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Lets suppose we want to investigate proposition "All Vatican anarchists are honest". We can transform this proposition into implication "If a citizen of Vatican is an anarchist then he/she is honest". If this implication is true for every citizen of Vatican, then proposition "All Vatican anarchists are honest" will be true.

Lets question the Pope. -- Hey Pope, are you an anarchist? -- No

Implication "If a citizen of Vatican is an anarchist then he/she is honest" is true in the Pope's case, because the Pope isn't an anarchist.

Lets assume we questioned every citizen of Vatican and none of them is an anarchist. So, the implication is true for every Vatican citizen. As a result, statement "All Vatican anarchists are dishonest" is false and "All Vatican anarchists are honest" is true.

Now lets pretend we have never questioned citizens of Vatican and begin from scratch. But this time we are going to test statement "All Vatican anarchists are dishonest". We can easily convert it into "If citizen of Vatican is an anarchist then he/she is dishonest". We interviewed all citizens and none of them is an anarchist, thus this implication is true for everyone. Thus statement "All Vatican anarchists are honest" if false and "All Vatican anarchists are dishonest" is true.

And now lets remember our previous survey. Holy cow, we have severe contradictions!

"All Vatican anarchists are honest" is true, thus "All Vatican anarchists are dishonest" is false.

BUT at the same time "All Vatican anarchists are dishonest" is true, thus "All Vatican anarchists are honest" is false.

KarmaPeasant
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2 Answers2

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In your first survey, you establish firmly that "All vatican anarchists are honest". However, this is not the negation of "All vatican anarchists are dishonest" - and this is why you get a contradiction. In fact, the negation of the first statement is, "There is a vatican anarchist who is dishonest" - which is false by your survey.

The second survey, done rightly, should establish, "All vatican anarchists are dishonest" and that its negation, "There is an honest vatican anarchist", is false. Thus, when we write out the "contradictory" statements formed correctly, we now have

There is no honest vatican anarchist.

And

All vatican anarchists are honest.

Both of which are true - the first premise tells us that no honest vatican anarchist can exist, and the second tells us that any extant vatican anarchist is honest - and taken together, they mean no vatican anarchist exists.

Milo Brandt
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  • "this is not the negation"
    I know. They are contrary, thus they can't be true at the same, but they can be both false.     – KarmaPeasant Dec 21 '14 at 05:16
  • That's not true either. Think about it this way, if you have "All Vatican Anarchists are Honest" and "All Vatican Anarchists are Dishonest", the way you'd wish to prove those contradict is "Choose some Vatican Anarchist. They cannot be both honest and dishonest, therefore one of the statement is false" - however, when we write "Choose some Vatican Anarchist", we are assuming there is one. If there is none, then our proof that the statements contradict each other does not function - and, as it happens, both statements are true. – Milo Brandt Dec 21 '14 at 05:21
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I have this same sort of complaint at the beginning of every sports season. The sports section begins the season by printing the statistics (thus far) of all teams in the league to be covered. Although there are $0$ wins and $0$ losses, as it should be, a third statistic is given -- games won divided by games played. At the start of the season, this third stat yields $\frac00$. The newspaper prints it as $.000$. The trouble with this is that while it is true that a team with this stat has LOST every game it has played thus far (average is $.000$), it is equally valid to say that the team has WON every game it has played thus far (average is $1.000$. That's why $\frac00$ is called an indeterminate form.

Senex Ægypti Parvi
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