Jonathan Dolhenty: Logic and Critical Thinking
This page is adapted and expanded from the original work by the late Dr. Jonathan Dolhenty whose websites and writings have been expiring and/or malfunctioning following his death - some of his work is only available through The Internet Archive's Wayback Machine now. Wikipedia links have been added as a starting point to various words, the first-time they are mentioned. The source links have been put at the beginning of each major section in this combined series of essays about Logic and Critical Thinking. Screenshots have been taken of the original tables where it's not been possible to duplicate the colouring here.
Truth is the object of thinking. Some truths are obvious; others are difficult to acquire. Some judgments we make are simple; some judgments are complicated. Some arguments, whether made by us or others, may be straightforward and easily understood; other arguments may be complex and consist of a series of smaller arguments, each needing to be critically examined and evaluated.
Almost every object of knowledge has a branch of knowledge which studies it. Planets, stars, and galaxies are studied by astronomy. Chemistry studies the structure, composition, and properties of material substances and the transformations they undergo. The origin, evolution, and development of human society is the object studied by sociology. Economics, biology, geography, and grammar all have objects of knowledge which they investigate, describe, and try to explain.
Critical thinking involves a knowledge of the science of logic, including the skills of logical analysis, correct reasoning, and understanding statistical methods. Critical thinking, however, involves more than just an understanding of logical procedures. A good critical thinker must also understand the sources of knowledge, the nature of knowledge, and the nature of truth. But first, what is the science of logic?
The object of knowledge involved in the science of logic is "thinking," but it is "thinking" approached in a special way. Generally speaking, logic is that branch of knowledge which reflects upon the nature of "thinking" itself. But this may confuse logic with other branches of knowledge which also have the nature of "thinking" as a part of their specific object of investigation. We need a more detailed and accurate definition to eliminate any confusion.
Logic doesn't just deal with "thinking" in general. Logic deals with "correct thinking." Training in logic should enable us to develop the skills necessary to think correctly, that is, logically. A very simple definition would be: Logic is the subject which teaches you the rules for correct and proper reasoning. For those of you who want a more complete and "sophisticated" definition of logic, you can define it this way: Logic is the science of those principles, laws, and methods, which the mind of man in its thinking must follow for the accurate and secure attainment of truth. Take your choice.
We need to be aware of a distinction between what some call "natural logic" or common sense and "scientific" logic. We all have an internal sense of what is logical and what is not, which we generally refer to as "common sense." This "natural" logic we have learned from the moment of birth, through our personal experiences in the world and through our acquisition of language. Scientific logic, on the other hand, is simply our natural logic trained and developed to expertness by means of well-established knowledge of the principles, laws, and methods which underlie the various operations of the mind in the pursuit of and attainment of truth.
We have referred to the "science" of logic but logic is really more than just a science. The science part is the knowledge of the principles, laws, and methods of logic itself. This is important, to be sure. But logic must be put into action or else the knowledge provided within the science of logic is of little use. We can, therefore, also speak of the "art" of logic, that is, the practical application of the science of logic to our everyday affairs. Logic is not intended merely to inform or instruct. It is also directive and aims at assisting us in the proper use of our power of reasoning. In this sense, we can speak of logic as both a science and an art, a practical art meant to be applied in our ordinary affairs.
We want to be sure that we don't confuse the science of logic with the science of psychology. Psychology also studies "thinking," but it is a separate, autonomous discipline of its own. And logic is not a branch of psychology, but a separate discipline of its own. How are logic and psychology different?
The most obvious difference is that psychology is a "descriptive" science while logic is a "prescriptive" science. The difference between a descriptive science and a prescriptive science can best be illustrated by an example.
Let's suppose we are scientists and have been asked to study the differences between the American form of government and the British form of government. We find that in the United States there are three separate branches in the central government: the executive branch which includes the president, the legislative branch which includes the Senate and the House of Representatives, and the judicial branch which includes the Supreme Court. We discover that the president is elected by vote of the people, as are the senators and representatives, and that the judges of the Supreme Court are appointed by the president with approval of the Senate. Furthermore, we find that the president is both the ceremonial leader and the chief executive of the nation.
Now we turn our attention to England. We see that the Queen of England is not elected and functions primarily as the ceremonial leader of the country. Instead of an elected Congress, England has a parliament system, consisting of a House of Commons, which is elected, and a House of Lords, which is not elected by the people. Furthermore, we find out that the prime minister, who is the real head of the government, is not elected by the people, but is elected by the leading political party in the House of Commons.
What we have done in the above example is simply "describe" and report on each form of government, noting any similarities and differences between them. We have been functioning as "descriptive" scientists, in this case, as political scientists since governments are an object of knowledge of a scientific discipline called political science.
Let's suppose now that we go on to argue that England should adopt the form of government we have in the United States. In this case, we are no longer describing or reporting on a state of affairs. We are now recommending or "prescribing" how England should conduct its affairs when it comes to government. We have ceased to be scientists at this point and have become political philosophers. We are no longer being "descriptive," we have become "prescriptive."
Psychology is also a descriptive science. It is not primarily interested in how we "ought" to think but in describing how we actually think. It is interested in questions such as: Do men think differently from women? Do members of a primitive society think differently from members of an advanced civilization? What is learning and how can it be measured? What goes into the processes of thinking and learning? These questions call for descriptive answers.
Logic, on the other hand, is a prescriptive science, usually considered a branch of philosophy. It is interested in formulating the general rules for correct reasoning, prescribing how we must proceed if we are to argue clearly, consistently and, yes, logically.
Critical thinkers must be intimately acquainted with the concepts and methods of logic in order to be successful in activities involving critical thinking skills.
Most of the terms that are commonly used in logic will be fully described and explained in other sections of these essays on logic and as they become important to the discussion.
It may help us, however, to have some preliminary definitions to guide us on our way and it is advisable to point out at this time some limitations on the use we will make of some common words in English which can cause confusion. We need to realize that some words have several meanings in ordinary discourse and we need to be specific about how certain ordinary words will be used.
A complete discussion of the nature of the idea is undertaken here. You will notice then that we use the word "idea" somewhat differently from its ordinary, and many times misleading, meaning.
The word "idea" will be used to mean the intellectual representation of a thing. We consider the word "thing" to be the same as "being," the most general word that can literally apply to any actual or possible existent. In our ordinary conversations, we tend to use the word "idea" in a very broad sense to denote several things which we link together. Here we will be using the word very specifically. An "idea" will represent a single "thing," "single being," or "single existent," actual or potential.
The word "term" will be used as a "name" for the "idea." As we will see later, neither ideas nor terms are "true" or "false." Ideas simply "are," and terms are used to express them. Terms are simply sensible conventional signs which we use to express an idea.
The term "judgment" will be used to mean an act of the mind pronouncing the agreement or disagreement of ideas among themselves. The terms "true" and "false" apply only to a judgment. It is possible for a judgment to be merely an opinion if its state of certainty is in question. We should really refer to it then as an opinion, and not as a judgment.
A "proposition" is a sentence which expresses a judgment, either in speaking or writing. A proposition may be true or false, a determination which is actually made by the judgment which it represents. Propositions differ from other types of sentences such as questions, commands, and exclamations. Only propositions can be asserted or denied. An "argument" consists of propositions.
An argument is not a mere collection of propositions. An argument has a structure. We use the terms "premise" and "conclusion" when we talk about the structure of an argument.
The conclusion of an argument is that proposition which is affirmed on the basis of the propositions in the argument. This is what we are trying to show is true. This is what we want someone to accept at the end of any argument we may present.
The premises of an argument are those propositions which are used to provide the support or reasons for accepting the conclusion. These are what we show to justify our conclusion. These are an essential part of any argument.
It should be noted that premise and conclusion are relative terms. One and the same proposition can be a premise in one argument and a conclusion in another argument. This is one reason why many arguments can become complex and sophisticated. One argument, using one or more of the same propositions, can lead to another related argument using the same propositions. But, never fear. We'll learn how to deal with these multiple arguments.
In our ordinary everyday conversations, we tend to get sloppy with words, using the same words but with different meanings scattered throughout our speaking. One of the things that critical thinkers must do is to take words seriously and define them accurately. The words "reasoning" and "inference" are so important to critical thinking we need to make sure we understand how they will used during this study.
Reasoning is, first of all, a process. When we are engaged in reasoning in its simplest form, we are comparing two doubtful ideas with a third idea which we already know. If both doubtful ideas agree with this third idea, they also agree among themselves. If one of our doubtful ideas agrees with the third idea, and the other doubtful idea does not, then they also disagree among themselves. In the first case of reasoning, we have an affirmative conclusion, that is, we have affirmed the conclusion. In the second case of reasoning, we have a negative conclusion, that is, we have denied the conclusion.
The process of reasoning means that from certain things we already know to be true, we can acquire another truth not already known but that follows necessarily from those truths already known. It may seem strange to see it described this way, especially since reasoning is something we are constantly doing all day long. It's just that we are not consciously aware of what we are actually doing.
Reasoning and inference are sometimes thought to be the same process. This is true if we are talking about what is called mediate inference. Reasoning and mediate inference, which include deduction and induction (to be described later), are the same thought process. But there is another kind of inference called immediate inference, which some think is a primitive type of reasoning, wherein we draw a conclusion about something immediately without going through the process of thinking it out. Self-evident truths are an example of immediate inference.
Many people are confused by the terms evidence and proof. During a discussion of the inductive method and empirical science, much more will be said about evidence and proof. For now, however, let's just consider a few general ideas regarding the use of these terms.
The term "evidence" we'll define as any grounds used to assert a proposition to be true. We can also say that evidence is any supposed fact which is considered as supporting the truth of a given proposition. There are obviously many kinds of evidence. There is what we commonly call "firsthand" evidence that we all use as grounds for stating propositions to be true. We say, "There are blue and white colored fish in the aquarium." How do we know? We look and see. If someone questions our statement, we invite that person to come look and see. We see, hear, smell, taste, and feel things "firsthand." Usually, this type of evidence doesn't cause much of a problem in our everyday life.
But suppose we were testifying in court about an incident we witnessed. Our evidence would be "firsthand." We saw the accident. We heard the fighting words. We smelled the smoke of the fire. Presenting "testimonial" evidence based on a "firsthand" account may involve us in some complications, however, especially from an attorney on one side of the case or the other. Later, we'll investigate testimonial evidence more in detail.
There is also the matter of "circumstantial" evidence we hear so much about these days in criminal trials. Circumstantial evidence involves those relevant circumstances or facts which enable us to draw legitimate inferences to some principal fact, which fact then explains the existence and presence of these relevant circumstances or facts. This is really "indirect" evidence, one or more steps removed from what we generally consider to be "firsthand" evidence. In criminal trials, the presence of and analysis of blood, DNA, fingerprints, and so on are considered circumstantial evidence if no "firsthand" or direct witness to the criminal event was present at the time of the event.
Empirical scientists are very concerned about evidence. They collect evidence to document and support their scientific principles, laws, theories, and so forth. How do we know dinosaurs roamed the earth even though they no longer exist? Well, paleontologists and geologists have found evidence of their skeletons, eggs, and so on. How do we know that water boils at sea level when it reaches 212 degrees F.? Well, because physicists and chemists have collected evidence that it does and expressed it in a general scientific law.
