What is Logic?
Logic is the science of necessary inference. An inference is the forming of a conclusion from premises by logical methods -- the conclusion itself. The adjective necessary innecessary inference or necessary consequence means there is no way to avoid the conclusion of an argument. We define an argument as one or more propositions in support of another proposition. The propositions in support of the other proposition are called premises; the proposition supported by the premises is called the conclusion. More aboutnecessary inference later, but first, what is a proposition?
Propositions
A proposition is a form of words in which the predicate is affirmed or denied of the subject of a declarative sentence. A proposition is the meaning of a declarative sentence. Declarative sentences are either true or false. Propositions are the premises and conclusions of arguments. Other sentences, in expressing commands, posing questions, or conveying exhortations are neither true nor false. Some questions, rhetorical questions, are intended as propositions; but if a question is not rhetorical, then it is neither true nor false.
Arguments
Arguments divide into two classes: deductive arguments and inductive arguments. This classification amounts to two different claims. The premises of Inductive Arguments claim to provide incomplete or partial reasons in support of the conclusion. The premises of Deductive Arguments claim to provide conclusive reasons for the conclusion. In Inductive Argument, the conclusion is said to be either probable or improbable. With Deductive Argument the conclusion either follows necessarily or it does not. That is to say, the conclusion is either a necessary consequence of the premises or it is not a necessary consequence of the premises. Another way of stating the same thing: A Deductive Argument consists of a conclusion presumably deduced from premises. The deduction of conclusions from premises is at the heart of logic.
Validity
The phrases necessary consequence and necessary implication mean necessary inference . The use of one or the other phrase depends on the emphasis. If one stresses that the premises imply a conclusion, one speaks of necessary implication. If one stresses the conclusion resulting from premises, one speaks of necessary consequence. With either phrase, the reference is a claim of necessary inference between premises and conclusion of a Deductive Argument. If the conclusion of a Deductive Argument is a necessary consequence of the premises, then the argument is valid; otherwise, invalid. Using other words: If the premises of a Deductive Argument necessarily imply the conclusion, then the argument is valid; otherwise, invalid.
Summarizing this section. Logic is the study of the relation between premises and conclusion in Deductive Arguments. If the conclusion follows from premisesnecessarily (that is, the conclusion is unavoidable), then the argument enjoys valid status; if not (that is, the conclusion can be avoided), then the argument is invalid. Every Deductive Argument is either valid or invalid.
Why study Logic?
There are at least three reasons.
First. To the question what is more basic than the three R's of Reading, wRiting, and aRithmetic, we answerTHOUGHT. To engage in any one of the three activities, you must think! Thinking, if it is correct, follows rules. Sometimes we think incorrectly, when we neglect the rules for correct thinking. Other times, we make mistakes in thinking or reasoning. The rules for correct thinking and methods for avoiding mistakes in reasoning belong to the subject of logic.
Second. The study of logic trains the mind to distinguish logical from emotional (psychological) appeals offered in support of a conclusion or a position. To opt for a course of action confusing an emotional appeal with a logical appeal is to fall victim to incorrect thinking. It is a fallacy to accept an emotional-inference as a necessary-inference. Logic is the irreplaceable means for correct thinking and avoiding fallacious reasoning.
Third. The structure of Man's mind is the same as his Creator's. God is not insane; He is a rational being; the structure of God's mind is logic. For these reasons, we say not only that logic is irreplaceable and universal, but logic is necessary and fixed. It is not one scheme of things among others. It is not something optional, for Man's mind was formed on the principles of identity, excluded middle, and contradiction.
Three Laws of Logic
The three laws of thought are universal, irrefutable, and true for reasons already stated. Without these laws, it is impossible to imagine how anything written or spoken could be intelligible. More to the point, the laws are the basis of necessary inference, for without them, necessary inference vanishes! To repeat, the laws of logic are universal, irrefutable, and true. By "universal," we mean allows for no exception. "Irrefutable" means that any attempt to refute them, makes use of them; thus, establishing them as necessary for argument. "True" means not only "not-false," but not-false because they are grounded in the Logos of God, the source and determiner of all truth. Moreover, the laws stand together as a trinity; to fault one, is to fault all, and to uphold one, upholds the others. Together, these laws establish and clarify the meaning of necessary inference for logic and all intelligible discourse.
Here is a brief statement of each.
| 1 | The law of identity states that if any statement is true, then it is true; or, every proposition implies itself: A implies A. |
| 2 | The law of excluded middle states that everything must either be or not be; or, everything is A or not-A. |
| 3 | The law of contradiction states that no statement can be both true and false; or, A and not-A is a contradiction and always false: thus, not both A and not-A. |
Without the first, identity or sameness is lost; without the second, confusion begins; and without the last, irrationalism is in full residence.
To recapitulate. Logic is the science of necessary inference. The basic elements are propositions in arguments. A proposition is the meaning of a declarative sentence. An argument is composed of propositions some of which are premises, one of which is the conclusion. The premises are reasons given to support the conclusion of an argument or a position. Arguments are classified as either inductive or deductive. With Deductive Argument, we ask: "Does this conclusion follow as a necessary consequence from these premises?" If the answer is affirmative, the Deductive Argument is valid; otherwise, the argument is invalid. Deductive Arguments are either valid or invalid. Also, if the argument is not invalid, then it is valid. If the argument is not valid, then it is invalid.
Three reasons for the study of logic are (1) correct thinking requires it; (2) discerning minds necessarily depend on it; and (3) man is a rational being in the image of his Creator. Logic is universal, necessary, and irreplaceable. Man's mind was formed on the principles of identity, excluded middle, and contradiction. These three laws are the basis for all intelligible thought. Without them, all rational discourse vanishes.
PROPOSITIONS
Propositions are either standard form or nonstandard. If a proposition is not standard form, it is classified as nonstandard. We first consider the four standard form propositions, then discuss nonstandard propositions in the last section of this Study.
Standard Form Propositions
There are only four standard form propositions. Each consists of a subject and a predicate. In each form, the subject and the predicate are joined together by is or are, the copula. The relation between the subject and predicate is identified by the use of: All, No, Some, or Some ... not.... If a and b stand for the subject and predicate terms, respectively, the four forms are: (1) All a is b, (2) No a is b, (3) Some a is b, and (4) Some a is not b.
The A Form
The proposition "All men are mortal" asserts a relation of inclusion between the class of men and the class of mortals. More plainly, it states that all members of the class men fall within the class mortal. The form of all such propositions is All a is b, or A(ab) where a stands for the subject term and b stands for the predicate term. Note that in Apropositions, the subject is included in the predicate, but not the predicate in the subject. For example, from "All men are mortals" (true) it does not follow that all mortals are men (false).
The E Form
The proposition "No Christian is an atheist" asserts a relation of exclusion between two classes, Christians and atheists. No member of the class Christians is a member of the class atheists, and conversely, no atheist is a Christian. The classes of E propositions are mutually exclusive. The form is No a is b, or E(ab), where a stands for any subject, and bstands for any predicate. Thus, with E propositions all members of one class are excluded from the other, and vice versa.
The I Form
The proposition "Some Americans are Calvinists" asserts a relation of partial inclusion between the class Americans and the class Calvinists. Something less than all members of the subject-class is included in the predicate-class, and conversely, some members of the class Calvinists are included in the class Americans. The form of the I proposition isSome a is b, or I(ab), where, as before, a stands for any subject, b for any predicate. Ordinarily, some can mean a few in number. In logic, the word can also mean as few as one or any number less than all.
The O Form
The proposition "Some men are not Christian" asserts a relation of partial exclusion between the two classes, men and Christians. Some men are entirely excluded from all of the class of Christians. Does it follow then that some Christians are not men? Perhaps some angels are Christian? No, the converse of an O proposition does not follow from the original. Its form is Some a is not b, or O(ab). Remember, there is no converse for an O proposition.
The following chart serves as a summary of the foregoing descriptions of the four forms. Do not be confused in that the letters a and b are used throughout, even when the propositions contain different subject matter. Recall that the letters, a and b, stand for any subject and any predicate, respectively. Indeed, we could have used x and y or any other pair of letters to stand for subjects and predicates.
Chart 1.1: Four Forms
| All men are mortal | All a is b. | A(ab) |
| No Christian is an atheist. | No a is b. | E(ab) |
| Some Americans are Calvinists. | Some a is b. | I(ab) |
| Some men are not Christian. | Some a is not b. | O(ab) |
The source of the letters for the four forms is of historical interest. From affirmo (I affirm), meaning affirmative in quality, we get A and I; E and O come from nego (I deny), meaning negative in quality.
Formal Properties of the Forms
The four forms share three important properties: distribution, quantity, and quality defined just below.
Distribution
The formal properties, quality and quantity, of A, E, I, and O forms depend on the definition of distribution. We distinguish a distributed term (subject or predicate) from an undistributed term in this manner: A distributed term is one modified by All or No. When a term is modified by "some," it is undistributed. Using the subscripts "d" for distributed and "u" for undistributed, the four forms distribute their terms as indicated below in Chart 1.2.
Chart 1.2: Distribution of Terms
| Forms | Subject Term | Predicated Term | |
| A | All sd is pu | Distributed | Undistributed |
| E | No sd is pd. | Distributed | Distributed |
| I | Some su is pu. | Undistributed | Undistributed |
| O | Some su is not pd. | Undistributed | Distributed |
Where, s = subject term; p = predicate term.
To recapitulate: With the A form, only the subject term is distributed; the predicate is undistributed, since, as noted previously, all of the predicate is not included in the subject. The E form distributes both subject and predicate terms, since No s is p; and No p is s. With the I form, some part of the subject term class is included in some part of the predicate term class; therefore, both terms are undistributed. Last, in the O form, some part of the subject term class is excluded from all of the predicate term class (Some s is not p); therefore, only the predicate term is distributed, the subject term, undistributed.
Quality
Previously we indicated that the A and I letters came from the Latin affirmo, and E and O from the Latin nego. Remembering the sources of the letters may help to recall that the A and I forms are affirmative in quality; E and O, negative in quality. An affirmative form is one that does not distribute its predicate. The A and I forms do not distribute the predicates; therefore, they are affirmative in quality. A negative form is one that does distribute its predicate. The E and O forms distribute the predicates; therefore, they are negative in quality.
Quantity
Each of the four forms is either universal or particular in quantity. If a form distributes its subject term, it is universal in quantity. The A and E forms are universal, since each distributes its subject term. On the other hand, a form is particular in quantity if its subject term is undistributed. The I and the O forms have undistributed subject terms; therefore, these are particular.
Chart 1.3: Distribution, Quantity, and Quality
| Forms | Quantity | Quality |
| All sd is pu | universal | affirmative |
| No sd is pd | universal | negative |
| Some su is pu. | particular | affirmative |
| Some su is not pd | particular | negative |
Chart 1.3 may serve the student as a memory device for reinforcing how quantity and quality is determined by distribution of terms in standard form propositions. The chart is no substitute for memorizing the definition of distribution and understanding what it means. The importance of distribution of terms cannot be overemphasized, for it not only serves as the basis for defining the quality and quantity of the four forms, but is the basis for some of the rules that test the validity of deductive inference.
Nonstandard Propositions
Only standard form propositions are candidates for the premises and conclusion for the type of argument (syllogism) discussed in Study Three. Most nonstandard propositions are easily translated to standard form. Others will require practice and careful attention to the meaning of the proposition in question. This may result in some awkward formulations of English. The goal is clarity of meaning, not elegant prose.
Use of Parameters
In the case of an English verb other than the present tense of the verb to be, change the verbs into predicate adjectives. For example, "All competent students know logic" becomes "All competent students are knowers-of-logic.
When the language of the sentence contains clauses or prepositional phrases as well as a verb other than the English copula, the use of parameters will help make the sense of the proposition clear. For example, "All persons-who-are-competent-students are persons-who-are-knowers-of-logic." Here the word, persons, appears in both the subject and predicate, and together with hyphens assists in reading the proposition as an A proposition. The purpose is to make the sense of the proposition crystal clear.
More effort is required with two other classes of propositions: exclusive and exceptive propositions.
Exclusive Propositions
How can we make clear the sense of this exclusive proposition? "Only atheists will be ejected." What does it mean? It means "All persons-who-are-ejected are persons-who-are-atheists." Thus the sense of exclusive propositions (only x is y) is the A form, the result obtained when subject and predicate are interchanged.
Exceptive Propositions
Exceptive propositions (all except x is y) are really two in one form. For example, "All except the soldiers gave up the fight" means (1) All persons who are non-soldiers (civilians) are persons who gave up the fight; and (2) No person who is a soldier is a person who gave up the fight. Not to anticipate the material of the next Study, let it be merely noted for now that neither one of these can be deduced from the other.
