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?

Answer 1

To represent the given argument in a logical form, you can use the valid argument pattern known as modus ponens. The argument can be broken down into premises and conclusions.

Step-by-step explanation:

To represent the given argument in a logical form, you can use the valid argument pattern known as modus ponens. The argument can be broken down into the following premises and conclusions:

1. 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: ¬(P & S) ⊃ R
2. If it's not the case that Reverend Green was the murderer, then Miss Scarlet was in the conservatory and Colonel Mustard was there too: ¬R ⊃ (S & *)
3. Conclusion: If Reverend Green is not the murderer, then Professor Plum was in the study and Colonel Mustard was there too: ¬R ⊃ (P & *)

Regarding the representation of 'Colonel Mustard was there too', you can use a single proposition, such as M, to represent Colonel Mustard's whereabouts in both cases. So, the final logical form would be: ¬(P & S) ⊃ R, ¬R ⊃ (S & M) ⊢ ¬R ⊃ (P & M).