The key is to distinguish a missing middle term from an undistributed middle term. They are related, but not exactly the same thing.
The statement:
“You cannot jump from the pagan sources over to the Bible because there is no middle term to link the two together.”
is describing an argument that lacks a justified connecting premise. It can illustrate the problem that the fallacy of the undistributed middle exploits, but strictly speaking, we need to reconstruct the argument to see the fallacy.
In a categorical syllogism, the middle term is the term that occurs in both premises but not in the conclusion. It is supposed to establish a connection between the two things you are trying to relate.
For example:
All M are P.
All S are M.
Therefore, all S are P.
Here M is the middle term. It connects S to P:
S → M → P
For example:
All mammals are animals.
All dogs are mammals.
Therefore, all dogs are animals.
“Mammals” is the middle term connecting “dogs” and “animals.”
A term is distributed when the proposition makes a claim about all members of the class represented by that term.
Consider:
All cats are animals.
“Cats” is distributed because the statement is about all cats.
But “animals” is not distributed, because we are not saying anything about all animals. We are only saying that cats belong to the class of animals.
So an undistributed middle occurs when the supposed middle term is used in both premises but never refers to the whole class, meaning that the premises haven't actually established the necessary connection.
Classic example:
All dogs are animals.
All cats are animals.
Therefore, all cats are dogs.
The middle term is “animals.”
The structure is:
Dogs → animals
Cats → animals
Therefore, cats → dogs
But merely sharing a category does not establish that the two things are identical or otherwise connected.
The problem can be visualized like this:
Dogs → Animals ← Cats
The arrows converge on “animals,” but there is no logical path from dogs to cats.
Suppose someone's argument is roughly:
Pagan source X says Y.
The Bible says Y.
Therefore, the Bible got Y from pagan source X.
The supposed connection might look like:
Pagan source X → Y ← Bible
The common term Y is functioning as the supposed “middle.”
But merely establishing that two sources contain the same idea doesn't establish a causal or historical connection between them.
For example:
Pagan mythology contains a flood story.
Genesis contains a flood story.
Therefore, Genesis borrowed its flood story from pagan mythology.
The common element is “flood story.”
But:
Pagan source → flood story ← Genesis
doesn't establish:
Pagan source → Genesis
There could be other explanations:
both accounts derive from a common earlier tradition;
the biblical account is independent;
the pagan account borrowed from an earlier tradition;
both preserve memories of a historical event;
the similarities are coincidental;
etc.
So the evidence establishes similarity, but not necessarily dependence.
This is where I would make an important distinction.
If someone says:
“There is no middle term linking the pagan sources to the Bible,”
they may mean that there is no established proposition connecting the two.
For example:
Pagan source X contains idea Y.
The Bible contains idea Y.
Therefore, the Bible derives Y from pagan source X.
What's missing is something like:
The Bible's account of Y is historically dependent upon pagan source X.
That additional premise would provide the necessary bridge:
Pagan source X → historical dependence → Bible
Without it, we're essentially jumping from similarity to dependency.
So the problem can be represented:
Pagan source X has Y.
Bible has Y.
∴ Bible got Y from X.
The conclusion doesn't follow because “has Y” doesn't establish “derived from X.”
Not necessarily. This is an important qualification.
The phrase “there is no middle term” is sometimes used loosely to describe a missing logical connection. But the undistributed middle is a specific formal syllogistic fallacy.
To demonstrate undistributed middle specifically, we need an argument in which a term actually appears in both premises but is undistributed in both.
For example:
Pagan sources contain stories about divine beings.
The Bible contains stories about divine beings.
Therefore, the biblical stories came from pagan sources.
Here “stories about divine beings” is the supposed middle:
Pagan sources → stories about divine beings ← Bible
But neither premise establishes that all such stories belong to one particular source or that the two bodies of literature are historically connected.
So the argument commits the same structural error as:
All A are C.
All B are C.
Therefore, all B are A.
The shared characteristic is being mistaken for a logical or causal connection.
The phrase you quoted is getting at this fundamental principle:
You cannot establish a connection between A and B merely by showing that A and B share some characteristic C.
That's the essence of the undistributed-middle problem:
A → C ← B
doesn't logically give you:
A → B
So if someone argues from:
“The pagan sources contain X, and the Bible contains X”
to:
“Therefore the Bible got X from the pagan sources,”
they need an additional connecting premise establishing historical dependence. Without that bridge, the argument has not established the conclusion.
And strictly speaking, if the argument doesn't actually have a middle term occurring in both premises, calling it “undistributed middle” may be imprecise; it may instead be better described as a non sequitur, affirming the consequent, or an unsupported causal inference, depending on the exact form of the argument.