Lesson 9: Demonstration


In our last two lessons we talked about the fundamental tool of the judging part of logic, the syllogism. St. Thomas told us that judgment implies certainty. The syllogism provides us with a conditional certainty by its form alone; that is, if the premisses of the syllogism are true, then the conclusion certainly is true. We want, however, a logical tool that will provide us with absolute certainty. That kind of certainty will be provided by the tool called demonstration, or the demonstrative syllogism. Demonstration uses the form of the syllogism, and adds to it a certain matter to produce an absolutely certain conclusion. This lesson will be a discussion of demonstration.

Aristotle discusses demonstration in his book called the Posterior Analytics. He begins that book by referring to the problem of learning. Since the classic statement of that problem is given before Aristotle, by Plato, we are first going to look at his dialogue called Meno to see what exactly the problem of learning is.

The Problem of Learning

We talked about Meno before in our discussion of the definition. Meno asks Socrates whether virtue is teachable, and Socrates responds that he cannot answer that question until he knows what virtue is. After Meno gives several bad definitions of virtue, he decides to give up the search for the definition of virtue, and poses the problem of learning as the cause. This is how the problem is stated in the dialogue:

The learner cannot search for what he knows since then there is no need to search. Nor for what he does not know, for he does not know what to look for.

For example, if the learner already knows what virtue is, then there is not need to learn it. But if he does not know what virtue is, then even if someone told him the definition he would not recognize it as the true definition of virtue. Therefore, it is impossible to learn what virtue is.

Plato himself, through the mouth of Socrates, gives as an answer to this problem his theory of recollection. Plato claims that before we were born our souls existed in another realm in which we knew everything that we later say that we are learning. When we are born, we forget that knowledge, and the process called learning is simply the recalling, or recollecting, of knowledge that we possess already but in a hidden way. We can restate this position in another way: we already know the answers to all the questions, but our own knowledge is hidden from us, and the process of learning is simply the process of uncovering that hidden knowledge.

Aristotle has several reasons for rejecting Plato's solution to the problem of learning. First, he does not think that the soul can exist before the body. Second, he thinks the notion of "hidden knowledge" is incoherent. We will give his reasons later in this lesson. Third, and most important for our immediate purposes, he thinks that he can solve the problem of learning without appealing to hidden knowledge. Aristotle believes he can use his doctrine of the syllogism to solve the problem of learning.

As we saw before, if the premisses in a syllogism are true, then the conclusion necessarily is also true. But recognizing the truth of something which one did not previously know is learning. The syllogism, then, solves the problem of learning. It gives a way for the learner to recognize the truth of the conclusion, namely, because it follows from the truth of premisses the truth of which he already knows. That is, he can use the knowledge that he already possesses in order to learn what he does not yet know.

In the Posterior Analytics Aristotle is talking about learning in the strongest sense of the term, acquiring certain knowledge of some new truth. And the logical tool that we use to acquire certain knowledge of a new truth is called demonstration. Our next task then is to look at Aristotle's account of demonstration.

The Definition of Demonstration

In Chapter Two of his Posterior Analytics, Aristotle gives the following definition of demonstration. He writes:

By demonstration I mean a syllogism productive of scientific knowledge. A syllogism, that is, the grasp of which is in itself such knowledge.

This definition, like every good definition, has a genus and a specific difference. The genus is obvious: it is "syllogism," and we have already given a lengthy account of what a syllogism is. Let us move right to the specific difference, "producing scientific knowledge."

In the modern world, the word "science" usually refers to knowledge of physical things that is acquired through systematic experimentation and that is expressed mathematically. A modern man would call physics, chemistry, or astronomy "sciences," but he would be reluctant to say that ethics or philosophy is a "science." After all, in ethics and philosophy there is no experimentation and no mathematical rigor, and these disciplines are not about physical things.

If we go back to ancient or medieval times, however, we find that the word "science" had a much different meaning. The ancients meant by "science" any kind of knowledge which was gained through the process of reasoning and which achieved a great degree of certainty. Consequently, modern men refuse to call science many of the disciplines which the ancients did. For example, the ancients and medieval called ethics the "moral science" because there is reasoned out certain knowledge about ethical matters, but few modern men would call ethics a science.

