Constructing an Argument:
The Technique of Finding the Middle Term

Contact: Dr. Jan Garrett

This page revised January 13, 2004

When we are trying to prove a statement, often we are trying to establish a link between two concepts (or between two simpler) statements.

We are trying to change the mind of those who may be doubtful or less than certain about the link. In order to succeed in this we must appeal to reasons or premises that our listeners or readers originally find more plausible than they do the conclusion. But the reasons themselves must be linked appropriately.

Example 1: A Simple Argument

Suppose you were trying to convince somebody, who already accepts that there is a God (Note), that God is wise. They are not quite ready to accept this conclusion, or at least they wonder why they should accept it. One solution would be to find a third concept that can be linked both to God and to wisdom in ways that are plausible to your audience. This is known as finding the middle term.

If we think carefully about our ideas concerning God, one such idea is that God is superior to us. Your audience might be willing to accept that God is so superior to us that God has all the very best qualities. Well, perhaps we could also link that notion to wisdom. (Surely wisdom is a lot better than its opposite or an intermediate state of opinion that happens to be right but is not accompanied with understanding.)

So we might have two links that we can use to support the link between God and wisdom we are aiming at:

God -- very best qualities
wisdom -- one of the very best qualities
We then proceed to construct the argument
(Premise)     God has all the very best qualities.
(Premise)     Wisdom is one of the very best qualities.
(Conclusion) Therefore God is wise.
Connecting three terms in a three statement argument, with the middle term shared between the premises, and the two terms of the conclusion shared with the two premises, does not always produce a valid or strong argument. Other conditions must also hold. Describing thoroughly how to ensure they are connected in the right way would require a long detour into the subject-matter of a course in Logic.

But there is an intuitive test. An intuitive test of the strength or validity of an argument is to ask yourself whether the conclusion really receives from the premises the degree of support the author is claiming.

In arguments that are deductive, as mathematical arguments and many philosophical arguments (including this one) are, the question becames: If, for the "sake of argument," we assume the premises true, does the conclusion have to be true?

(The preceding argument is in fact valid, that is, if the premises were true, the conclusion would have to be. It could still be unsound, and would be if either of the premises is false.)

Example 2: A More Complex Argument

Plato once (in his Laws, book X) tried to prove that the gods exist by proving first that the stars were gods. It is obvious that if the stars are gods, then gods exist (even if the stars may not be the only gods).

What he did was to find some attributes common to gods and stars. The idea of gods must be familiar: indeed the idea of gods would have been familiar to Greeks of Plato's time, since they had been raised on Homer's tales of gods and heroes.

Now, the gods were typically portrayed as superior to human beings. They were sometimes given credit for planning the universe so they had to be thought of as intelligent. People sometimes made sacrifices to the gods in thanks for what they thought the gods did for them, so the gods were conceived as beneficial. And the gods' superiority to humans seemed to imply that they were immune to weaknesses humans have, even immortal or everlasting.

gods -- intelligent
gods -- everlasting
gods -- beneficial
We might express these links as follows
(G): Whatever is intelligent, everlasting, and beneficial is a god.
Note that (G) might be true even if there were no gods. All (G) commits one to is that if anything has all three attributes, then it is a god.

Now, can we connect these attributes of the gods to stars?

Plato appealed to observation: it seems as if the stars do not change: so far back as we can remember, there have always been the same stars. (He was wrong about this: new stars were in fact reported by the ancient Chinese observers; but it is understandable that even a moderately good observer might have found Plato's claim plausible.)

Thus we might try the following two links:

stars -- appearance of not changing
appearance of not changing -- everlasting nature
From which one might construct the argument:
1, P) The stars aren't seen to change.
2, P) Whatever isn't seen to change is an everlasting being.
3, IC) They are everlasting beings.

3 is labeled an intermediate conclusion because it will be used later as a premise.

Plato argued for the intelligence of the stars by connecting intelligence
to the pattern circular movements of the stars. In other words, the middle term had to do with perfectly patterned movements.
stars -- perfectly patterned movements
perfectly patterned movement -- intelligence
From which one might construct the argument:
4, P) The stars move in perfectly patterned ways.
5, P) Whatever moves in a perfectly patterned way displays intelligence.
6, IC) The stars are intelligent beings
And from the part played by regular stellar movements in navigation and time-keeping, one can see the beneficial nature of the stars. (If that were not persuasive, one could refer to the special role of the sun, another heavenly body often grouped with the stars, and refer to causing the seasons, so important for agriculture.)
7, P) The movements of the stars make us able to tell time accurately.
8, P) The capacity to tell time accurately is beneficial
9, IC) The movements of the stars are beneficial.
Thus we have three links
stars -- intelligence
stars -- beneficial nature
stars -- everlasting
Now we're ready to pull it all together:
3, IC) The stars are everlasting beings. (repeated)
6, IC) The stars are intelligent. (repeated)
9, IC) The movements of the stars are beneficial. (repeated)
10, IC) The stars are everlasting, intelligent, and beneficial.
  (based on 3, 6, 9)
Now put (10) together with (G) (note that the middle term is "everlasting, intelligent and beneficial"):
10, IC) The stars are everlasting, intelligent, and beneficial. (repeated)
11, P) Whatever is everlasting, intelligent, and beneficial must be a god. (= G)
12, FC) The stars are gods. (10, 11) ("FC" stands for final conclusion.)
Now, we can express this argument more prosaically, i.e., informally, perhaps as follows:
Hear me now and believe me later. The stars are gods, for whatever is everlasting, intelligent and beneficial must be a god. And the stars have all three qualities. They don't seem to change, as anyone can observe. So they must be everlasting. They move in perfectly patterned ways, and whatever moves in perfectly patterned way displays intelligence. So they must be intelligent. What's more, the stars are beneficial because we are able to tell time accurately thanks to stellar movement.

Note. You can think through this example with "the gods" or "the goddess" instead of "God" if you wish. In fact, my original example was "the gods," but I changed the example so that its subject-matter would be more familiar to most students. The ancient Stoics appear to have reasoned about the supreme deity in this way, though technically they were still polytheists.