Propositions lacking Quantifiers
Some logic books label propositions with proper names, singular propositions. We make no distinction between singular and other universal propositions. All propositions using proper names are either Form A or Form E, depending on the quality. The name Socrates, in "Socrates is mortal" is the entire subject class, which happens to have only one member. An example of an E form is "Socrates is not immortal," or, "No Socrates is immortal." These are not the only propositions wherein "all" or "no" is implied. Some propositions appear to name only some members of a class, when all members of a class are either included or excluded. Example: "Dinosaurs are extinct" does not mean that some are or some may not be extinct. The sense of the statement is that all dinosaurs are extinct. In other words, the "all" is implied, and when the context calls for "all," or "no" the result is either an A Form or an E Form, depending on the quality of the original. Also, an implied "some" proposition is translated as the I Form or O Form, depending on quality.
Logical versus Grammatical Subjects
The grammatical and logical subjects of some propositions sometimes need to be distinguished, if one is to achieve the correct sense of a proposition. An example cited in a logic book is: "You always squirm out of an argument." The grammatical subject, "you," is not the logical subject. Rather, always meaning "every time you get into an argument" is the logical subject. The sense of the original is "All times-you-get-into-an-argument are times-you-squirm-out-of-it." (The statement may appear to be awkward, but the meaning is accurately worded and that's what matters!)
Application of tests to determine the validity of inference depends on the clear sense of standard form propositions. However, the job of re-wording nonstandard propositions into standard form A, E, I, and O has benefits beyond the requirements of deductive inference. Where testing for validity is not an issue, rewording nonstandard into standard forms will avoid misunderstandings, mistakes, and confusion. If you can't reword a nonstandard proposition into standard form, you probably do not know what it means. Therefore, it is essential that you develop translation skills to achieve clarity of thought and to avoid misunderstanding or mistakes in reasoning.
Summation
Standard form propositions consists of subject and predicate terms joined by the copula "is" or "are" and qualified by "All," "No," "Some," or "Some ... not ...." These requirements yield four forms: (1) All a is b, (2) No a is b, (3) Some a is b, and (4) Some a is not b known as A, E, I, and O forms, respectively. (The forms are also expressed as A(ab), E(ab), I(ab), and O(ab).) The formal properties of distribution, quality, and quantity of the four standard forms were explained and illustrated. A distributed term is one modified by "all" or "no." If a term is modified by "some," it is undistributed. If a proposition's predicate term is distributed, the proposition is said to be negative in quality; if the predicate of a proposition is not distributed, then it is affirmative in quality. This definition of quality distinguishes E(ab) and O(ab), both negative, from A(ab) and I(ab), both affirmative. If a proposition distributes its subject term, it is universal in quantity. On the other hand, if a proposition's subject term is undistributed, it is particular in quantity. By this definition, we distinguish A(ab) and E(ab), both universal, from I(ab) and O(ab), both particular. Finally, some guidelines for translating nonstandard propositions into standard form were described.
Immediate Inference
Valid Inference
An inference is valid whenever the form of the conclusion is true every time the forms of the premise are. If the form of the conclusion is not true every time the forms of the premise are true, then the inference is invalid. The significance of this definition is the subject of the paragraphs that follow.
One way of illustrating the meaning of the definition makes use of circles representing the four standard forms. (Another is by the Method of Deduction, discussed in the next study.) The five sets of circles in the chart below correspond to five ways in which two terms (subject and predicate terms) can be related in the four forms. The circles are numbered as cases 1 through 5 for easy reference. (Read A(ab) as All a is b; E(ab) as No a is b; and so forth.) To repeat. If the form of the conclusion is true every time the forms of the premise are true, then the conclusion follows validly from the premise. One inspects the circles to see if a particular immediate inference does not violate the definition for validity. In a sense, the chart circles and lines "operationalize" the definition of validity. Descriptions of each case in the chart follow, after which we offer some examples.
Forms, Circles, and Lines
| CASE #1 | One sense of A(ab) form is where All a is b and All b is a. Count the number of cases when this sense of A(ab) is true. Inspection reveals this sense A(ab) is true in one of the five sets of circles: case 1. Note Line A covers this case. |
| CASE #2 | Another sense of A(ab) is where All a is b, but not All b is a. If you count the number of cases where this sense of A(ab) is true, the count is two of the five circles: cases 1 and 2. Line A covers the two cases. |
| CASE #3 | The third set of circles, corresponds to Some a is b and Some b is a. I(ab) is true in four cases: cases 1 through 4. (Count them!) Note Line I includes the four cases. |
| CASE #4 | The fourth set of circles, corresponds to O(ab) form, i.e., Some a is not b. O(ab) is true three times, in the 3rd, 4th, and 5th cases, as shown by line O. (Count the number of cases!) |
| CASE #5 | The fifth set of circles, corresponds to No a is b, and No b is a. E(ab) is true only once, in the 5th case, as shown by line E. |
Chart 2.1 Four Forms Diagrammed
The lines labeled A, I, O, and E mean All, Some, Some ... is not, and No, respectively.
The test for validity compares the conclusion to the premise with reference to the circles to determine if the conclusion is true every time the premise is. For example, one may ask, does A(ab) imply I(ab), or, what comes to same thing, is I(ab) a necessary consequence from A(ab)? With either question, we are asking whether A(ab) and I(ab) form a valid inference. Inspection of Chart 2.1, shows that I(ab) is true every time A(ab) is true. Check it out for yourself. I(ab) is true in cases 1-4; A(ab) is true in cases 1 and 2. Thus, A(ab) and I(ab) make a valid inference. Similarly, E(ab) and O(ab) form a valid inference, since O(ab) is true every time E(ab) is true. This time check out the lines as well as the circles. Note that Line O includes the whole of Line E. To provide a contrast, does O(ab) imply E(ab)? Is E(ab) true every time O(ab) is? E(ab) is true in case 5 (Line E); O(ab) is true in cases 3, 4, and 5 (Line O). Therefore, one cannot validly infer from O(ab), the form E(ab), since E(ab) is not true every time O(ab) is true; O(ab) is true three times, E(ab) only once. By the same reasoning, I(ab) and A(ab) do not form a valid inference, since A(ab) is not true every time I(ab) is true.
Square of Opposition
The square of opposition incorporates the valid inferences just mentioned above. Pairs of the forms stand in opposition to each other as contraries, subcontraries, subalternation, or contradiction as depicted in Chart 2.2.
Chart 2.2: Square of Opposition
The square of opposition's contraries, subcontraries, subalterns, and contradictories require identical subject terms in each of the four forms as well as identical predicate terms in each. Definitions follow in the order listed.
Contraries
A(ab) and E(ab) are opposed as contraries. By definition, contraries cannot both be true together; however, both may be false. For example, if No Christian is an atheist is true, then it is false that All Christians are atheists. Why? Because the forms are contraries and they cannot both be true together. This appeal to the definition of contraries is supported by Chart 2.1 analysis of the corresponding lines. Lines A and E do not overlap which means they cannot both be true in any instance. Suppose we assume that an A Form is true. Applying the definition for contraries, we conclude that the E Form will be false. How can contraries be both false? Observe that Lines A and E do not exhaust all five cases. The significance of this fact is they could both be false together. One example: Some Christians are Calvinists is true, then the corresponding A Form (All Christians are Calvinists) and E Form (No Christians are Calvinists) are both false.
Subcontraries
I(ab) and O(ab), are subcontraries, meaning that they cannot both be false together, but they could both be true. Consider, for instance, the I Form, Some atheists are Christian is a false proposition by all accounts. If so, it follows that the related subcontrary, the O Form, Some atheists are not Christian is true. A similar line of reasoning applies when one begins with a false O Form. The corresponding I Form is true by the definition of subcontraries. The definition also states that subcontraries can both be true together. Here is an example: Some Christians are Calvinists, and Some Christians are not Calvinists. Again, Chart 2.1 analysis of the lines supports the definition of subcontraries. Lines I and O exhaust all 5 cases, and overlap each other to show that they can both be true together.
Subalterns
Subalterns is the name for two pairs of forms: A(ab) & I(ab); and E(ab) & O(ab). Subalterns may both be false together or both true together. For example, if the E Form No men are angels is true, then the A Form (All men are angels) and the I Form (Some men are angels) are both false. A similar analysis applies to the second pair of subalterns. If,All men are sinners is true, then the corresponding E Form (No men are sinners) and O Form (Some men are not sinners) are both false. Under what conditions are subalterns both true? Follow this reasoning: what is true of All will also be true of Some, assuming, of course that we use the identical subject and predicate terms. To stick with our example, if All men are sinners is true, then what is true of All is also true of Some (or any portion of the class). Likewise, if No men are angels is true, then it may be pointless, but nevertheless true that Some men are not angels.
Examination of Chart 2.1 once again supports the definition of subalternation. Lines A and I are both true under cases 1 and 2, and both false in case 5. Lines O and E are both true under case 5, and both false under cases 1 and 2. To "see" the relations between lines and cases may require some effort and more practice to acquire the necessary expertise. In the meantime, rely on the definitions and the square of opposition to determine what are valid inferences and which are not permitted (invalid). Definitions are indispensable, if one seeks to understand correctly or be understood accurately.
Contradiction
The strongest form of opposition is contradiction. The definition of contradiction incorporates some aspects of the properties of contraries and subcontraries. Contradictoriescannot both be true together and cannot both be false together. A(ab) & O(ab), and E(ab) & I(ab) are contradictories. If the A Form is true, then the contradictory O Form is false. If the O Form is true, then the contradictory A Form is false. A similar reasoning applies to the other pair, the E Form and the I Form. To illustrate. If All men are sinners is true, then the contradictory, Some men are not sinners is false. Another example. If Some men are Christian is true, then the contradictory E Form, No men are Christian is false.
Observe the lines of Chart 1.2. Lines A and O, and I and E can be seen to meet without overlapping and, at the same time, each pair exhausts all cases. These characteristics apply only to contradictions. Thus it is a mistake to think of contraries or subcontraries as contradictories. Contraries and subcontraries are opposed, but their opposition is not total. Only contradictories represent total opposition. Some men are liars and Some men are not liars are not contradictories; although they are opposed. They are opposed as subcontraries. Recall the definition.
A useful analogy for the square of opposition is to treat it as a simple computer that calculates results based on two values, true or false (1 or 0). Given the value of any form or corner of the square, one can calculate the values of the other forms in the square of opposition. If, for example, we assume the value of True for the A Form corner, what are the values of the other three forms? The contradictory O Form corner has the value of False, since contradictories cannot both be true together and both be false together. The contrary E Form corner has the value of False, based on the definition of contraries. One can arrive at the value of the I Form corner in three ways. First, it has the value of True, since it's contradictory (E Form) is False. Second, it has the value of True, as the subaltern of the A Form corner which we assumed has a value of True. Third, it has a value of true as the subcontrary of a false O Form.
Similar calculations can begin with True or False from any other corner of the square with similar results -- almost always. As with any computer, there are limitations. You will receive an error message: "Can't Compute; Insufficient Information," when one of two initial conditions hold. You plug in False in either of the upper corners (A Form or E Form). Or, you plug in True in either of the lower corners (I Form or O Form). Either of these two related conditions will result in a pair of forms for which no value can be calculated. We know the result is either true or false, but which one?. Our simple computer becomes stuck. It cannot choose between the two: Not enough information. One example should suffice. Suppose we assume True for the O Form corner. The A Form corner has the value of False, since it is the contradictory of the O Form. What about the other pair, the E Form and I Form corners? There is no way to calculate the values of this pair of forms. For instance, from the truth of the subcontrary O Form, the other subcontrary I Form's value is either true or false. Recall the definitions of subcontraries. From the falsity of the contrary A Form, the other contrary E Form's value is either true or false. Recall the definition of contraries. The result is a pair of Undetermined Forms, in this case, the E Form and its contradictory, the I Form.
Given the initial conditions described above, it is always the case that our simple computer calculations will produce a pair of forms with Undetermined values. The pair of Undetermined Forms will always be a diagonal pair, never any other pair.
A couple of comments on the square of opposition relations and propositions. Some texts display a leaner model of our simple computer. It amounts to a stripped-down model, consisting of only the cross of contradiction. We opt for the original, for it illustrates, as we have shown, a number of useful immediate inferences not possible with the stripped-down version. A reason often offered for the stripped-down model is the claim that subcontraries assert or assume the "existence" of members of the subject classes. This is considered by some to be problematic. Let it be noted here that logic alone does not assert the existence or the nonexistence of anything. The existence or nonexistence of men, sinners, or angels in propositions is a matter for history or biology, anthropology, or some other discipline. "Some a is b" does not assert the existence or nonexistence of "a's" or "b's."
Invalid Inferences
We have shown the value of Chart 2.1 in testing the validity of the immediate inferences depicted in the Square of Opposition. Obviously, use of the same methods proves invalidity as well, for if an inference is not valid, then it must be invalid -- the only other possibility.