And on the other hand, certain things that modern men call science, the ancients might be reluctant to give that label to. For example, modern men would tend to call some of the more conjectural theories of modern physics advances in "scientific knowledge," but the ancients would not because these theories do not achieve the certainty which they considered requisite to science. Thus when Aristotle says that demonstration is a syllogism which produces "scientific knowledge," he means it produces reasoned-out and certain knowledge on any subject.

In the Posterior Analytics Aristotle gives us a more precise account of what he means by scientific knowledge. He writes:

We suppose ourselves to possess unqualified scientific knowledge of a thing when we think that we know the cause on which the fact depends as the cause of that fact and of no other, and further that the fact could not be other than it is.

Perhaps we could rephrase Aristotle's definition this way: I have scientific knowledge of a fact when I know that some particular fact absolutely must be true, and when I know exactly why that particular fact is true.

We can use the example from the last lesson to illustrate what Aristotle means. The following syllogism also constitutes a demonstration.

All three-sided figures have angles that add up to 180 degrees.
All triangles have three sides. 
Therefore all triangles have angles that add up to 180 degrees.

This syllogism constitutes a demonstration first because it proves that a certain fact cannot be otherwise than it is. That is, we know both that every triangle has to have three sides and that a three-sided figure has to have angles that add up to 180 degrees. Since those two propositions are necessarily true and the conclusion necessarily follows from them, the conclusion is also necessarily true. Thus, our syllogism shows that this fact is necessarily true.

Furthermore, it gives the reason why this particular fact is true. When we study geometry, we find that the sum of angles in any geometrical figure varies according to the number of sides that such a figure has. For instance, the sum of angles in a four-sided figure adds up to 360 degrees, that of a pentagon is 540 degrees. When someone asks us why a triangle has angles that add up to 180 degrees, our answer is "because a triangle has three sides." Thus, our syllogism not only shows that the conclusion has to be true, it also shows precisely why this particular conclusion is true. It is a demonstration in the strict sense of that term.

It is clear now what Aristotle means when he defines demonstration as a syllogism which produces scientific knowledge. But Aristotle gives another definition of demonstration, a definition that St. Thomas in his commentary says is a definition through the matter of the syllogism rather than through its end.

When St. Thomas distinguishes definition through end and definition through matter, he is making a subtle philosophical point. Every tool has two aspects, an end and a matter. The end is the purpose of the tool, that is, what the tool is made for, while the matter is what the tool is made of. Since different tools are made of different materials because they have different purposes, we can deduce the matter of a tool from its purpose. Then in the definition of the tool we can take either its purpose or its matter as the specific difference.

For instance, I can define the saw as "a carpenter's tool for cutting large pieces of wood." Such a tool has to be made of a handle and a metal blade with teeth, because only something with such a handle and such a blade could cut large pieces of wood. Thus I can give a second definition of the saw in which the specific difference is taken from the matter rather than from the purpose: a saw is a carpenter's tools made of a handle and a metal blade with teeth.

Likewise, there are two definitions of the logical tool which is demonstration. After giving a definition from its purpose, Aristotle defines demonstration through the nature of its matter, that is, its premisses. Because demonstration has as its purpose achieving scientific knowledge, it has to be made of premisses which can produce scientific knowledge. Thus Aristotle gives the following account of the premisses of demonstration. He writes:

Assuming then that my thesis as to the nature of scientific knowing is correct, the premisses of demonstrated knowledge must be true, primary and immediate, better known than and prior to the conclusion, which is further related to them as effect to cause.

Here Aristotle has given a list of the four prime characteristics of demonstrative premisses.

First, the premisses of a demonstration have to be true. We recognize the truth of the demonstrated conclusion because it follows from the truth of the premisses. But if the premisses of the syllogism are false, they cannot help us recognize the truth of the conclusion, even if the conclusion happens to be true. It is possible to draw a true conclusion from false premisses, but it is impossible to know that a conclusion is true through false premisses. Thus the premisses of a demonstration must be true.

