1

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 entered Switzerland.

I have trouble whenever “either or” comes into play. Is this argument valid or invalid, and why is that the case? The “either or” phrase confuses me and complicates the argument to no end.

  • What do you find confusing about "either or"? – Eliran Feb 9 at 4:26
  • 2
    In basic argument forms, nothing. But when it’s “either or” between three different events, coupled with a bunch of other premises (as in the example above), I lose the ability to determine validity. – A. Delarge Feb 9 at 4:29
  • The title edit doesn't make sense. The argument is not about "either or" in any meaningful way. – Eliran Feb 11 at 5:41
3

The argument is valid, and happily this doesn't depend on whether the disjunction is exclusive or inclusive.

Firstly, 'unless' has the same truth table as 'if not', so we have

  1. If it is not the case that something weird has occurred, either the basketball game is cancelled, the football game is cancelled, or Churchill never entered Switzerland.

We are given that

  1. Nothing weird occurred.

So from 1 and 2 we can detach to get

  1. Either the basketball game is cancelled, the football game is cancelled, or Churchill never entered Switzerland.

We are also given

  1. If any game is cancelled, then something weird has occurred.

But this together with 2 gives by modus tollens

  1. No game was cancelled.

3 and 5 now assure us that

Churchill never entered Switzerland.

2

I agree with @Bumble's answer. I offer this as a different way to approach the problem.

Since validity depends on the form of the argument and not the meaning of the English language sentences, the meaning of those sentences may become distracting. It may be worthwhile to abstract away that meaning by creating a symbolization key for the sentences and then checking the validity with a proof checker.

This introduces some risk since the symbolization key may be a source of error. Hopefully it avoids the distraction that paying too much attention to the English language meaning provides.

Here is the argument:

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 entered Switzerland.

Here is a proposed symbolization key:

W : Something weird occurs.
B : The basketball game occurs.
F : The football game occurs.
C : Churchill entered Switzerland

Here is the argument using these symbols:

Premises:

  1. ¬B → W If any game is cancelled, then something weird has occurred, that is, if basketball is cancelled, then something weird occurred.

  2. ¬F → W If any game is cancelled, then something weird has occurred, that is, if football is cancelled, then something weird occurred.

  3. ¬W → [¬B ∨ (¬F ∨ ¬C)] Unless something weird has occurred, either the basketball game is cancelled, the football game is cancelled, or Churchill never entered Switzerland. As Bumble notes: "'unless' has the same truth table as 'if not'".

  4. ¬W Nothing weird has occurred.

Conclusion:

¬C Churchill never entered Switzerland.

The following proof shows that the argument is valid:

enter image description here

Conditional elimination (→E) can be found in section 15.3 of forallx. Contradiction introduction (⊥I) occurs if two lines contradict each other. Negation introduction (¬I) can be found in section 15.7. Disjunctive syllogism (DS) can be found in section 16.2.

Although in this particular example, using a proof checker may be more difficult than keeping the English language meanings in mind, it provides a way to check one's answer if one is only concerned about the validity of the argument.


Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

  • 1
    I agree with your answer too. The reason I decided not to render the sentences directly into the propositional calculus was that the disjunction present in the first sentence could easily be read as exclusive; indeed the poster in the original version of their comment says that they do so. The point of my proof is that I do not depend on an inclusive reading of the disjunction. I suppose I could have done two versions. – Bumble Feb 9 at 16:02
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“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 entered Switzerland.”

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))
  • ~W
  • ∴ 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))
  • ~W0
  • ∴ 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.

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