Every time any of us states a proposition which we assert to be true, we try to give evidence supporting the truth of the proposition. This evidence constitutes the grounds for saying the proposition is true. We gather facts which we consider to be supportive of the truth of the proposition we assert. The next chapter will discuss some ways by which we attempt to discover and provide various types of evidence.
The word "proof" does not designate the same thing as the word "evidence," and proof is what we are more concerned with in the science of logic. Logic may be said to be concerned with the question of the adequacy or probative value of different kinds of evidence. Traditionally, however, logic has devoted itself mainly to the study of what constitutes proof, that is, complete or conclusive evidence. Proof is essentially a process, an act of testing to determine the validity of an argument which will hopefully support the truth of a proposition presented as a conclusion.
"Proof" is not a simple matter, particularly in situations where evidence has to be weighed in favor of one conclusion or the other. In deductive logic, the matter of proof is fairly straightforward and rules have been made to help us determine the validity of a deductive argument. In inductive logic and processes using scientific method, the situation is not as clear and decisive. Here we enter the world of probability, partial evidence, probable inference, and the problem of the weight of evidence. It is here, also, where arguments become controversial and, to some people at least, most exciting.
There are thousands of languages spoken throughout the world. No one knows exactly when or how human beings began to use what today we call ordinary language, but a number of theories have been offered by early linguists.
There is the "ding-dong" theory which assumed that there was some necessary and logical connection between the sound of a word and the thing referred to. This theory has largely been discarded. Then there is the "poo-poo" theory which states that primitive languages were a result of exclamations of surprise, fear, and so forth. There was a group of linguists which put forth the "bow-wow" theory which insisted that language developed from the imitation of natural sounds.
However and whenever human beings first began to use language and how such language developed will probably never be known. We do know, however, that all languages, even so-called primitive ones, are really a very subtle and complicated instrument of human communication. We also know that there is more to language (particularly spoken language) than words, sentences, and paragraphs. Human beings use language in many ways and for many purposes.
Most of you have probably learned in school that a sentence is defined as a unit of language which expresses a complete thought. Additionally, you were probably taught that sentences can be divided into four categories: declarative, interrogative, imperative, and exclamatory. But it should be noted now that these four grammatical categories do not coincide with those of assertions, questions, commands, and exclamations. This brings up the difference between form and function.
A declarative sentence, for example, is a form of sentence but its function may vary depending on its purpose in a conversation. Not every declarative sentence states an assertion which may be considered true or false. It is a mistake to confuse declarative sentences with the informative function. If someone says, "I really enjoyed your lecture," this is a declarative sentence but it need not be informative at all. It could be ceremonial or expressive, exhibiting a feeling of appreciation or a sense of good manners.
An interrogative sentence does not have to be a question asking for information. It could, for instance, be a command to hurry like in the sentence "Do you realize we'll be late for the party?" The form of the sentence is interrogatory but the function may be imperative. Let's look at another interrogative sentence: "Isn't it true that the State Department was riddled with Communists after the Second World War?" This may be a question asking for information or it may not be. Such a sentence could also be an attempt to evoke a feeling of hostility in the listener or express a feeling of hostility in the speaker. Its form is interrogatory but its function may be expressive.
We run into the same situation with imperative sentences. The imperative sentence "Let us pray" may not be functioning as a command at all but is simply being used as ceremonial or expressive language. Sentences which are exclamatory may serve functions other than expressive. "Heavens to Betsy, it's late!" may function as a command to hurry. "What a good book!" uttered by a salesman in a bookstore may function more directively than expressively.
The important point to remember is that conversations, whether spoken or written, may serve more than one function. There may be portions of the conversation which serve an informative function and are to be evaluated as true or false. There may be passages which serve the directive function and be evaluated as right or wrong, proper or improper. There may be a passage which is expressive and needs to be evaluated as sincere or insincere, valuable or otherwise.
To properly evaluate a given passage in a conversation it is important to recognize the function or functions it is intended to serve.
Logic and critical thinking are primarily concerned with matters of truth and falsehood and with correctness and incorrectness of argument. Logic is more concerned with the informative function of language. But it is important to be able to distinguish this informative function from other functions which the same passage may be serving in any given conversation. The grammatical structure (the form) may serve as a clue to the function of a particular passage in a conversation, but there is no necessary connection between function and grammatical form.
We must also be aware that a passage taken in isolation, that is, a group of words taken out of a conversation and treated independently, may cause problems in determining the function of the selected passage. This is because context is very important in determining the meaning of and the function of a given passage. What is serving an informative function in one context may be serving a directive function in another.
Again, what we are primarily concerned with in the science of logic is the informative function of language.
Generally speaking, we can say that there are at least three basic uses of language that we encounter virtually every day.
Much of our speaking and writing is devoted to using language to communicate information. This is sometimes called the informative function and is usually accomplished by using what are called propositions, sentences which affirm or deny something. This use of language involves the concepts of truth and falsity. A proposition may be true or it may be untrue.
A second basic use of language is called the expressive function. Here we are using language to communicate feelings, emotions, and attitudes. There is no problem of truth and falsity when using language in this way. Feelings, emotions, and attitudes may be right or wrong, proper or improper, appropriate or inappropriate, but ordinarily we don't say they are true or false.
Language may also be used in a directive way. Here we are using language for the purpose of causing or preventing some overt action. Ordinarily we call such sentences commands or requests and we don't apply the concepts of truth and falsity to such sentences. Whether or not a command should be obeyed or a request granted is, of course, quite another matter and doesn't concern us here.
Categorizing the above three basic functions of language as informative, expressive, and directive may help us begin to understand the complexities of linguistic communication but this threefold division is really an oversimplification. Our ordinary conversations are much too diverse and complicated and these three functions are intermixed and modified in actual practice. Then there are some other ways in which we use language which do not neatly fall into one of the three categories, such as using language for a ceremonial or performance function.
We need to become aware of the ways in which we use language for certain specific purposes.
We have to be careful about arguments that involve words which are not descriptive but "emotive." Emotive words express an attitude or feeling and can have an emotional impact on readers and listeners. Some words, however, can have both a descriptive function and an emotive one. We have to be careful to differentiate between both functions.
Let's take the word "bureaucrat." A bureaucrat is a government functionary, a person who works in a bureau of the government. This is a "descriptive" definition. But the word "bureaucrat" can also have an "emotive" function. Many people think of a "bureaucrat" with resentment and disapproval, a person who causes them harm or difficulty.
It is important for the student of logic to realize that a word may have both a descriptive or literal meaning and an emotional meaning. In logic, we are not concerned with the emotional meaning and its impact. Many arguments go awry because these two functions are confused.
Think, for instance, of the following terms: "pervert," "maniac," terrorist," "unnatural," "abnormal," and "antisocial." These words have descriptive definitions. But they also have emotional import. As far as logic is concerned, we are not interested in the emotive function these words may have. As a matter of fact, most people who use these words in their everyday conversation would have a difficult time defining what they mean by them if they were challenged by a good logician. We, as good logicians, will not use these words in our arguments without clearly defining them and making sure they are descriptive, not emotive.
Poetry is very important in our lives as human beings. The language of poetry, however, carries with it some problems when it comes to logic. The poet's language is, in a sense, descriptive. It is, however, descriptive in a special sense, an emotive sense. The poet is trying to move us emotionally or he may be trying to persuade us of something.
The poet is permitted by custom and convention to have a certain "license" with language (called poetic license, in fact) to permit poetry to draw "word pictures" that may not, in fact, represent reality as it is. Words may be used to elicit emotions or feelings. Sentences may be constructed to send us beyond what we know as reality. "Flights of fancy" are perfectly proper for the poet.
We must realize, however, that poetic language, no matter how beautiful it may be, has no place in logic. Logic is emotionless and non-feeling. Poetic language, like emotive words, has no place in propositions we use in logic.
Here we have another possible pitfall in language. We all use figures of speech in our daily conversations. We use metaphors, similes, and analogies without sometimes realizing we are doing so. In ordinary circumstances these figures of speech are perfectly all right and actually add color and interest to our speaking and writing.
The problem we can have with figures of speech occurs when we take them literally, that is, when we think the figure of speech actually represents the real thing or event. The man who "roars like a lion" is not really a lion, after all. The woman who "looks like an old crow" is not really a crow. A famous old fable may seem to represent an actual case, but it may not be true or applicable except by analogy.
Sentences which contain figures of speech are not acceptable as propositions to be used in logic. Metaphors, similes, and analogies may be good for illustrating things and events in a "poetic" sense; they are not, however, acceptable for use in propositions.
The importance of definitions cannot be overemphasized, particularly in analyzing and evaluating arguments in everyday contexts. We need to know the meanings of words in a sentence to determine whether or not they express a proposition and can, therefore, be logically analyzable. Also, the vagueness of everyday speech can create problems in arguments which demand accuracy of thought and expression.
It is important to distinguish between symbol and object. The object is that which the symbol marks and the object is that to which the symbol points. For instance, "dog" is a symbol whereas the class of dogs to which it points is the object. "John" is a symbol whereas the class of persons to which it points is the object. "Tornado" is a symbol whereas the class of natural events to which it points is the object.
There is also a distinction between natural symbols and conventional symbols.
A natural symbol marks and signals a meaning relationship that we find in nature, and it is a relationship over which human decisions of linguistic usage have no effect. For example, certain atmospheric conditions mean a storm is approaching. The connection between atmospheric conditions and the storm is not decided by us, for it is found in nature. The symptoms indicating an approaching storm are symbols which signal a state of affairs in the world of nature, not in the world of conventional linguistic discourse.
On the other hand, an arbitrary or conventional symbol is one that has been established through a deliberate decision or linguistic convention. The English language, for example, is a set of conventional symbols. There is no natural law which establishes the relationship between a language and the objects to which it relates. Conventional symbols are labels made by man.
We can see, then, that objects in our world do not have anything like a "natural" name. The names of the various breeds of dogs and cats, for instance, were not "discovered," but were "invented" and "assigned" by human beings. It is a matter of usage and convention and these common uses should not be ignored if we want to communicate successfully.
This is one reason why we say that "names" of objects or "definitions" of words are not, strictly speaking, true or false. At least not in the same sense that we say a statement is true or false. What we really mean when we say a definition or meaning of a word is "false" is that it is not being used in the ordinary, common, or conventional way.
It is important to distinguish between the use of a term and the mention of a term. In the statement "Boys are strong," the word "boys" is used in the conventional way. The object of the term "boys" is the class of all boys. In this situation, we say that we use the word "boys."
Consider, however, the following statement: "The word 'boys' refers to young people of the male gender." In this case, the object of "boys" is not the class of all boys; the object is the word "boys" itself. We are talking about the word-object. So we say that we mention the word "boys."
Writers generally indicate when a word (or phrase) is being mentioned, rather than used, by setting such a word (or phrase) off in quotation marks or by italicizing the word (or phrase). It is more difficult to indicate the distinction when speaking.
It should be noted there is an informal logical fallacy called the Use/Mention Fallacy which refers to an argument that fallaciously persuades by confusing the mentioning of a word with the use of it. One commits this fallacy when:
One example will be given to illustrate the Use/Mention Fallacy. This is a "word game" attributed to the ancient Greeks:
"You can't say the word wagon because whatever you say must come through your mouth; but a wagon is far too big to come through your mouth."
It should be obvious that the word "wagon" is being merely mentioned in one part of the sentence and then used in another part. The argument could be made clear by writing it this way:
"You can't say the word 'wagon' because whatever you say must come through your mouth; but a wagon is far too big to come through your mouth."
