If A then B.
B.
Therefore, A.
Affirming the consequent (the name of this fallacy) doesn’t automatically imply that you took cyanide, as there are other causes of death. In terms of necessary and sufficient conditions, the error here is the same as the error of denying the antecedent; it is the fallacy of assuming that p. 304 a sufficient condition is a necessary one. Cyanide is a sufficient condition of dying, not a necessary one.
Back to the Cat: All cats are detached. But the Cheshire Cat is more detached than most. He is probably a very direct symbol of ideal intellectual detachment. He can disappear because he can abstract himself from his surroundings into himself. He can appear as only a head because he is almost a disembodied intelligence. He can appear as only a grin because he can impose an (unsettling) atmosphere without being (entirely) present.
According to Martin Gardner, a former editor of Scientific American, the phrase “grin without a cat” is probably not a bad description of pure mathematics. Although mathematical theorems often can be usefully applied to the structure of the external world, the theorems themselves are abstractions built on assumptions that belong to another realm “remote from human passions.” Bertrand Russell once put it as, “remote from the pitiful facts of nature . . . an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world.”
Gardner’s idea is probably hokum. But I will admit it has endearing features. The origin of the idea of the Cheshire Cat has been discussed ad nauseam, especially by Katsuko Kasai. The origin I like the best is Kasai’s interesting conjecture that Cheshire cheese was once sold in the shape of a grinning cat. One would tend to slice off the cheese at the cat’s tail and end up with only the grinning head on the plate.
On another level, my idea is probably more basic: Carroll was probably aware of the use of cats in the logical discussion of the principal relations among classes. In dividing the universe of creatures into cats and not-cats, logicians used the defining form
x ∈ cat
which defines the class of cats. This, in turn, was seen to determine the class of not-cats, which, I suppose, started with “x is unfeline.” If this was used as the class of not-cats, and if not-cats is ~C, then “cats” is C. Using this concept, let’s work out the truth tables to see if a cat can logically exist without a smile and vice versa:
Let C = cat
p. 305 Let S = grin (think of S as a smile, which is a kind of grin.
S looks less like a C than a G does, so S will serve better
as a symbol for grin than G.)
& is the logical connective signifying conjunction meaning both.
∨ is the logical connective signifying disjunction meaning either, or, or both.
Therefore, ∼C means not cat; ∼S means no grin; (C & S) means a cat with a grin; (C & ∼S) means a cat without a grin; (∼C & S) means a grin without a cat; and (∼C & ∼S) means no cat and no grin. Thus, truth tables for such simple and complex statements would be:
C
S
∼C
∼S
(C&S)
(C & ∼S)
(∼C & S)
(∼C & ∼S)
Case 1
T
T
F
F
T
F
F
F
Case 2
T
F
F
T
F
T
F
F
Case 3
F
T
T
F
F
F
T
F
Case 4
F
F
T
T
F
F
F
T
The table lists all possible combinations of the statements involving the simple terms C and S. There are only four cases. They correspond to the conditions (1) cat with a grin, (2) cat without a grin, (3) no cat but a grin, (4) no cat and no grin. Column 3 is the no-cat column, and its truth values are the opposite of column 1. Column 4 is the no-grin column, which is just the opposite of the grin column (2). The complex statements follow directly. For example, (C & S) is true in case 1 because only in case 1 are C and S true. In the cases 2, 3, and 4, C is false, S is false, or both are false. If either C or S is false, then the conjunction (C & S) must also be false.
So what?
Hold your horses. I do have a point in mind, and that will come out with the analysis of the more complex statements [(∼C & S) & (C & ∼S)] and [(∼C & S) ∨ (C & ∼S)].
p. 306
(∼C & S)
(C & ∼S)
[(∼C & S) & (C & ∼S)]
[(∼C & S) ∨ (C & ∼S)]
Case 1
F
F
F
F
Case 2
F
T
F
T
Case 3
T
F
F
T
Case 4
F
F
F
F
Thus, we prove that you can’t have a cat without a grin and no cat with a grin because all the cases of that conjunction are false. Such a statement is a contradiction. But the last column proves that we can have a grin without a cat or a cat without a grin. The complex statement reflecting those assertions, [(∼C & S) ∨ (C & ∼S)], is true in case 2 and in case 3. Situations that are true in some cases and false in others are contingent on the circumstances for their truth and are logically possible, as indicated in the last column.
In the same way with the same tables, I could prove to you that a cat without a grin or a cat with a grin is just a cat, and a grin without a cat or a grin with a cat is just a grin. Carroll probably worked out these tables (just for grins?), and that is why the eminently logical Alice mentions the truth value of [(∼C & S) ∨ (C & ∼S)], although she claims (contrary to fact) that she has never seen a grin without a cat.