Second, the premisses of a demonstration have to be first and immediate. Aristotle means by "first and immediate" not being proved by some prior syllogism. That is, the demonstrations have their ultimate source in a starting point which is self-evident. For example, the premiss "All triangles have three sides" is first and immediate: geometry states this truth but does not try to prove it. How we come to know the self-evident premiss is something he discusses later, when he completes his solution to the problem of learning.

Third, Aristotle says that the premisses of a demonstration have to be better known than and prior to the conclusion. That is, the premisses are prior to the conclusion in the order of knowledge. Recall what we said about the prior and posterior in knowledge in the Categories: A comes before B in the order of knowledge when I can know A without knowing B, but I cannot know B without knowing A.

This is clearly going to be the relationship between the premisses of the demonstrative syllogism and its conclusion. We can know the premisses of a demonstrative syllogism individually, and fail to put them together in a syllogism. Then we know the premisses but we do not know the conclusion. But we cannot know the truth of the conclusion without beforehand knowing the truth of the premisses. Therefore premisses are prior to and better known than the conclusion in the order of knowledge.

For instance, we can know that a triangle has three sides without knowing that its angles equal 180 degrees. We cannot know that its angles equal 180 degrees without knowing that it has three sides. Thus, in the science of geometry the fact that a triangle has three sides is better known than and prior in the order of knowledge to the fact that a triangle has angles that equal 180 degrees.

The last characteristic of the premisses of a demonstrative syllogism is that they are the cause of the conclusion. What Aristotle means by this is that the premisses of the syllogism indicate the cause, the reason why the conclusion of the syllogism is true. And this clearly follows from the definition that we gave before for scientific knowledge. If demonstration is to produce scientific knowledge, it has to be a syllogism which gives us the reason why the conclusion is true. Now the only thing in the syllogism that can do that are the premisses. The premiss "triangles have three sides", gives us the reason why the conclusion "triangles have angles that add up to 180 degrees" is true. Thus, the premisses tell us why the conclusion is true, that is, give the cause for the conclusion.

We have then two definitions of this most perfect tool of reasoning, the demonstrative syllogism. The first definition takes its specific difference from the end. Demonstration is a syllogism whose purpose is to give us certain and reasoned-out knowledge, or scientific knowledge. The second definition is taken from the matter of this tool. Demonstration is a syllogism the premisses of which are true, first and immediate, better known than and prior to the conclusion, and the cause of the conclusion.

A Second Meaning of the Term "Demonstration"

But there is a problem. Remember, at the end of the Categories Aristotle distinguishes four main meanings of the terms prior and posterior. We noticed that there is a difference between being prior in being and prior in knowledge. For instance, God comes before creatures in being because God can exist without creatures, but creatures cannot exist without God. This is true because God is the cause of creatures. But creatures come before God in the order of knowing because we can know creatures without knowing about God, but we cannot know about God without knowing something about creatures. In this case the effect, creatures, is more known to us than the cause, God.

If we apply what we have said about the premisses of demonstration to this example, it wreaks havoc Aristotle's definition. Our definition said that the premisses of a demonstration must be both better known than the conclusion and the cause of the conclusion. But there are many cases in which what is prior in the order of knowledge is the effect, and what comes posterior in the order of knowledge is the cause. It seems, then, that it is impossible to have demonstration about these matters.

Distinction of Knowledge of Fact and of Reasoned Fact

Aristotle did not overlook this difficulty, and he gives the following solution in Chapter 13 of the first book of the Posterior Analytics. He writes:

Knowledge of the fact differs from knowledge of the reasoned fact. To begin with, when the premisses are immediate, but instead of the cause, the better known of the two reciprocals is taken as the middle, for of reciprocally predicable terms, the one which is not the cause may quite easily be better known and so become the middle term of demonstration.