It can clearly be seen that the argument is false. We can certainly say the word "wagon" because here we are merely mentioning it and the object of the word is the word itself and not the class of actual wagons. The second time the word appears in the argument, it is being used and not merely mentioned. In this second appearance, the word indeed refers to the class of actual wagons.
The point to be made is that it is important to keep in mind that ordinary words can be used to confuse and distract. Stay alert!
In logic and critical thinking, definitions are used primarily to define words, not concepts or ideas. The term to be defined is called the definiendum, and that part of the total expression which clarifies the definiendum is called the definiens. For example, in the statement "A parrot is a tropical bird," "parrot" is the definiendum and "is a tropical bird" is the definiens.
When framing a definition, it is important to indicate the context in which the definiendum is being used. There are several kinds of definitions and how a definition is classified will depend on the type of definiens provided.
This is a definition in which the definiens consists of only one word which, in suitable contexts, can be used interchangeably with the definiendum. For example, "hit" is synonymous with "strike" in an appropriate context. And "gang" can be synonymous with "pack" in certain contexts. And "cold" can be synonymous with "chilly" in the right context.
The best place to locate synonyms is, of course, a thesaurus, and you should become familiar with one if you haven't done so. A good dictionary and a good thesaurus are musts for any Super Thinker. See the list of resources in the back of this book for suggestions.
This is a definition in which the definiens lists words referring to or presents actual examples of things, properties, relations, concepts, and so forth, to which the definiens can be properly applied. Enumerative definitions are divided into two classes depending on the nature of the definiens.
The first class of enumerative definitions is called the ostensive definition. This type of definition provides example of things to which the definiendum can be applied. This can be done, for instance, by pointing. A foreigner shopping in a land whose language he does not speak may have to avail himself of ostensive definitions to make himself understood. He may have to point to an object to get his message across to a store clerk. Babies seem to learn a great deal in their early stages of linguistic growth through the use of ostensive definitions. I can define the word "animal" by pointing at a dog, a cat, a cow, a horse, or any other kind of animal that is present.
Ostensive definitions have both advantages and disadvantages. They are easy to formulate, can teach concepts previously unknown, and do not depend on preexisting language. But, on the other hand, they depend on the actual presence of the definiens and they are prone to misinterpretation.
The second class of enumerative definitions is called the denotative definition. This type of definition does not require the physical presence of the definiens. It lists examples of things, or types or classes of things, to which the definiendum applies. The list constituting the definiens indicates what is called the "denotation" or extension" of the definiendum. For example, we could define the term "skyscraper" by listing the Empire State Building, the Chrysler Building, the Woolworth Building, The World Trade Center, and so forth.
Denotative definitions are usually easy to formulate and are generally understood. One disadvantage, however, is that many terms cannot have their extensions completely enumerated for various reasons. For instance, the extension of the term "number" is infinite and all examples of the term could never be listed. The extension of the term "star," referring here to the heavenly body and not the Hollywood body, would also be very difficult to complete, so a partial enumeration is all that could be hoped for.
This kind of definition pinpoints the meaning of the definiendum by listing a set of properties common to all the things to which the definiendum can be correctly applied, and common only to those things. The sum total of a definiendum's essential properties is called the connotation of the definiendum.
For example, "television" can be defined as "an optical and electric system for continuous transmission of visual images and sound that may be instantaneously received at a distance." In this definition, the essential attributes of "television" are listed.
You need to simply note here that an increase in the intention of a term, that is, adding additional properties, will either decrease the denotation (extension) or leave it unchanged, and a decrease in the intention of a definition, that is, eliminating some of the properties, will either increase the denotation or leave it unchanged.
These definitions are part of a theory of the meaning of scientific concepts. The view is that all physical concepts are to be defined by indication of the operations that are required to measure them. The definiens provides a test or a formal procedure which is to be followed in order to determine whether or not the definiendum applies to a certain thing.
For example, an operational definition for the term "buoyant" would be: "If you place an object in water, it does not sink to the bottom."
Operational definitions are dependent to a large extent on the scientific concept or theory which includes them. For instance, the meaning of the term "electron" will vary depending on the specific scientific theory under discussion. The meaning of "electron" in the electromagnetic theory of Lorentz has quite a different meaning and designates quite a different scientific object from what is designated by the same word in the theory of quantum mechanics.
The same concern applies to the term "instinct," which has both a common, conventional meaning in ordinary discourse, and a technical operational meaning in the science of psychology. When we ask the question, "Do human beings have instincts?", it is vitally important as to whether we are using the term "instinct" with the common meaning ordinary used in non-scientific discourse, or whether we are using it with the scientific meaning that psychologists attach to it. While most people would probably say that human beings do have instincts, according to the scientific definition I learned in advanced psychology classes in college, human beings definitely do not have instincts.
Here is another question which has intrigued a great many people in the past: "If a tree falls in the forest, and no living creature is within a hundred miles, is there any sound?" Well, here again, we need to be careful how we are using the term "sound." The ordinary person in common discourse usually uses the term "sound" to mean that which is "heard" by a sensing organism. But that is not the meaning attached to it by the physicist. "Sound," to the physicist, refers to vibrations along the electromagnetic spectrum. Therefore, the answer to the question above would be "yes," because there would be vibrations set off by the falling tree once it landed on the ground. To the physicist there would be a sound even if no living creatures were around to hear it.
We have discussed above some of the different kinds of definitions there are and some of the different methods by which a word can be defined. It is also important to be aware of some of the uses to which definitions can be put. Some of these will be briefly discussed below.
These definitions are used to report the meaning of a term as it is used and understood by a particular group of people. When someone states that a certain definition is "false," it is usually to a lexical definition he is referring. While, strictly speaking, definitions are neither true nor false in themselves, there is a sense in which they are true or false if we are speaking of the way in which a particular group of people use a specific word. In this case, a lexical definition is reporting a definition or meaning that a word already has and, in this sense, the definition can be said to be true or false.
Therefore, the statement that "The word 'mountain' means a large mass of earth or rock rising to a considerable height above the surrounding country," is true. It is true because it is a true report of how English-speaking people use the word "mountain." Of course, we could redefine the word "mountain" and if the new meaning was accepted by everyone, then the above definition for "mountain" may not then be true.
When you use an ordinary dictionary to get at the meaning of a word, you are receiving, for the most part, a reportive or lexical definition in common use. Some dictionaries, of course, give more than the commonly-accepted definition, and may also provide technical definitions which differ from those used in ordinary discourse. These may be limited reportive definitions, that is, the meaning may be limited to a certain context such as science, philosophy, or law. Two such examples are theoretical definitions and legal definitions.
It is in connection with theoretical definitions that most "disputing over definitions" occurs. A theoretical definition defines a word in terms of the meaning which it carries in a particular scientific or philosophical theory. Since some words have both ordinary and theoretical lexical definitions, it is easy to see how disputes over definitions can occur. The examples given above in the discussion of operational definitions (the meaning of "sound" and "instinct") illustrates the problem. "Sound" and "instinct" have both ordinary and theoretical lexical definitions and one must specify which meaning is being used in a particular discussion.
One of the most serious debates in the history of philosophy has been over the status of "ideas." Students of philosophy attempting to compare Plato's notion of "idea" with Aristotle's notion of "idea" have usually indulged in a wasteful argument because they have not understood the meaning of "idea" in each philosopher's technical theory. For those of you familiar with the problem or who have studied philosophy, let me briefly outline it for you.
The term "idea" means something different in Plato's theory from what it means in Aristotle's theory and, unless the meaning of the term within each theory is understood and clarified, useless argumentation occurs. An Aristotelian "idea" is really a Platonic "sensible" given an immortal persistence, whereas a Platonic "idea" is not even in part a "sensible." "Ideas" and "sensibles" are totally different things. For Plato, "sensibles" are nominalistic and purely transitory. This is why Plato says that the sense world is a world of "becoming" and not a world of "being." For Aristotle, on the other hand, the sense world is a world of "being" and the process of "becoming" is explained through the use of Aristotelian "forms."
A legal definition is one that is specified in laws as formulated by a legislative, judicial, or executive body. It serves a limited reportive function when it is used in reference to a definition which is generally accepted within an existing legal system. Again, the difference between an ordinary lexical definition and a legal definition may give rise to verbal disputes, unless which meaning is being used is made clear.
Consider, for a moment, the term "statutory rape." Ordinarily, we think of rape as an act committed through force, against the will of the victim, without the victim's consent, and associated with some degree of physical violence. But, legally, this may not be so.
Sexual activity with someone under the designated age of consent constitutes "rape" in many jurisdictions even though no force is used and no violence is present and the "victim" may have consented to the activity. This is "rape as defined by statute" or statutory rape. And, furthermore, what is considered statutory rape in one jurisdiction may not be considered rape at all in another jurisdiction. It may depend on the defined "age of consent," which may differ from place to place.
Another example of a legal definition is the definition of "family" by the U.S. Census Bureau, a definition which differs from the ordinary meaning most people attach to the term. The term "blindness" may have a specific legal definition for purposes of receiving public assistance and this definition may differ from the common meaning attached to it by ordinary people.
A stipulative definition is that which is given to a brand-new term when it is first introduced, or a word which has a generally accepted meaning but is used in a new sense.
Anyone who "coins" a new word or introduces a new symbol has complete freedom to stipulate what meaning is to be given to it. The assignment of meanings is a matter of choice and, therefore, stipulative definitions cannot be "true" or "false" in any sense. Of course, a stipulative definition can be judged as being good or not so good, depending on whether or not it achieves the purpose for which it was introduced.
An interesting characteristic about stipulative definitions is that, once the term and definition become absorbed into general usage, that is, they become part of ordinary, common discourse, they cease to be stipulative and become reportive or lexical definitions.
Stipulative definitions are an important part of science and philosophy. There are many advantages to introducing a new and technical symbol defined to mean what would otherwise require a long sequence of familiar words for its expression. This helps to economize space and time. Also, the emotive suggestions of familiar words are often disturbing to a scientist or philosopher interested only in their literal or informative meanings. Stipulative definitions may help to keep terms free from emotional overtones, bias, and prejudices.
Neither stipulative nor lexical definitions can serve to reduce the vagueness of a term. A vague term is one for which borderline cases may arise, such that it cannot be determined whether the term should be applied to them or not.
Precising definitions are used to eliminate ambiguity or vagueness. Any of the kinds of definition previously discussed &emdash; synonymous, enumerative, etc. &emdash; can be used as a precising definition.
If a term is vague or its meaning may be misunderstood, it is always useful to present a precising definition to help others grasp what is being proposed. For example, we might want to say: "In this discussion (or argument), the term 'poor' will refer only to those with incomes under $20,000 per year."
The purpose of a persuasive definition is to influence attitudes. Their function is expressive and usually reflect the beliefs or persuasive intent of the speaker or writer. Persuasive definitions are usually intended to affect someone's evaluation of the definiendum, usually in hopes of affecting other people's behavior in some way.
Consider, for example, the following definitions for "pot smokers":
"Pot smokers" are "enlightened experimenters, victims of a puritanical and hypocritical society, who have challenged age-old patterns of behavior.
