01527--Write a note on the eight fallacies.

If we rely on experiences as evidence for our inferences and explanations, we must screen the ways in which we handle those that offer themselves so as to avoid making unwarranted presumptions about them and to avoid exploiting their ambiguity in various tempting ways. Otherwise, we may be guilty of fallacies of presumption and ambiguity in a variety of ways.  Examining them will help us avoid egregious errors in our thinking. Although thinking that commits such fallacies is common, it is always misleading.

I

If we rely on experiences (or anything else for that matter) as evidence for our inferences and explanations, we must screen the ways in which we handle them in order to avoid making unwarranted presumptions about them and equivocating over what they mean.

A. These cautions are also prerequisites for useful inference drawing.

B. When we presume, in one way or another, facts that are not in evidence, and when we play fast and loose with the meanings of our locutions, we are (once again) “not even in the ballpark, much less in the game.”

II

Here are descriptions and examples of eight forms that such bad reasoning can take.

A. Petitio principii

This fallacy amounts to inferring a conclusion from premises that are, in fact, indistinguishable from the conclusion itself. This fallacy is also called circular reasoning and question begging.  Example: I know that God exists because the Bible says so. And I know that everything in the Bible is true because it is God’s word and God wouldn’t lie. 

B. Complex question

This fallacy amounts to presuming without evidence that a certain state of affairs obtains, then shaping one’s inquiry in terms of that presumption.  Example: The classic is “Have you stopped beating your wife?” but it is equally clear in “Don’t you want to be a good boy and go to bed?” 
 

C. Equivocation

 This amounts to exploiting ambiguities of words. Some are simple plays on an everyday noun or adjective. Some exploit the subtleties of dispositional and episodic participles.   Examples: When mother asks, “Were you good at the party, Susie?” Susie responds, “Well, Miles said I was.” When father asks, “Are you smoking [these days], Fred?” Fred responds, “No I’m not [right this moment].”

D. Amphibole

 This fallacy amounts to exploiting ambiguities of syntax.  Example: A subway rider explains why he lit a cigar right next to the sign that said “No Smoking Allowed” by noting the two ways that sign can be read: “Smoking is forbidden” and “Refraining from smoking is permitted.” 

E. Accent

 This fallacy amounts to exploiting ambiguities of emphasis, including selective data use.  Example: Story positioning in the media, headline writing, and small print on a box of Broccoli Rice Surprise show just what accent can do. 

F. Category mistake

 This fallacy amounts to exploiting ambiguities of classification. The term comes from Gilbert Ryle’s The Concept of Mind.  Example: Not seeing the forest for all the trees, the parade for all the marchers, and the university for all the buildings and greens are all examples of confusing things and systems. “If we have minds, then where are they?” is a more telling case in point.
    

G. Composition and division

These fallacies amount to exploiting ambiguities between the properties of individuals and the properties of the sets that they compose.  Example: “Everyone in my gymnastics class is tiny. There’s no one there over 80 pounds. I can’t see why the instructor is complaining that the class is too big.”

H. False dilemma

 This fallacy amounts to exploiting ambiguities of complementarity.  Example: “Well, Ali was certainly no hero, so he must have been a coward.”

01525--Write a note on types of the fallacy of non sequitur.

If we rely on experiences as evidence for our inferences and explanations, we must sift through those that offer themselves so as to focus on ones that are relevant to the conclusions that we seek to draw. Inferences that rely on irrelevant “evidence” fail, being guilty of the fallacy of non sequitur.

I

There’s another kind of logic, usually called informal logic, commonly covered in books or courses about critical thinking.   Although it’s less technical and less demanding, it is no less important than the formal matters of logical inference.  Informal logic concerns the standards that need to be satisfied in order for us to get formal reasoning underway. 

II

If we rely on experiences (or anything else, for that matter) as evidence for our inferences and explanations, we must sift through those that offer themselves and focus on the ones that are relevant to our enterprise.   Evidential relevance is a prerequisite for useful inference drawing. Unless our purported evidence is relevant to the inferences we are trying to draw, we are not even in the ballpark, much less in the game.

III

Inferences that rely on irrelevant “evidence” commit non sequitur in one form or another. Here are descriptions and examples of seven forms that such bad reasoning can take: 

A. Ad vericundium. This fallacy amounts to an appeal to an improper authority (often due to some equivocation over the notion of authority itself).  Example: “Don’t question the President. He is the highest authority in the land.” 