Carroll’s point in this is (I believe) to demonstrate that in the analogical matrix known as fiction the grin can exist without a cat and that in the analogical matrix known as logic the same grin can also exist without a cat. In nature such a thing is also possible, if the class of no-cat is defined simply as the class of all things that exist that are not cat. At least I think that’s true. Who knows? I do know that my wife, Ethel, has an excellent endearing cat grin, which she puts on when it suits her. In that case, E! (S & ∼C) might be justified since the grin in this particular case is attached to a human and not a cat.
In math, √-1 exists, but in nature this can’t exist.
Grins without cats and math without reality: This is a distinction with a difference, a difference of which we should not lose sight. Math is one thing; the real world is another. Only sometimes do the twain, the two great realms, meet. At other times, the two may be far, far apart. Mathematical proofs can describe or not describe the real world, depending on how closely the assumptions, on which the proofs always depend, relate to the real world.
Math may be the queen of the sciences, but the history of mathematics proves that the queen is often in error (though rarely ever in p. 307 doubt) because she has been contradicted by reality. The history of mathematics is a graveyard of reasoned “proofs” once thought perfect and later found defective.
Lesson: Do not give undue weight or too much attention to supposed mathematical proofs. Mathematical proofs are only as good as the assumptions that underlie them and may not reflect reality or truth.
Example: When a mathematical analysis by AT&T scientist John Carson was quoted as conclusive proof that FM radio was not possible, the technical community committed the kind of error then that, unfortunately, con
tinues to be quite common now. Someone proves a statement based on certain assumptions; others forget those assumptions and remember only the conclusions. People then tend to apply such conclusions to all cases, even ones that do not satisfy the original assumptions. This is what had happened to FM, and it was tantamount to a prejudice against FM.
Edwin Armstrong, a Columbia University professor who was always suspicious of mathematical proofs of the impossible, in a brilliant moment of lateral thinking decided to challenge that wisdom and find out what would happen if he used instead a wide band of frequencies for his FM signal. The rest, as they say, is history. Armstrong built an FM receiver and transmitter that was far, far better than any AM setup then available. Having done that, Armstrong was convinced that FM radio would succeed. The only thing that he thought could temporarily slow it down was, in his words, “those intangible forces so frequently set in motion by men, and the origin of which lies in vested interests, habits, false mathematical proofs, customs, and legislation.”[27]
“Have some wine,” the March Hare said in an encouraging tone.
Alice looked all round the table, but there was nothing on it but tea.
[Look again, Alice. There is a milk jug on the table. We know this because later on in the tea party, the March Hare upsets it. Failure to observe carefully is a cause of incorrect conclusions. Failure to see the obvious is common. Many of our failures result from the mental set governing the search procedure at the time. In this case, Alice is looking for wine, not milk. She sees no wine, but she misses the milk. Milk would have been a more appropriate drink for a girl seven and one-half years old. But perhaps Alice was more interested in wine.]
p. 308 “I don’t see any wine,” she remarked.
“There isn’t any,” said the March Hare.
“Then it wasn’t very civil of you to offer it,” said Alice angrily.
“It wasn’t very civil of you to sit down without being invited,” said the March Hare.[28]
In effect, the March Hare is saying because Alice is guilty of doing the same thing that she criticizes him for (i.e., being uncivil), her argument is no good. Thus, with this counterattack on Alice, the March Hare avoids the obligation to explain his uncivil behavior. This is a violation of the relevance criterion. That some other person engages in a questionable practice is irrelevant to whether such a practice merits acceptance. “Practice what you preach” is OK advice but not a logical argument. Two wrongs don’t make a right. Shortcomings of your position cannot be defended by pointing out the errors or shortcomings of the opposition. Tu quoque (you do it, too) is a fallacy.
Tu quoque thinking is a common and very powerful psychological response, which most of us have experienced since childhood, to the inconsistent behavior of a critic. We often feel no obligation to respond to criticism under the circumstances. Because such thinking is so emotionally convincing, its fallacious character is usually not fully recognized until it is pointedly brought to one’s attention. And that, of course, dear reader, is your job.
“Take some more tea,” the March Hare said to Alice, very earnestly.
“I’ve had nothing yet,” Alice replied in an offended tone: “so I ca’n’t take more.”
“You mean you ca’n’t take less,” said the Hatter: “it’s very easy to take more than nothing.”[29]
True. And an excellent explanation of the null case (the set with no members) known as nothing. It’s hard to get less than nothing but easy to get more because nothing is the lowest you can go in the real world of material objects. About nothing, we and Lewis Carroll shall talk more (not less) later.
“You can draw water out of a water-well,” said the Hatter; “so I should think you could draw treacle [British for molasses] out of a treacle-well—eh, stupid?”
“But they were in the well,” Alice said to the Dormouse, not choosing to notice this last remark.
p. 309 “Of course they were,” said the Dormouse: “well in.”
This answer so confused poor Alice, that she let the Dormouse go on for some time without interrupting it.
“They were learning to draw,” the Dormouse went on, yawning and rubbing its eyes, for it was getting very sleepy; “and they drew all manner of things—everything that begins with an M—”
“Why with an M?” said Alice.
