“Unless something weird has occurred, either the basketball game is
cancelled, the football game is cancelled, or Churchill never entered
Switzerland. If any game is cancelled, then something weird has
occurred. Nothing weird has occurred. Therefore, Churchill never
Let us translate the argument into its propositional form :
Unless something weird has occurred. Whenever you have "P unless Q", translate into "if not Q then P", i.e : (~Q ⊃ P) .
So, the first sentence is (W stands for "Weird") : (~W ⊃ P). What is P here? everything that comes after "occurred," until the end of the sentence.
P = either the basketball game is cancelled, the football game is
cancelled, or Churchill never entered Switzerland.
Now, your main difficulty is with "Either ... or ...", I agree that this is tricky, often logicians use it to mean inclusive "or".
But as the common language is ambiguous here, let us try to see whether it makes more sense with an exclusive or inclusive or. Mainly, is it possible for both disjuncts to make sense together:
- The basketball game is cancelled, the football game is cancelled.
- Churchill never entered Switzerland
The two disjuncts can be true together, there is nothing that prevents that, and if this colloquial argument meant an exclusive or, it would state it clearly. So, it is more likely that this is just an inclusive or
P= ((B v F) v C)
So, the first premise in the argument is :
(~W ⊃ ((B v F) v C))
The second premise is ~W (nothing weird has occurred).
The third premise is ((B v F) ⊃ C)) (if any game is cancelled then Churchill never entered Switzerland), only B and F stand for cancelled games.
The conclusion is C (Churchill never entered Switzerland).
I did not translate "never" to not-.. since the proposition is used in the negative in both times it occurred in the argument, for simplicity we use C
The argument :
- (~W ⊃ ((B v F) v C))
- ((B v F) ⊃ C))
- ∴ C
Suppose that we set the premises to true (1) and the conclusion to false (0)
- (~W ⊃ ((B v F) v C)) = 1
- ((B v F) ⊃ C)) = 1
- ~W = 1
- ∴ C = 0
If the premises are true and the conclusion false, then C and W have to be 0, we put 0 next to C and W.
If C is 0 then (B v F) has to be 0 for the conditional in the second premise to be 1, for the second premise to be 1. So we put 0 next to (B v F)
- (~W0 ⊃ ((B v F)0 v C0))
- ((B v F)0 ⊃ C0))
- ∴ C0
But the first premise then is false, since the antecedent ~W is 1 (because W is 0), and all the disjunction in the consequent has the truth value of 0. Which makes the first premise false in this case.
So, the argument is valid, since we could not make the conclusion false and the premises true.