B. Post hoc ergo propter hoc. This fallacy amounts to the inference that because one thing follows another in time, the later of the two must have been caused by its predecessor.   Example: Keeping Mark Twain’s story in mind, any wino with good teeth will serve.

C. Ad populum. This fallacy amounts to inferring that a point of view or opinion must be true on the grounds that it is widely held.  Example: “Fifty million Frenchmen can’t be wrong!” “Join the swing to Dodge!”

D. Ad baculum. This fallacy amounts to inferring that a point of view or opinion is true (or false) on the grounds that the one who holds it has (or lacks) the power to impose it on others.  Positive Example: “You do exactly what I said, young man, or else!”   Negative Example: “And exactly how many tanks does the pope have?” 

E. Ad misericordiam. This fallacy amounts to inferring that a point of view or opinion must be true on the grounds that those who hold it deserve (or are, at least, natural targets for) our sympathy.  Example (a defense lawyer at the sentencing hearing after a conviction for matricide): “Please be lenient with my client. He is, after all, a motherless child.”

F. Ad hominem. This fallacy amounts to inferring that a point of view or opinion must be true (or false) because of the character and/or the position of those who hold it.  Positive Example: Teresa must have been right about her visions coming directly from God. She was a good and virtuous person.  Negative Example: Bill Clinton’s improper liaisons prove the illegitimacy of his political policies. 

G. Accident and converse accident (hasty generalization). These fallacies amount to inferring that a member of a group has certain characteristics on the grounds that they are common to the members of the group or that all the members of a group must have certain characteristics on the grounds that some of its members do.  

Example: Any case of stereotyping will do for the accident fallacy: “White men can’t jump.” Any case of jumping to conclusions will do for the converse accident fallacy. Where, after all, do the stereotypes come from?

IV

We should not be misled by the fact that such fallacies are common, by the fact that some of them “sound OK” to careless ears, or by the fact that contrived examples of them can be amusing. They are always dangerous. They never settle an issue.

01524--Write a note on the ' Limits of Sense Experience'.

What content for thought does sense experience, by itself, provide? Do we sense the structure or pattern of events or just unconnected bits? As cases in support of the latter view, David Hume argued that we have no sensations of causation as such (with the result that all of our causal claims amount to interpretations of what we sense), that every generalization of particular experiences relies on the notion that nature is uniform (a notion that cannot be demonstrated without circularity), and that our accounts of experience involve the association of ideas according to principles that are habitual and not justificatory. In his view, experiencing is more than “picking berries off the bush” with our senses. Put my own way, it involves “construals” of our sensations at the very least, and construals are a contribution of the subject, not part of the object out there. Should this lead us in the Cartesian direction of demanding necessary truths as the basis for our ratiocination, or can we achieve something more reliable than “naïve realism” without taking that poison pill? 

If we set the benchmark for knowledge too high, we may conclude that we never have any knowledge at all.   Many philosophers now think that only analytic truths are a priori and that the search for synthetic a priori truths is doomed to fail.  If so, the Cartesian “quest for certainty” may, in fact, open the door to radical skepticism rather than the door to solid knowledge.

What does sense experience—taken by itself—actually provide as fodder for thought? A. Do we sense the structure or pattern of events or just “blooming buzzing confusion”?  What would sense experience be like if we could take it “neat”—that is, without the contributions of the following:  a. Memory takes us from sensation to sensation.   Patterns, whether discerned or imposed by us, are an essential part of the usable content of the experience that we have.   Associations have enormous implications, for example, in how we classify and react to people we meet.  Habits channel and shape what we discern. e. Presumptions, too, channel and shape what we discern.  Vast established mind sets (blicks, theoretical frameworks, paradigms) provide ways of thinking for whole cultures.  Absent such contributions, sense experience amounts to unconnected bits.

As cases in support of the view that experiencing does not amount just to “picking berries off the bush” with our senses, David Hume argued (in the 18th century) that:

1. We have no experience of causation as such; consequently, our notions that one thing is caused by another and that all explanations should be governed by some causal version of a “principle of sufficient reason” (the principle that nothing just happens) presume relationships that are not evident to the senses;

2. Every generalization of particular experiences relies on the notion that nature is uniform—a notion that cannot itself be demonstrated without circularity; and

3. Such generalizations involve the association of ideas according to principles that are habitual rather than justificatory.

a. We habitually “associate” ideas in terms of similarity, contrast, proximity of one sort or another, inertia, mimicry, and so on.

b. Such principles are not “justificatory” because there are examples of each and every one of them that are obviously false or misleading. As we have learned from the post hoc, ergo proctor hoc fallacy, there are many observable regularities that are not causal.

