Thinking Logically About Logic

Thinking Logically

You’ve heard the word before, but what does it mean? Here’s the lowdown on the second leg of the trivium.

Introduction

The best way to answer the question “What is logic?” is with a definition. But that is easier said than done. Throughout history, many people have thought and written about the subject of logic, and many people have offered definitions. Some of them are useful and some are not.

Josiah Royce, an American philosopher, defined logic as “the science of order,” but this definition is so general that it really could include things outside of what we would properly call logic, and so it really doesn’t tell us much.

Other definitions are a little too simple. The writer Oliver Wendell Holmes said, “Logic is logic. That’s all I say.” That obviously won’t help us.

The writers of a book on fallacies defined logic as “the defense against trickery.” That’s one thing logic is, but certainly not all.

Much better is the definition given by Raymond McCall: “Logic in general is the science of right thinking.” Jacques Maritain, a very famous philosopher, had a similar definition: “Logic,” he said, “studies reason as the tool of knowledge.”

Irving Copi, who wrote a book on logic still used in many colleges and universities, gets even a little more specific: “The distinction between correct and incorrect reasoning is the central problem with which logic deals.” As you proceed in this book, you will see that this is so.

Where does Logic fit in?

The field of philosophy is divided into three recognized divisions: The first division of philosophy is theoretical philosophy, or philosophy proper. The sciences in this branch of philosophy are employed solely for the pleasure of knowledge. These include the philosophy of mathematics, which studies the being of things by virtue of their quantity (ens quantum); the philosophy of nature, which studies the being of things by virtue of their sensible properties (ens mobile); and, finally, metaphysics, which studies the being of things by virtue of their being (ens in quantum ens). The formal object of theoretical philosophy is the being of things.

The second division of philosophy is practical philosophy. While the object of theoretical philosophy is a knowledge of the first principles of the theoretical order, the object of the study of practical philosophy is a knowledge of the first principles of the practical order. These would include the philosophy of art, which has to do with man’s ability to make or create and ethics or moral philosophy, which has as its object the absolute good of man. The formal object of practical philosophy is human acts.

Logic is the third division of philosophy, although it is considered the introduction to the rest of philosophy, since it provides the methods used in the other two branches. In this sense, logic is less a division of philosophy than the science or art of which the rest of philosophy makes use. The division of logic is divided further into formal and material logic (these are treated more in depth below). Logic studies the conceptual being (ens rationis) and directs the mind toward truth.

The two main branches of logic, one called formal or minor logic, the other material or major logic, are quite distinct and deal with different problems.

Material logic is concerned with the content of argumentation. It deals with the truth of the terms and the propositions in an argument.

Formal logic is interested in the form or structure of reasoning. The truth of an argument is of only secondary consideration in this branch of logic. Formal logic is concerned with the method of deriving one truth from another.

The distinction between these two branches of logic was nicely described by G. K. Chesterton:

Logic and truth … have very little to do with each other. Logic is concerned merely with the fidelity and accuracy with which a certain process is performed, a process which can be performed with any materials, with any assumption. You can be as logical about griffins and basilisks as about sheep and pigs … Logic, then, is not necessarily an instrument for finding out truth; on the contrary, truth is a necessary instrument for using logic—for using it, that is, for the discovery of further truth … Briefly, you can only find truth with logic if you have already found truth without it.

This last remark of Chesterton’s is important. It is not the purpose of formal logic to discover truth. That is the business of everyday observation and, in certain more formal circumstances, empirical science. Logic serves only to lead us from one truth to another.

That is why it is best to study formal logic first. In formal logic you study the form of an argument apart from or irrespective of its content, even though some content must be used in order to show the form. Maritain put it this way:

To study any complicated machine, a reaper for instance, we must begin by making it work in the void, while we learn how to use it correctly and without damaging it. In the same way, we must first of all learn how to use reason correctly … without damaging it.

Deductive Arguments vs. Inductive Arguments

An important distinction between arguments according to their form is that between deductive arguments and inductive arguments. At the most fundamental level, the difference between the two is that in a valid deductive argument, the conclusion asserts no more than what is contained in the premises, while in an inductive argument, more is asserted in the conclusion than is contained in the premises. That is why in a valid deductive argument, the truth of the premises guarantees the truth of the conclusion, while in a valid inductive argument, the truth of the premises only makes the conclusion probable.