Consider the following matrix of forms. It represents the number of ways two forms can be combined to form inferences or implications. For example, in the first cell "AA," the implication to be tested is: Does A imply A? Of course, by the Law of Identity, since every proposition implies itself, the answer is affirmative. Similar reasoning for the other combinations in the diagonal (upper left to lower right: EE, II, and OO) show these to be valid implications as well.
Chart 2.3: Matrix of Four Forms
| FORMS | A | E | I | O |
| A | AA | AE | AI | A0 |
| E | EA | EE | EI | EO |
| I | IA | IE | II | IO |
| O | OA | OE | OI | OO |
We eliminate the combinations of the other diagonal (upper right to lower left: OA, IE, EI, and AO), because each is a pair of contradictories. No form implies its contradictory. There are a total of 16 combinations in the matrix (4 rows x 4 columns = 16). Thus far we can account for eight of the combinations, four in each of two diagonals. What of the remainder, listed below?
Using, "< " for "implies," we cast the remaining 8 combinations of forms as implications to be tested for validity using the measures we have previously described. When dealing with implications, as we have, the proposition before the "< " is the antecedent; the proposition after "< " is the consequent.
Chart 2.4: Test of Implications (1st Figure)
| 1.1 | A(ab) < E(ab) | 1.5 | I(ab) < A(ab) |
| 1.2 | A(ab) < I(ab) | 1.6 | I(ab) < O(ab) |
| 1.3 | E(ab) < A(ab) | 1.7 | O(ab) < E(ab) |
| 1.4 | E(ab) < O(ab) | 1.8 | O(ab) < I(ab) |
Assessment of the first and the last implications from the list above, we conclude each to be invalid. A(ab) < E(ab), is invalid because, as Chart 1.2 shows, E(ab) is not true every time A(ab) is true; E(ab) is true once in the fifth case; A(ab) is true twice in the first and second cases. O(ab) < I(ab), is invalid, since I(ab) is true four times (cases 1, 2, 3, and 4), while O(ab) is true three times (cases 3, 4, and 5), and for a valid inference the form of the conclusion must be true every time the form of the premise is true. 1.2 through 1.7 should receive similar treatment as shown with the first and last.
The list of implications above (1.1-1.8) are in the First Figure. Reordering the terms of the consequent produces another set of implications said to be in the Second Figure. There are only two figures for immediate inferences. In 1.1, the conclusion E(ab) is in the First Figure; E(ba) arrangement is Second Figure. It may be expressed the other way around: E(ba) is in the First Figure; E(ab) is in the Second Figure. The list of implications below are in the Second Figure. It should be an easy task to determine the validity of each using Chart 1.2 information.
Chart 2.5: Test of Implications (2nd Figure)
| 2.1 | A(ab) < E(ba) | 2.5 | I(ab) < A(ba) |
| 2.2 | A(ab) < I(ba) | 2.6 | I(ab) < O(ba) |
| 2.3 | E(ab) < A(ba) | 2.7 | O(ab) < E(ba) |
| 2.4 | E(ab) < O(ba) | 2.8 | O(ab) < I(ba) |
Other Immediate Inferences
Definitions of the three immediate inferences that follow are clear and simple and, for these reasons, should be committed to memory. The tests provided by Chart 1.2 are available for those who think it difficult to memorize definitions. The definitions for conversion, obversion, and contraposition follow.
Conversion
The converse of a proposition is formed by interchanging the subject and predicate of the E Form and the I Form. The A Form is a special case, described below and the O Form has no converse.
Chart 2.6: Conversions (Valid & Invalid)
| 3.1 | E(ab) < E(ba) | valid |
| 3.2 | I(ab) < I(ba) | valid |
| 3.3 | A(ab) < I(ba) | valid ** |
| 3.4 | O(ab) < O(ba) | invalid |
**special case
3.1 and 3.2 are valid on the basis of the application of the definition for valid inference and the information provided in Chart 1.2. The consequents are true every time the antecedents are, as Chart 1.2 will show. A(ab) < I(ba) is known as conversion per accidens. It is a valid implication since I(ab) is true every time A(ab) is true. The implication in 3.4 is invalid because the consequent is not true every time the antecedent is true.
To illustrate each of the above. The E Form, No Christians are atheists has a valid converse No atheists are Christian. The I Form, Some men are believers converts to Some believers are men. The familiar A Form All men are mortal converts by limitation to the I Form Some mortals are men. The O form Some vegetables are not carrots (True) has no converse, since to permit it yields the false consequent Some carrots are not vegetables.
Note also that the distribution of the subject and predicate terms is unchanged in conversion of E Form and I Form propositions. The A Form to I Form conversion is permitted by limiting or reducing the distribution of the subject term of the A Form where the subject term is Distributed to Undistributed subject term in the I Form. Moreover, we know that A(ab) implies I(ab) through the square of opposition's subalternation. We also know that I (ab) implies I(ba) by conversion. Therefore, A(ab) implies I(ba). Follow? Finally, notice that to permit conversion for the O Form changes the distribution of both terms, thus changing the meanings of the original proposition. It is true that Some mortals are not men (cats and dogs, for examples), but to assert that it follows that Some men are not mortal is false. The distribution of the terms have undergone a complete reversal:mortals from undistributed in the first to distributed in the second, and men from distributed in the first to undistributed in the second.
Obversion
To obvert a proposition, change the quality of the proposition and replace the predicate by its complement. Each of the four forms has an obverse. Before showing these, a comment about the notion of complement.
A term and its complement (some use "contradiction" in place of "complement") are said to exhaust the universe of objects. Thus, if "a" stands for the class of objects, its complement class, non-a, contains everything else not included in the class a. The whole, a and non-a totally exhausts the universe of entities, since everything in the universe must fall either into one class or its complement class. The complement or contradictory term of a class is symbolized by the use of the prime character ( ' ). Thus, " aa' " means the class "a" and its complement or contradictory, a-prime, or "a'." It is customary to use the word "non" with the declarative statements and the symbol " ' " with the forms when symbolized.
Chart 2.7: Obversion, Valid Implications
| 4.1 | A(ab) < E(ab' ) | valid | And | E(ab' )< A(ab) | valid |
| 4.2 | E(ab) < A(ab' ) | valid | And | A(ab' ) < E(ab) | valid |
| 4.3 | I(ab) < O(ab' ) | valid | And | O(ab' ) < I(ab) | valid |
| 4.4 | O(ab) < I(ab' ) | valid | And | I(ab' ) < O(ab) | valid |
Every form has an a valid obverse. The A Form obverts to an E Form and vice versa. The I Form obverts to an O Form and vice versa. The definition of obversion proves to be a superior criterion for validity than the use of Chart 1.2. To make use of the chart entails labeling all of the complement classes for every term in each of the diagrams, an awkward and somewhat complicated, although workable solution. The more practical method is to simply make use of the definition of obversion. (A similar recommendation will apply with the last inference in this section: contraposition.) Some examples of obversion for each form follow.
Chart 2.7a: Obversion Examples
| Propositions | Obverts |
| All men are mortal | No men are immortal (non-mortal). |
| No men are perfect | All men are imperfect (non-perfect) |
| Some men are practical. | Some men are not impractical (non-practical). |
| Some men are not saved. | Some men are unsaved (non-saved). |
The reverse of each of the above is valid as well. When you obvert each of the obverses above, one has the original proposition. The definition of obversion must be applied consistently and completely in order to have a valid obverse. Also, there are a number of other combinations possible. To cite one: "Some non-Christians are not immoral." The obverse is: "Some non-Christians are moral." One more: "If some things are not excusable, then some things are inexcusable."
Contraposition
The contrapositive of a proposition is one in which the complement terms of the subject and predicate are interchanged. Contraposition can be thought as first obverting the original, converting the resulting proposition, then obverting the resultant one. The A Form and the O Form have straightforward contrapositives. The E Form is a special case. There is no contrapositive for the I Form.
Chart 2.8: Contraposition, Valid & Invalid Implications
| 5.1 | A(ab) < A(b'a') | valid | & | A(b'a') < A(ab) | valid |
| 5.2 | O(ab) < O(b'a') | valid | & | O(b'a') < O(ab) | valid |
| 5.3 | E(ab) < O(b'a') | valid | & | O(b'a') < O(ab) | valid |
| 5.4 | I(ab) < I(b'a') | invalid | |||
Before providing the reader with some examples, consider 5.3 and 5.4 above. As mentioned earlier, contraposition can be thought of as first obversion of the original, then converting the result, then obverting again. So, E(ab) obverts to A(ab') which (being an A Form) converts by limitation to I(b'a). Applying obversion once again yields O(b'a'). The critical step is the conversion per accidens of the A Form. Follow a similar process with 5.4. I(ab) < O(ab') by obversion. The resulting obverse, O(ab'), has no converse, for O Form propositions have no converses. The process aborts, therefore, I Form propositions have no contrapositives.
Chart 2.8a: Contraposition Examples
| Propositions | Contrapositives |
| All humans are mortal. | All non-mortals are non-human. |
| Some pleasant things are not worthy. | Some unworthy things are not unpleasant. |
| No ungraciousness is excusable behavior. | Some inexcusable behavior is not gracious. |
| Some human beings are irreverent | None |
Check out the last example. Does Some human being are irreverent mean that Some reverent persons are non-human? Hardly! The I form has no contrapositive since to permit it warrants drawing a false proposition from a true one.
Each of the operations of conversion, obversion, and contraposition refer to the forms in the Original Column of Chart 2.3.
Chart 2.9: Summary of Three Immediate Inferences
| Original | Conversion | Obversion | Contraposition |
| A(ab) | I(ba)* | E(ab' ) | A(b'a' ) |
| E(ab) | E(ba) | A(ab' ) | O(b'a' )* |
| I(ab) | I(ba) | O(ab' ) | (none) |
| O(ab) | (none) | I(ab' ) | O(b'a' ) |
*Special Cases
The definitions of obversion and contraposition serve admirably. As previously suggested, the use of Chart 2.1 for obversion and contraposition, is impractical and unnecessary given the simplicity of the definitions. Memorize the definitions. Nevertheless, all of the immediate inferences can be tested using the Chart 2.1. Should the student be inclined to carry out these procedures, it should be noted that for each case, everything outside of circle a is a'; everything outside of circle b is b', and vice versa. These additional notations must be filled in the 5 sets of circles correctly for accurate results.
Three Additional Inferences
The remaining immediate inferences, reflexive, symmetrical, and transitive, apply to relationships, like "is greater than," or "is less than," when speaking of numbers or quantities. One or more may apply to other types of relationships; for example, family relationships, "the son of" or "the sister of," and so forth. Some relationships exhibit one or more; some none of these. They are included here for the sake of completeness.
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Summation
The wealth of immediate inferences available from a set of four standard form propositions may come as a surprise to some. Memorizing and understanding the various definitions is essential which when applied correctly will distinguish valid from invalid inferences. Here is an opportunity to apply what has been explained. Consider what valid inferences follow from this passage?
| "There is therefore now no condemnation to them which are in Christ Jesus, who walk not after the flesh, but after the Spirit." |
Reworded so that the sense of the A form is clear:
| All persons-who-are-in-Christ Jesus-who-walk-not-after-the-flesh-but-after-the-Spirit arepersons-for-whom-there-is-now-no-condemnation. |
The related I form is true by subalternation; the E form is false by contraries; the contradictory O form is false. The valid converse per accidens is an I Form. It has a valid obverse and contrapositive. The importance of definitions and the impressive power of the logic of immediate inference should be obvious.
We turn in the next study to the power of mediated inference, the syllogism.
The Syllogism
While immediate inference contained two propositions, a premise and a conclusion, and thus, two and only two terms, a standard syllogism contains more. The familiar syllogism of men, mortals, and Socrates will again prove its value, providing the basis for introducing new terms and new definitions. The "\ " is read as "therefore."
| All men are mortal. |
| Socrates is a man. |
| \Socrates is mortal. |
The Basic Elements
The standard syllogism must contain three and only three propositions, two of which are premises; the other is the conclusion. The two premises and the conclusion share three and only three terms. In the syllogism above, the three terms are men, mortals, and Socrates. Socrates in the conclusion and the second premise; mortal in the conclusion and the first premise; and men (or man) in the two premises. Each appears twice, but never twice in the same proposition. Each term must mean the same thing, that is to say must be univocal. For example, "mortal" in the conclusion and the premise must mean the same thing. Thus, a syllogism is an argument having two premises and a conclusion with the subject term of the conclusion in one of the premises, the conclusion's predicate term in the other premise, and a third term in both premises. The third term of the premises must never appear in the conclusion.
The Terms of the Syllogism
The syllogism above or any other standard syllogism, can be expressed as an implication.
A(mp) A(sm) < A(sp),
where, s stands for Socrates; p stands for mortal; m stands for man; and "< " stands for implies. The subject term of the conclusion is the minor term. The predicate of the conclusion is the major term. The term that appears in both premises, not the conclusion, is the middle term. The premise that contains the major term is the major premise, and is usually placed first. The premise that contains the minor term is the minor premise; it is placed after the major premise.