Example: Mark Twain joked that his perfect teeth after a lifetime of drinking whiskey proved the teeth-preserving properties of whiskey.

Experience is not passive. We make contributions to it and hence to our experiential understanding of the world. a. The presumption of the uniformity of nature is something that we cannot generalize from the experiences we’ve already had. We supply it.  b. The principle of causation is a way of making sense of the patterns that we discern through the filters that we bring to the actual sensory experiences as they occur. c. Gestalt or field experiences—which are something more than what we merely sense—are another contribution that we bring to experience

My own way of putting this point is to note that experience (as opposed to sensation) always involves “construal,” and that construal is a contribution by the subject who construes, not by the object that is construed. 1. This means that experience is inevitably “subjective,” that is, it includes a contribution made by a subject or involves a transaction between a subject and object. 2. This also means that its content is not “logically certain” or “necessarily accurate.”

Should this lead us in the Cartesian direction of demanding a priori truths as the basis for our ratiocination, or can we achieve something more reliable than “naïve realism” without taking that (possibly) poison pill? A. The Cartesian offer of “certainty” is attractive, but perfect and indubitable truths are a will-o’-the-wisp. B. Naïve realism simply takes things “at face value,” and we all know the cost of saying “who cares?” or simply giving up. C. As we shall see in the lectures ahead, modern rational empiricism offers us a third route.

[Courtesy: Professor James Hall]

01523--Write a note on the relationship between reason and the “revelations”.

The early moderns elevated human reason, downplaying the epistemic role of revelation. Some also questioned the safety of relying on sense experience. In the 17th century, René Descartes proposed a method of systematic doubt to clear away every basis for thought that could be called into question. His aim was to find an a priori basis for it instead. Although we may question the certainty of the foundation that he claimed to find (the famous cogito ergo sum), we can recognize the cogency of his demand for reliable foundations for thinking. Here, we will recapitulate some of the reasons for calling sense experience into question, examine the alleged need for “certain” foundations for thought, and show how that quest for certainty can have radically skeptical results.

Many early modern Western thinkers elevated the epistemic role of reason over revelation.  Modern rationalism, a part of the Enlightenment, is identified with such thinkers as Descartes, Leibniz, and Spinoza.  The Enlightenment was not “new.” It came from a genuine renaissance and harked back to an ancient Greek epistemic model that the medieval “era of faith” had subordinated.

Some early moderns also questioned the safety of relying on sense experience as a basis for our rational exercises.  This kept a good “Christian” mistrust of fleshly things center stage.  However, it also made this aspect of the Greek revival more Platonic than Aristotelian.  Nevertheless, revelation was not the thing either.   But if a reliable basis for thought is not provided by revelation or by sensation, where can it come from?    Early modern rationalists, such as Descartes, were foundationalists who believed that there are basic or foundational items of knowledge that we have from which we can reason to the full array of knowledge that we seek. 

In the 17th century, René Descartes proposed a method of systematic doubt to clear away every basis for thought (every sort of content source) that could be called into question.  

His aim was to find an a priori basis for it instead. What he arrived at is the famous cogito ergo sum (which he wrote in French—je pense, donc je suis).  The cogito is supposed to be beyond doubt. Doubt itself is a form of thought, and whatever thinks is.  Everything else can be doubted. Sense experience, as we have seen, is notoriously unreliable. Revelation, by the same token, is also unreliable, because what may seem to be a revelation from God might be the result of the interference of an “evil genius.”  Even if we agree with Descartes in believing that the occurrence of doubt entails the occurrence of thought and that the occurrence of thought entails the occurrence of a thinker, his claim that “I think” entails “I am” does not necessarily follow. As Bishop George Berkeley later showed, alternatives to the “I” (the substantial ego) are readily available (for example, the mind of God).  Although we may question the self-evidence, certainty, or intuitive necessity of the cogito, we can understand Descartes’ mistrust of sensation and recognize the cogency of his demand for an unshakable foundation for thinking. 

