## Lesson 7: The Logic of Demonstration

A crucial consideration of both the taped lectures and of these lessons is the way in which sciences are distinguished from one another. The following two lessons will particularly deal with aspects of this issue. Both will presuppose certain things about the logic of science or of demonstrative proof. In this lesson I will set down the bare bones of that doctrine.

Thomas distinguishes a number of different mental attitudes vis-a-vis a proposition, that is, something complex that is susceptible of truth or falsity. In order to understand him, we must see that each of these attitudes is seen as bearing on one side of a contradiction. That is, if I think that is true, my attention centers on p and not on -p its contradictory opposite. Here are the different mental acts Thomas compares.

I have the opinion that p.

I doubt that p.

I know that p.

When I have an opinion about p I do not wholly exclude the possibility that -p is true. I may have stronger or weaker grounds for thinking what I do, but if it is an opinion it leaves the door open to its contradictory. If I doubt that p is true, I may be said to have the opinion that -p is true, so there is a symmetry between these two. When I say that I know something, I wholly and definitely exclude its contradictory.

The internal structure of the value of is, minimally, S is P -- this is a simple proposition (as opposed, for example, to "If p, then q"). Now there are two ways in which I may be said to know p. First, it may be such that I know straight off that it is true because of the meaning of its constituent terms. "Equals taken from equals leave equals." If I know what equals are and what "take from" means, I have all I need to see that the proposition is true. Such a proposition is said to be known in, of, through itself, per se. An example of this is had when the predicate of the proposition is part of the definition of its subject. Second, a proposition may be known to be true because it follows from other true propositions; then it is said to be known through those other truths, per alia. How can other truths ground the truth of the proposition in question?

Just as "S is P" is the simplest form of the proposition, so the simplest form of discursive reason looks like this. We want to establish that "S is P." Now, M is P and S is M, it follows that S is P. Such discourse links the terms of the conclusion by finding premisses in which a third term occurs in such a way that the conjunction of the premises yields the conclusion.

Let us say that I wonder why I should hold that "Man is risible." If I should know that A rational being is risible and that man is a rational being, I have linked Man and risible and assert its truth on the basis of these premises.

This arrangement is called the syllogism, but all "syllogism" means is discursive knowledge. Knowing something on the basis of something else. In the syllogism, something is held to be true because other things are true. Those other things, the premisses, must be of a certain kind and arrangement in order for the conclusion to follow. The figures of the syllogism are distinguished on the basis of the location of the middle term in them. The arrangement mentioned above gives us the first figure -- the figure in which the middle term seems more manifestly in between the predicate and subject of the conclusion.

It is because the constituent propositions can be universal or particular, affirmative or negative, that there can be different modes of each of the three valid figures of syllogism. Not all such combinations permit inference, needless to say, and the logician will help us to see which do and which do not and why.

Once the syllogism is understood, we have in hand the basic form of argumentation. Arguments are of all kinds, of course. Sometimes a conclusion follows from its premises but we feel no compulsion to accept it. If the premises are probable, the conclusion will be probable. Science is had when we have a conclusion which establishes the necessary truth of the conclusion.

The distinction just made makes it clear that it is one thing for a proposition to follow necessarily on premises and quite another for the conclusion to be a necessary truth. In scientific argument, both necessities are in play. At this point, the logician will examine the requirements of the premises in an argument in which the conclusion not only follows necessarily but is also a necessary truth. The premises will have to be themselves necessary truths in order for this to result. And there will be other requirements as well. But perhaps we have enough to clarify what is meant by the subject of a science.

Such clarity is needed because much time will be spent establishing how the subject of metaphysics differs from the subjects of the philosophy of nature and mathematics. Obviously, such discussions will only make sense if we have at least a preliminary grasp of what is meant by the subject of a science.

Science will be had from a syllogism of a certain kind; scientific knowledge is knowledge of the conclusion of a demonstrative syllogism. The conclusion can be called the object of the science. The subject of the science is the subject of the conclusion of a demonstrative syllogism. If I prove that the sum of the internal angles of a plane triangle equals 180 degrees, I will do so by using the definition of triangle for my middle term. The predicate of the conclusion, shown to belong to the subject because of what it is, as expressed by its definition, is a property. The triangle does not just happen to have it; it necessarily has it. It is not true of the triangle because it is a plane figure nor because it is a scalene triangle, but simply because it is a triangle. The most manifest example of a demonstrative syllogism is one in which the predicate is shown to be a property of the subject because it belongs to it thanks to the subject thanks to what it is.

One further point. If you were asked for a definition of science, you would probably begin by saying that it is a body of knowledge . . . . Thus far, we have been speaking of science in terms of one argument, one syllogism, but as the allusion to plane geometry indicates, a science is a concatenation of arguments. How do they all fall to the same science?

Much will be said of this in the following chapters, but this much must be said now. When I said that the property of the triangle belongs to it as triangle and not as plane figure or scalene triangle, I might have put it in another way. The property belongs to the subject, not because of its species, not because of its species, but because of what it is. Of course what is proved of a plane figure will be true of triangle as one species of plane figure. And what is proved of triangle will be true of the species of triangle. Thus it can be said that one way it is clear that many arguments belong to the same science is because their subjects are related as genera and species.

This should suffice to follow the next two lessons profitably.

Commentary on Metaphysics, Bk. 4, lessons 1 and 2.

#### Suggested Writing Assignment

Show how the above account of proof is exhibited in Selection 11 from the Penguin book.

#### Desideratum

Consult John A. Oesterle's classic logic text published by Prentice-Hall more than a quarter century ago and is still in print.

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