Thus, the conclusion of our syllogism is an inference from the major premise through the mediation of the minor premise.
Moods
The mood of an argument is an individual case of an inference, a mediated inference. For example, each of the propositions of the syllogism above are of the form All a is b -- theA Form. The mood, we say, is AAA; the first letter denotes the major premise, the second letter denotes the minor premise, and the third letter denotes the conclusion. Thus, the mood of a syllogism refers to the forms of the syllogism and the order of the forms beginning with the major and ending with the conclusion. Every standard syllogism has a mood of three and only three forms, but there is more.
The Figure of a Syllogism
The figure of a syllogism refers to the position of the middle term in the premises. Omitting any reference to the conclusions, there are four possible positions as shown below. It may be helpful to think of Figures 1 and 4 as mirror images of each other, as are Figures 2 and 3. (m stands for the middle term; p stands for the major term; s stands for the minor term.)
| 1st Premise | M-p | p-M | M-p | p-M |
| 2nd Premise | s-M | s-M | M-s | M-s |
| Figure | 1 | 2 | 3 | 4 |
By figure, then, we indicate the relative positions of the one term shared by both premises -- the middle term.
The Frame of a Syllogism
The frame of a syllogism is a name assigned to the combination of the mood and the figure of a standard syllogism. Thus, when we speak of the form of a syllogism, we mean the frame -- its mood and figure together. Our syllogism above has this frame: AAA-1.
Valid syllogistic frames were given names by logicians. In part, their purpose was the development of a system of frame-names in verses as a memory device to aid in identifying the different valid moods and figures of the syllogism. Some other characteristics of these names will be discussed in due course.
With four forms and four figures, there are 256 frames. (Four forms can be combined in pairs results in 16 different sets of premises; each pair has one of the four forms as a conclusion for a total of 64 (16 x 4); factor in 4 figures, for a total of 256 frames.) Of course, not all of these were named, only the valid ones. There are 24 valid frames, hardly as intimidating as 256, if one had to rely on memory. Fortunately, there are other means more reliable than memory.
Validity
As has been stated, valid is a quality of arguments in which the conclusion necessarily results from the premises. An argument is valid if the form of the conclusion is true every time the forms of the premises are true. This means essentially that if an argument is valid, then it is impossible for the premises to be true and the conclusion false. Obviously, the conclusion of a valid syllogism must not contain a term not in the premises.
The validity or invalidity of a syllogism can be determined by either the application of rules or the Method of Deduction. We start with rules since they are quite easy to apply once an argument's frame has been made explicit and leave the deductive method for the next section. The rules themselves are derived from the valid frames. There are five rules for testing the validity of syllogisms.
The Five Rules
Is a syllogism valid? It is if it does not violate these five rules.
| Rule 1 | Two premises in both of which the middle term is undistributed do not imply a conclusion. |
| Rule 2 | Two premises with undistributed terms having a conclusion which distributes those same terms do not imply a conclusion |
| Rule 3 | Two affirmative premises do not imply a negative conclusion. |
| Rule 4 | Two negative premises do not imply a conclusion. |
| Rule 5 | An affirmative and negative pair of premises do not imply an affirmative conclusion. |
First, symbolize the syllogism. The choice of subject and predicate letters is arbitrary only take care to use them in consistent fashion. There must be only three such letters, since a standard syllogism contains three and only three terms each used univocally. The letters s, p, and m, used previously will serve. As before, s stands for "Socrates" and is the minor term; p stands for "mortal" and is the major term; and m stands for "man" and is the middle term. The subscripts "d" and "u" stand for distributed and undistributed, respectively.
| Major Premise | All m d is p u | A(mp) |
| Minor Premise | All s d is m u | A(sm) |
| \Conclusion | \ All s d is p u | \ A(sp) |
Second, apply each of the rules. The syllogism must satisfy each and all of the rules if it is to count as valid. The first rule states that the middle term must be distributed in at least one of the premises. Observe, the middle term is distributed in the major premise. The second rule compares terms in the premises with terms in the conclusion. If a term is undistributed in the premise, it must not be distributed in the conclusion. Check the major term P in the major premise. It is undistributed and it remains undistributed in the conclusion. The third rule states that two affirmative premises do not imply a negative conclusion. This syllogism has two A Form (affirmative) propositions as premises. The conclusion is also affirmative. The fourth rule makes reference to negative premises. The premises of this syllogism are affirmative in quality. The fifth rule says that an affirmative premise and a negative premise do not imply an affirmative conclusion. This syllogism ends with an affirmative conclusion, but it does not contain a negative premise. Therefore, this syllogism is valid. Indeed, all syllogisms having this form are valid. The frame AAA-1 is a valid frame, since it satisfies all of the rules.
The rules themselves are both sufficient and necessary. They are sufficient since they leave untouched the 24 syllogisms proved valid by the deductive method, and prove the remaining ones invalid. The rules are also necessary since each applies to at least one invalid syllogism for which none of the others apply.
A study of the rules alone will eliminate a number of invalid frames. For example, by Rule #4 the syllogisms with premises EE, EO, OO, and OE (all negative) are invalid. Rule #1 declares invalid the syllogisms with premises II; OI, figures 1 and 3; and IO, figures 3 and 4, since these arrangements leave the middle term undistributed. A systematic study of the rules should eliminate as invalid all but 24 of the 256 frames. Thus the significance of necessary and sufficient rules is no mere detail.
The Method of Deduction
It is an unavoidable fact, though many try to skirt it, that every system of thought, philosophy, theology, or body of knowledge has starting points without which the system could not get off the ground. To put it another way: every system of thought or knowledge has an axiom or a set of axioms which are indemonstrable within that system. An axiom is a first principle or premise which cannot be demonstrated precisely because axioms themselves are used to demonstrate or prove other statements which we call theorems. Atheorem is a proposition deduced from an axiom. Thus, first principles or axioms are the basis of all argument and demonstration.
The deductive method proves valid frames as theorems. To that end, seven of the 24 valid frames will be proved as theorems. These proofs should suffice to introduce a beginner to the significance of the deductive method in syllogistic reasoning. We state first the two axioms, then two (operational) rules. These are applied to the axioms to deduce theorems. They may also be applied to theorems to deduce additional theorems.
Axiom 1: A(ba) A(cb) < A(ca) Axiom 1 reads: All b is a & All c is b implies All c is a.
Axiom 2: E(ba) A(cb) < E(ca) Axiom 2 reads: No b is a & All c is b implies No c is a.
| Rule I DM | If in any valid implication the premise and the conclusion are interchanged and contradicted, the result is a valid implication. ("DM" stands for Deductive Method.) |
| Rule II DM | If in any valid implication its premise be strengthened or its conclusion be weakened, then a valid implication will result. |
Application of Rule I above is easily accomplished. It can be applied as often as necessary, first to one premise and conclusion, then to the other premise and conclusion, in any order. Application of Rule II may raise some concern about the meanings of strengthened form of the premise and weakened form of the conclusion. So let us define these. The premise of a valid mood and figure can be said to be a strengthened form of the conclusion, and the conclusion a weakened form of the premise. More explanation follows.
Some examples applying the rules might be helpful. From the square of opposition, we know that A(ab) implies I(ab) is a valid inference. If we interchange the premise and conclusion and contradict each (Rule I), we prove that E(ab) implies O(ab). Of course, the latter implication is valid based on subalternation, but here we illustrate the application of the Rule I. Now apply Rule II to the valid implication, A(ab) implies I(ab). A(ab) is the strengthened form of I(ab); and I(ab) is the weakened form of A(ab). Also, I(ba) is the weakened form of I(ab). By weakening the conclusion of A(ab) implies I(ab), we prove A(ab) implies I(ba). Of course, the latter is valid per accidens, but here we illustrate the application of Rule II. Confused? Well, just below starting with the Law of Identity as an axiom, we deduce two theorems to illustrate further the use of Rules I and II in proving theorems.
| 1 | A(ab) < A(ab) | Using the Law of Identity as an Axiom |
| 2 | A(ab) < I(ab) | Theorem One, by Rule I, replacing the conclusion of #1, above, by its weakened form, I(ab). |
| 3 | E(ab) < O(ab) | Theorem Two by Rule II, interchanging and contradicting the premise and conclusion of Theorem One. |
Deduction of Theorems
It should be obvious now that the process of deducing or proving theorems from axioms is not a difficult operation; although, it requires careful attention at each step. The order of the deduced theorems may vary from one person to another. For example, one person may apply Rule I to the minor premise and conclusion of an axiom first, then secondly to the major premise and conclusion of the axiom. Another may reverse the process. Whether you start with the minor and conclusion, or the major and conclusion is arbitrary. Likewise, whether you apply Rule I first, then Rule II, or vice versa is optional.
| Axiom 1 | A(ba) A(cb) < A(ca) | |
| Axiom 2 | E(ba) A(cb) < E(ca) | |
| Theorem 1 | A(ba) O(ca) < O(cb) | by Rule I on Axiom 1: Interchange and contradict Axiom 1'sminor premise and conclusion. |
| Theorem 2 | O(ca) A(cb) < O(ba) | by Rule I on Axiom 1: Interchange and contradict Axiom 1'smajor premise and conclusion. |
| Theorem 3 | I(ca) A(cb) < I(ba) | by Rule I on Axiom 2: Interchange the contradictories of the conclusion and major premise. |
| Theorem 4 | E(ba) I(ca) < O(cb) | by Rule I on Axiom 2: As above, only this time with the conclusion and the minor premise. |
Theorems 1-4 are not in conventional format, and they must be if one is to ascertain the correct mood and figure. (By conventional format we mean a certain order based on a "ca" conclusion.) So, let us stipulate that "c" is the minor term; "a" is the major term and "b" is the middle term. Applying these conventions to Theorems 1-4, we obtain conventional format for each and thereby the correct mood and figure.
| Theorem 1 | A(ab) O(cb) < O(ca) | AOO-2 |
| Theorem 2 | O(ba) A(bc) < O(ca) | OAO-3 |
| Theorem 3 | I(ba) A(bc) < I(ca) | IAI-3 |
| Theorem 4 | E(ab) I(cb) < O(ca) | EIO-2 |
The conclusions of the two axioms are universal. A universal conclusion validly implies the corresponding particular. By weakening the form of the conclusion of Axiom 2, we deduce two additional theorems. Note that weakening the form of a conclusion and strengthening the form of a premise, function as replacements of one form by another logically valid form. Thus, the conclusion E(ca) can be weakened by replacement with E(ac) or O(ca). Axiom 2's premise, E(ba), can be strengthened by replacing it with E(ab).
| Axiom 2 | E(ba) A(cb) < E(ca) | |
| Theorem 5 | E(ba) A(cb) < E(ac) | by Rule II on Axiom 2: Weakened form of the conclusion. [E(ac) counts as weakened form of E(ca). One is the converse of the other.] |
| Theorem 6 | E(ba) A(cb) < O(ca) | by Rule II on Axiom 2: Weakened form of the conclusion, E(ca). |
| Now deduce Theorem 7 from Theorem 5 still using Rule II | ||
| Theorem 7 | E(ba) A(cb) < O(ac) | by Rule II on Theorem 5: Weakened form of the conclusion, E(ac). |
Theorem 6 is in conventional format, 5 and 7 are not. This operation will require the re-ordering of the premises in Theorems 5 and 7. Recall that the premise with the major term (the same as the predicate term of the conclusion) is the major premise and is placed first; the minor premise, i.e., the premise with the minor term (the same as the subject term of the conclusion) is placed second.
| Theorem 5 | A(ab) E(bc) < E(ca) | AEE-4 (mood & figure) |
| Theorem 6 | E(ba) A(cb) < O(ca) | EAO-1 (mood & figure) |
| Theorem 7 | A(ab) E(bc) < O(ca) | AEO-4 (mood & figure) |
The Deductive Method has proven seven theorems from two axioms all of which can be used to deduce additional theorems using the rules and the definitions provided. Theorems may be used at any stage together with the original axioms and rules to prove additional theorems. The problem now is to deduce the remaining theorems for a total of twenty four. Your deductions may prove theorems previously deduced, but keep trying until you have twenty four unique theorems each in conventional format. If you deduce one which is doubtful, appeal to the set of Five Rules to check your proof.
Frame Names
The valid frames of syllogistic logic were named and may be of more historical interest than practical. The vowels indicate the mood. Other lower case letters stand for certain operations we shall briefly describe in due course, but first, the names:
| 1st Figure | Barbara, Celarent, Darii, Ferio. |
| 2nd Figure | Cesare, Camestres, Festino, Baroko. |
| 3rd Figure | Darapti, Disamis, Datisi, Felapton, Bokardo, Ferison. |
| 4th Figure | Bramantip, Camenes, Dimaris, Fesapo, Fresison. |
These names designate nineteen valid frames. Five others are available on the basis that the universal conclusion of a valid frame implies the corresponding particular. Thus, from Barbara-1 or AAA-1, application of Rule II yields AAI-1, the Weakened Form of Barbara. Similarly, from Celarent-1 or EAE-1, Rule II application yields EAO-1, the Weakened Form of Celarent.
