# The Hebrew (Jewish) Calendar Paradox

Uri Geva
MathVentures, a Division of Ten Ninety

Document Version 0.2 (draft)
This work comes with absolutely no warranty!

## Abstract:

The Hebrew (Jewish) Calendar Paradox is, as far as I know, the only true logical paradox that is the direct outgrowth of a legal system. It is caused by the fact that in Jewish legal system, there is a law that defines time measurement and time measurement is used to rule whether a law is valid or not. This gives rise to circular reference and, more specifically, to what I call the vicious Mobius reference.

## The Paradox Definition -- The Conditions that Set Forth the Paradox:

The Hebrew Calendar Paradox is the result of a specific combination of Jewish laws:
1. The Hebrew day starts at the evening and spans the night and the following daytime till the next evening.

2.
3. On any day, a judge can declare that the current day is the first day of a Hebrew month if at least two witnesses testify that they did not see the moon during the previous night. (We can assume that the law presumed that, if the moon were visible, the witnesses would have seen it; that is, the skies were not obscured by clouds and the witnesses were in a position and alert to see the moon throughout the night.)

4.

It is important to keep in mind the following fact: This legal discussion takes place during the day time after the beginning of the day in question. In other words, the judge has to determines weather or not an even took place during the previous evening and, if it did, it signaled that the current day is the first day of a new month. That is, the judge's decision is retroactive.

5. A single witness, who testifies that he did see the moon, is sufficient to refute these two witnesses. In this case the judge may not declare the day as the first of the month.

6.
7. A person is a valid witness if he is an adult man. [Note that masculinity has in fact nothing to do with this paradox.]

8.
9. An adult man is one who is at least thirteen-year old.

## Effects:

1. The effect of law #1 and law #2, taken together, is that the beginning of a new month can be determined only in retrospect. That is, if on a certain day a judge declares that this day is the first day of a month, then the new month started on previous evening.

2.
3. Considering the following scenario some ancient wise rabbis declared a stalemate (In Hebrew they said: "Tyku", the acronym for "Tishbi Yitaretz Kooshiot Ve’ba’ayot", which roughly translates as "the Messiah will explain questions and problems.")
One morning two witnesses come to court and each of them declare to the judge that he did not see the moon during the previous night. A boy comes forward and testifies that he did see the moon. The judge asks the boy for his age and the boy replies: "On the first of the month I will be thirteen." If the judge accepts the testimony of the two witnesses, then the boy is thirteen-year old and thus his testimony is valid. Therefore, he refutes the two witnesses. Rejecting their testimony means that it is not the first day of the month and thus the boy is too young to qualify as a witness. So he cannot refute the two witnesses and the judge must accept their testimony that a new month has started. In which case the boy is of age and...
In modern times it is clear that root of the paradox is a circular dependency: A matter of law defines a matter of time and a matter of time defines a matter of law. Evidently the wise rabbis did not recognize that such reasoning is not valid.

The paradox cannot be avoided by altering the number of witnesses or counter witnesses or by changing the qualifying age to be a witness. For example, If one must be at least thirteen-year old and one day or at least one day less than thirteen-year old there may be a case in which a boy witness who is exactly a day older or a day younger than thirteen on the first of the month. In other words, this paradox is independent of the number of witnesses and counter witnesses and the age at which a the witness' testimony becomes valid.

A negating twist must exist amongst a finite set of proposition for such a paradox to occur. Indeed, this set of propositions contains such a negation twist. The Mobius vicious circle principle, as I call it, is a necessary and sufficient condition to generate such paradoxes as Russell's paradox and the liar's paradox.

## Open Questions:

• All of the information I have about the Hebrew Calendar Paradox is based on childhood recollections. I do not know for a fact that it truly exists. Therefore, I am looking for any information that can confirm or refute its existing, shed more light or provide any additional information about it.