Formalising Definite Description (Russell)

Question

So, we've been doing a lot of formalising of definite descriptions according to Russell, where we take the Definite Description as a predicate rather than designator.

So, I've formalised sentences like "The major of london ......" with this kind of predicate:

Mx: x is a major of London

But, my question is if I can formalise this even more?

On a recent test I, out of habit, made this key (because I was once told "the more we formalise, the better"):

Mxy: x is a major of y

l: London

But in hindsight I fear this may have been a mistake ...

I translated the sentence "The major of London is happy" as

$$ \exists x \Big(Mxl \land \forall y(Myl \rightarrow y=x) \land Hx \Big)$$ but, as said, I fear it that this is instead correct: $$ \exists x \Big(Mx \land \forall y(My \rightarrow y=x) \land Hx \Big)$$ where Mx stands for 'x is a major of London'.

Any ideas? I can't find anywhere online that says anything about this. The only examples I can find is of the second style, which is why I am worried.

Thanks :)

Answer 1

When you modified the predicate $M$ to mean "$x$ is a mayor of $y$" (in the second gray box), the notation should have become $Mxy$ instead of $Mx$. Apart from that, your proposed formalization looks fine to me.

Answer 2

A good principle in formalizing is expose just as much logical structure as is relevant to the case in hand.

Suppose you were interested only in the argument "Boris is the mayor of London. If Boris is the mayor of London, then business won't be scared away. So business won't be scared away." All the structure you need to expose to show this argument is valid is revealed by $P, P\to Q \therefore Q$.

Suppose next you are interested in the argument "Boris is blonde, Boris is the current mayor of london, so whoever is the current mayor of london is blonde". Here we do need to expose some more structure, but talk of mayors only appears in the context "is the current mayor of London". Then you might as well treat "is the current mayor of London" as a fused whole, with no locally significant internal structure, and symbolise "Boris is the current mayor of London" as $Mb$.

But suppose that you want to formalize the argument "Boris is the mayor of London, Bill is the mayor of New York; different cities have different mayors, London is not New York, hence Boris is not Bill". Then you will evidently need to expose the relational structure in the claim "Boris is a mayor of London", and formalize that as $Mbl$.

So there is no one "right" formalisation for "Boris is a mayor of London" or "Boris is the mayor of London": how much logical structure to expose in your formalisation will depend on context.