuel Clemens wrote Huckleberry Finn
2. Frege's theory of sense and denotation
Frege believed that in addition to having a denotation, terms and names also have a sense, which accounts for their cognitive significance and the way that they are conceived.
Then, the sense of 4, \(s[4]\) is different from the sense of \frac{8}{2}, \(s[\frac{8}{2}]\). So, the sense of \(4=4\) is different from \(4=\frac{8}{2}\).
2.1. Denotation
Consider the sentence 'John loves Mary'. Then,
- the denotation \(d[j]\) is the real individual John
- similarly, \(d[m]\) is the real individual Mary
- \(d[L]\) is the function that maps individuals to concepts of the form, e.g. 'loves Mary'
- \(d[Lm]\) is the function that maps individuals to truth values if they satisfy the predicate 'loves Mary'
- \(d[jLm]\) is The True if the real individual John falls under the concept 'loves Mary', otherwise it is The False
2.2. Sense
- the sense \(s[j]\) is the sense of the name John
- \(s[m]\) is the sense of the name Mary
- \(s[L]\) is the function that maps the sense of names to the sense of the predicate, e.g. 'loves Mary'
- \(s[Lm]\) is a function that maps, e.g. \(s[j]\) to the sense of the whole sentence
Frege calls the sense of the whole sentence the thought. And while the denotation of a sentence is one of two truth values, Frege says there can be an infinite number of senses for a sentence.
To resolve the propositional attitude report problem, Frege says that anything that follows a propositional report verb does not have it's usual denotation, but instead denotes the sense that it would usually have. So, in 'John believes that Mark Twain wrote Huckleberry Finn', 'Mark Twain' does not denote the real individual named by 'Mark Twain'. Instead, it denotes the sense of the name 'Mark Twain'.