Under the heading "Defining Abortion a Tricky Business" appeared the following story:
Amidst the emotional debate on the abortion issue at the State Legislature, humor still lives. Anonymous legislative staffers this week drafted and circulated to legislators a proposed "general response to constituent letters on abortion." It goes like this:
You ask me how I stand on abortion. Let me answer forthrightly and without equivocation.
If by abortion you mean the murdering of defenseless human beings; the denial of rights to the youngest of our citizens; the promotion of promiscuity among our shiftless and valueless youth and the rejection of Life, Liberty and the Pursuit of Happiness&emdash;then, Sir, be assured that I shall never waver in my opposition, so help me God.
But, Sir, if by abortion you mean the granting of equal rights to all our citizens regardless of race, color, or sex; the elimination of evil and vile institutions preying upon desperate and hopeless women; a chance to all our youth to be wanted and loved; and, above all, that God-given right for all citizens to act in accordance with the dictates of their own conscience&emdash;then, Sir, let me promise you as a patriot and a humanist that I shall never be persuaded to forego my pursuit of these most basic human rights.
Thank you for asking my position on this most crucial issue and let me again assure you of the steadfastness of my stand.
Mahalo and Aloha Nui.
(from: "Thanks and Love," The Honolulu Advertiser, February 14, 1970)
We all get into arguments. Sometimes these arguments can be heated and our emotions come into play. Sometimes arguments lead to quarrels. Arguments can, unfortunately, even break up longtime friendships and split apart families. This is a sad reflection on us as rational human beings.
When we use the word "argument" here, however, its use has nothing to do with quarreling, fighting, or breaking up relationships. For our purposes here, an argument is a piece of reasoning in which one or more statements are offered as support for some other statement. An argument is simply a set of statements, one of which is designated as a conclusion and the remaining statements, called premises, are asserted as being true and are offered as evidence that supports or implies the conclusion.
An argument should be distinguished from a debate. A debate is really a series of arguments usually, but certainly not always, about a single topic or a set of related topics. Think of the famous "Presidential Debates" held every four years during the U.S. presidential campaign. The debate always consists of many arguments about many topics.
A debate may be a formal debate such as the presidential debate or the debate held between two scholastic debating teams. A debate may be an informal debate during, for instance, a dinner party or a public hearing where rules of debate are not enforced.
The first step in recognizing an argument for the purposes of understanding and evaluating the argument is to identify the premises and the conclusion which make up the argument. There are certain "indicator" words which may help.
Conclusions are commonly preceded by these words: thus, therefore, accordingly, it follows that, implies that, hence, consequently, so, we may infer that, we may conclude that, and in conclusion.
Premises are commonly preceded by these words: since, because, for, given that, due to, insofar as, inasmuch as, in view of, as shown by, can be inferred from, and on the ground that.
Some arguments may not contain indicator words and you will have to seek out the premises and conclusion within the context of the argument, seeking the relationship of the sentences in the argument to each other. It is helpful, in cases like this, to ask yourself questions like "What is being argued for in this argument?" or "What is this person trying to persuade me of?" This will help to point out the conclusion.
To find the premises in the argument, it is helpful to ask yourself questions like "What evidence is being provided to support the conclusion?" or "What reasons are being given as grounds for the conclusion?" or "What facts are cited as justification for the conclusion?"
Another thing to be aware of is that the conclusion and the premises in an argument do not necessarily occur in any particular order within the argument. It would be nice if in all arguments the conclusion was stated first with the premises following to support it or the premises stated first as support for the conclusion to follow. But, sometimes, arguments are not as orderly as we would like them to be, especially in oral discussions where the structure may be rather loose.
HINT: If this occurs in an argument you're having with someone, ask them to slow down and organize what they're saying, stating clearly what conclusion they're trying to support and what evidence or reasons they're presenting to support the conclusion.
Arguments presented in books and other written material, especially if offered by a seasoned writer, are usually well organized and easy to follow. The same tends to be true of experienced speakers who are presenting lectures.
One type of argument you may not normally consider an argument has to do with advertising. Many television commercials, as well as other forms of delivering advertisements, are actually arguments. An advertisement may contain a conclusion such as "This is the best product you can buy," along with reasons (expressed as premises) why you should buy the product. Many advertisements, however, are guilty of committing what are called "logical fallacies." You'll learn a lot more about logical various fallacies as you continue on in this program.
REMEMBER: Critical thinkers must be able to recognize an argument when one is presented. Critical thinkers must also be able to identify the premises and conclusion of a complete argument.
An argument is a set of statements containing a conclusion and one or more premises used to support that conclusion. The conclusion is a claim made by one of the parties in an argument for which that person provides evidence justifying the claim. The claim is supported by "reasons" and, therefore, the whole process is an act of reasoning.
If there is no conclusion supported by reasons, there is no argument. An announcement is not an argument. A command is not an argument. An apology is not an argument. A list of questions is not an argument.
For an argument to be present, there must be some claim, expressed as a conclusion, supported by evidence or reasons, expressed as premises. Anything else is a non-argument.
It can sometimes be difficult, however, to determine whether some passage in written or spoken material is a genuine argument. Some written or spoken material may offer explanations which look like arguments, but are not genuine arguments. There may even be "reasons" given to explain something.
The best way to deal with these is to ask yourself what the primary intention of the writer or speaker is. If the intention is simply to explain, it is probably not an argument. But if something is being asserted (a conclusion) and reasons are given to justify the assertion, then it is an argument.
Many times people fall into the trap of arguing over what are called value-claims. Value-claims must be distinguished from fact-claims. A conclusion that is a value-claim may appear like an ordinary conclusion in an ordinary argument, but it is not. A conclusion in a genuine argument is always a fact-claim. What is the difference between a value-claim and a fact-claim?
A value-claim never has the element of objective truth in it because it does not deal with a fact. It is a matter of taste, not of objective truth. For instance, if I prefer chocolate ice cream and you prefer strawberry ice cream, we are expressing matters of taste. It is useless to attempt to argue about our preferences regarding ice cream flavors. In fact, no real argument is possible.
A fact-claim has the element of objective truth or, as may be the case, it can be shown to be false. A fact-claim can be argued about. There is a way to determine whether a fact-claim is true or false, although sometimes we may not be entirely able to do so at a given point in time.
Ethical or moral claims present a special problem. And we probably spend a great deal of our time "arguing" over moral issues. The status of ethical "arguments" is controversial. Some authorities have argued that ethical and moral claims are merely a matter of "taste," "personal preferences," or "feelings." Others have argued that at least some moral claims can be expressed as fact-claims.
The position I take is that moral "principles" are, indeed, fact-claims even though many moral "rules" may reflect value-claims. Rather than get into this controversial issue here, I refer the reader to my essays concerning ethics and moral philosophy.
REMEMBER: Critical thinkers can tell the difference between a genuine argument and a non-argument. Critical thinkers do not get into arguments over value-claims, but only over fact-claims.
Sometimes it can be difficult to tell whether a real argument is taking place. For instance, a set of statements may be asserted as true but no conclusion is offered or they may not provide support for a conclusion. What is presented here is not an argument but an exposition or simply an explanation.
Some of the indicator words given above may be present in some statements even though no argument is taking place. Some of these words have more than one meaning and do not always identify conclusions and premises. "It is raining because a storm is passing through" is not an argument even though the indicator word "because" is contained within the sentence.
A "conditional" statement may sometimes be mistaken for an argument when it is not. For instance: "If you get plenty of sleep (a conditional statement), you will wake up well-rested." While this may be used as part of a genuine argument, it is not an argument itself. The only thing being asserted here is that a relationship exists between "plenty of sleep" and "well-rested." It is not being argued that "If you get plenty of sleep" is either true or false.
Sometimes an argument is not fully stated. A conclusion or one or more of the premises may be left out. These arguments are called enthymemes. For example:
All human beings are mammals; therefore, Mr. Jones is a mammal.
Something is missing here and it is one of the premises: "Mr. Jones is a human being." This premise fills the argument in and completes it.
When you come across an argument that seems to be missing one or more premises, or even a conclusion, a rule called the principle of charity suggests that you supply the missing premise, premises, or conclusion to make the argument as good as possible.
REMEMBER: Critical thinkers are aware of the problems involved in recognizing arguments and take their time to evaluate as to whether or not a genuine argument has been presented.
A real disagreement occurs when the statements of one person's position are logically inconsistent with the statements of the opposing person's position. In other words, it is logically impossible for the statements of both positions to be true at the same time. For example:
Here there is a real disagreement. Both statements cannot be true at the same time.
An apparent disagreement (sometimes called a pseudo-disagreement) occurs when the statements of one person's position are not logically inconsistent with the statements of the opposing person's position. In other words, it is logically possible for the statements of both positions to be true at the same time. For example:
Here there is only an apparent disagreement. It may be true what each believes and each statement is logically possible.
In many arguments, a key word or phrase used by the opponents may be used with different meanings. Here we have merely a verbal disagreement. This is not a real disagreement, although it may appear to be so. It is really a type of pseudo-disagreement where the opponents are applying different meanings to the same word or phrase. For example:
Here we have a merely verbal dispute. Person 1 considers anyone who weighs more than 150 pounds to be obese and Person 2 considers anyone who weighs more than 250 pounds to be obese. The word "obese" is not being used with the same meaning.
Upon closer examination, many so-called disputes are not arguments at all, but merely apparent disagreements or verbal disagreements.
REMEMBER: Critical thinkers know the difference between a real disagreement and a pseudo-disagreement and can respond appropriately. Critical thinkers must identify and label these pseudo-disagreements and refuse to engage in them. If the disagreement is merely verbal, discuss the meanings of the words involved in the discussion.
There are basically two types of argument. Deductive arguments are arguments in which the conclusion is presented as following from the premises with necessity. Inductive arguments are arguments in which the conclusion is presented as following from the premises only with probability.
Consider this classical deductive argument:
The premises in this argument are the first two statements. The conclusion is the last statement. The conclusion follows by necessity from the two premises; the conclusion follows with certainty from the premises. It can't be any other way. The premises fully support the conclusion.
Now consider this inductive argument:
The premises in this argument are the first four statements. The conclusion is the last statement. Note that the conclusion follows only with some degree of probability from the premises. The conclusion does not follow from the premises by necessity or with certainty.
The difference between deductive and inductive arguments can be seen by noting that all the information needed to reach the conclusion in the deductive argument above is contained in the premises. It is not necessary to go outside the argument for any additional information.
On the other hand, in the inductive argument above, the conclusion is not contained by necessity in the premises given. The conclusion requires us to go beyond the information contained in the premises. If at some later time a human being is discovered who is not mortal, the argument will have to be reevaluated. Inductive arguments do not give us absolute certainty because the premises cannot provide absolute support.
Of course, with some inductive arguments, we can get pretty darn close to certainty. For instance:
But as sure as we are that the conclusion is true, the conclusion does not follow logically from the premise. It is still only probable, although highly probable, that the sun will rise tomorrow.
Deductive arguments prove or fail to prove their conclusions with certainty. A deductive argument is either valid or invalid. In a valid deductive argument, if the premises are true, the conclusion must be true. It is impossible for the premises to be true and the conclusion false. The validity of a deductive argument is determined by its logical form, not by the content of the argument.
Inductive arguments are neither valid nor invalid as the terms are used in deductive arguments. Inductive arguments, since the conclusion is only probable, are said to be good or bad, strong or weak, or better or worse. It depends on the strength of the supporting premises.