Aristotle is saying that besides knowledge of the reasoned fact, a knowledge of the reason why, there is another kind of knowledge, knowledge of the bare fact. Knowledge of the reasoned fact is the knowledge that some fact is necessarily true through the causes of that fact. Knowledge of the fact is simply knowledge that the fact is necessarily true, but not through the causes of that fact. It tells us what is true, but not why it is true.

This results in two meanings of the term "demonstration." In textbooks of scholastic philosophy and theology they are called demonstration propter quid and demonstration quia. Demonstration propter quid is demonstration in the strictest sense, which gives us knowledge both of the necessity of the fact and of the reason why the fact is true. Demonstration quia is demonstration in a weaker sense, which only tells us that the fact must be true. It starts from the effect and deduces the cause. For example, our knowledge of God's existence comes through a quia demonstration: seeing the created world, we deduce the existence of an intelligent and all-powerful Creator. The quia demonstration solves the problem of how we can have demonstration in natural theology.

Aristotle gives a very clear example of demonstration of the fact. Let us go over that example briefly. He lays out the following syllogism:

All planets do not twinkle.
Every light that does not twinkle is a relatively near light.
Therefore, every planet is a relatively near light.

First we will examine how he comes to this particular syllogism and second why this syllogism is a quia demonstration and not propter quid.

If we observe the night sky, we see that most of the stars keep a fixed position relative to each other. But there are a few stars which move around in the sky, that is, have different relative positions on different nights. The ancients called those stars planets, or wanderers. The planets have another property: they do not twinkle when we look at them. The other stars, called the fixed stars, do twinkle when we look at them. This is the source of Aristotle's first premiss: all planets do not twinkle.

We also can notice that even on earth night nearby lights do not twinkle, while distant lights do. For instance, the streetlight in front of my house does not twinkle to my sight but rather gives off a steady light, while the lights outside of homes across the valley from our home do twinkle. This is the source of Aristotle's second premiss: every light that does not twinkle is a relatively near light.

Putting those two premisses together in the aforementioned syllogism, we get the conclusion that every planet is a relatively near light; or, more precisely, that the planets are nearer to us than the fixed stars. But this syllogism is a demonstration of the fact, a quia demonstration, and not a demonstration of the reasoned fact, or propter quid demonstration. It gives us only knowledge of the fact that the planets are near, it does not tell us why the planets are near. In fact, in this case the order of reality is the opposite of the order of our knowledge. The planets do not twinkle because they are near. That is, this demonstration takes an effect, that the planets do not twinkle, and makes it the premiss of our syllogism, and from that premiss we deduce the cause, that the planets are near. We must proceed in this way because the effect, the non-twinkling of the planets, is better known to us than the cause, the relative position of the planets. Thus the argument that the planets are nearer than the fixed starts is example of a demonstrative syllogism in this secondary sense of demonstration, demonstration of the fact, or quia demonstration.

These are the very basic issues involved in the subject of demonstration. Our discussion of the difference between quia and propter quid demonstrations shows us that solving the problem of learning is not going to be as easy and uncomplicated as it might have seemed from our first discussion of demonstration. But before we get to the next lesson, I would like to talk about one more thing in Aristotle's Posterior Analytics, the last chapter. We saw earlier that demonstration needs first and immediate principles. We now need to ask how we can learn those self-evident principles.

Return to the Problem of Learning

Since the syllogism is not going to solve the problem of how we learn the truth of the self-evident principles of all knowledge, we might be tempted at this point to adopt a modified form of Plato's solution to the problem of learning. We might claim that we are born with a knowledge of the first principles, and we merely need to recollect them in order to use them. Later philosophers took up the same idea as the doctrine of innate ideas. Aristotle explicitly rejects this position. He writes:

Now it is strange if we posses them [that is, the first principles] from birth, for it means that we possess apprehensions more certain than demonstration and fail to notice them.

Remember that the premisses of a demonstrative syllogism are always better known than the conclusion. Therefore, the truths that we know best are the first principles. No one who possess demonstration of a truth is unaware that he possesses this knowledge. How then could we possess knowledge even more certain, knowledge of the first principles, and be unaware that we possess it? According to Aristotle, a Platonic account of our knowledge, even our knowledge of the first principles, is incoherent.