There is a mathematical model at work here again: Conclusions are to be rationally derived from necessarily true axioms (emulating Euclid’s theorems that were said to be grounded in the necessarily true axioms of geometry).  This model is, once again, strongly reminiscent of Plato.  The model is put directly in play when Descartes uses Euclidian geometry to show how his ontological proof of the existence of God works (which proof, please note, appeals directly to reason, not to religious experience and/or revelation, however “theological” its topic). The proof of God, if it works, eliminates the possibility that our reasoning is being manipulated by an “evil genius.”  This model requires that its axioms (starting points) actually be self-evident, necessary, or logically true.  But are statements about matters of fact (for example, the statements of applied geometry) ever selfevident, necessary, or logically true? Or, conversely, do logically true statements (for example, the statements of formal geometry) necessarily have any factual content or application? Isn’t their applicability to the world contingent?    Non-Euclidean geometries—in which the interior angles of a triangle don’t add up to 180°—apply very well in an area of intense gravitation where, according to Einstein’s theories of relativity, space itself is distorted or bent, and triangles drawn there are not Euclidian.

Descartes’ rational reconstruction of knowledge is based on what he takes to be a self-evident and necessary truth from which he aims to reconstruct a full understanding of the world around us, but the self-evident truth that he starts with is not necessarily self-evident. The model that he follows is based on a conventional and arbitrary starting point, and its applicability is a contingent matter of fact, not a matter of logical necessity. 

[Courtesy: Professor James Hall]

01522-- Write a note on Subject classes and Venn Diagrams.

Does every claim assert that its subject class has members? If so, what is a claim’s truth value when its subject class is empty? For instance, are all claims about my daughters false if I have no daughters? If so, then the rules of contradictories and subcontraries fail, and any attempt to maintain the rule of contradictories succeeds only if we abandon contraries and subalternation along with subcontraries. Modern syllogistic logic, following George Boole, adopts a different interpretation of A and E statements to deal with this. Graphically represented in Venn diagrams, this interpretation provides a convenient way to determine the validity of three-term syllogistic arguments, but is still severely limited in its scope of application.

I

Some classes have no members. For instance, the class of unicorns, the class of round squares, and the class of my daughters are all null. This creates problems because we don’t always know whether the classes we are discussing are populated or not. We need a logical apparatus that can be relied on, either way. 

Since any particular (I or O) claim about a null class clearly asserts that its subject class is populated, then they must all be false—for example, “Some of my daughters are blonde” and “Some of my daughters are not blonde.” But then, if we insist that the law of contradictions holds, their contradictory universals (A and E) must both be true—for example, “All of my daughters are blonde” and “None of my daughters is blonde.”

Modern syllogistic logic, following the 19th-century mathematician/logician George Boole, recognizes that particular (I and O) claims assert that their subject classes are populated but reads universal (A and E) claims differently, so as to preserve the law of contradiction. 

·         All S are P is read in obverse—No S are ~P, or S outside of P is null—which is clearly true when Some S are not P is false. 

·         No S are P is read straightforwardly as asserting that the intersection of S and P is null, which is clearly true when Some S are P is false. 

·         In this analysis, both of the “contraries” are true of a null class because they truly assert that certain sets are empty, and both of the “subcontraries” are false because they falsely assert something to exist that does not.

·         Consequently, the rules of contraries, subcontraries, and subalternation disappear from the modern square of opposition, and a further syllogistic rule is established: No valid categorical syllogism can have a particular conclusion (I or O) unless it has at least one particular premise.

·          A convenient way to represent categorical propositions, so interpreted, is in terms of null forms. Here, we represent the intersection or overlap of two classes by placing the class names side by side. Every class S has a complement, written ~S, and read “curl S” or “tilde S.” For example, the intersection of S and P is written SP, and the intersection of S and ~P is written S~P, and whether that intersection is populated or null is indicated by saying it is, or is not, equal to zero. Thus, “All S are P” can be represented with “S~P = 0,” which is read as “The intersection of S and non-P is empty” or “S outside of P is empty.” 

Null forms help us work with Venn diagrams.

II

Venn diagrams provide a graphic way to test the validity of three-term syllogistic arguments, by shading out empty areas and placing an X in populated areas. We know that a syllogism is valid if, upon inspection, it is evident that diagramming its premises is all it takes to provide a complete diagram for its conclusion. 

A valid AAA-1 syllogism. 

 Anything meritorious (M) is praiseworthy (P). 

All scholarship winners (S) are meritorious (M). 

Therefore, all scholarship winners (S) are praiseworthy (P).

A valid AII-1 syllogism. 

Everyone who is meek (M) is polite (P).

Some sophomores (S) are meek (M).

Therefore, some sophomores (S) are polite (P).

