Logic is the science and art of correct thinking. We have already seen that the kind of thinking that logic is principally concerned with is inference, which is expressed externally by oral or written argumentation, and that this is likewise called inference. The following syllogism has already been given as a typical example:
Every dog is an animal; but every hound is a dog; therefore every hound is an animal.
The main task of logic is to establish norms for determining whether or not an alleged conclusion is actually implied in the premises from which it is said to follow. The main task of logic, in other words, is to formulate general norms, laws, or rules that will help us answer questions like this: Supposing that every dog is an animal and that every hound is a dog, does it follow from, this that every hound is an animal?
Inference, according to its broadest meaning, signifies any process to which the mind proceeds from one or more propositions to other propositions seen to be implied in the former. However, in its strictest and proper sense, inference signifies the operation by which the mind gets new knowledge by drawing out the implications of what it already knows. The distinction between inference in the broad sense and inference in the strict and proper sense will be clearer after we have contrasted intermediate and mediate inference.
The word "inference" is also applied to any series of propositions so arranged that one, called the CONSEQUENT, flows with logical necessity from one or more others, called the ANTECEDENT. Sometimes the name is given to the consequent alone, viewed in relation to the antecedent from which it flows.
Etymologically, "antecedent" (derived from the Latin antecedo) means "that which goes before"; it is defined as "that from which something is inferred." "Consequent" (derived from the Latin consequor) means "that which follows after"' it is defined as "that which is inferred from the antecedent.
The antecedent and consequent of a valid inference are so related that the truth of the antecedent involves the truth of the consequent (but not vice versa); and the falsity of the consequent involves the falsity of the antecedent (but not vice versa). In other words, if the antecedent is true, the consequent (if it really is a consequent) is also true; and if the consequent is false, the antecedent is false. However, if the antecedent is false, the consequent is indifferently false or true (and therefore doubtful); and if the consequent is true, the antecedent is indifferently true or false (and therefore doubtful).
The connection by virtue of which the consequent flows with logical necessity from the antecedent is known as consequence or simply SEQUENCE. The sequence (which is signified by "therefore," "consequently," "accordingly," "hence," "thus," "and so," "for this reason," and so on) is the very heart of inference; and when we make an inference, our assent bears on it directly.
A genuine sequence is called valid; a pseudo sequence is called invalid. Notice that an invalid sequence is really not a sequence at all but is merely called a sequence because it mimics one (just as counterfeit money is called money, although it really is not money but only make-believe money).
Valid sequence springs either from the form of inference or from the special character of the matter or thought content. If the sequence springs from the form of inference, the sequence is FORMAL and the argument is said to be formally valid or formally correct; if the sequence springs from the special character of the thought content, the sequence is MATERIAL and the argument is said to be materially valid.
The LOGICAL FORM of inference is the order that the parts of an inference have towards one another. We refer primarily to the order of concepts and propositions in the mind. The connection, however, between our thoughts and their written expression is so close that we can represent the logical form of inference by the arrangement of terms and propositions on a printed page. For instance, the following example illustrates one kind of inference that is formally valid:
Every S is a P; therefore some P is an S.
We can substitute anything we want to for S and P, and the consequent will always be true if the antecedent is true. If we substitute "dog" for S and "animal" for P, we get:
Every dog is an animal; therefore some animal is a dog.
If we substitute "voter" for S and "citizen" for P, we get:
Every voter is a citizen; therefore some citizen is a voter.
[Note: We may not proceed from an antecedent that does not assert the actual real existence of a subject to a consequent that does assert its actual real existence. For example: Every unicorn is a horse; therefore, some horse is a unicorn.]We shall now examine an inference that is formally invalid but materially valid:
Every triangle is a plane figure bounded by three straight lines; therefore every plane figure bounded by three straight lines is a triangle.
In this example the consequent does not flow from the antecedent because of the form; but it does flow because of the special character of the thought content. "Plane figure bounded by three straight lines" is a definition of "triangle" and is therefore interchangeable with it. Suppose, however, that we retain the same form but substitute "dog" for "triangle" and "animal" for "plane figure bounded by three straight lines." Suppose we argue:
Every dog is an animal; therefore every animal is a dog.
This inference is obviously invalid; yet it has exactly the same form as the materially valid inference given above. The form is:
Every S is a P; therefore every P is an S.
As we shall see when we study the conversion of propositions, an inference with this arrangement is always formally invalid.
Whenever we use the terms "sequence," "inference," "validity," "correctness of argumentation," and so on, without qualification, we shall understand them in their formal sense unless it is clear from the context that we are speaking of material sequence.
