Kashrut Through the Lens of Information Theory: The Hidden Logic of Redundancy

The Torah is often admired for its brevity.

Yet in some places, it appears surprisingly redundant.

Consider the two most familiar signs of kashrut.

Regarding fish, the Torah states:

These you may eat of all that are in the waters: whatever has fins and scales in the seas and in the rivers, them you may eat. But whatever has not fins and scales… shall be an abomination unto you.

These you may eat of all that are in the waters: whatever has fins and scales in the seas and in the rivers, them you may eat. But whatever has not fins and scales… shall be an abomination unto you.

At first glance, this seems perfectly straightforward.

But the Mishnah records an important biological observation:

Every fish that has scales also has fins, but there are fish that have fins and do not have scales.

Every fish that has scales also has fins, but there are fish that have fins and do not have scales.

Maimonides codifies the same principle:

Fish have two signs: fins and scales… Every fish that has scales also has fins.

Fish have two signs: fins and scales… Every fish that has scales also has fins.

— Mishneh Torah, Forbidden Foods 1:24

In other words, the requirement for fins is logically redundant.

If every fish with scales necessarily has fins, why didn’t the Torah simply say:

All fish with scales are kosher.

The question becomes even stronger when we turn to land mammals.

Whatever parts the hoof, is cloven-footed, and chews the cud—among the animals—you may eat.

Whatever parts the hoof, is cloven-footed, and chews the cud—among the animals—you may eat.

It then lists four apparent exceptions:

Three chew their cud but lack split hooves.

One—the pig—has split hooves but does not chew the cud.

Remarkably, both the rabbinic tradition and modern observation have long regarded the pig as uniquely occupying this category among the land mammals familiar to the Torah’s original audience.

Maimonides goes even further. He states as a general principle:

Every animal that has split hooves chews the cud except the pig.

Every animal that has split hooves chews the cud except the pig.

— Mishneh Torah, Forbidden Foods 1:2–3

He presents this not as a legal rule but as a fact of nature used to identify kosher animals.

All split-hoofed mammals are kosher except the pig.

That would be both shorter and logically equivalent.

The rabbis recognized this apparent redundancy long ago.

Regarding fish, the Talmud itself raises the question: why does the Torah mention fins if every fish with scales necessarily has fins?

Over the centuries, several........

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