It might be noted here that most of the arguments we run into in our daily lives are of the inductive type. Therefore, it is especially important to learn as much about the inductive method as you can.
REMEMBER: Critical thinkers can tell the difference between a deductive argument and an inductive argument. Critical thinkers know that a valid deductive argument gives certainty, while an inductive argument gives only probability.
A good deductive argument, wherein the conclusion can be trusted to be true, is said to be sound. To be a sound deductive argument, three things are necessary:
The validity of a deductive argument is determined by its form, not by the content of the argument.
The truth of the premises is of little concern for the logician as a logician. The logician is primarily interested in the form of the argument. But truth is important to people and ways of establishing truth are the subject of a philosophical discipline called epistemology, which deals with what is referred to as the "problem of knowledge."
A circular argument is one wherein the conclusion is merely a restatement of one of the premises. In other words, the argument goes around in a "circle." The fallacy here is sometimes called begging the question and you may already be familiar with it.
Inductive arguments are more difficult to evaluate than deductive arguments. Inductive arguments are not considered to be sound or unsound, valid or invalid. Recall that the premises of an inductive argument do not provide absolute support for the conclusion.
The truth of the premises of an inductive argument should, of course, be reasonably well established.
In any inductive argument we must consider the relative strength that the premises provide to support the conclusion. The stronger the support of the premises (assuming them to be true), the more probable the conclusion is true.
REMEMBER: Critical thinkers are aware of the general criteria for good deductive and inductive arguments.
There are three concepts which are commonly confused by a lot of people: truth, validity, and soundness. But taken together, these three concepts provide a solid foundation for evaluating any argument. While these concepts are considered in-depth later in this book, a brief discussion of them now will be valuable.
Truth is generally considered to be that which is in accord with a state of affairs. A statement is true if it is in accord with the facts. Truth is more concerned with the content of an argument, rather than with its form.
Validity refers to the correctness of the reasoning involved in an argument. A conclusion has been correctly inferred from the premises in an argument if the conclusion follows from them.
For the conclusion of an argument to be considered sound, we need to know that the premises are true, and we need to know that the inference made on the basis of the premises is valid, that is, the conclusion follows from the premises.
Consider the following four possibilities of interaction between the concepts of truth and validity:
The following chart may help in understanding these interactions.
REMEMBER: Critical thinkers know the difference between the concepts of truth, validity, and soundness. Critical thinkers can use these concepts to aid them in evaluating any argument.
Ideas are the raw materials of knowledge but ideas are not in themselves true or false. There is no truth or falsity until we take two or more ideas, compare them, and express an agreement or disagreement between them. Only then can we speak of truth or error.
Truth and error lie in the judgment, not the idea. A judgment is an act of the mind pronouncing the agreement or disagreement of ideas among themselves. It is an act in which the intellect affirms or denys one idea of another.
There are three things necessary for making a judgment. First, the mind must understand the two ideas about which it intends to make a judgment. Second, the mind must compare the two ideas under consideration. Third, the mind must express mentally the agreement or disagreement between two ideas. This latter act constitutes the essence of the judgment.
Put in the simplest terms, we take one idea, let's call it the subject, and we say something about it (with another idea), let's call this part the predicate, and we compare the two ideas. We then pronounce agreement or disagreement between the two ideas.
But how do we determine if a judgment is true or false? The discussion of this question does not, strictly speaking, belong to the science of logic. It belongs to a branch of philosophy called epistemology, which is the philosophic study of knowledge in its most general sense. Logic deals with the validity of an argument, not specifically with the truth of an argument.
Nevertheless, a brief discussion of truth and falsehood may be appropriate. We have said that an idea is fundamentally a representation of a thing as it is in itself, independent of the mind. Since judgments are constituted of ideas, the judgment is also a representation of things as they are in themselves, independent of the mind. When our minds compare two ideas with each other and pronounces an agreement or disagreement between them, it actually compares two things with each other and judges about their agreement or disagreement among themselves as they are in reality. If a judgment coincides with reality, it is true and, if not, it is false.
The "test" of truth is, therefore, agreement of the judgment with reality. We verify a judgment by comparing it with the reality it is supposed to represent. We refer to this as objective evidence and this is the criterion of truth for us.
Ideas are expressed in words which we call "terms." Judgments, the agreement or disagreement between ideas, are expressed in sentences we call "propositions." All propositions are sentences but not all sentences are propositions.
There are different kinds of sentences in our language. We ask questions and these are expressed in interrogative sentences. We issue a command or make a request and this is expressed in an imperative sentence. We express joy, surprise, or some other emotion, and these may be expressed in exclamatory sentences. These types of sentences are of no concern to logic.
Propositions are a special kind of sentence for they must contain a judgment. A proposition may be defined as a judgment expressed in a sentence. And three elements enter into the construction of a proposition: the subject, the predicate, and the copula.
The subject is the term designating the idea about which the pronouncement is made. The predicate is the term designating the idea which is affirmed or denied of the subject. The copula is the term expressing the mental act which pronounces the agreement or disagreement between subject and predicate. The copula is usually expressed with a term such as "is" or "is not."
It should be noted that the copula always expresses the present act of the mind and will always be represented by the present tense of the verb "to be." Every proposition can be reduced to this present tense even though the proposition may refer to some past or future event. Example: "The Republicans did not win the last election" can be restated as "The Republican party is not the party which won the last election." The meaning of the proposition has not changed, merely the form has changed.
Sometimes the verb "to be" is hidden in a sentence. A sentence like "The cat bites," which appears not to contain a form of "to be," should be restated as "The cat is biting," which does contain a form of "to be." The meaning has not changed, merely the form has changed.
Many times in ordinary language, a judgment will be expressed in a form that is unsuitable for logic. We have the right to change the wording of a proposition to meet the needs of logic as long as the original meaning of the judgment remains the same. Sometimes the form of a proposition may appear clumsy or unusual when converted to a proposition useful in logic, but we are not concerned here with beautiful prose but with the substance of the thought expressed.
We are so use to excess verbiage and pompous speech, particularly in the political arena, that it may appear impossible to deal logically with complex judgments and complicated arguments. It doesn't matter, however, how complex a sentence is; if it expresses a judgment it can be reduced to a simple proposition including a subject, a predicate, and a copula. Complicated arguments may have to be reduced to set of simple propositions in order to make sense of them logically. But it can done.
|THREE ELEMENTS OF A PROPOSITION|
|The boy||is||a student.|
Truth and falsity are found in the judgment and proposition. A knowledge of the various types of propositions is necessary and there are general types and special types.
The general types of propositions are based on the quality, quantity, and the relation of subject and predicate found in the proposition and it is to these general types we now turn our attention.
The quality of a proposition affects the copula, making the proposition either affirmative or negative. The predicate is either affirmed or denied of the subject.
Consider the following propositions:
Both of these propositions are affirmative. The copula affirms the predicate of the subject.
Consider these propositions:
Both of these propositions are negative. The copula denies the predicate of the subject.
Sometimes a sentence will have two copulas, one in the main proposition and the other in a qualifying clause. Here are two examples: [the clause is within brackets]
In both these sentences, the clause affects the subject "man." Are these propositions affirmative or negative? If the copula of the main proposition is negative, it is a negative proposition. If the copula of the main proposition is affirmative, it is an affirmative proposition. It is clear that the first sentence is negative because of the copula "is not," which is negative. The predicate "healthy" is being denied of the subject "man." The second sentence is affirmative because of the copula "is," which is affirmative. The predicate "healthy" is being affirmed of the subject "man."
When we run across sentences such as the above, which contain qualifying clauses, we must look to the meaning of the sentence. The meaning can usually be discovered by some slight change of the words (but be careful not to destroy the original meaning).
For example, the first sentence could be restated, "A sick man is not healthy," and the second could be restated, "A not-sick man is healthy." This clears up the proposition without changing its meaning.
Since the predicate affirms or denies something of the subject, how does this affect the comprehension and extension of the predicate? Does the comprehension and extension remain the same or are they changed in any way?
This is an important question and has a vital bearing on the validity of an argument. The relation of predicate to subject from this viewpoint needs to be well understood. Here are the rules to follow:
In an affirmative proposition the predicate is always affirmed of the subject according to the whole of its comprehension and according to a part of its extension.
If we affirm, for example, that "Dogs are mammals," what do we mean to assert by applying the predicate "mammals" to the subject "dogs"? Of course, we assert an identity between the two ideas of "mammal" and "dog." Therefore, the comprehension of the idea "mammal" must be found in the idea "dog." And that, in fact, is the case. We are applying the whole of the comprehension of "mammal" to the subject "dog," because the definition of "mammal" is contained in the definition of "dog."
It's different now when we consider the extension. We don't mean to assert by the proposition that the whole of the extension of "mammal" applies to "dog," since that would mean that "dog" would fill out the whole extension of "mammal." There wouldn't be any other things contained in the class of "mammal" except "dogs." But we know this isn't true since human beings are mammals, as are cats, mice, and raccoons.
In an affirmative proposition we intend to assert merely that the subject forms a part of the extension of the predicate. In an affirmative sentence the predicate is taken only as a particular term (a universal term taken partly and indeterminately with regard to its extension). Another way of saying this is: the predicate in an affirmative proposition is not distributed and therefore not used as a universal. Note the words "not distributed," as these will become very important later.
In a negative proposition the predicate is always denied of its subject according to only a part of its comprehension and according to the whole of its extension.
If we state, for example, that "Dogs are not reptiles," we deny the identity between the predicate "reptiles" and the subject "dogs." The comprehension of "reptiles" contains something which is not found in the comprehension of "dogs." By denying that the whole of the comprehension of "reptiles" is found in "dogs," we realize that part of the comprehension may be found in the subject. For instance, the ideas "animal" and "vertebrate" are found in the comprehension of "reptile" and also of "dog." In a negative sentence, therefore, the whole of the comprehension of the predicate never applies to the subject, but a part of the comprehension does.
Also, in a negative proposition, the predicate is always taken according to the whole of its extension and denied of the subject. When we state that "Dogs are not reptiles," we intend to assert that "dogs" do not belong at all to the class of "reptiles." They stand completely outside the class, because every one of them (all dogs) do not have all the characteristics that "reptiles" have.
In a negative sentence, therefore, the predicate is always taken according to its whole extension as a universal and then denied of the subject. Both subject and predicate belong to totally different classes and neither one belongs to the class of the other.
The quantity of a proposition affects the whole judgment as a judgment and it expresses the number of individuals to whom the judgment or proposition applies.
Since the predicate is referred to the subject, the proposition will be true of all the individuals contained in the extension of the subject. From the viewpoint of quantity, propositions will be universal, particular, singular, or collective, depending on the way the subject is taken.
A proposition is universal if the subject is a universal term applied distributively to each and all of the class. The quantifiers "all" or "every" coming before the subject indicate the universality of the proposition. Consider the following propositions:
There can be no doubt about the term "every." But the term "all" may be ambiguous. Does "all" mean "all taken collectively," and apply to each and every member of the class?
If we say "All members of the club were present at the meeting," we are using the term "all" distributively. We mean that "Every member was present." But if we say "All members of the club filled the room," we are using the term "all" collectively. We don't mean that "Every member filled the room." We have to look to the meaning.
The universal negative proposition is expressed by putting "no" in front of the subject, as in:
A proposition is particular when the subject is a universal term used partly and indeterminately. It is indicated by the term "some" or "not all."