Aristotle's answer is that we learn the first principles through sensation. He says we granted before that learning was possible only because there was some pre-existent knowledge from which you could produce the new knowledge. We knew the truth of the premisses, and therefore we could learn the truth of the conclusion from them. Our knowledge of the first principles is the first kind of intellectual knowledge we have, but it is not absolutely first. Sensation pre-exists intellectual understanding. Thus the knowledge which is sensation provides us with the pre-existent knowledge we need in order to learn the first principles.

In order to explain how we get knowledge of the first principles from sensation, Aristotle makes a comparison to a battle. Let us examine that comparison, and then apply it to the process of coming to know the first principles. Aristotle writes:

It is like a rout in battle stopped by first one man making a stand and then another, until the original formation has been restored. The soul is so constituted as to be capable of this process.

A civilized nation organizes the men in its armies into fighting units, formations with officers and ordered ranks. But when a fighting unit is overwhelmed by a more powerful force, the unit is routed. The individual men are the same, but after the rout they are simply a fleeing mob. But at a certain point, one man may stop, look back, and see that the enemy force is not pursuing their unit. He takes a stand. Another, seeing him taking a stand, looks back and takes a stand as well. The whole mob coalesces around these men. They then reconstitute themselves as a fighting unit.

This process of going from a fleeing mob to a fighting unit parallels the process of going from sense knowledge to a knowledge of the first principles. The mob of fleeing men is like sensation because sensation is a constant flow of new perceptions. Some of these sensations remain in us, however, and become memories. When those many memories are like each other, they can gather together and constitute one experience in our memory, just like the mob coalesces around the men who have taken a stand. And from those many memories collected in experience, the first universal comes to be in the soul, just as when those men retake their positions, and become again a fighting unit.

After Aristotle has compared the process of coming to know the first principles to the rout in battle, he uses an example from medical practice to make the process even clearer. Doctors want to cure fevers and in ancient times would try various herbs as remedies. Suppose that a doctor cures Socrates' fever with a particular herb, and Plato's and Aristotle's fevers as well. Each case initially constituted a sensation, a perception of a fever and a giving of the herb. Each incident remains in the doctor's mind after each of the patients is gone and thus constitutes a memory. When the doctor takes these many memories and groups them together, saying to himself, "Socrates, Plato, and Aristotle were all cured of their fever by this herb," he has one experience. And the experience takes us to the threshold of intellectual knowledge because at this point the doctor draws the following conclusion: "This herb cures fever in all men." At that point the doctor has arrived at the universal. His knowledge is now intellectual because universal and he has perceived something that can be a first principle in the science of medicine. Aristotle's contention, then, is that we learn the first principles of demonstration through sensation.

All of these considerations show us that we can look at the whole Posterior Analytics in two different ways. In one way we can look at it as the book which teaches us about the most perfect tool of discursive reasoning, the demonstrative syllogism. It is through the demonstrative syllogism that we can have certain knowledge of the truth and that kind of knowledge of the truth is the goal of logic. But we can also look at it as the answer to Meno's problem of learning. In the first part of the book, Aristotle shows us how we can use the demonstrative syllogism to solve Meno's problem with regard to our knowledge of conclusions. In the second part of the book, Aristotle shows us how we can solve Meno's problem with regard to the first principles of demonstration.


If we read only the Posterior Analytics, we might get an oversimplified version of how to solve the problem of learning. We know how to construct a demonstrative syllogism, but not how to discover one. Moreover, we know that the mind goes from experience to the universal principle, but we don't know what logical tools it uses in that process. It is thus clear that we will need more than the demonstrative syllogism to make sure that we acquire certain knowledge of the truth. The rest of this course is going to be devoted to talking about the discovering part of logic, which gives us tools of reasoning which, while less perfect, are just as necessary as the demonstrative syllogism.


Short Essays (250 words)

1. Explain how "demonstration" is used analogously of demonstration propter quid and demonstration quia.

2. How does the Posterior Analytics solve Meno's problem of learning?


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