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Guarded With a Twist of Logic

Consider this "logical" argument....adapted from W.V. Quine's Methods of Logic (1950)....

If the guards searched all who entered the premises except those who were accompanied by members of the firm, and if some of Fiorecchio's men entered the premises unaccompanied by anyone else, and if the guard searched none of Fiorecchio's men; then some of Fiorecchio's men must have been members of the firm.

Your Task: Determine if this argument is logical (i.e. is the conclusion "some of Fiorecchio's men must have been members of the firm" warranted).

 

Source: C. Denbow & V. Goedicke. Foundations of Mathematics. 1959, pp. 242, 256-57


Hint: Can you draw Venn Diagrams to represent the situation and argument...you will need four closed regions?

 


Solution Commentary: Denbow & Goedicke suggest: Draw three overlapping circles, labeled G (those searched by guards), P (those who entered the premises), and A (those accompanied by members of the firm).

Now add an irregular closed region that intersects each of the other eight regions, calling it F (Fiorecchio's men).

To claim that the guards searched all who entered the premises except those who were accompanied by members of the firm (i.e. all P are G or A), shade out regions in P outside of G and A.

Now, jump to "the guard searched none of Fiorecchio's men" (no F are G), so shade out region which lies in both G and F

Now, we know that some of Fiorecchio's men entered the premises (some F in P), put an X in the only remaining unshaded region common to F and P.

Finally, since the X lies in a region interior to A, we can conclude that "some of Fiorecchio's men must have been members of the firm."

But, since Fiorecchio's men entered the premises unaccompanied by anyone else, the members of the firm who accompanied them must have been part of their own group, that is in F.

Thus, the conclusion is valid!