## study four: additional argument forms

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:

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 Preview 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 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. (Recall:

Substituting proposition for the variables, reveals this argument: If Judas stole the money, then he has a guilty conscience. He stole the money. Therefore, he has a guilty conscience.

*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 Preview 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. (Recall:

**.:.**stands for "therefore.")- a implies b
- a is true
**.:.**b is true

Substituting proposition for the variables, 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:

- a implies b
- b is false
**.:.**a is false

Substituting propositions for the variables yields this argument:

- If this beaker contains acid, then it will turn litmus paper red
- It does not turn litmus paper red.
- Therefore, this beaker does not contain acid.

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

**Two 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.

**1. 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.

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."

**2. 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.

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.

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 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
- b implies c
**.:.**a implies c

*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.*

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:

- "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."
- In "she studies 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.
- In "the Gospel or Good News" one has the synonymous sense of "or" in mind.

The argument form is:

- either a or b
- a is false
**.:.**b is true

*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.

(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 must conform to the forms for each type of dilemma. 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 dilemmas 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.

First, the Constructive Dilemma: (a < b) (c < d) (a + c) < (b + d).

- a implies b
- c implies d
- a or c
**.:.**b or d

The " ' " or prime when attached to a letter, is a denial of the proposition represented by that letter

- a implies b
- c implies d
- b' or d'
**.:.**a' or c'

- If you do nothing, you will be considered an accomplice by your silence.
- If you resist, then you will be labeled a trouble-maker.
- Now, either you do nothing, or you resist.
- Thus, either you will be considered an accomplice or trouble-maker.

*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.

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.

**1. 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 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) [Note: Double negation yields the original propositional variable.]

**2. 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."

**3. 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 Preview. 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.

**Summary**

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) < (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 next Study.

**Review**

- What are the basic differences and similarities between the argument forms modus ponens and modus tollens?
- 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.)
- Construct an example of the fallacy of denying the antecedent in which the premises are obviously true and the conclusion obviously false.
- 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.)
- 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."

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