“Why not?” said the March Hare.[30]
The confusions of meanings for in the well and well in and drawing pictures and drawing treacle have been covered. Note that it is the March Hare, not the Dormouse, who answers Alice’s question. He has a vested personal interest in the matter because his name starts with M and he wants to be part of the story. He wants to be drawn for the same reason that most wealthy and privileged members of Edwardian society in England in the nineteenth century wanted to have their portraits painted by John Singer Sargent. Those aristocrats had a vested interest in seeing themselves well portrayed (pun intended).
Like most people with vested interests (that need defending), the March Hare’s defense is irrelevant. In this case, it is an irrelevant appeal to absence of a reason. Failure to know or to have a reason to justify a statement (such as March Hare’s “Why not?”) is not a reason supporting the statement. Absent evidence, ignorance itself, or no reason whatsoever never justifies anything.
A word about unbirthdays (celebrated by Humpty Dumpty in Through the Looking Glass): the concept of contradictions has occupied philosophers from day one. Unbirthday is a Carrollian logical extension of birthday. If birthday exists, then its negation would be not-birthday. The day you were born is your birthday. All days that you were not born are your unbirthdays. In this context, un is just another way of indicating denial. Yes or no; true or false; 0 or 1; + or −; go or no go; not, and, or; NOT, AND, OR; she loves me, she loves me not. Unbirthday is the denial or negation of birthday. It is another way of saying “not birthday.” The concept is trivial, but the results are not trivial and underlie much of our Western civilization’s quest for logical certainty. You have one birthday a year, and all the other days are unbirthdays. This could be formulated B not B, or we could abbreviate the statement as logicians do by placing the sign for negation, the tilde, ∼, before the second B. Thus, ∼B stands for unbirthday.
p. 310 Now, it is obvious that when a statement is true, its denial is false, and when a statement is false, its denial is true. Using the shorthand, we can symbolize this information in a truth table:
B
∼B
T
F
F
T
The truth table tells us when it’s a birthday, it is not an unbirthday and vice versa, the same thing the Hatter has told us but in symbolic form. Truth tables like this have had a major effect on the development of computerized information processing and underlie most computer logic. The probability of today being your unbirthday is 364/365, and the probability of this being your birthday is 1/365. Therefore, the probability of ∼B/B is 364 to 1. By the truth table, it is obvious that any statement of B is inconsistent with its negation, ∼B. In other words, B and ∼B are mutually exclusive. If it is B, then it can’t be your ∼B, and if it is ∼B, it can’t be your B. The Hatter explains this in so many words. But he could have just given Alice the equation, Probability (B or ∼B) = 1, and therefore, Probability (1 - ∼B) = B.
By the way, the Hatter isn’t entirely reasonable. He thinks it is much better to celebrate unbirthdays than celebrate birthdays because there are so many more unbirthdays than birthdays. This reasoning assumes celebrating is desirable. It might be from his perspective or the perspective of the kids, but it might not be true for others, especially adults. The Hatter’s reasoning disregards the utility factor, which must be included in any value judgment. Any mother of a two-year-old knows, especially if she has suffered through just one birthday party, that some birthday parties are no fun for adults. In fact, some parties for two-year-olds that I have attended have been pure torture.
Warning! We now pass on to t
he Queen of Hearts, a character who probably represents the embodiment of ungovernable passion in a person of power—a blind and aimless Fury, a Hitler or Stalin type. Her constant orders for beheading are shocking to those modern critics of children’s books who feel that juvenile fiction should be free of all violence. As far as I know, there have been no empirical studies of how children react to such scenes and what harm, if any, is done to their psyche. Absence of evidence, however, is not evidence of absence. So the question must remain open.
p. 311 My guess is that the normal child finds it all very amusing and is not damaged in the least. However, I do feel that this stuff is not entirely suitable for adults. It especially should not be permitted to circulate indiscriminately among adults who are undergoing psychoanalysis. The depiction of royalty here set forth contains an enormous amount of dignity, arbitrariness, and paraded prestige as necessary to bolster up the absurd pretensions of incompetent leaders, something that has modern resonances, something we should think about when we hear “Hail to the Chief” or read about the shenanigans of the Royals of England.
The reason that the Queen of Hearts and the subsequent trial of the Knave of Hearts can be so terrifying to adults is that most realize that they live in a slapstick modern world under an inexplicable sentence of death. When they try to find out what the castle authorities want them to do, they are shifted from one bumbling bureaucrat to another, receiving no reasonable answers.
Franz Kafka’s The Castle (which I believe was inspired by AAW) represents the stratified, organized, controlled, completely bureaucratized society, in which the individual is a number and has lost specific and distinct individual dignity, integrity, freedom, and all appearance of such. Yes, Kafka is true, but often we don’t recognize Kafka as such because when we go through the Kafkaesque experience, we are kicking, biting, fighting, trying to survive, doing lots of things, but not reading Kafka.
Truth, Knowledge, or Just Plain Bull: How to tell the difference Page 37