The following examples are particular propositions:
Be cautious, however, about some sentences. Words can be deceiving. The sentence "All men are not drunkards" seems at first to be universal (because of the term "all" in front of the subject). But if such a sentence was universal, it would mean "No men are drunkards" and this is clearly not intended. What is meant is probably "Not all men are drunkards," which is the same as saying "Some men are not drunkards," and which makes this a particular (not a universal) proposition.
A proposition is singular when the subject applies to a single individual only.
Consider the following propositions:
Singular propositions have the same value as universal propositions and are treated the same way. The subject is taken according to the whole of the extension, which in this case is one.
A proposition is collective when the subject is a collective term, applying to all taken together as a class, but not to the individuals composing the class.
Consider the following propositions:
In these propositions we mean the Germans as a nation, the flock as a group, and all his books as a set of books. A collective term represents many considered as one. It is taken according to the whole of its extension and it, too, is treated as a universal.
There is one more type of proposition we need to watch out for. This is the indefinite proposition. An indefinite proposition has no definite sign of quantity attached to the subject.
Consider the following propositions:
These propositions indicate no definite quantity. They evidently mean "some" or "all" and are either particular or universal propositions. To determine the exact quantity, the propositions must be evaluated from the sense of the statement or the context in which they appear.
Since singular and collective propositions are equivalent to universal propositions, all judgments have the value of either universal or particular propositions. And as all propositions will be either affirmative or negative, we arrive at the following results:
|A||Universal Affirmative||Every man is mortal.|
|E||Universal Negative||No man is an angel.|
|I||Particular Affirmative||Some men are kind.|
|O||Particular Negative||Some men are not content.|
Another general division of propositions is based on the relation between subject and predicate.
The subject and predicate of every proposition have the relation of agreement or disagreement among themselves. This relation, however, may be either necessary or contingent. This means that the connection between the subject and predicate is either absolutely necessary and unchangeable or it is contingent and changeable.
Consider these propositions:
We can tell just by looking at these propositions that the connection between the subject and the predicate is absolutely necessary and unchangeable.
The subject "whole" is related to the predicate "greater than any of its parts" by a necessary and unchangeable relation. We cannot say that the "whole" is "equal to" or "smaller than" any of the parts which makes up the "whole." We know this simply by analyzing the meanings involved. The predicate must belong to the subject. The same holds true for "man is an animal" and "a square is a quadrangle."
On the other hand, it is possible for the subject-predicate relation of propositions to be contingent and changeable.
Consider these propositions:
We can tell that the predicate "inexpensive mineral" is related to the subject "salt," but it is not necessarily related to it. Under certain circumstances, salt could be or become expensive. We only know the truth of the proposition from experience. A mere analysis of the subject and predicate terms is not sufficient.
The same is true of the other two propositions. There is no absolutely necessary relation between "Alaska" and "largest state." Another state may be admitted to the United States and be larger in area. "Cats" are not absolutely necessarily playful all their life.
The difference between the two types of propositions, absolutely necessary (unchangeable) and contingent (changeable), is easy to see.
The first set of propositions involves something essential. By essential we mean the whole or part of the essence (species, genus, differentia) or something necessarily resulting from the essence (property). The relation between the subject and the predicate is such that the one is the species or genus or differentia or property of the other. One of the terms is contained in the comprehension of the other.
For example, a "quadrangle" is a plane figure with four sides and a "square" is a plane figure having four equal sides with four right angles. A "quadrangle" is the genus of the "square" and is contained in its comprehension. An analysis of "square" reveals the predicate "quadrangle" as part of the comprehension and and essence of the subject "square."
The second set of propositions, in which the relation between subject and predicate is contingent and changeable, presents us with something different. This set contains only accidental attributes. The subject is not contained in the comprehension of the predicate nor is the predicate contained in the comprehension of the subject. The relation between the two is one of contingent fact only and while it may be actually so, it could be otherwise.
Salt may be an inexpensive mineral but it is not necessarily so. It could be otherwise. Cats may be playful all their life but not necessarily. It could be otherwise. These attributes of salt and cats are merely accidental and not part of the comprehension or essence of salt and cats.
We now come to some new words which will be used to designate these different relations between the subject and the predicate of a proposition.
You may recall that the relation between the subject and the predicate in the first set of propositions could be seen to be absolutely necessary and unchangeable. We could actually determine this relationship by an analysis of the terms. By analyzing the subject "square" and the predicate "quadrangle," we could determine that there was an absolutely necessary relation between them. A square will always be a quadrangle. It cannot be otherwise.
If the relation of subject and predicate is necessary and unchangeable, we say the proposition is analytic (from, of course, the word analysis). Another term you may hear is a priori. This means the same thing. Analytic propositions are necessary, essential, and a priori. Knowledge is said to be a priori when it is obtained by reasoning from the whole to the parts.
We may, therefore, define an analytic proposition (or a priori proposition) as one in which either the predicate is contained in the comprehension of the subject, or the subject is contained in the comprehension of the predicate.
Now let's consider the second set of propositions. These, as you recall, contained a relation of the subject and predicate which was contingent and changeable. The predicate was only accidentally (not essentially) related to the subject. We cannot determine this relation by analysis. We can do so only from experience. There was no absolutely necessary relation between the subject and the predicate; the relation was merely contingent and changeable. When this is the case, we say the proposition is synthetic. Another term you may hear is a posteriori. This means the same thing. Synthetic propositions are contingent, accidental, and a posteriori. Knowledge is said to be a posteriori when it is obtained by reasoning from the parts to the whole.
We may, therefore, define a synthetic proposition (or a posteriori proposition) as one in which neither the subject nor the predicate is contained in the comprehension of the other.
The function of language is to convey thought and truth from one mind to another. The complexity of language, however, tends to cover up the truth of a judgment with words. Language is only an imperfect medium of expression. It is the imperfection of language which forces the mind to weave it into intricate textures of words and many times the truth is almost more hidden than revealed.
Since we tend to express ourselves in complicated sentences in a variety of ways using words which may be subtle and involve nuances of meaning, the task of the person who wants to think logically can become difficult. It is our job, then, if we want to be good logicians, to resolve these complicated sentences into simpler forms, so we can uncover the hidden truth of their meaning.
Truth, as we've already learned, lies ultimately in the judgment and the proposition. It becomes necessary, therefore, for us to learn to classify and analyze the various types of propositions. The two main divisions of propositions we will be concerned with are the single and the multiple, the categorical and the hypothetical.
The single proposition is one that contains one subject and one predicate. Examples: "Man is a rational animal," "The car is blue," "Jack is a tall boy."
The multiple proposition is one that contains two or more propositions united into one. Examples: "The car and the truck are blue," "Jack is a tall boy and a good student," "The lawn is white, because it snowed."
A categorical proposition is one in which a predicate is attributed to its subject outright, without restriction or condition. Examples: "The car is blue," Jack is a good student and an excellent athlete," "Gold and silver are valuable ores."
A hypothetical proposition is one which does not attribute a predicate to its subject directly, but asserts the dependence of one judgment on another. Examples: "If it rains, you will wear a raincoat," "A statement cannot be true and false at the same time from the same point of view," "An animal is either in motion or at rest."
When we look at the relationship between the two main groups of propositions, we will notice immediately that they are not exclusive of each other. Of course, single and multiple propositions do exclude each other and categorical and hypothetical propositions do exclude each other. But while the single propositions are always categorical and the hypothetical propositions are always multiple, the categorical propositions may be either single or multiple.
The categorical proposition makes a direct assertion of agreement or disagreement between the subject and the predicate.
The single categorical proposition contains only a single sentence in its construction. It contains one subject, one predicate, and the copula.
If these elements of the sentence are without any qualification or composition, it is a simple categorical proposition. Examples: "Gold is an ore," "Jack is tall," "Man is rational." There should be no difficulty with these types of propositions since they are so simple.
If, however, a qualification or composition enters into the subject or predicate or copula, we have a composite single categorical proposition, and these may be of two types: complex propositions and modal propositions.
The complex proposition is a composite single sentence in which both the subject and the predicate or either one is a complex term. Comparative examples: "Man" is a simple term. But "Good man" or "Learned man" or "Intelligent man" are complex terms. These latter terms used a subjects or predicates, or any verb which expresses past or future time used as a predicate, makes the sentence a complex proposition.
Consider the following propositions:
In the above propositions, there is only one judgment, with one subject and one predicate. But the subject or the predicate in each proposition is a qualified (complex) term.
The modal proposition is a composite single sentence in which the copula is so modified as to express the manner or mode in which the predicate belongs to the subject. The qualification does not affect the subject or the predicate. It affects the copula itself. It states whether the objective connection between the subject and the predicate, expressed by the copula, is necessary, impossible, possible, or contingent. There are, then, four different modes, each producing a different type of proposition.
The necessary modal proposition states that the predicate belongs to the subject, and must belong to it. Examples: "A circle is round," "Man is an animal," "Two plus two equal four."
The impossible proposition states that the predicate does not and cannot belong to the subject. Examples: "A square has five sides," "A dog is a rational animal," "Four plus four equal nine."
The possible proposition states that the predicate is not actually found in the subject, but it might be. Examples: "A diabetic may go into a coma," "My truck can go over 100 miles per hour," "Paul may be a good student."
The contingent proposition states that the predicate actually belongs to the subject, but it need not. Examples: "These students need not attain good grades," "It is not necessary that the dog keep on barking," "I need not go to the dance Friday night."
Multiple categoricals are propositions which contain two or more sentences in their very construction. Some of these are overtly multiple and some are covertly multiple. The covertly multiple propositions are called exponibles.
The overtly multiple categoricals are plainly composed of two or more propositions. There are five types of these.
1. The copulative proposition is a multiple categorical proposition which has two or more subjects, or two or more predicates, or two or more subjects and predicates. Examples: "Richard Nixon and Gerald Ford were presidents of the United States," "Peter is president and chairman of the board of his company," "Jack and Jill went up the hill and slid back down again."
Each of these sentences can be resolved into as many single propositions as there are different subjects and predicates. Examples: "Richard Nixon was president," "Gerald Ford was president," "Peter is president of his company," "Peter is chairman of the board." The truth of copulative categoricals depends on the truth of all the single sentences which compose the multiple proposition.
2. The adversative proposition is a multiple categorical proposition which consists of two propositions united in opposition to each other by conjunctions such as "but," "yet," "although." Examples: "The dog was barking, but no one cared," "Jack lost his business, yet he was not depressed," "The woman bought a purse, although she did not need one."
To be true, each sentence must be true, and the opposition must be true. Example: It must be true that "The dog was barking" and it must be true that "no one cared."
3. The relative proposition is a multiple categorical proposition which expresses a relationship of time or place between two sentences. Examples: "Before beginning to eat his salad, he told me the secret," "After hurrying home, Jack called Jane on the telephone."
In order to be true, the single statements must be true and the relation of their sequence must be true. It must be true, for instance, that Jack called Jane on the telephone and it must be true that he did it after hurrying home.
4. The causal proposition is a multiple categorical proposition which combines two statements in such a way that the one is given as the reason or cause of the other. The words "because" and "for" commonly appear in this type of proposition. Examples: "The times are good, for people have a lot of money to spend," "Jack is happy, because he got elected class president."
The truth of the causal proposition depends on the truth of each categorical proposition contained in it and on the causal connection that is declared to exist between them. It must be true that "Jack is happy" and it must be true it is "because he got elected class president."