An invalid OOO-1 syllogism. 

Some moderates (M) are not politically savvy (P).

Some Senators (S) are not moderates (M).

Therefore, some Senators (S) are not politically savvy (P).

III

Using diagrams to show graphically that a categorical syllogism is valid—or that it’s not—was a wonderful advance over the more traditional ways of handling syllogisms but perfectly consistent with that ancient system.  (Note to readers: Even with these embellishments, however, the logic of categorical syllogisms is still severely limited in its scope of application. It will not comfortably handle categorical arguments with more than three terms, and it does not reveal the relationship between syllogistic logic—which is part of a larger realm known as predicate logic—and sentential logic. Those gaps will be partly filled in Lectures Twenty through Twentytwo.)  

[Courtesy: Professor James Hall]

01521--Write a note on Categorical Syllogisms.

A categorical syllogism consists of three categorical propositions—two premises and a conclusion. To be tested for validity, it must be stated in standard form. Standard-form categorical syllogisms can be sorted in terms of mood and figure into 256 possible arrangements, only some of which pass muster as valid in the system. To be valid (that is, for its conclusion to “follow” from its premises), it must satisfy certain formal restrictions on the number of terms that may occur in it; on the positions in which those terms may occur; on the “distribution” of the middle, major, and minor terms; and on the occurrence of negative statements

I

A categorical syllogism consists of three standard-form categorical statements: two serving as premises and the third as the conclusion.  Each of the individual statements must be unambiguous.   Example: “All undergraduates aren’t philosophy majors” might be either a universal negative claim that “No undergraduate students are philosophy majors” or a much less sweeping particular negative that “Some undergraduates are not philosophy majors.”

To be in standard form, a syllogism must have exactly three terms, each of which occurs in two of its three propositions.

1. The predicate term of the conclusion (called the major term) will also occur as either the subject or predicate of the premise that is stated first (hence, the major premise).

2. The subject term of the conclusion (called the minor term) will also occur as either the subject or predicate of the premise that is stated second (hence, the minor premise).

3. The third term (called the middle term) will occur in both of the premises (it can be the subject or the predicate term of either one) and will not occur in the conclusion.

Major premise:                                   middle term, major term (in either order)

Minor premise:                                   middle term, minor term (in either order)

Conclusion:                                          minor term, major term (in this order only)

Some nonstandard-form syllogisms can be reduced to standard form by reducing their number of terms to three—for example, if the syllogism seems to have more than three terms because of synonymy or because of the use of complementary terms—by manipulating their constituent propositions by means of immediate inferences, or by placing their premises and conclusion in proper order.

 Example: The argument “All Athenians are mortal because they are all Greeks and no Greeks are immortal” can be reduced to standard form by obverting “No Greeks are immortal” to “All Greeks are mortal,” by specifying the reference of “they,” and by placing the statements in proper order (major premise, minor premise, conclusion): 

All Greeks are mortal. 

All Athenians are Greeks. 

Therefore, all Athenians are mortal.

Note:  A syllogism with irreducibly more than three terms is not valid in this system.

II

For a standard-form categorical syllogism to be valid, it must comply with rules that restrict its structure in terms of (a) the “distribution” of the middle, major, and minor terms and (b) the occurrence of negative statements.

Restrictions on distribution: The distribution of a term has to do with whether or not the proposition in which it occurs conveys some information about all, or only part of, the class it names. No term can be distributed in the conclusion that is not distributed in the premise in which it occurs; that is, a conclusion cannot say more than the premises support. 

1. If the major term is distributed in the conclusion but not in the major premise, the argument fails due to “illicit process of the major term.” If the minor term is distributed in the conclusion but not in the minor premise, the argument fails due to “illicit process of the minor term.”

2. The middle term must be distributed in at least one of the two premises. If it is not, the argument fails due to “undistributed middle.”

There are also restrictions on negative propositions. 

1. If one of the premises is negative, the conclusion must be negative.

2. If both of the premises are negative, no valid conclusion can be drawn. 

Again, there must be exactly three terms and exactly three propositions, not four or more.

III

Standard-form categorical syllogisms display both mood and figure.  The mood of a syllogism is captured by listing the quality/quantity of each of its propositions in order (AAA, AEO, EIA, and so on).

The figure of a syllogism depends on where its middle term resides. 

Figure One: The middle term (M) is the subject of the major premise and predicate of the minor.  Figure Two: The middle term (M) is the predicate of both premises. 

