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The Jonathan Dolhenty Archive |
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DEDUCTION: Part 1 by Jonathan Dolhenty, Ph.D.
Human beings are the only living things on earth that can think and reason. Reasoning is the supreme operation of man as man. It is the power to reason which lifts man above and beyond every other being in nature. The power to reason is at the center of the essence of man and provides him with his most valuable tool of survival. And it all begins with ideas. The human mind forms ideas of things. Different ideas are compared and then united in a judgment expressing their mutual agreement and disagreement. And truth lies in the judgment. All knowledge begins in the senses and in the presence of sense objects the human mind automatically goes into action. First, a sense-image of the sensed object is formed. This sense-image is then fashioned into an idea, which may be vague and very general at first. As the mind continues to observe and study, distinctions are made between what is essential to the object and what are merely accidental attributes. of the object. Eventually, the mind has honed the idea and developed it to the point where the mind has a relatively full knowledge of the object in its comprehension and extension. The function of the mind is to attain truth. Truth, however, is not found in the ideas for they contain only the elements of truth. Trues lies in the judgment. The mind, after developing ideas to a relatively sophisticated level, now holds them up for inspection, compares them, and recognizes their agreement or disagreement with each other. The mind then forms a judgment, pronouncing their identity or non-identity. The two ideas the mind is comparing are expressed in propositions containing a subject and a predicate, each representing one of the two ideas, and a copula which represents the act of pronouncing agreement or disagreement. The judgment expresses an agreement or disagreement between things as they exist in themselves, independent of the mind. If the mind's judgment corresponds with reality, it is true, and if not, it is false. Immediate Inference Many times the agreement or disagreement between the ideas of a judgment is self-evident. This is a process of immediate inference, a primitive form of reasoning. It is an essential and foundational part of the reasoning process, but it is very basic and doesn't get us very far on the road to advanced knowledge. Consider, for example, the three "laws of thought" that we discussed previously: the Principle of Identity, the Principle of Contradiction, and the Principle of Excluded Middle. These principles are judgments made by the mind and they are self-evident. We can't "prove," strictly speaking, the Principle of Identity (A is A or A = A). It is simply, a self-evident truth. The same holds for the Principle of Contradiction. There is no way to prove the principle. These principles must be accepted as true, however, because they are foundational to all knowledge and we can't reason or participate in argumentation without accepting them as true. Attempting to reason without acceptance of the three principles would result in total nonsense. The truth of the three principles is arrived at by merely analyzing the ideas involved in the judgment, a form of immediate inference. Sense perception can also lead to true judgments made on the basis of immediate inference. For example, we look outside and it is raining. We form the judgment, "It is raining outside." How do we "prove" the truth of the judgment? If someone should ask for proof, we ask them to look outside for themselves or, in a worst case scenario, we throw them outdoors and let them get wet! A great many of our daily judgments are commonplace ones of this type. The "proof" is simply in the seeing, the hearing, the touching, the tasting, or the smelling. In other words, we form judgments based on immediate inferences using our physical senses as the arbiters of truth. The problem with immediate inference as a form of reasoning is that it doesn't get us very far along the road to advanced knowledge. Mere analysis and direct observation do not satisfy our innermost yearnings to know about things beyond our immediate observations. We are, after all, when making immediate inferences, only comparing two ideas and making a single judgment about their agreement or disagreement. The human mind, not being content with just this process, seeks knowledge about things which are not readily apparent to the physical senses or decided by mere analysis of the ideas involved in the judgment. When the mind cannot perceive the agreement or disagreement between two ideas by analysis or sense perception, the mind is in a state of doubt. Human beings tend to dislike being in a state of doubt. We like to "know" things and to "know" whether our judgments about these things are true or false. We will remain in a state of doubt until the mind compares these ideas of ours with a known third idea and sees their identity or non-identity with this third idea. If both ideas are identified with and in this third idea, then they are also identical with each other. If, however, one of the two ideas is not identical with the third idea, even though one of them is so identified, we have a disagreement of the ideas among themselves. The resulting judgment is one of affirmation or of denial depending upon the agreement or disagreement. This process is more advanced than making an immediate inference from mere analysis of ideas or direct observation through the physical senses. This is really the process most of mean when we talk about "reasoning." And this brings us to the topic of mediate inference. Mediate Inference In its simplest form, what we ordinarily mean by reasoning consists of three judgments. We have two judgments expressing two separate ideas, stated in propositions called premises, and we compare each idea separately with a known third idea, resulting in a judgment which we call the conclusion. There are at least three ideas and three judgments involved in the process of reasoning. This process is called mediate inference for the following reason: the agreement or disagreement between two original ideas is inferred through the "mediation" of a third idea with which both original ideas are compared. A general definition of mediate inference would be that it is that process by which, from certain truths already known, the mind passes to another truth distinct from these but necessary following from them. Types of Mediate Inference There are two main types of reasoning or mediate inference: deduction and induction. Deduction is the process of reasoning in which we conclude from a general law or principle to a particular instance falling under the general law or principle. Let's consider an example of deduction. Let's suppose that it has been established that "All soldiers are men." This becomes our general law or principle. This is our first judgment and our first premise in the deduction. Let's suppose further that it has been established that "All heroes are soldiers." This is our second judgment and our second premise in the deduction. From these two premises, we can now draw a conclusion: "All heroes are men." Here's how this argument can be formally stated:
This argument is valid according to the rules of logic. Notice it says nothing about the "truth" of the argument, only its validity. We know today that women have become soldiers in the United States Army, serving on the front lines in many cases. The first premise could be objected to on the grounds it is no longer "true." The second premise could also be objected to on the grounds that it is untrue. We could further object to the conclusion as being untrue. Be that as it may, the deduction is still valid. If the first two premises are true, the conclusion is true. If the first premise is untrue and the second premise is true, the conclusion is still valid, even though it is untrue. It's important to keep in mind that logic is concerned with the validity of an argument and not necessarily with the truth of an argument. We can also show the deductive process with a hypothetical proposition. For example:
The first premise, acting as a hypothetical, has the effect of a general principle. If we accept it as true and we know the second premise is true, then the conclusion necessarily follows. It is a valid argument. Note that in deductive reasoning, we argue from a general law or principle to a particular case: from the general to the individual.
The other type of mediate inference is called induction. It can be defined as the process of reasoning in which we conclude from the individual cases to the existence of general laws or principles. This is the reverse of deduction. We go from particular cases to general laws and principles. Let's consider a simple example. Scientists discovered that the boiling of point of water at sea level is 212 degrees F. Every time that the temperature of water at sea level reached 212 degrees F., water would begin to boil. From this observation, scientists concluded that a general law could be formulated that would apply to all cases of water heated at sea level. This became a universal law based on all the individual or particular observations made. This is how we can formally state the argument:
This is a valid argument formulated through induction, going from the particular cases to a general law or principle.
This, in general, is what mediate inference is all about. But there is one important point to be made regarding deduction and induction. There must exist a logical connection between the judgments and ideas, so that the conclusion follows necessarily from the first two premises. A mere listing of unrelated ideas and judgments, even though appearing in the proper logical format, does not constitute a valid inference. The conclusion must have a logical connection to the two judgments leading to the conclusion. We can now see that the matter of mediate inference consists in the various ideas and judgments of the inference and the form consists in the special arrangement of ideas and judgments by which the conclusion follows necessarily from the original ideas and judgments. There must be a logical connection between the ideas and judgments and this constitutes the consequence or consistency of the inference. The conclusion of an argument, if drawn with consistency from true judgments, must always be true. But the conclusion of an argument, if drawn with consistency from false judgments, may be true or false. The only time we can be certain of the truth of a conclusion, thereby getting rid of our state of doubt, is when the initial judgments from which the conclusion is drawn are true in themselves and there is a logical connection between the ideas and judgments. The relation must be consistent. The greatest challenge in any inference is to be found in discovering whether the initial judgments, the first two premises, are true or false. And this is the "stuff" of which scientific investigation in all its forms is made. The Argument A term is the verbal expression of the idea and that a proposition is the verbal expression of the judgment. What is the verbal expression of a mediate inference? This is called the argument. An argument can be defined as a discourse which logically deduces one proposition from other propositions. As you already know, the propositions from which the conclusion is drawn are called the premises of the argument. Another term used for premises is antecedents, because they come before a conclusion. The form of argument we are presently considering is deduction and it is most commonly expressed in a syllogism. A syllogism is an argument which generally contains two premises and a conclusion. This is not an exact definition but will suffice for our purposes here. Every syllogism is composed of three sentences: two premises and a conclusion. Each sentence of the syllogism contains four parts: a quantifier, a subject, a predicate, and a copula. A subject term refers to that thing or entity about which we assert something. The predicate term designates that which is asserted of the subject. The copula will always take some form of the verb "to be" (usually "is" or "are"). The function of the quantifier is to indicate the extent to which we refer to the individuals denoted by the subject term (usually "all," "some," "none," "no," or "nothing"). When a sentence in a syllogism contains all four parts, we say that the sentence is in standard form. In an argument, the subject and predicate of the conclusion are called the extremes. The subject of the conclusion is called the minor extreme or minor term, and the predicate of the conclusion is called the major extreme or major term. The third term or idea with which the minor and major terms are compared in an argument is called the middle term. The reason why we call them "extreme" and "middle" is because of their relative position. The middle term functions as a mediating idea between the subject and predicate of the conclusion: in other words, it establishes a connection between them. It occupies a middle position between the extremes. The predicate of the conclusion is called the major term because it generally represents a class whose extension is larger than the subject: in this situation, larger means the same thing as major. The subject generally refers only to one of the members which belong to the class as represented by the predicate in the sentence. Its extension is smaller and therefore minor. The premise which contains the major term is called the major premise. The premise which contains the minor term is called the minor premise. But we have to be careful in looking at a syllogism; the major premise may come first in the argument (and usually does), but it does not have to. The minor premise can come first and the major premise second. There is, however, an easy way to determine which of the premises is the major premise and which is the minor premise. Start with the subject of the conclusion and find the premise which contains this term. This will be the minor premise. The other premise will contain the predicate of the conclusion and be the major premise. The middle term, of course, will occur in the major and minor premises but not in the conclusion since it is the term of comparison.