Chart 3.1 lists the names in the order of Theorems 1-7 already deduced.
Chart 3.1: Theorems and Frame Names
| Theorems | Mood & Figure | Names | |
| 1 | A(ab) O(cb) < O(ca) | AOO-2 | Baroko |
| 2 | O(ba) A(bc) < O(ca) | OAO-3 | Bokardo |
| 3 | I(ba) A(bc) < I(ca) | IAI-3 | Disamis |
| 4 | E(ab) I(cb) < O(ca) | EIO-2 | Festino |
| 5 | A(ab) E(bc) < E(ca) | AEE-4 | Camenes |
| 6 | E(ba) A(cb) < O(ca) | EAO-1 | Celarent-1, weakened form |
| 7 | A(ab) E(bc) < O(ca) | AEO-4 | Camenes-4, weakened form |
The vowels of the names, as mentioned above, stand for the mood of the syllogism. The other lower case letters in the names of the first figure do not have any special meaning, but the "s," "p," and "k" of the names in figures two, three, and four do.
"s" stands for simple conversion of the preceding proposition. For example, if in Camenes you convert the conclusion E(ca) to E(ac) and change to conventional format, which in this case requires a reordering of the premises, you get Celarent, EAE-1. Similar conversions hold for Cesare, Camestres, Festino, Disamis, Datisi, Ferison, Dimaris, Fesapo, and Fresison, all having at least one "s" preceded by a letter standing for a standard form.
"p" means to convert the preceding proposition by limitation or per accidens. If you apply this operation to Fesapo (EAO-4), you get Festino (EIO-2). Other frames that qualify are Darapti, Felapton, and Fesapo.
"k" stands for reductio ad absurdum (RAA) or assuming the conclusion to be false as part of the premise set in order to deduce by valid inferences, step by step, a contradiction. In this manner, one demonstrates that the assumption of a false conclusion as premise was unwarranted, and the original implication, valid.
To illustrate, let us show that Bokardo (OAO-3) is valid by RAA proof.
O(ba) A(bc) < O(ca) Bokardo-3
| 1. | O(ba) | true premise | \ O(ca) | ||
| 2. | A(bc) | true premise | |||
| Assume | 3. O(ca) is false | RAA method | |||
| Then | 4. A(ca) is true | contradictory of 3 | |||
| Then | 5. A(ca) A(bc) < A(ba) | 4 & 2; Barbara-1 | |||
| But | 6. A(ba) cannot be true | contradictory of 1, O(ba) | |||
| So | 7. A(ba) must be false | 1 & 6 contradictory | |||
| But if | 8. A(ba) is false | Step 7 | |||
| Then | 9. A(ca) or A(bc) is false. | 5 & 8; Barbara-1 | |||
| Option 1 Assume | 10. A(ca) is false | From Step 9 | |||
| Then | 11. O(ca) in 3 can't be false | 3 & 10 contradictories | |||
| Then | 13. O(ca) is both true & false | 3 & 11; Impossible! | |||
| Option 2 Assume | 14. A(bc) is false | From Step 9 | |||
| Then | 15. A(bc) is both true & false | 2 & 14; Impossible! | |||
16. So, in assuming that the true premises imply a false conclusion, we have deduced by valid inferences contradictions in Steps 13 and 15. Therefore, O(ca) must be true. The original implication (Bokardo-3) is valid.
Nonstandard Syllogisms
A syllogism may fail to be in standard form in a number of ways. The first pair of examples below are syllogisms containing more than three but not unrelated terms. Also, their propositions are not in the proper order: major premise, minor premise, then conclusion. The second set of examples discusses syllogisms with a suppressed premise or conclusion (enthymemes). Last, a third type of nonstandard syllogism, sorites, is described.
Syllogisms Containing More Than 3 Terms
1st Argument:
| All inexpensive things are poorly constructed. |
| All German cars are expensive. |
| \ No poorly constructed things are German cars. |
The terms are inexpensive things, poorly constructed (things), German cars, and expensive (things).
2nd Argument:
| Some of the stolen books are not replaceable. |
| No irreplaceable things are deductible. |
| \ Some of the stolen books are non-deductible. |
The terms are stolen books, replaceable (books), irreplaceable things, deductible (items), and non-deductible (items).
Both arguments, above, have more than three terms each. So, the first task is to reduce the number of terms to three, if possible, making certain that each term is used in the same sense. This can be accomplished quite easily by obverting the second premise of the first argument and the first premise and the conclusion of the second argument.
Phase 1, First Example:
| All inexpensive things are poorly constructed. |
| No German cars are inexpensive. (by obversion) |
| \ No poorly constructed things are German cars. |
The terms have been reduced to three, each used in the same sense.
Phase 1, Second Example:
| Some of the stolen books are irreplaceable. (by obversion) |
| No irreplaceable things are deductible. |
| \ Some of the stolen books are not deductible. (by obversion) |
Again, the terms have been reduced to three univocal terms.
Now change the order of the premises in each argument.
Phase 2, First Example:
| Major | No German cars are inexpensive. |
| Minor | All inexpensive things are poorly constructed. |
| Conclusion | \ No poorly constructed things are German cars. |
INVALID, EAE-4, Rule #2 (The minor term, poorly-constructed-things, is undistributed in the premise but distributed in the conclusion.)
Phase 2, Second Example:
| Major | No irreplaceable things are deductible. |
| Minor | Some of the stolen books are irreplaceable. |
| Conclusion | \ Some of the stolen books are not deductible. |
VALID, EIO-1, Ferio-1. The tests of Five Rules are met in this example.
Enthymeme
An otherwise perfectly valid categorical syllogism may appear not to be so when one of its propositions is suppressed or understood but not explicitly stated. Such an argument is known as an enthymeme. The first enthymeme has a suppressed major premise, the second, a suppressed minor premise, and the third, a suppressed conclusion.
Suppressed Major Premise
Some NFL quarterbacks are good passers because some NFL quarterbacks have strong throwing arms.
Identify the conclusion first, then classify the premise as either the major or minor. In this case, the premise is the minor premise, since it contains the minor term.
| Missing Major | All persons with strong throwing arms are good passers. A(ba) |
| Minor | Some NFL quarterbacks have strong throwing arms. I(cb) |
| Conclusion | \ Some NFL quarterbacks are good passers. I(ca) |
Complete Syllogism: A(ba) I(cb) < I(ca). Valid: AII-1, Darii.
Suppressed Minor Premise
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No one in his right mind claims infallibility, for only perfect persons can claim infallibility.
| Major | All persons claiming infallibility are perfect persons. A(ab) |
| Missing Minor | No person in his right mind claims to be a perfect person. E(cb) |
| Conclusion | \ No person in his right mind claims infallibility. E(ca) |
Complete Syllogism: A(ab) E(cb) < E(ca). Valid: AEE-2, Camestres.
Suppressed Conclusion
No fair-minded person is capricious and some capricious people are irresponsible.
| Major | No fair-minded person is capricious. E(ab) |
| Minor | Some capricious people are irresponsible. I(bc) |
| Missing Conclusion | \ Some irresponsible people are not fair-minded. O(ca) |
Complete Syllogism: E(ab) I(bc) < O(ca). Valid: EIO-4, Fresison.
Sorites
Nonstandard categorical syllogisms may contain more than the required three forms. A sorites consists of a series of propositions in which the predicate of each is the subject of the next. The conclusion consists of the first subject and the last predicate. The chain of propositions is arranged in pairs of premises to make explicit the suppressed conclusion, thereby revealing the syllogism. The validity of the entire chain will depend on the validity of each syllogism in the chain. In this example, a = atheists; t = theologians; n = nihilists; s = scholars; and u = unreasonable (people). What can be concluded, given the following four propositions?
| i | All atheists are nihilists. | A(an) |
| ii | All nihilists are misologists. | A(nm) |
| iii | All misologists are unreasonable. | A(mu) |
| iv | All unreasonable ones are fools. | A(uf) |
One interpretation takes "nihilists" in the first two propositions as the middle term and rearranging the premises yields the first syllogism.
| Major | (ii) | All nihilists are misologists. | A(nm) |
| Minor | (i) | All atheists are nihilists. | A(an) |
| 1st Conclusion | \ All atheists are misologists. | A(am) (made explicit) |
Using the 1st Conclusion as a premise in conjunction with the third proposition and rearranging the premises yields the second syllogism.
| Major | (iii) | All misologists are unreasonable. | A(mu) |
| 1st Conclusion (Minor) | All atheists are misologists. | A(am) | |
| 2nd Conclusion | \ All atheists are unreasonable. | A(au) (made explicit) |
Using the 2nd Conclusion as a premise in conjunction with the fourth proposition and rearranging the premises yields the third syllogism.
| Major | (iv) | All unreasonable ones are fools. | A(uf) |
| 2nd Conclusion (Minor) | All atheists are unreasonable. | A(au) | |
| 3rd Conclusion | \ All atheists are fools. | A(af) (made explicit) |
As stated earlier, for a sorites to be valid each syllogism forming a part of the sorites must be valid; otherwise the sorites is invalid. Each syllogism above is an instance of AAA-1, Barbara. Therefore, the sorites as a whole is valid.
In evaluating a sorites, keep in mind these requirements:
| 1 | If a conclusion is negative, then one and only one of the premises must be negative. |
| 2 | If the conclusion is affirmative, all of the propositions must be affirmative. |
| 3 | If the conclusion is universal, all of the premises must be universal. |
| 4 | A particular conclusion calls for not more than one particular premise. |
Premise and Conclusion Indicator Words
You may have noticed that some of the arguments in this Study included such phrases as "because," "for," "so," etc. These words are known as indicator words or phrases. They introduce or otherwise indicate the presence of a premise or premises and a conclusion. Thus, the two lists of indicator words that follow.
| Premise Indicators | Conclusion Indicators |
| ... and ... | so |
| ... but ... | thus |
| since ... | hence |
| because ... | therefore |
| however .. | consequently |
| assuming that ... | accordingly |
| inasmuch as ... | it follows that |
| nevertheless ... | which implies that |
| this is why ... | which means that |
| implied by ... | one can conclude that |
Summation
Mediated inferences, that is, syllogisms, their elements, the arrangement of their forms and terms to determine their moods and figures have all been the subject matter of this Study. Next, the Five Rules for evaluating syllogistic frames as either valid or invalid were described. The Method of Deduction proved seven of the twenty four valid frames, using two axioms and two deductive method rules. Of more historical than practical interest are the names of the valid frames. The significance of lower case letters in some of the frame names was described. The RAA proof was illustrated in detail. Indicator words provide means for identifying premises and conclusions in arguments. Last, nonstandard syllogisms were described and methods for evaluating them were introduced. Of these, perhaps the most important is the enthymeme, since much of contemporary argumentation consists of enthymematic reasoning.
Of course, there is more. The use of diagrams for showing the validity of syllogisms is left for advanced study. Other aspects of syllogistic reasoning have been reserved for the last two studies.
Review
All of the syllogisms below are invalid. Each invalid argument illustrates the violation of one of the five rules for determining the validity of a syllogism. What rule is violated in each case? Does each example violate one and only one rule? Understanding the particular rule violated should suggest corrective strategies to convert an invalid syllogism into a valid one.
| 1 All hedonists are irrational. All irrationalists are misologists. \ Some misologists are not hedonists. ANSWER:______________ | 2 All men are intelligent. All men are bipeds. \ All bipeds are intelligent ANSWER:______________ |
| 3 Some fruit is not sweet. All pears are sweet. \ Some pears are fruit.ANSWER:______________ | 4 No dictators are benevolent. Some kings are not dictators. \ Some kings are not benevolent. ANSWER:______________ |
| 5 All men have two legs. All apes have two legs. \ All apes are men.ANSWER:______________ | |
Formal Fallacies
A fallacy is a mistake in reasoning. A formal fallacy is a mistake in the form of the argument itself; it is an invalid argument. There are two formal fallacies sometimes mistaken for Modus Ponens and Modus Tollens. These are known as (1) the fallacy of denying the antecedent; and (2) the fallacy of affirming or asserting the consequent.
Fallacy of Denying the Antecedent
An implication as premise, and denial of its antecedent as a another premise do not imply a conclusion. To claim that such premises do imply a conclusion is a fallacy. Here's it formulation and an example.
a implies b and a is false; therefore, b is false
| If Jane is a good speller, then she can spell "syllogism." |
| Jane is not a good speller. |
| Therefore, Jane cannot spell "syllogism." |
Even though Jane is a poor speller, she may, nevertheless, happen to know how to spell "syllogism." For example, because Jane's father was a logic teacher,syllogism was the third word she learned as a toddler after "Mama" and "Daddy." Unfortunately for Jane, while she mastered logic at an early age, her trainingin phonics was deficient. So, she had trouble with spelling for the rest of her life. And ... well, we'll leave off Jane's life story there. But you see the point.
