Natural language into logic and proof

Question

I'm working through Logic by Paul Tomassi, and there is one particular problem I'm stumped with. The problem is on pg 186 and involves representing an argument in English as a sequent and then constructing a proof for it:

'If it’s not the case both that Professor Plum was in the study and Miss Scarlet was in the conservatory then the murderer was Reverend Green. But if it’s not the case that Reverend Green was the murderer then Miss Scarlet was in the conservatory and Colonel Mustard was no doubt there too. So, if Reverend Green is not the murderer then Professor Plum was in the study and Colonel Mustard was there too.'

The first part seems fairly straight forward.

P – Professor Plum was in the study. S – Miss Scarlet was in the conservatory. R – Reverend Green was the murderer.

This seems to lead to an argument with the following form:

¬(P & S) ⊃ R, ¬R ⊃ (S & *) ⊢ ¬R ⊃ (P & *)

Where * represents Colonel Mustard's whereabouts.The tricky bit seems to involve the ambiguity of the phrase 'Colonel Mustard was there too'. I've tried different ways to represent this, but I can't seem to get it to work. Which, if any of these could do the trick.

One proposition for both cases: M – Colonel Mustard was there too.

Two propositions for each case: M1- Colonel Mustard was in the conservatory. M2 - Colonel Mustard was in the study.

Or maybe one proposition which is then negated in the conclusion.

Is there another way to represent this argument that I'm overlooking?

Thanks so much for any answers.

Answer 1

It would be a good idea to use two different proposition symbols, one for Mustard is in the conservatory and one for Mustard is in the study. Otherwise the expression "there too" is ambiguous.

But the text is very odd and I wonder if you have it right. If Rev Green is not the murderer then the second premise tells us Mustard is in the conservatory, but the conclusion tells us Mustard is in the study. If we help ourselves to the unstated premise that Mustard cannot be in both places at once, then these can only both be true if Rev Green is the murderer.

Hence the argument is not valid, since a countermodel would exist when Rev Green is not guilty, Plum is in the study, Scarlet is in the conservatory and Mustard is in the conservatory.

Maybe the purpose of the example is to get you to recognise that "there too" is ambiguous, otherwise the example is flawed.