Logical truth consists in the conformity of our minds with reality. A proposition, as we explained above, is true if things are as the proposition says they are. Logic studies reason as an instrument for acquiring truth, and the attainment of truth must ever remain the ultimate aim of the logician. Still, in the chapters on formal inference, we shall study only one part of the process of attaining truth. We shall not be directly concerned with acquiring true data but rather with conserving the truth of our data as we draw inferences from them. In other words, we shall aim at making such a transition from data to conclusion that if the data (antecedent, premises) are true, the conclusion (consequent) will necessarily be true. Formal validity, correctness, rectitude, or consistency will be our immediate aim. We shall not ask ourselves, Are the premises true?, but, Does the conclusion flow from the premises so that IF the premises are true, the conclusion is necessarily true?
The following syllogism is correct in this technical sense although the premises and the conclusion are false:
No plant is a living being; but every man is a plant; therefore no man is a living being.
This syllogism is correct formally because the conclusion really flows from the premises by virtue of the form, or structure, of the argument. IF the premises were true, the conclusion would also be true.
The following syllogism is not correct formally although the premises and the conclusion are true:
Every dog is an animal; but no dog is a plant; therefore no plant is an animal.
This syllogism is not correct because the conclusion does not really flow from the premises. Its invalidity will be obvious if we retain the same form but change the matter by substituting "cow" for "plant."
Every dog is an animal; but no dog is a cow; therefore no cow is an animal.
In this syllogism (it is really only a make-believe or apparent syllogism), an obviously false conclusion comes after obviously true premises, so the syllogism must be incorrect, for in a correct or valid syllogism only truth can follow from truth.
Inference is either immediate or mediate. Immediate inference consists in passing directly (that is, without the intermediacy of a middle term or a second proposition) from one proposition to a new proposition that is a partial or complete reformulation of the very same truth expressed in the original proposition. Except for terms prefixed by "non-" and their equivalents, immediate inference has only two terms, a subject term and predicate term (at least in most of its forms), and, strictly speaking, involves no advance in knowledge. Since immediate inference involves no advance in knowledge, it is inference only in the broad, or improper, sense. [Note: Some logicians object to calling it inference at all. In Basic Logic: The Fundamental Principles of Formal Deductive Reasoning. New York: Barnes & Noble, Inc. Raymond J. McCall considers the term "immediate inference" a "terminological monstrosity of the first water" because "inference" is classically defined as "mediate judgment"; thereby making "immediate inference" an "immediate mediate judgment" (p. 114).]
Mediate inference, on the other hand, draws a conclusion from two propositions (instead of one) and does involve an advance in knowledge. Consequently, mediate inference is inference in the strict, or proper, sense. It is mediate in either of two ways. In the categorical syllogism it unites, or separates, the subject and predicate of the conclusion through the intermediacy of a middle term; in the hypothetical syllogism the major premise "causes" the conclusion through the intermediacy of a second proposition.
The goal of mediate inference is not only a new proposition but also a new truth, for in mediate inference, as we have seen, there is an advance in knowledge. This advance is either in the order of discovery or in the order of demonstration or explanation.
The advance in knowledge is in the order of discovery when we proceed from a known truth to a new truth that we did not hitherto know to be true. The new truth, of course, must be contained somehow or other in the premises; for if it were not in the premises, it could not be gotten out of them. Still, it was in the premises only virtually and implicitly, whereas in the conclusion it is stated actually and explicitly.
The advance in knowledge is in the order of demonstration when we already knew the truth of what it stated in the conclusion but now either accept it for a new reason or have come to understand why it is true. Previously we may have accepted it on authority or as an object of opinion or natural certitude. But now we see its connection with more basic principles, and thus we possess it in a more perfect manner.
First we shall treat of immediate inference, and then of mediate inference.
Deduction is the process by which our minds proceed from a more universal truth to a less universal truth, as in the syllogism "All men are mortal; but Peter is a man; therefore Peter is mortal." Induction, on the other hand, is the process by which our minds proceed from sufficiently enumerated instances to a universal truth, as in the example "This ruminant (a cow) is cloven-hoofed; this one (a deer) is cloven-hoofed; and this one (a goat) and this (an antelope) and this (an elk); therefore all ruminants are cloven-hoofed."
Induction precedes deduction. It is principally by induction that we get the universal principles that constitute the premises of deductive arguments; it is by induction, too, that we grasp the rule-governing deduction as well as the principles underlying them. Nevertheless, it is customary in logic courses to treat of deduction before induction.
All formal inference and many instances of material inference are deductive. All induction, however, is material inference. At present, we shall content ourselves with merely mentioning the division of inference into deduction and induction; in later chapters we shall treat of each of them in detail.
In inference the mind proceeds from one or more propositions to another proposition so related to the original propositions that if they are true it must also be true. Consequently argumentation must have two or more propositions as its component parts. Propositions, in turn, are made up of terms (like “dog,” “animal,” and “hound” in the example given above). Hence, before we can determine the conditions of valid inference, we must know something about propositions; and before we can understand the nature of propositions, we must know at least a little about terms.