Fallacy of Affirming the Consequent
An implication as premise, and affirming the consequent as another premise do not imply the antecedent of the implication as a conclusion. It can be symbolized in the following way, followed by an example of the fallacious argument.
a implies b and b is true; therefore, a is true.
| If he is honest, he will not lie. |
| He will not lie. |
| Therefore, he is honest. |
Note that the consequent is negative in quality, just as the second premise. The variable letter b stands for the proposition "he did not lie." This argument could have this symbolic formulation: a implies not-b and not-b is true, therefore, a is true. It remains an instance of the Fallacy of Affirming the Consequent.
Transitive Hypothetical Syllogism
Some logic books cite this argument form by as Hypothetical Syllogism. To avoid the confusion of assigning the same name to different argument forms, this argument form's name includes "transitive" in its name. As with all of these argument forms, the order of the premises is of no consequence. Its form is:
a implies b and b implies c; therefore, a implies c
To illustrate this argument form and, at the same time, show that valid arguments can be unsound, having one or more false propositions, consider the following.
| If students cheat on exams, this means that the exams are too difficult. |
| If the exams are too difficult, the instructor should be disqualified. |
| Therefore, if students cheat on exams, the instructor should be disqualified. |
Really? Perhaps the students who cheat should be disqualified? In any event, though unsound, it is valid, for the premises, if true, necessarily imply the conclusion. Of course, we should always use valid arguments. Better yet, make sure your arguments are sound which means that an argument is not only valid but contains nothing but true propositions as premises and conclusion.
Disjunctive Hypothetical Syllogism
A disjunction is, of course, an either ..., or ... statement. As we all know, "or" has more than one sense. It has three: (1) "Heaven or Hell" as the title of a sermon means one to the exclusion of the other but not both. This is called the exclusive sense of "or." (2) In "she studied logic or she is a home-schooler," we illustrate the inclusive sense of "or." Of course, Jane may have both studied logic and been home-educated. The inclusive sense of "or" means at least one, not requiring, but permitting both. (3) In "the Gospel or Good News" one has the synonymous sense of "or" in mind.
The "or" of Disjunctive Hypothetical Syllogism is the inclusive sense; it serves well for all logical purposes. The argument form is:
either a or b, and not-a ; therefore, b
Once again, substituting propositions in consistent fashion for the variables yields this argument:
| Either he is afraid to tell the truth or he enjoys telling falsehoods (perhaps both?). |
| He is not afraid to tell the truth. |
| Therefore, he enjoys telling the falsehoods. |
In a sentence: An inclusive sense disjunction and a denial of one of its disjuncts implies the other disjunct as a conclusion. Other variations are:
either a or b and not-b; therefore, a
either a' or b and a; therefore, b
(The exclusive sense of "or," as a special case of the inclusive sense, means "either a or b, but not both a and b.")
The Dilemma
The argument form known as the dilemma is perhaps the most complex of the five argument forms. It consists of two conditionals and a disjunction as premises and a disjunction as conclusion. The disjunctions as premise and conclusion must conform to the formats outlined below. There are two varieties of this argument form: the constructive variety and the destructive variety, shown below in that order.
Using " < " for implies and "+" for the inclusive sense of "or," the constructive dilemma can be expressed as an Implication of implications and disjunctions. The "< " is used below to separate the antecedent-premises from the consequent-conclusion of the Implication. The parentheses indicate the individual conjuncts of the premise set. The premise set is a conjunction of three premises.
The destructive variety closely resembles the constructive except for the disjunctions. Of course, the letters represent propositions, and the " ' " or prime when attached to a letter, is a denial of the proposition represented by that letter. Using the notation we introduced for the constructive variety, we have this formulation of the destructive dilemma.
Some examples follow.
| If you do nothing, you will be considered an accomplice by your silence. [and] |
| If you resist, then you will be labeled a trouble-maker. [and] |
| Now, either you do nothing, or you resist. |
| Thus, either you will be considered an accomplice or trouble-maker. |
This is a no-win situation, or is it? In a paragraph or two, we cite two possible mistakes associated with dilemmas. But first, an illustration now of the destructive variety.
| If I lie, I will be considered an accomplice, and if I protest, I'm labeled a trouble-maker. |
| Either I'm not an accomplice or I'm not a trouble-maker. |
| Thus, either I am not a liar or I am not a protestor. |
This appears to be a no-lose situation, or is it? The answer to both questions depends on several important considerations.
| First | Is each of the first two premises a valid inference? If one or the other of the "if ... then" premises is an invalid inference, the dilemma fails as a valid argument form. |
| Second | The third premise, the inclusive sense disjunction, must be a complete disjunction. That is to say, there must not be a third alternative or possibility. If the disjunction is incomplete, the dilemma fails as a valid argument form. |
| Third | As already noted, the various letters of the dilemma are propositional variables. The identical proposition must be substituted for the letter a, for example, wherever it appears in the dilemma, and similarly for each of the variables. Otherwise, the dilemma doesn't so much fail, as never gets started. |
Obviously, in a valid implication (argument), the conclusion should contain no proposition not in the premises. If any proposition in the conclusion is missing in the premises, the argument is invalid. Apart from this, most often, dilemmas fail in that one or more of the conditionals is not a valid inference. Occasionally, this defect is coupled with a faulty disjunction as a premise. Some disjunctions are considered incomplete disjunctions on rejection of the theology underlying the disjunction. For example, the disjunction: "You are either saved and going to heaven, or You are lost and going to hell" is a complete disjunction for Calvinists, but not so considered by non-believers. Careful attention to the requirements listed should be useful for evaluating the soundness of dilemmas.
We turn now to three important types of logically equivalent expressions: (1) the relation between conjunction and disjunction; (2) implication and conjunction; and (3) implication and disjunction.
Conjunction and Disjunction
The rule that covers the relation between conjunction and disjunction is simply this: The denial of a conjunction is equivalent to (equal to) a disjunction of the denials of the propositions. And, the denial of a disjunction is equivalent to a conjunction of the denials of the propositions. The relation is symmetrical; so, the order is of no consequence. As before, " + " is inclusive disjunction; "(ab)" or "(ab) (cd)" are conjunctions; " ' " stands for the denial of the letter to which it is attached; and " = " stands for "is equivalent to" or " is equal to. "
The denial of a conjunction:
(ab)' = (a' + b')
In a sentence, the denial the conjunction, a and b, is the disjunction of the two separately denied, not-a or not-b. In turn, the disjunction above is equivalent to a denial of the conjunction.
The denial of a disjunction:
(a + b)' = (a' b')
The denial of a disjunction is the conjunction of the two propositions separately denied. And, the conjunction above is equal to the denial of the disjunction. There are more complicated formulations of the relation between conjunction and disjunction, but the basic principle remains unaltered. For example, consider these slightly more complicated versions.
(a' + b')' = (a b)
(a'b')' = (a + b)
Double negation yields the original propositional variable on the right hand side of the expressions above. (In other words, the expression "(a ' ) ' " is logically equivalent to the expression "(a).")
Implication and Conjunction
The rule is: An implication is equivalent to a denial of a conjunction of the antecedent and the denial of the consequent. The formula is as follows:
(a < b) = (ab' )'
It reads: The implication, "if a then b" is equal to "it is not the case that a and not-b." The following propositions mean the same thing:
"If you are a good student, then you will master logic."
"It is NOT the case that you are a good student and you will not master logic."
Implication and Disjunction
The rule is: An implication is equivalent to a disjunction consisting of the denial of the antecedent as one disjunct and the consequent of the implication as the other disjunct. It has this form:
(a < b) = (a' + b)
It reads: The implication, "if a then b" is equal to "either not-a or b." The following two propositions mean the same thing.
"If you are a good student, then you will master logic."
"Either you are not a good student, or you will master logic."
These relations between conjunction and disjunction, implication and conjunction, and implication and disjunction have names which at this stage constitute extra baggage for the student. The important lesson here is to realize that conjunction, disjunction, and implication are interdefinable. The fact that a conjunction can be expressed as a disjunction, an implication as a conjunction, and an implication as a disjunction may come as a surprise to the beginning student. Perhaps not, but then the student should at least have come to a deeper appreciation of the power of symbols for the expression of complex meanings.
How many lines of English do you think are necessary to express the relations in this form?
(a < b) = (a' + b) = (a b' )'
We are now in a position to express in more definitive language the relations between the laws of logic mentioned in the Introduction. There, it was noted that the Law of Contradiction encompasses the other two. We could have said "contained the other two." The ambiguity of the verbs "to encompass" or "to contain" is eliminated by this:
(a a' )' = (a' + a) = (a < a)
Given the logical equivalence of the three -- to deny one is to deny all; to uphold one is to uphold all.
Summation
In this Study, the aim has been to introduce and provide some examples of additional argument forms. The five additional argument forms are standard versions. We did not illustrate the nonstandard varieties. For example, Hypothetical Disjunctive Syllogism can be expressed like this: (a + b') (b) É (a). If this is confusing, reread the definition. You will find that it does not eliminate nonstandard versions of the standard argument form. This Study closes with important truths about the relations between conjunction and disjunction, and each of these with implication. These relations form the basis for showing that the Law of Contradiction "contains" the other two laws; indeed, each contains the others, but the Law of Contradiction is supreme. These laws together with the other argument forms will serve as foundation for the truth table analyses of arguments in the final Study.
Review
| 1. What are the basic differences and similarities between the argument forms modus ponens and modus tollens? |
| 2. Construct an ordinary language argument illustrating the fallacy of affirming the consequent in which the premises are obviously true and the conclusion obviously false.(For example: "If the President resigns for misconduct, then he will not be impeached. The President will not be impeached. Therefore, the President will resign for misconduct.) |
| 3. Construct an example of the fallacy of denying the antecedent in which the premises are obviously true and the conclusion obviously false. |
| 4. What is the disjunctive form and the conjunctive form of this implication: (a' + b) < c'? (It reads: IF either not-a or b, THEN not-c. Recall that an implication can be expressed as a disjunction. An implication can be expressed as a conjunction, also.) |
| 5. Evaluate this form according to the first two possible mistakes that were listed about dilemmas. "If I vote the liberal ticket, I shall encourage socialism. If I vote conservative, I encourage unemployment. But I must vote either liberal or conservative. So I am forced to encourage socialism or unemployment." |
Truth Tables
Truth tables may be defined as schemata for the analysis of forms and relations among them. Their use requires additional information about logical connectives and the construction of truth tables.
Logical Connectives
The logical connectives or logical constants for "and," for "or," for "implies," and for "not"" are used to connect propositional variables to form compounds of propositional forms. The letters p, q, r, etc. stand for any propositions whatsoever. The symbol usually assigned for "not" is the ~ (tilde). We have used the prime symbol, " ' " attached to the variable letter (or logical expression) that is negated. Conjunction is sometimes assigned a centered dot ( · ) or the ampersand (&). Our notation simply places one propositional variable next to another to show conjunction. The customary symbol for a disjunction is the wedge (v), but we use the + sign to join disjuncts. Implication is most often indicated by a horseshoe symbol É but we shall continue to use the "< " sign to stand for "if ... then" relation. These logical constants are said to be truth-functional connectives because by means of their truth table definitions, we calculate the truth or falsity of compound propositional expressions. We first define negation, then show truth table definitions of conjunction, disjunction, and implication.
Negation
| p | p' |
| T | F |
| F | T |
A proposition is either true or false. When a proposition is true, its denial is false; when a proposition is false, its denial is true. Since there are only two possibilities, there are only two rows in this truth table. The next three truth table definitions consist of four rows each because in each case there are four possibilities given two values (True and False) and two propositions.
Conjunction
| p | q | (p q) |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
A conjunction is true if and only if both conjuncts are true; or, if a conjunction consists of more than two, then it is true only if each conjunct is true. This condition is met only in the first row of the truth table; the other three rows fail to meet the condition for a true conjunction.
Disjunction
An inclusive disjunction is false in one and only one set of circumstances where both disjuncts are false. If the disjunction consists of more than two disjuncts, then a disjunction is false only when each and every one of the disjuncts is false. Otherwise, the disjunction is true, as the following truth table shows.
| p | q | (p + q) |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
The fourth row depicts the meaning of a false disjunction. The first three rows, where one or the other or both disjuncts are true, complete the full meaning of inclusive disjunction.
Implication
One combination of values is fatal to an implication and that is the condition displayed in the second row of the next truth table. There the antecedent is true and the consequent false. The other rows show that an implication is true. Thus, an implication is false if and only if the its antecedent is true and its consequent is false.
| p | q | (p < q) |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Ordinarily the combination of truth values in the first two rows causes no perplexity. The truth value combinations of the last two rows, however, sometimes raise mild objections. Can a false statement imply a true one? Yes it can. Indeed, a false statement can imply another false statement. Try this. Using the expression "if x is less than 2, thenx is less than 4" substitute values for "x" to obtain the truth values of rows 3 and 4 of the truth table. In both cases, the implication "p < q" is true.