Valid deductive arguments offer sufficient proof for their conclusions, whereas valid inductive arguments only offer good grounds for believing in the conclusion. In fact, because induction is a weaker form of proof than deduction, many people do not even use the term “valid” for a good inductive argument, because validity has the sense of necessary proof, which is absent from even a good inductive argument. They say instead that a good inductive argument is “cogent,” a term which means convincing, rather than demonstrative.

One of the most recognizable characteristics of deductive arguments is that they argue from the general to the specific, or from the more general to the less general, by way of a middle term. Inductive arguments, on the other hand, reason from the specific to the general, or from the less general to the more general and have no middle term that firmly connects one truth to another. Deduction relies on the acceptance of a general principle and reasons from that general principle through an iron chain of reasoning to a conclusion. Induction reasons from repeated particular observations (which are usually observable) to more general truths through statistical generalizations and analogies which are sometimes unobserved (and which are considered stronger by virtue of the number of confirming instances that are appealed to in the premises).

This distinction—between deductive and inductive reasoning—is often misunderstood in common language. Arthur Conan Doyle, for example, has his character Sherlock Holmes refer to his own style of reasoning as “deduction,” when, in fact, Sherlock Holmes is not notable for his deduction, but for his induction. Holmes reasons from particular observations to more general conclusions.

Deductive arguments are more common in theoretical fields, such as philosophy and mathematics, while inductive arguments are more common in the field of the natural sciences.

Systems of Deductive Logic

There are several major systems of formal deductive logic: the first is traditional or syllogistic logic; the second is modern propositional logic; and the third is modern predicate logic. Traditional logic concerns itself with the relationships between terms in an argument, using the “to be” verb (am, is, or are) as the connector. An example would be:

All men are mortal
Socrates is a man
Therefore, Socrates is mortal

This argument deals with the relationships between and among the terms “men,” “mortal,” and “Socrates.”
Modern propositional logic deals with the relationship between propositions in an argument without taking the interior structure of the statements into account. It uses logical operators such as “if … then,” “and,” “or,” “only if,” or “if and only if” as the connectors. An example would be:

If all men are mortal, then Socrates is a mortal
All men are mortal
Therefore, Socrates is a mortal

This argument deals with the relationships between and among the statements “All men are mortal” and “Socrates is a mortal.”
Modern predicate logic deals with the relationship between and among both terms and propositions. It can use many kinds of connectors. An example would be:

Some angels are evil. Furthermore, some animals are rational.
If there are any angels, then animals are sinful if they are rational.
Therefore, some animals are rational.

In this article, we are concerned primarily with traditional syllogistic logic.

Truth, Validity, and Soundness

The form of an argument is found in its argumentative structure; the matter of an argument is found in the statements. Statements of fact, for example, cannot be called logical or illogical, since these labels refer to form; they can only be properly called true or false, labels which refer to matter. Likewise, an argument cannot be called true or false, only valid or invalid. Only arguments are valid or invalid, and only statements are true or false.

Validity is the term we use when we mean to say that an argument is logical. The term “soundness,” however, is a term that refers both to the form and the content of an argument. It is applied to an argument to say something about both its truth and its validity.

Truth means the correspondence of a statement to reality. An argument is valid when its conclusion follows logically from its premises. The term soundness is used to indicate that all the premises in an argument are true and that the argument is valid.

An argument can contain true premises and still be invalid. Likewise, it can be perfectly valid (or logical, if you prefer) and contain false premises. But if an argument is sound, its premises must be true and it must be valid.

The Components of an Argument

An argument contains several components. In order to illustrate what these components are and how they work in the reasoning process, let us begin with a simple argument:

All men are mortal
Socrates is a man
Therefore, Socrates is mortal

The first two statements are premises and the last is the conclusion. All arguments must have at least two premises and one conclusion.

On the face of it, this argument contains a number of words making up three statements which fit together into what looks and sounds like an argument. But there is more here than meets the eye.
In formal logic, we recognize three kinds of logical processes. We recognize that each of these originates in a mental act, but that each also manifests itself as (and is known to us in the form of) a verbal expression.

Term

The mental act involved in the first of these three logical processes is called simple apprehension. We call the verbal expression of simple apprehension the term. A simple apprehension occurs when we first form in our mind a concept of something. When we put this concept into words, we have put this simple apprehension in the form of a term.

At the point of simple apprehension, we do not affirm or deny anything about it. We just possess or grasp it.