Figure Three: The middle term (M) is the subject of both premises. 

Figure Four: The middle term (M) is the predicate of the major premise and subject of the minor.

In all four figures, the subject of the conclusion (S) appears in the minor premise (the second premise) of the syllogism and is known as the minor term. The predicate of the conclusion (P) appears in the major premise (the first premise) of the syllogism and is known as the major term. C. There are 256 possible moods and figures for standard form syllogisms, from AAA-1 to OOO-4. Very few are valid. 

Example:

AAA-1 (Valid). 

All Greeks are mortal. 

All Athenians are Greeks 

Therefore, all Athenians are mortal.

Example:     

AAA-2 (Invalid, undistributed middle). 

All Rastafarians are bearded. 

All billy goats are bearded. 

 Therefore, all billy goats are Rastafarians.

By Aristotelian standards, 24 of the 256 possible categorical syllogisms are valid.

Problems with null classes remain, as will be seen.

[Courtesy: Professor James Hall]

01519--Write a note on immediate inferences.

Standard-form categorical propositions with the same subject and predicate terms can be aligned in a traditional square of opposition that shows certain immediate inferences that can be drawn from the truth or falsity of one of them, based on such intuitive relationships as contradiction, contrariness, subcontrariness, and subalternation. Statements about empty classes, however, generate problems with this traditional square. All of the legitimate immediate inferences can be used to manipulate the various propositions in an extended argument so as to help put the argument itself in standard form.

I

Standard-form categorical propositions with the same subject and predicate terms can be aligned in a traditional square of opposition. 

II

Each of the four corners can be obverted (change quality and replace predicate with its complement).  The I and E corners can be converted (switch subject and predicate).  The A and O corners can be contraposed (replace subject and predicate with their complements).

                                                                

                                                                          III                                                                             

The square of opposition highlights relationships between standard-form categorical propositions with the same subject and predicate terms that enable certain further immediate inferences to be drawn from the truth or falsity of one of them, as long as none of the classes mentioned is empty (null).  

Contradictories (A and O, E and I) differ from each other in both quality and quantity, with the result that contradictories always have opposite truth values. If a statement is true, then its contradictory is false (and vice versa, this being a two-way street).

Contraries (A and E) are universal claims that differ from each other in quality but not in quantity, with the result that while they can both be false, they cannot both be true. However, null classes—those that do not have any members at all, for example, round squares—lead to problems. 

Subcontraries (I and O) are particular claims that differ from each other in quality but not in quantity, with the result that while they can both be true, they cannot both be false. Null classes, however, present problems.

Subalternation is the relationship between a universal claim and its dependent or hanging particular claim. Thus, a statement and its subalternate differ from each other in quantity but not in quality, with the result that if a statement is true, its subalternate is also true (but not vice versa, this being a one-way street). 

1. However, null classes again present problems.

2. The view of ancient logicians that you could convert a universal proposition “by limitation”—by moving to the subalternate of the universal proposition—does not hold if the class is empty. 

3. For example, “All round squares are truly remarkable,” seems true, but “There is at least one truly remarkable round square” is clearly false.

IV

The various immediate inferences can be used to modify the propositions in an argument so as to (a) have each one begin properly with its quantity indicator and (b) reduce the number of terms that occur in it to the three that a syllogism can handle.

A. If one of the statements in an argument amounts to the denial of a standard-form proposition (e.g., “Not all Greeks are Athenians”), one may appeal to the rule of contradiction to replace that denial with the assertion of its contradictory (in this case, “Some Greeks are not Athenians”).

B. If two of the statements in an argument employ complementary terms (e.g., heroes and non-heroes), one may obvert one of them, thus ensuring that both propositions are about the same set.

C. If one of the statements in an argument amounts to the denial of an I proposition (e.g., “It is not the case that some wolves are strict vegetarians”), one may appeal to the rule of subcontraries (or to the rules of contradiction and subalternation) to replace that denial with the assertion of its corresponding O proposition (“Some wolves are not strict vegetarians”). This is highly problematic, however, if the subject term names a null class (as in “It is not the case that some unicorns are carnivores”). We shall examine this problem in Lecture Eight.

D. The same thing can be done with the denial of an O proposition, to replace it with the assertion of its corresponding I, but this is also problematic when null classes are in play.  E. Once the statements in an argument are cleaned up, the argument itself can be put in standard form and assessed for validity in terms of formal rules, as we shall see in Lecture Eight.

[Courtesy: Professor James Hall]