Row 3: let p = "3 is less than 2," (false antecedent) and q = "3 is less than 4" (true consequent).
Row 4: let p = "4 is less than 2," (false antecedent) and q = "4 is less than 4" (false consequent).
Truth Table Construction
The instructions for a truth table construction are easy to follow. Symbolize the argument using letter variables to stand for particular propositions. The conjunction of the premises constitute the antecedent of an implication, and the conclusion becomes the consequent. Now, count the number of distinct propositional variables. Note, "p" and "p' " are not "distinct variables; "p" and "q" are distinct, in our sense of the word. If an implication contains two distinct propositional variables, then the number of rows is four. A single proposition can be true or false, two truth-values, but a compound proposition of two simple propositions has four possibilities: both can be true; the first true and the second false; the first false and the second true; and both can be false. If an expression contains three distinct propositional variables, then the number of rows is eight. The formula for calculating the number of rows is R = 2n, where R stands for rows, and n stands for the number of distinct variables. For 3 distinct propositional variables, the number of rows is 23, or 2 raised to the third power: 2 x 2 x 2 = 8 rows.
The arrangement of the two values, true and false, is governed by two practical concerns: (1) Does the truth table contain all possible combinations of true and false? (2) Does the arrangement of truth values, "T" and "F," depict, in consistent and identical fashion, a truth table that all can use without confusion. Follow these steps:
| STEP 1 | Count the number of distinct variables in the expression to be analyzed. Suppose you count 3 distinct variables. |
| STEP 2 | Determine the number of rows required for the truth table using the formula R = 2n. R = 23 = 8 rows. |
| STEP 3 | Divide the first column in half and place T's in the first half, F's in the second half. See the first column of the truth table below. |
| STEP 4 | Divide the second column into two's place alternate 2 T's and 2 F's in the rows as shown in the second column of the truth table |
| STEP 5 | The third column has alternate T's and F's for the full number of rows in the truth table, as shown in the third column of the truth table. |
| p | q | r |
| T | T | T |
| T | T | F |
| T | F | T |
| T | F | F |
| F | T | T |
| F | T | F |
| F | F | T |
| F | F | F |
| i | ii | iii |
| STEP 6 | Once you have entered all possible combinations of truth-values into your truth table rows and columns, then deal column by column and row by row with the expression to be analyzed. Begin with the simplest parts of the expression working toward the major logical connective. If the expression to be analyzed is a disjunction, the major logical connective is the " + " sign; if an implication, the " < " sign. Assign T's and F's under each part, according to the truth table definitions of contradiction, conjunction, disjunction, and implication. |
As a simple example, let us analyze this expression: (p q) + r. It reads, "either p and q, or r." It is a disjunction, and the first disjunct is a conjunction. (The order of the variables p, q, and r do not matter.)
| ROWS | p | q | r | [(p q) | + | r ] |
| 1 | T | T | T | T | T | T |
| 2 | T | T | F | T | T | F |
| 3 | T | F | T | F | T | T |
| 4 | T | F | F | F | T | F |
| 5 | F | T | T | F | T | T |
| 6 | F | T | F | F | T | F |
| 7 | F | F | T | F | T | T |
| 8 | F | F | F | F | F | F |
| i | ii | iii | (2) | (3) | (1) |
The numbers in parentheses indicate the order of operations. Column (1), the simplest, is identical to column (iii); its the identical variable. Next, enter the T's and F's for the conjunction (p q) in column (2). Last, enter the values for the disjunction in column (3). When is a disjunction false? When all of its disjuncts are false, and that obtains in Row 8.
| STEP 7 | Answer questions by inspection of the rows of your truth table with T's and F's. For example, does the truth of (p q) depend on the truth of (r)? You will have to examine the truth table to realize that it does not matter. (Hint: Find rows where (p q) is true; compare with r-values.) |
Suppose we wanted to display the relations between (p < q), (pq' )', and (p' + q). The first reads: if p then q; the second: it is not the case that p and not-q; the third: either not-p or q. There are 2 distinct variables; using R = 2n, the number of rows is 4. The first column will contain 2 T's and 2 F's. The second column will consist of T's and F's, for 4 rows as shown in the truth table below. In every case, the use of parentheses or brackets are intended as punctuation devices to indicate accurately the sense of the expression.
| Rows | p | q | (p < q) | (p q' )' | (p' + q) |
| 1 | T | T | T | T | T |
| 2 | T | F | F | F | F |
| 3 | F | T | T | T | T |
| 4 | F | F | T | T | T |
| i | ii | iii | iv | v |
The truth values (T's and F's), beyond the first two columns were assigned according to the definitions of the logical connectives. What can we conclude? If we had started with English sentences, the inferences would prove to be more interesting. But for now, examine column iii through v. The expressions have identical truth values in these columns which means that they are logically equivalent. If one is true, the others are true also; and if any one is false, the others are false too. This truth table shows the interdefinability of implication, conjunction, and disjunction described in the previous Study.
Symbolizing Implications
Students sometimes encounter difficulty in symbolizing more complicated expressions of implications. The following list contains some of the more common expressions of implications.
| p only if q | p < q |
| p thus q | p < q |
| p therefore q | p < q |
| p hence q | p < q |
| p if q | q < p |
| p since q | q < p |
| p because q | q < p |
| p for q | q < p |
| p when q | q < p |
Other ways of expressing implications may not have the "If ..., then" formulation. We have used the word "implies" in "p implies q;" also, "q is implied by p." Another example: "Saving faith means belief in an understood proposition" is an implication made plain as, "If you possess saving faith, then you possess belief in the understood propositions of the Gospel." The key word is the verb, means. Thus, x means y is a formula for an implication: if x, then y. The "if" introduces the antecedent of an implication, but "only if" introduces the consequent as in: "You are saved, only if you believe the Good News of the Bible." Careful attention to the sense of a proposition is required for accurately symbolizing a proposition.
Other Symbolizing Difficulties
Difficulties in symbolizing conjunctions may occur when the word "and" is absent but implied. Other conjunction words used instead of "and" are: "but," "yet," "however," "although," "whereas," "nevertheless," and sometimes "plus," or only a comma or semicolon. Is there a difference between "not both p and q" and "both not-p and not-q?" Yes, there is! The first is a denial of a conjunction, (pq)'; the second is a conjunction of denials, (p' q'). To complicate matters, sometimes "and" is used but the proposition is not a conjunction as in "1 and 1 is equal to 2," or "Peter and Paul were contemporaries."
Symbolizing disjunctions proves difficult when it is not clear which sense of or is the intended sense. Using the phrase "and/or" distinguishes the inclusive sense from the others; the phrase "but not both" signals the exclusive sense. The trouble is that these phrases are often implied, not explicitly stated. Of course, "+" stands for the inclusive sense; we have no symbol for the exclusive sense having determined that the inclusive serves our purposes well. Nevertheless, suppose the exclusive sense is intended as in "Either you are regenerate or you are forever lost." One or the other, but not both. Symbolized, it is: (r + l) (rl)'. Another minor difficulty: "Neither p, nor q" is not (p' + q'); the correct symbolization is (p + q)', a denial of the disjunction. A less difficult case is the use of "unless" in "Unless you study logic, you will believe propaganda." This proposition means "Either you study logic, or you'll believe propaganda." Again, careful attention is required to achieve the correct translation of the intended sense of a proposition.
A Demonstration
This demonstration will serve as a summary of truth table construction and application. The purpose is not only to show the advantages of symbolizing propositions, but to indicate how truth table methods may assist in understanding relations among propositions.
Either Planck was a successful physicist or Einstein was in some respects a failure. If Einstein was in some respects a failure, then Hawking is a failure. Either Planck was not a successful physicist or Hawking is not a failure. If Einstein was a failure, then Hawking is not.
Let p stand for "Planck was a successful physicist." Let e stand for "Einstein was a failure in some respects." Let h stand for "Hawking is a failure. Each of the variables, in this case, stands for a simple, that is to say, single proposition. Symbolizing the compound propositions, we have:
| p + e | Either Planck was a successful physicist or Einstein was a failure. |
| e' < h | If Einstein was not a failure, Hawking is a failure. |
| p' + h' | Either Planck was not a successful physicist or Hawking is not a failure. |
| e < h' | If Einstein was a failure, Hawking is not a failure. |
| Rows | p | e | h | (p + e) | (e' < h) | (p' + h' ) | (e < h') |
| 1 | T | T | T | T | T | F | F |
| 2 | T | T | F | T | T | T | T* |
| 3 | T | F | T | T | T | F | T |
| 4 | T | F | F | T | F | T | T |
| 5 | F | T | T | T | T | T | F |
| 6 | F | T | F | T | T | T | T* |
| 7 | F | F | T | F | T | T | T |
| 8 | F | F | F | F | F | T | T |
| i | ii | iii | iv | v | vi | vii |
*Rows 2 and 6 are the only ones that have values of true for columns iv-vii, inclusive. Notice the contradictory values for p column i, rows 2 and 6; nothing can be said about Planck. But e is true in rows 2 and 6, and h is false in the same rows. So it is true that Einstein was a failure, but false that Hawking is a failure, according to this truth table analysis. The compound propositions are not offered in support of a position; so, no argument is involved and questions about validity do not apply.
Modus Ponens Revisited
Once an argument has been symbolized, the premises become a conjunction of premises. This conjunction forms the antecedent of an implication, and the conclusion of the argument becomes the consequent. We thus transform an argument into a truth-functional expression (an implication) suitable for truth table analysis.
Accordingly, Modus Ponens, expressed as an implication is: [(p < q) (p)] < (q). The major logical connective is the second "< ." The premise set, enclosed by brackets, is the antecedent; the conclusion is the consequent of the implication. Will a truth table analysis reveal that Modus Ponens is a valid argument form?
| Rows | Variables | Premise #1 | Premise #2 | Conclusion | ||
| p | q | (p < q) | p | < | q | |
| 1 | T | T | T | T | T | T |
| 2 | T | F | F | T | T | F |
| 3 | F | T | T | F | T | T |
| 4 | F | F | T | F | T | F |
| i | ii | iii | iv | v | vi | |
| (3) | (1) | (4) | (2) | |||
The values of columns (iv) and (vi) are identical to columns (i) and (ii), respectively, being the identical variables. (The numbers in parentheses indicate the order of operations from the first to the fourth.) If the argument is invalid, one would expect to find at least one row in which the premises are both true and the conclusion false. Inspection of Rows #2 and #4 shows "q" is false, but in both cases one of the premises is false. Only in Row #1 are the premises true; and, the conclusion is true also. In a valid argument form it is impossible for the premises to be true and the conclusion false because the form of the conclusion will be true every time the forms of the premises are true. Thus, Modus Ponens is shown to be valid by truth table methods.
The truth table reveals all T's under the major logical connective, "< " of column (v) and the last operation (4). Recall the truth table definition for implication. It is false where the antecedent is true and the consequent false. This condition does not obtain in any of the four rows. Check it out yourself! Under all possible assignments of T's and F's to the distinct variables of this valid argument form, the result reveals all T's under the major logical connective, an additional confirmation that Modus Ponens is a valid argument form.
Let us now analyze the fallacy associated with Modus Tollens.
Fallacy of Affirming the Consequent Revisited
Symbolizing the fallacy as an implication we have: [(p < q) (q)]< (p), again with the second "< " as the major logical connective. As before, the premises form a conjunction within the brackets and constitute the antecedent of the implication. "(p)," of course, is the conclusion or the consequent of the implication.
| Rows | Variables | Premise #1 | Premise #2 | Conclusion | ||
| p | q | (p < q) | q | < | p | |
| 1 | T | T | T | T | T | T |
| 2 | T | F | F | F | T | T |
| 3 | F | T | T | T | F | F |
| 4 | F | F | T | F | T | F |
| i | ii | iii | iv | v | vi | |
| (3) | (1) | (4) | (2) | |||
Columns (ii) and (iv) are identical; columns (i) and (vi) are also identical. (The numbers in parentheses indicate the order of the steps with "(4)" being the last step.) Again, if the argument is invalid, one would expect to find at least one row in which both of the premises are true and the conclusion false. Inspection of Row #3 shows that both of the premises are true and the conclusion false. Notice also that at Row #3, column (v) you have a "F" under the major logical connective because there you have a case of true antecedent and false consequent. Therefore, the argument form is invalid, as we knew it to be.
Don't be confused by the truth values of Row #1: the premises are true and the conclusion is true also. This only indicates the possibility of an invalid argument with true propositions. In a valid argument form, true premises imply a true conclusion --always. "Always" means in each and every row of a truth table. To repeat. The only "F" in column (v), row #3, confirms that we are in the presence of an implication with true premises and a false conclusion: an invalid inference. Thus, the Fallacy of Affirming the Consequent is shown to be an invalid argument form by truth table methods. A similar outcome of invalid would obtain with the formal fallacy associated with Modus Tollens, the Fallacy of Denying the Antecedent. One row in the truth table will reveal true premises and a false conclusion.