5. The comparative proposition is a multiple categorical proposition which compares the relation between a subject and predicate with the same relation between another subject and predicate, and expresses the degree of this relationship as being either less or equal or greater. This may sound complicated, but it really isn't.
Example: "As you live, so you shall die." Three statements are actually contained in this proposition. The first one is "Your life has a certain character." The second one is "Your death has a certain character." The third statement is not as obvious but is implied, and states "The character of your death is the same as the character of your life." Now, that wasn't too hard, was it? Let's consider another proposition.
Example: "Harry Truman was a greater president than Richard Nixon was." Here again, we have three statements. One, "The quality of Truman's presidency was great." Two, "The quality of Nixon's presidency was great." Three, "The quality of Truman's presidency was the greater of the two."
The truth of a comparative proposition depends on the truth of the two separate sentences and on the truth of the degree mentioned as existing between them.
The covertly multiple categorical propositions have the appearance of single propositions but are really multiple. Their composition lies concealed in some word and this needs an exposition to show the multiple character of the proposition. This is why these sentences are called exponibles (from expressed). We can resolve these multiple propositions into individual sentences called exponents.
The truth of exponible propositions is determined by looking at the exponents and the logical connection between them. If any parts of an exponible proposition are false, the entire proposition is false. There are four types of exponible propositions.
1. The exclusive proposition is a multiple categorical proposition which contains some word or words such as "only," "alone," "solely," or "none but," which indicates the exclusion of any other predicate from this subject or any other subject from this predicate.
Example: "Only the better students will go on the field trip." This proposition may appear to be a single categorical proposition but it is not. This proposition can be resolved by means of a copulative proposition, in which one sentence is affirmative and the other is negative. It becomes the proposition "The better students will go on the field trip and no others will go on the field trip." Now it is obvious it is indeed a multiple categorical proposition.
Another example: "None but the brave will receive medals of honor." This appears to be a single categorical proposition. It can, however, be resolved into the following: "The brave will receive medals of honor and no others will receive medals of honor." We can now see clearly that it is really a multiple categorical proposition.
2. The exceptive exponible proposition is a multiple categorical proposition which contains a word such as "except" or "save," to indicate that a portion of the extension of the predicate does not apply to the subject, or a portion of the extension of the subject does not apply to the predicate.
Example: "All the athletes except John won an event in the tournament." The resolution of this is not difficult. It becomes, "All athletes in the tournament won an event, and John did not win an event in the tournament." Clearly a multiple categorical.
Another example: "All the animals save one were killed in the fire at the zoo." Resolution: "One animal was not killed in the fire at the zoo, and all the other animals were killed." Again, clearly a multiple categorical.
3. The reduplicative exponible proposition is a multiple categorical proposition which contains an expression which duplicates the subject or predicate, giving it special emphasis, such as "as such" or "as a," and implies the reason or cause for the connection between subject and predicate. Confused? Don't be. Examples will help you understand this ponderous definition.
Example: "Man, as man, is endowed with free will." This proposition can be resolved into: "Man is endowed with free will, because he is man." Now it appears as a multiple categorical proposition. It is reduplicative because it "reduplicates" itself, that is, the idea of "man" is intrinsically joined with the idea of "free will." You can't have one without the other.
A reduplicative proposition is true when both the plain statement and the reduplicative substatement are true.
4. The specificative exponible proposition appears similar to the reduplicative proposition but really is quite different. It is a multiple categorical proposition which contains an expression which duplicates the subject or predicate, giving it special emphasis, such as "as such" or "as a," but merely implies the time element or condition of this connection.
Example: "Peter, as a student in high school, took part in the debating society." The fact that Peter is a student has nothing to do with his participating in the debating society as a student. The resolution of this proposition is: "Peter is a student in high school," and "Peter took part in the debating society."
Another example: "The professor, as a historian, was a very easy grader." The fact that the professor is a historian has nothing essential to do with the fact that he is an easy grader. He could have one characteristic without the other. The resolution is: "The professor is a historian," and "The professor is an easy grader."
Specificative propositions are true when both the plain statement and the substatement are true.
This ends our discussion of single and multiple categorical propositions. Just remember that categorical propositions always make a clean-cut assertion; they always affirm or deny the predicate to the subject outright with no qualifications.
How does the hypothetical proposition differ from the categorical proposition? While the categorical proposition makes a definite and unqualified assertion, the hypothetical proposition does not declare an unqualified affirmation or denial. It expresses the dependence of one affirmation or denial on another affirmation or denial. There are three types of hypothetical propositions.
The conditional proposition is a hypothetical proposition which expresses a relation in virtue of which one proposition necessarily flows from the other because a definite condition is verified or not verified. Sometimes these are called the "if" propositions.
Examples: "If the barometer falls, there will be a storm." "If Peter is a good boy, he will be able to go on the trip." "If I make a lot of money, then I will be able to buy a mansion."
The part of the proposition containing the "if" is called the "condition" or the "antecedent." The other part is called the "conditioned" or the "consequent." Notice there is a strict relation expressed in a conditional hypothetical proposition. The "antecedent" must be true before the "consequent" can follow.
The truth of conditional hypothetical propositions does not depend on the truth of the statements taken by themselves or individually. The truth depends on the relation between the statements.
For example, take the proposition "If the barometer falls, there will be a storm." We are not asserting that the barometer is falling. We are not asserting that a storm is coming. We are simply saying that the coming of a storm is dependent on low atmospheric pressure which is indicated by the falling of the mercury in a barometer.
In a conditional hypothetical proposition, it is the dependence of one idea on the other that is affirmed or denied. The truth of the whole statement rests on the truth of the dependence.
The disjunctive hypothetical proposition is one which contains an "either-or" statement, indicating that the implied judgments cannot be true together nor false together, but one must be true and the other must be false.
Examples: "Either the sun or the earth moves in an orbit." "An automobile is either in motion or at rest."
This type of proposition should present no difficulty.
The conjunctive hypothetical proposition is one which expresses a judgment that two alternative assumptions are not or cannot be true at the same time.
Examples: "An automobile cannot be in motion and at rest at the same time." "A person cannot be a saint or a sinner at the same time."
For the truth of such a proposition, it is necessary that they be really irreconcilable at the same time. If we can prove that they may be present together, we would prove the statement to be false.
We have learned something about the general and special types of propositions in a previous essay. It's time now to investigate certain properties of these propositions as they are compared to one another. This brings us to the matter of logical opposition. Propositions are said to be logically opposed to each other when they have the same subject and predicate but with a change in quality or quantity or both.
We have already learned that all truth is based on the three laws of thought known as the Principle of Identity, the Principle of Contradiction, and the Principle of the Excluded Middle. These three principles are the foundation for all human knowledge. They are self-evident and need no proofs or demonstrations. In fact, they cannot be "proved" in the ordinary sense of the term. If we reject them, however, we are at the end of rational discussion since we need to accept them as true in order to initiate and continue any rational discussion.
There are three important laws of thought that every critical thinker needs to know. Without them, we would find it very difficult to reason correctly.
The Principle of Identity states that "A is A." Other ways of saying the same thing are "What is, is," "Everything is what it is," and "A thing is identical with itself." Can we seriously challenge such a principle?
The Principle of Contradiction states that "A cannot be A and not A at the same time in the same respect." It can also be stated as "Whatever is, cannot at the same time not be under the same circumstances," or "It is impossible for the same thing both to be and not to be at the same time from the same point of view." From the standpoint of logic, the Principle of Contradiction can be read as "The same attribute cannot at one and the same time be both affirmed and denied of the same thing in the same respect."
The Principle of Excluded Middle can be stated in different ways: "A thing either is or is not," "Everything must either be or not be," and "Any attribute must be either affirmed or denied of any given subject." For the purpose of the study of logic, the principle can be stated: "If we make an affirmation, we thereby deny its contradictory; if we make a denial, we thereby affirm its contradictory."
The logical opposition of propositions is the relation which exists between propositions having the same subject and the same predicate, but differing in quality, or in quantity, or in both.
There are four possible way in which a proposition having the same subject and the same predicate may appear:
1. as a universal affirmative (A)
2. as a universal negative (E)
3. as a particular affirmative (I)
4. as a particular negative (O)
These four propositions represent the four types of opposition. They can be diagramed, together with their mutual relations as opposites, in what is called a Square of Opposition.
Here are explanations of the four types of opposition and the four relations resulting from the opposition:
Subalternation: the opposition existing between a universal and particular affirmative (A and I), and between a universal and particular negative (E and O). Both propositions, the universal and the particular, are called subalterns. The universal is the subalternant (A and E). The particular is the subalternate (I and O).
Contradiction: the opposition existing between a universal affirmative (A) and a particular negative (O), and between a universal negative (E) and a particular affirmative (I).
Contrariety: the opposition existing between a universal affirmative (A) and a universal negative (E).
Subcontrariety: the opposition existing between a particular affirmative (I) and a particular negative (O).
Here is a diagram showing the logical opposition of propositions (with examples provided):
We can now formulate certain laws of truth and falsity regarding propositions which contain these various relations.
The Law of Subalternation: A--I and E--O. This law has two phases, depending on whether we begin with the truth or the falsity of one of the subaltern propositions.
Beginning with the truth of one of the subaltern propositions (A--I, E--O), the first rule states: The truth of the universal involves the truth of the particular (A to I, E to O); but the truth of the particular does not involve the truth of the universal (I to A, O to E).
In other words:
There are, therefore, two sections to this first rule.
The first section states: "It is always logical to conclude from the truth of the universal to the truth of the particular." After all, what is true of all individuals of a class must also be true of some of these individuals. What is true of the whole must be true of every part of the whole.
Examples: If "All men are mortal," then surely "Some men are mortal." If "No men are dogs," then "Some men are not dogs, either."
The second section states: "The truth of the particular does not involve the truth of the universal; the truth of the universal will always be doubtful." What is true of some need not be true of all. What is true of a part of a class need not be true of the whole of the class.
Examples: If it is true that "Some men are content," we cannot conclude, on the basis of the proposition alone, that "All men are content." If it is true that "Some men are not content," we cannot conclude that "No men are content."
We can see from the examples that the truth of the particular propositions (I and O) does not involve the truth of the universals (A and E). Although the particular propositions I and O are true, their respective universals A and E are false.
It could happen, of course, that what is true of some is also true of all and what is true of a part is also true of the whole. In this case, both the particular propositions (I and O) are true, and their respective universals (A and E) are also true. But we are never permitted to conclude from the truth of the particular to the truth of the universal. It may be so, but it need not be so. We cannot validly argue from some to all and from the part to the whole.
The second rule of the Law of Subalternation states: The falsity of the particular involves the falsity of the universal; but the falsity of the universal does not involve the falsity of the particular. Here we begin with the falsity of one of the subaltern propositions (I to A, O to E). The rule states:
There are also two sections to this second rule.
The first section states: "We can always validly conclude from the falsity of a particular proposition to the falsity of the universal." This makes sense. For something to be true of all, it must be true of every individual that belongs to the all. For something to be true of the whole, it must be true of every part contained in the whole.
Example: The particular proposition I, "Some men are dogs," is false. Actually it would be true to say that "Some men are not dogs." In order, however, for the statement to be true that "All men are dogs," it could not be true to say that "Some men are not dogs," because all must include some, and the whole must include every part.