If in your mind, for example, you think of your computer (the one you’re using right now), you are performing this first logical process. You are having a simple apprehension. And if you speak or write anything about it, you will have to use a term, the term “computer.”

In the argument above (the one about Socrates), there are three terms representing three simple apprehensions. The first is “men”; the second is “Socrates”; and the third is “mortal.” Each one of these represents in our mind a concept that we have transformed into a word. The concept we call the simple apprehension and the word we call the term.

Mental Act Verbal Expression
Simple Apprehension Term

Proposition

The mental act involved in the second of these three logical processes is called judgment. The verbal expression of a judgment is called a proposition. We perform a judgment any time we think in our mind that something is something else (which we call affirmation), and also when we think that something is not something else (which we call denial). To judge is to affirm or deny.

If you think that this computer is complicated, then you are performing a judgment. If you verbally express this judgment, you will have to do it in the form of a proposition, the proposition “This computer is complicated.” The judgment is the mental act you have when you think that this computer is complicated and the proposition is the statement you make to express that thought.

In the argument above, there are three propositions expressed. The first is “All men are mortal”; the second is “Socrates is a man”; and the third is “Socrates is mortal.” Each one of these represents in our mind a thought that something is something else: that “all men” are “mortal”; that “Socrates” is a “man”; and that “Socrates” is “mortal.”

We should point out that some people use the term “statement” instead of proposition. They mean the same thing, but to be consistent, we will use the term proposition.

Mental Act Verbal Expression
Judgment Proposition

Syllogism

The mental act involved in the third of these three logical processes is called deductive inference. We call the verbal expression of deductive inference the syllogism. A deductive inference occurs when we make the logical connections in our mind between the terms in the argument in a way that shows us that the conclusion either follows or does not follow from the premises. When we verbally express this in an argument, we have put this deductive inference in the form of a syllogism.

It is at this point that we are said to make progress in knowledge. It is through the process of deductive inference, as expressed in a syllogism, that we can say, as we explained above, that we have gone from one truth or set of truths to another truth.

Let’s say the reason you think this computer is complicated is because you think all computers are complicated. If this were true, you would be performing a deductive inference. You would be thinking to yourself, all computers are complicated, and this is a computer. Therefore, this computer is complicated. And if you verbally expressed this deductive inference, you would do it in the form of a syllogism. The judgment expressed by “All computers are complicated” and “This is a computer” are different than the judgment “This computer is complicated.” Through deductive inference, however, you can go from these first two to the last one. In this way, you have gone from one set of truths to another truth (if indeed they are true).

We would say that the argument above (the one about Socrates), in its entirety, is a syllogism. It expresses a deductive inference that logically connects certain simple apprehensions that are parts of three judgments. And this process has been expressed in the form of a syllogism.

Mental Act Verbal Expression
Deductive Inference Syllogism

If we now put this all together, keeping our distinction between mental acts and verbal expressions, it would look like this:

Mental Act Verbal Expression
Simple Apprehension
Judgment
Deductive Inference
Term
Proposition
Syllogism

In order to give ourselves a mental picture of these three logical processes, let us think of a man walking. In order to get from, say, one room to another, he has to pick up his foot and take several steps in order to get to the room that is his destination. The initial act—picking up his foot—is like the initial logical act of simple apprehension. Taking a full step is like making a judgment. And stringing all the steps together into one movement is like deductive inference—we move from one place to another.

Conclusion

We started out by defining logic as “the science of right thinking.” We said logic was one of three divisions of philosophy, the others being theoretical philosophy and practical philosophy. We said there are two main branches of logic, one called formal or minor logic, the other material or major logic. Material logic is concerned with the content of argumentation, while formal logic is interested in the form or structure of reasoning.

We said that, broadly speaking, there are two kinds of formal logic: deductive logic and inductive logic. The conclusion of deductive logic, we said, asserts no more than is contained (explicitly or implicitly) in its premises, while the conclusion of inductive logic asserts more than is contained in its premises. Therefore, we concluded, while the conclusion of a deductive argument is conclusive, the conclusion of an inductive argument is only probable.

We defined truth as correspondence with reality. We said an argument is valid when its conclusion follows logically from its premises. And we said that soundness indicates that all the premises in an argument are true and that the argument is valid.

We said also that all arguments must contain two premises and a conclusion. And we said, finally, that there are three mental acts that make up the logical process: simple apprehension, judgment, and deductive inference. These three mental acts correspond to three verbal expressions: term, proposition, and syllogism.

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