Summation
Truth tables are schemata for analyzing the relations between different propositions, simple and compound. In this Study, the definitions of the logical connectives for conjunction, disjunction, implication, and negation were described using truth table methods. Thereafter, we set down instructions for constructing truth tables. A demonstration for the implementation of truth table methods served to illustrate what can be inferred by these techniques. Finally, to further illustrate the usefulness of truth table methods, two argument forms, one valid and one invalid, were subjected to truth table analyses. The results demonstrated that with a valid argument form, expressed as an implication, no single row shows true premises and a false conclusion. On the other hand, the invalid argument form, expressed as an implication, revealed a row with true premises and a false conclusion. As a heuristic method, truth table analyses not only confirm validity and invalidity of argument forms, but provide a practical method for illustrating both.
Review
Either the cat is meowing or the baby is crying. If the baby is not crying, then the wind is blowing. Either the cat is not meowing or the wind is not blowing. If the baby is crying, then the wind is not blowing. (Modified from Gordon H. Clark, Logic, p. 111)
The propositional variables are "c," "b," and "w." c = cat is meowing; b = baby is crying; w = wind is blowing.
| (c + b) | The cat is meowing, or the baby is crying. |
| (b ' < w) | If the baby is not crying, then the wind is blowing. |
| (c' + w ') | The cat is not meowing, or the wind is not blowing. |
| (b < w ') | If the baby is crying, then the wind is not blowing. |
Is the cat meowing? Is the baby crying? Is the wind blowing?
Informal Fallacies
What is a fallacy?
A fallacy is a blunder in reasoning. It is "false" reasoning, that is to say, reasoning with illogical argument or misleading argument. Reasoning means drawing inferences or conclusions from known or assumed facts or premises. The premises and conclusions of arguments should qualify as propositions, i.e., the meanings of declarative sentences which possess the essential characteristic of being either true or false. Recall that an argument is defined as a series of connected declarative sentences (premises) in support of another statement (conclusion) or a position. A fallacy consists of invalid or unwarranted inference of a conclusion from premises some of which may not qualify as propositions. Commands, exhortations, or exclamations do not possess the quality of truth or falsity and must be reworded into propositions if they are to serve as either premises or conclusions.
How are fallacies classified?
Fallacies may be broadly classified as either formal or nonformal fallacies. Formal fallacies are invalid inferences of conclusions from premises, the invalidity being due to theform of the argument. Nonformal, known more as informal fallacy, as a category includes a multiplicity of mistakes in reasoning, some of which involve careless use of language.Informal fallacy can be thought of as counterfeit argument, i.e., a type of argument that may seem to be correct but which proves on examination not to be correct. Informal fallacies, unlike formal fallacies, are not fallacies of form. Extralogical or emotional appeals usually constitute one of the sources of persuasion. In other cases, informal fallacies are deceptive pieces of "bad" English or mistakes due to ambiguity or vagueness of a term or phrase, or an entire sentence. In any case, the pretense of logical relevance, we could say, is the source of fallacy.
Fallacies of Form
Fallacies of form render arguments invalid irrespective of the content of the argument or the truth or falsity of its propositions. If the form of an argument allows an inference of a false conclusion from all true premises, then we know the argument to be invalid, for a valid argument will never result in the deduction of false conclusion from true premises. Indeed, if it so happens that the conclusion of a particular argument is known to be false, and the argument is valid, then we know that one or more of the premises is false as well. (See Study 4 for fallacies of form and some of the variations.)
Informal Fallacies
In reasoning that "If X looks like a Z, walks like a Z, talks like a Z, and even reasons like a Z, well then, X must be a Z," one runs the risk of accepting a counterfeit for the real thing. Informal fallacies are counterfeit arguments in that they consist of premises and conclusions which are related, it is claimed, as conclusions drawn logically from true premises. But herein lies the error. While there is a connection between premises and conclusion in such arguments, the connection is a psychological one masquerading as a logical relation between premises and conclusion in a particular argument. The mistake is one of confusing an emotional appeal for a logical one. An emotional appeal seeks to persuade you to accept a conclusion solely on its appeal to feelings of attraction or aversion to a particular object or set of objects or events. Many commercials make use of attraction and/or aversion for things or circumstances to persuade one to avoid something by buying a product, or to attain an attractive status by using a product. When their effect is to cause you to confuse an emotional appeal for a logical one, we can say that they make use of counterfeit argument to achieve acceptance of a conclusion. One should not ignore the context in determining whether a piece of language is functioning as counterfeit argument.
Informal fallacy classifications abound. Perhaps the simplest consists of two categories of the most common types: (1) fallacies of ambiguity and vagueness; and (2) fallacies of irrelevant conclusion.
Fallacies of Relevance
Fallacies of irrelevant conclusion are those for which the premises are not relevant to the truth of the conclusion. With such, the label non sequitur, meaning literally that the conclusion does not follow from the premises is often used. With these fallacies the premises are incapable of establishing the conclusion because they are logically irrelevant to the conclusion. Some of the more common informal fallacies have been given Latin names which have become part of the English language.
| Argumentum ad hominem abusive (AH) | when irrelevancies of character, circumstances, the beliefs or prejudices of the person are used for rejection a position. |
| Argumentum ad baculum (AB) | when one appeals to force or the threat of force instead of reasons to cause acceptance of a conclusion. |
| Argumentum ad misericordiam (AM) | when one appeals to pity instead of sound reasoning to gain acceptance of a conclusion. |
| Argumentum ad populum (AP) | when one attempts to gain popular assent to a conclusion by arousing the feelings and enthusiasms of the multitude. |
| Argumentum ad verecundiam (AV) | when instead of sound argument one appeals to the feeling of respect people may have for the famous to win assent to a conclusion. |
| Argumentum ad ignorantiam (AI) | whenever it is argued that a proposition is true solely on the basis that it has not been proved false, or that it is false because it has not been proved true. |
| False Cause (FC) | when one infers that because one event follows another, the first event caused the second. |
| False Dilemma (FD) | when one calls for a conclusion based on the assumption that two and only two mutually exclusive alternatives are possible when in fact more than two are possible or the two are not mutually exclusive. |
| Accident (A) | when it is argued that what is true in general is true universally and without qualification. |
| Hasty Generalization (HG) | when in argument one considers only exceptional or too few cases and generalizes to a rule that fits them alone. |
| Circular Reasoning (CR) | when one assumes as a premise for an argument the very conclusion that is intended to be proved. |
| Complex Question (CQ) | when in argument one treats a plurality of questions as if it were a simple one demanding a single answer. |
The following example has limited value. The intent is to describe communication with potential for counterfeit argument. While these statements are not explicit formulations of premises or conclusions, they are intended to illustrate possible instances of informal fallacies. Readers should use their imaginations to round out a context with the appropriate elements of oral expression. Imagine this setting: a young couple is discussing an accumulation of incidents that perhaps signals the end of their honeymoon.
| She | "You're just like your father--lazy and sloppy!" (AH) |
| He | "You wouldn't talk that way in front of my dad!" (AB) |
| She | "Maybe not, but I get no help from you; after 40 hours of hard work, I do all, I mean all the house cleaning and cooking while you watch TV?" (AM) |
| He | "All women who want both career and marriage seem to feel very happy about doing both! That's what all successful women say." (AP) |
| She | "Would Mr. Rogers, your hero, hold the opinions you have about marriages and careers?" (AV) |
| He | "Mr. Rogers has never argued that career and marriage are incompatible for women. Therefore, he must believe that they are compatible." (AI) |
| She | "Don't talk to me about Mr. Rogers. The last time you brought him into our discussion, we had a terrible fight!" (FC) |
| He | "I didn't bring Rogers into our discussion, you did! You either like him or you don't; I see you don't!" (FD) |
| She | "It's not a question of liking Rogers or not. It's the way you use what he says. One should never take what other people say out of context, as you do with Mr. Rogers." (A) |
| He | "This is the third time you have accused me of using what Mr. Rogers' says. When you run out of good arguments, you always say this about me." (HG) |
| She | "This is just like you! You conclude that you are innocent of any wrong-doing because you are innocent of any wrong-doing!" (CR) |
| He | "When will you stop hassling me?" (CQ) |
Now, that a counterfeit argument may fit into more than one category is not disputed. In such cases, perhaps more information about the context is required. However, even with exhaustive information about the context, one may find it difficult, if not impossible, to classify a particular counterfeit argument into one and only one category. Language serves multiple purposes. Reasoners sometimes operate with hidden agendas or motives. In cases where there exists the possibility of classifying a counterfeit argument into more than one known category, we can agree that it is one or the other, and possibly both. This should not preclude effort to classify counterfeit arguments based on the defining characteristics of known categories.
Fallacies of Ambiguity
Fallacies of ambiguity occur in formulations of argument that use ambiguous words or phrases. This is a smaller class of fallacies that include the fallacies of equivocation, amphiboly, composition, and division. Definitions follow.
| equivocation(EN) | when one confuses in argument the different meanings a single word or phrase may have. |
| amphibology(AY) | when in argument a statement's meaning is unclear because of the loose or awkward way in which the words are combined. |
| accent(AT) | when in argument words or phrases of a statement are emphasized or stressed producing quite different meanings from the original. |
| composition(CN) | when in argument one reasons fallaciously from the properties of the part or parts to the properties of the whole itself. |
| division (DN) | when in argument one reasons fallaciously that what is true of a whole must also be true of each of the whole's parts. |
It is not difficult to come up with examples of each of the above. Some uses are mere play on words as in "Good steaks are rare these days, so don't order yours well-done" where the equivocation turns on the meanings of "rare." More serious perhaps, is this one:"The end of a thing is its perfection; death is the end of life, Therefore, death is the perfection of life." (EN) (Examine the meanings of "end." )With this next example, the ambiguity lies in the structure or syntax of the sentence: "Leaking badly manned by a starved and thirsty crew one infirmity after another overtakes the little ship." (AY) Obviously, the phrase "manned . . . crew" needs to be relocated, perhaps at the end of the sentence to achieve clarity of meaning. The statement, "We should not speak ill of our friends," when quoted as: "We should not speak ill of OUR FRIENDS" (AT) stresses words not emphasized in the original conveying different meaning(s) from the original.Top of Page Composition and division are closely related. For example, if one argues that based on the properties of the elements of NaCl, the compound must be highly toxic (CN), one might suspect that the person knows little or nothing about chemistry or does not know that the compound is table salt, perhaps both. On the other hand, if someone argued that since salt possesses a class of salutary properties; therefore, the salt's elements (sodium and chloride) must be salutary, instruction in chemistry and perhaps more would seem to be necessary.
How can one avoid informal fallacies?
It was suggested above that one should not ignore the context in determining when to label something as informal fallacy. For example, when there is no attempt to disguise an emotional appeal as a logical appeal, there may be no point in accusing someone of using informal fallacy. Or, when all logical appeals have failed to convince a perverse arguer who knowingly and willfully disregards truth for error, what else remains but ad hominem (not the abusive variety) or even some ad baculum language? Silence? No doubt there are occasions where the use of threatening language may be the only alternative; for example, a police officer confronted by an armed felon. Ad hominem must not be confused with the abusive kind. Ad hominem is a form of argument that assumes the propositions of another for the sake of deducing contradictions or conclusions unacceptable to the person holding the position. There are special circumstances where it is quite appropriate to direct a complex question to another; for example. "Where did you hide the body?" or "Do you know the penalty for perjury?" are questions that are not conceived as instances of informal fallacy, once the groundwork has been set down for their use. The context is important.What can one do to avoid informal fallacy as counterfeit argument? It should be evident that telling someone that he or she is engaging in ad hominem abusivereasoning may not have the desired effect of causing the person to pause and reflect on his or her thinking. The person may not know what you mean by ad hominem, or informal fallacy. What then? Nevertheless, identification of the counterfeit argument by correct label is an important first step. A second step requires clear definitions of ambiguous or vague terms. A third step constructs a counterexample, analogous in every respect with the informal fallacy in which the premises are obviously true and the conclusion obviously false.
For example, suppose someone argues: "If President Kennedy was assassinated, then he is dead. Now, all acknowledge that he is indeed dead. Therefore, President Kennedy was assassinated."This argument is fallacious, guilty of the fallacy of affirming the consequent. Constructing a counterargument to make explicit the fallacious reasoning requires that (1) the propositions be of the same form as the original, (2) the format be identical to the original, and (3) the premises be true and conclusion be false. An appropriate response could be worded in this way:"You may just as well argue that if President Johnson was assassinated, then he is dead. President Johnson is dead. Therefore, President Johnson was assassinated."Obviously, the conclusion of the counterargument does not follow from the true premises. Similarly, the conclusion of the previous argument is not necessitated by its premises.
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