Before I begin, this is really a general summery of what is going on in the text. It is not comprehensive, but is meant to give you a rough idea on Bertrand Russels project. It should not be substituted for actual scholarly work or lecture notes.
A peculiarity of language
Before we begin, let us explain the following concept:
By ‘denoting phrase’, Russell means a phrase that indicates something. For example, ‘a man’, ‘some man’, ‘some men’, ‘every man’, ‘all men’, ‘the present king of France’, ‘the mass at the centre of the solar system’, etc. It does not have to actually exist for the phrase to indicate something. Such phrases can denote certainly or ambiguous, the one or the many, and so on. It does not matter for Russell, so long as it indicates.
For Russell, we may have no immediate acquaintance with the object, by which he means direct sensory experience of it. Similarly, we might not have any knowledge about it. Yet none the less, we still use such phrases. Hence it is perculiar.
Russell argues that denoting such as ‘a man’, ‘this tree’, ‘many things’, etc. only have meaning if they exist as a part of a proposition which consists in other parts. For Russell, a denoting phrase such as ‘a man’ is not a singular utterance or thought. It is useless on its own, but very handy if it is a part of a wider union of utterances. Russell gives us an example by showing how denoting phrases can only exist in a much larger framework.
If I say ‘I met a man’, what I am really saying is ”I met x, and x is a man’ is not always false’. This leave ‘a man’ by itself lacking in meaning because it has nothing to relate too. When we use phrases like ‘a man’, it is packed full of indirect associations. If we abstract these associations and focus on the pure in itself of the phrase ‘a man’, then it becomes empty. The meaning then, is not given to the phrase ‘a man’ by itself, but rather the complete holistic proposition in which ‘a man’ is nothing more than a constitute, a cog in a machine.
The Present King of France is Bald
If I were to say ‘the king of France is bald’, many people would think the sentence is nonsense. However, Russell argues that it is not nonsense but instead, merely false. The law of the excluded middle states that either ‘the present king of France is bald’ or ‘the present king of France isn’t bald’. But since there is no present king of France, most people think the sentence is not false but nonsense. However, Russell argues the proposition can be divided into two parts. Firstly, ‘there is a present king of France’ and secondly, ‘the king of France is bald’. Since it is false that ‘there is a present king of France’, the other constitute, which depends on the first one is also false.
Scott wrote Waverley
Russell claims that the utterances ‘Scott wrote Waverley’ and ‘the author of Waverley’ are treated as synonymous. However, Russell claims they refer to two different things. If I say ‘Scott was a man’, then I am saying ‘x was a man and x was human’. If I say ‘the author of Waverly was a man’, I am not saying the same thing as ‘x was a man and x was human’. In the utterance ‘the author of Waverley’, the gender of the author is unspecified.
By dividing propositions into their constitutes, we can see if the proposition is true or false. If I say ‘Zeus threw lightning bolts at me’, what I am saying is ‘There is an entity called x who exists’ and ‘x threw lightning bolts’ and ‘these lightning bolts were aimed at me’. Since ‘there is an entity called x who exists’ is false, then the following utterances are false as well. Russell’s project then, demonstrated how propositions which appear to be nonsense are not actually so, but rather false. There falsity is shown however when we separate the proposition into parts.