Another example: The particular proposition O, "Some men are not mortal," is false. We should say "Some men are are mortal." But if proposition E, "No men are mortal," is true, it would follow that the same some are and are not mortal at the same time.
If it is false that "Some men are dogs," it is all the more false to state that "All men are dogs." If it false to say that "Some men are not mortal," it is also false to say that "No men are mortal."
From the falsity of the particular proposition (I or O), we must conclude to the falsity of the respective universal proposition (A or E).
The second section of this rule states: "If A is false, I need not be false; if E is false, O need not be false." In order that a universal be true, every individual of the class and every part of the whole must be included in the truth of the universal. The universal, therefore, will be false if not every individual of the universal and not every part of the whole is included in the truth of the universal statement.
This means that if a universal proposition is false, some of its individuals must also be false, but some of the others may be true. But if some may be true, even if the universal is false, it is obvious we cannot validly conclude from the falsity of the universal to the falsity of the particular.
We can now see the truth of the Law of Subalternation. The truth of the universal involves the truth of the particular, but the truth of the particular does not involve the truth of the universal. The falsity of the particular involves the falsity of the universal, but the falsity of the universal does not involve the falsity of the particular.
The Law of Contradiction: A--O and E--I. This law has two phases.
The first rule states: "Contradictories cannot be true together."
In an affirmative universal (A) proposition, it is asserted that the predicate is affirmed of each and every individual belonging to the subject, as in, for example "All men are mortal." If this is true, then it must be false to deny this statement of some of the individuals. Therefore, the statement that "Some men are not mortal" (O) cannot be true.
In a negative universal (E) proposition, it is asserted that the predicate must be denied of each and every individual belonging to the subject, as in, for example "No men are dogs." If this statement is true, then it must be false to say that "Some men are dogs" (I).
What is true of all, must be true of every one of the class. To state at the same time that all are and some are not, and that none are and some are, would violate the Principle of Contradiction.
The second rule states: "Contradictories cannot be false together."
Example: If it is false that "All men are content," it must be true that "Some men are not content" (A--O). If it is false that "No men are content," it must be true that "Some men are content" (E--I).
Another Example: If it is false that "Some men are not mortal," it must be true that "All men are mortal." If it is false that "Some men are dogs," it must true that "No men are dogs."
We now can state the following conclusions:
The Law of Contrariety: A--E. There are two rules to be considered.
The first rule states that "Contraries cannot be true together." If A is true, E is false and if E is true, A is false. If one of the contraries is true, the other contrary must be false.
Example: If "All men are mortal" (a universal affirmative proposition--A) is true, then "No men are mortal" (a universal negative proposition--E) must be false. If A is true, E is false.
Another example: If "No men are dogs" (a universal negative proposition--E), then "Some men are dogs" (a particular affirmative proposition--I) must be false. The universal affirmative proposition--A) "All men are dogs," must also be false.
The second rule states: "Contraries may be false together." If one contrary is false, the other contrary may also be false, although it need not be false, and may be true.
Example: Consider the proposition "All men are content." This is a universal affirmative proposition (A) and let's consider this false. Since this statement is false, its contradictory, a particular negative proposition (O) "Some men are not content," must be true. But the Law of Subalternation states that the truth of the particular proposition does not involve the truth of the universal. Therefore, although it is true that "Some men are not content," we cannot validly conclude from this that its universal ("No men are content") is also true. "No men are content" (E) may be true or false. Therefore, both contraries may be false.
The second rule is established. From the truth of one contrary we can conclude to the falsity of the other; but from the falsity of one contrary we cannot conclude to the truth of the other.
The Law of Subcontrariety: I--O. There are two rules to this law.
The first rule states: "Both subcontraries cannot be false together."
The first rule says:
Example: Consider the statement "Some men are dogs." Let's say that this particular affirmative (I) proposition is false. Since this is false, its contradictory (a universal negative--E) must be true, that is, "No men are dogs." If a universal proposition is true, its particular proposition is also true (the Law of Subalternation). So, since E is true, O must also be true and must state that "Some men are not dogs." If I is false, O is true.
Another example: Let's say that O is false, that is, "Some men are not mortal." Its contradictory A, that is, "All men are mortal," must be true (the Law of Contradiction). But if A ("All men are mortal") is true, then I ("Some men are mortal"), must also be true (the Law of Subalternation). If O is false, I must be true.
We can see now the truth of the first rule regarding subcontrary propositions (I and O). Subcontraries cannot be false together, at least one of the two must be true.
The second rule of subcontraries (I and O) states: "Both subcontraries may be true together."
Example: Let's suppose it's true that "Some men are content," (a particular affirmative proposition--I). The contradictory of this proposition, "No men are content" (a universal negative proposition--E), must be false. We know, however, that the falsity of the universal does not involve the falsity of the particular (the Law of Subalternation). Therefore, even though E ("No men are content") is false, we cannot conclude that O ("Some men are not content") is false. This proposition may be true.
Another example: Let's suppose that "Some men are not content" (a particular negative--O) is true. It's contradictory, "All men are content" (a universal affirmative--A), is false. We cannot conclude, however, from the falsity of the universal to the falsity of its particular (the Law of Subalternation), so it does not follow that I ("Some men are content") is also false. The statement "Some men are content" may be true.
The two rules regarding subcontrary propositions (I and O) have now been established. Both subcontraries cannot be false together, but both subcontraries may be true together.
We can now state the following conclusions:
The summary diagrammed:
|If A is true||False||True||False|
|If A is false||Undetermined||Undetermined||True|
|If E is true||False||False||True|
|If E is false||Undetermined||True||Undetermined|
|If I is true||Undetermined||False||Undetermined|
|If I is false||False||True||True|
|If O is true||False||Undetermined||Undetermined|
|If O is false||True||False||True|
The Square of Opposition refers solely to categorical propositions which do not contain a mode affecting the copula. The way we treat modal propositions is similar to ordinary categorical propositions, but the logical opposition affects the mode itself. (Remember there are four modes we learned about.)
Obviously, the subject of the modal proposition may be either a universal term or a particular term (all, no, or some).
The opposition becomes more complicated thereby, but the general scheme must be carried out according to the logical opposition intended.
If the logical opposition intended affects both the quantity and the mode of the propositions, the Square of Opposition would be as it appears here. Thus,
On the other hand, if the logical opposition affects only the mode, but not the quantity, of the propositions, the quantity (all or some) will remain the same and only the mode will change.
From these relations of opposites it will be clear that we are often entitled to conclude from the truth or falsity of one proposition to the truth or falsity of another. This method of concluding from the truth or falisty of one statement to the truth or falsity of another is called immediate inference; it is called immediate because we can pass directly from the one to the other, without the necessity of adducing any other idea or judgment as proof.
The Square of Opposition and the Three Laws of Thought are sufficient to make their truth or falsity evident, provided we know beforehand that one of these opposites is true or false.
The Square of Opposition (with its relations of subalternation, contradiction, contrariety, subcontrariety) will act as a powerful aid toward correct thinking.
Besides the immediate inference of logical opposition, we have the immediate inference of eduction.
Eduction is a mental process whereby, from any proposition taken as true, we derive another proposition implied in it, though differing from the first proposition in subject or predicate or both. There are three main forms of eduction: obversion, conversion, and contraposition.
The purpose of this technique is to transform certain sentences into other sentences which are equivalent in meaning, but may have a different logical form. The advantage of this is that an argument which may not be in strict syllogistic form can be transformed into a syllogism.
Consider the following argument:
This argument appears to be valid but we can't test it by the rules already discussed because it contains more than three terms. It appears, in fact, to have five terms: unwise people, trustworthy people, wise people, unaggressive people, and aggressive people.
But look at the second premise. It means the same thing as "All aggressive people are unwise." So, if we substitute this latter sentence for the original sentence, we get the following argument:
The argument now contains only three terms (unwise, trustworthy, aggressive) and is a syllogism. It can now be tested using the standard rules for testing the validity of syllogisms. We can then see that the argument is valid.
Obversion is an eduction in which the inferred judgment, while retaining the original subject, has for its predicate the contradictory of the original predicate.
The original proposition is called the obvertend and the inferred proposition is called the obverse.
In obverting a given sentence we do two things:
Example: Consider the sentence "All men are mortal." First, we change the quality, and the sentence becomes "No men are mortal." Then we negate the predicate and the sentence becomes "No men are non-mortal." The sentence "No men are non-mortal" is equivalent to the sentence "All men are mortal."
Every A, E, I, and O sentence can be obverted. Study the following diagram:
|A||All men are mortal.||No men are non-mortal.|
|E||No men are mortal.||All men are immortal.|
|I||Some men are mortal.||Some men are not immortal.|
|O||Some men are not mortal.||Some men are immortal.|
Care must be exercised in obverting sentences in ordinary language since some negative English terms can be confusing and some terms, which may appear to negate, do not negate at all. For instance, "large" is not the negation of "small." Certain prefixes ("im," "un," "in") do not always express negation. Logicians prefer, therefore, to use the prefix "non" in order to negate the predicate. Therefore, the negation of "rich" is not "poor," but "non-rich."
When obverting, there must be a change in the quality of the sentence only. Do not change the quantity. Then a universal sentence remains a universal sentence and a particular sentence remains a particular sentence.
Conversion is an eduction in which the inferred judgment takes the subject of the original proposition for its predicate, and the predicate of the original proposition for its subject. In other words, when we convert we merely interchange subject and predicate.
The original proposition we call the convertend and the inferred proposition we call the converse.
Example: Consider the sentence "No dogs are horses." This sentence is equivalent to the sentence "No horses are dogs."
Conversion is unlike obversion because not every standard sentence has an equivalent converse. Only the E and the I sentences can be converted.
Example: The O sentence cannot be converted. From "Some woman are not nuns," we cannot infer "Some nuns are not woman."
Example: The A sentence cannot be converted simply. From "All dogs are animals," we cannot infer that "All animals are dogs." It is possible, however, to partially convert the A sentence, using a technique logicians call Conversion by Limitation. When we convert a true A sentence, we can transform it into a true I sentence. The sentence "All dogs are animals" can be partially converted into "Some animals are dogs." Partial conversion, however, does not give us a sentence which is exactly equivalent in meaning to the original. This is because the quantity of the original sentence is changed.
Study this diagram of permissible conversions:
|E||No men are mortal.||No mortals are men.|
|I||Some men are mortal.||Some mortals are men.|
|A||All men are mortal.||Some mortals are men. (Partial Converse)|
Contraposition is an eduction in which the subject of the inferred proposition is the contradictory of the predicate of the original proposition. It is (read carefully!) the obverse of a converted obverse.
In order to obtain the contraposition of a sentence, three operations must be performed:
Example: Consider the sentence "All dogs are animals."
Contraposition cannot be applied to all four standard sentences. The A sentence and the O sentence have contrapositives. The I sentence has no contrapositive and the E sentence has only a partial contrapositive. Usually, contraposition is only applied to A sentences.
You should now be able to transform large parts of ordinary language into arguments of a syllogistic form, making it possible to test the validity of the argument.
The chart below will help you see how to form equivalent sentences.
|A||All S is P||No S is non-P||Some P is S||All non-P is non-S|
|E||No S is P||All S is non-P||No P is S||Some non-P is not non-S|
|I||Some S is P||Some S is not non-P||Some P is S||None|
|O||Some S is not P||Some S is non-P||None||Some non-P is not non-S|
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