There are a number argument forms which may now be added to a student's basic equipment. This Study will describe five argument forms and a number of useful logical equivalences. Two of the five argument forms are best known by their Latin names: Modus Ponens and Modus Tollens. Two fallacies follow, since they are associated with Modus Ponens and Modus Tollens. Brief discussion of the remaining three argument forms will bring us to the last sections of this Study. There, we describe a number of relations between conjunction, disjunction, and implication.

Heretofore, our study of logical inference focused on the internal structures of the four standard form propositions. A systematic study of the forms of argument required that the propositions be analyzed into subject terms and predicate terms. For this reason, perhaps a more appropriate title for the subject of Traditional Logic is The Logic of Terms. In this study and the last, however, the variable-letters we shall use do not stand for the terms of a proposition. Instead of term-variables, we use propositional-variables, i.e., variable letters to represent propositions. It may not have escaped notice, that the Laws of Logic in the Introduction were stated by means of propositional variables. Thus the Law of Contradiction (or Non-Contradiction) can be stated as "Not both a and not-a." "If a, then a" expresses the Law of Identity. The Law of Excluded Middle can be expressed as "Either a or not-a." The letters, usually lower case, stand for entire propositions. Of course, once a letter is assigned to stand for a particular proposition in an argument, that same letter must be used for other instances of the same proposition. Remember, a proposition is the meaning of a declarative sentence.

Modus Ponens

Modus Ponens is also known as hypothetical syllogism, or the constructive hypothetical syllogism to distinguish it from another form of hypothetical syllogism described in the next section. Modus Ponens, as with all of the other argument forms except the dilemma, consists of two premises and a conclusion.

The structure of Modus Ponens consists of an implication (hypothetical) as a premise, and the antecedent of the implication as another premise from which the consequent of the implication follows as a conclusion. The order of the premises is of no consequence; although, the implication is usually placed first in the order of premises.

Substituting propositions for the premises and conclusions above reveals this argument:

If Judas stole the money, then he has a guilty conscience. He stole the money. Therefore, he has a guilty conscience.

In symbolic form: a < b and a, therefore, b.

Modus Tollens

Modus Tollens or destructive hypothetical syllogism has this form:

Restating the argument form, we have the following. With a hypothetical (or conditional) as a premise and the denial of its consequent as another premise, it is valid to infer the denial of the antecedent of the hypothetical as conclusion.

Substituting propositions for the premises and conclusion above yields this argument:

In symbolic form: a < b and not-b, therefore, not-a.

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

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.

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.

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:

 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.

(a < b) (c< d) (a + c) < (b + d)

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.

(a < b) (c < d) (b' + d' ) < (a' + c' )

Some examples follow.

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.

This appears to be a no-lose situation, or is it? The answer to both questions depends on several important considerations.

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