Russell’s Theory of Descriptions

English phrases that begin with ‘the’, such as ‘the man’ and ‘the present king of France’, are called definite descriptions (or descriptions, for short). So far, we have no adequate logical notation for descriptions. It is possible to translate ‘the man is bald’ by choosing a constant c for ‘the man’, a predicate letter P for ‘is bald’, and writing: Pc. However, treating the description as if it were a constant will cause us to classify some valid arguments as invalid.

For example, it should be clear that (1) entails (2).

1. (1) Aristotle is the philosopher who taught Alexander the Great.

2. (2) Aristotle taught Alexander the Great.

If we choose the constants: a for Aristotle, and g for Alexander the Great, we might notate (2) as (2′).

1. (2′) Tag

If we treat ‘the philosopher who taught Alexander the Great’ as a constant g, then (1) is notated by (1′).

1. (1′) a≈g

However, there is no logical relationship between the atomic sentences (1′) and (2′) that would cause us to recognize that the argument from (1′) to (2′) is valid. Clearly we need a way to notate the internal structure of ‘the philosopher who taught Alexander the Great’ if we are ever to show that (1) entails (2) in logic.