The Distribution of Terms The idea of distribution is a very important one in deductive logic but, fortunately, it is a very simple idea. We say that a term is distributed when the term refers to all the members of the class denoted by the term. Let's illustrate this with a few examples. If we say, "All Americans are patriotic," the term "Americans" is distributed because we have referred to all Americans. It is "distributed" to all members of the class called "Americans," and we say the term is distributed. If we say, however, that "Some Americans are patriotic," the term "Americans" is not distributed because we are referring only to part of the class of Americans. We say the term "Americans" is undistributed. Let's take a look at how all the A, E, I, O sentences distribute their terms. Since each of these sentences contains two terms, a subject term and a predicate term, we will have to determine for each type of sentence whether both, one or none of its terms are distributed. The "A" Sentence "All Americans are patriotic." In the universal affirmative sentence, the subject term is obviously distributed since we are referring to all members of the class. But the predicate term is not distributed. We are not referring to all patriotic persons when we say "All Americans are patriotic." We are merely saying that the class of "Americans" is included in the class of "patriotic" persons. We are not speaking about all patriotic persons and hence the predicate term is not distributed. "Dogs are carnivorous." In this universal affirmative sentence, the subject term is obviously distributed since we mean to include all dogs in the class of dogs. The predicate term is not distributed, however, since we are not referring to the class of all carnivorous things. The predicate term is not distributed. This can easily be seen by reversing the subject and predicate in a true "A" sentence. If we say, "All dogs are carnivorous," this sentence is true. If, on the other hand, we say, "All carnivorous things are dogs," we know this sentence is false. We are not, then, referring to all carnivores in the sentence "Dogs are carnivorous." So the predicate term is not distributed. The "E" Sentence "No elves are unhappy." In this universal negative sentence, both the subject and predicate terms are distributed. We are saying that the class of elves is entirely excluded from the class of unhappy people. We are referring to all elves and to all unhappy people. Both terms are distributed. "No American preadolescents are voters." In this universal negative sentence, both the subject and predicate terms are distributed. We are saying that the class of American preadolescents is entirely excluded from the class of American voters, since you have to be eighteen years of age to vote in elections. The "I" Sentence "Some birds are white." In this particular affirmative sentence, both the subject and the predicate are undistributed. Here we are asserting that the class of birds and the class of white things have a common member. We are not, however, talking about all birds and about all white things. We are only talking about some birds and some white things. Therefore, both the subject and the predicate terms are undistributed. "Some Americans are hearing-impaired." In this particular affirmative sentence, both the subject and the predicate are undistributed. We are asserting that the class of Americans and the class of hearing-impaired have a common member. We are not talking about all Americans nor about all hearing-impaired persons. Both the subject and the predicate are undistributed. The "O" Sentence "Some dogs are not friendly to strangers." In this particular negative sentence, we see that the subject term is undistributed, but the predicate term is distributed. It is easy to see that the subject is undistributed since it refers only to part of a class: some (but not all) dogs. But why is the predicate distributed? This is because we are saying that some dogs are excluded from the entire class of "friendly to strangers." We are speaking about the whole class of things friendly to strangers and saying that some dogs are not members of that class. "Some mountains are not steep." In this particular negative sentence, we again see that the subject term is undistributed since we are speaking only about some mountains. The predicate is distributed, however, since we are speaking about the entire class of things "not steep." We can now see that both universal sentences distribute their subject terms. The universal affirmative sentence, however, does not distribute its predicate, while the universal negative sentence does distribute its predicate. Both particular sentences do not distribute their subject terms. The particular affirmative sentence, however, does not distribute its predicate term, while the particular negative sentence does distribute its predicate.
NEXT - Part 2: The Categorical Syllogism
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