Lecture 3, MATH 210G.03, Spring 2013

Lewis Carroll and the Syllogism

Charles Dodgson aka Lewis CarrollOnce master the machinery of Symbolic Logic, and you have a mental occupation always at hand, of absorbing interest, and one that will be of real use to you in any subject you may take up. It will give you clearness of thought - the ability to see your way through a puzzle - the habit of arranging your ideas in an orderly and get-at-able form - and, more valuable than all, the power to detect fallacies, and to tear to pieces the flimsy illogical arguments, which you will so continually encounter in books, in newspapers, in speeches, and even in sermons, and which so easily delude those who have never taken the trouble to master this fascinating Art

1. All, who neither dance on tight ropes nor eat penny-buns, are old.2. Pigs, that are liable to giddiness, are treated with respect.3. A wise balloonist takes an umbrella with him.4. No one ought to lunch in public, who looks ridiculous and eats penny-buns.5. Young creatures, who go up in balloons, are liable to giddiness.6. Fat creatures, who look ridiculous, may lunch in public, provided they do

not dance on tight ropes.7. No wise creatures dance on tight ropes, if liable to giddiness.8. A pig looks ridiculous, carrying an umbrella.9. All, who do not dance on tight ropes, and who are treated with respect are

fat.

A — balloonistsB — carrying umbrellasC — dancing on tight ropesD — eating penny-bunsE — fatF — liable to giddinessG — looking ridiculousH — may lunch in publicJ — oldK — pigsL — treated with respectM — wise.

1. All, who neither dance on tight ropes nor eat penny-buns, are old.2. Pigs, that are liable to giddiness, are treated with respect.3. A wise balloonist takes an umbrella with him.4. No one ought to lunch in public, who looks ridiculous and eats penny-buns.5. Young creatures, who go up in balloons, are liable to giddiness.6. Fat creatures, who look ridiculous, may lunch in public, provided they do not dance on tight ropes.7. No wise creatures dance on tight ropes, if liable to giddiness.8. A pig looks ridiculous, carrying an umbrella.9. All, who do not dance on tight ropes, and who are treated with respect are fat.

1. (~C∧ ~D) ∧ ~J = ;2. (K ∧ F) ∧ ~L = ;3. (M ∧ A)∧~B = ;4. (G ∧ D) ∧ H = ;5. (~J∧ A) ∧ ~F = ;6. (E ∧ G ∧ C) ∧~H = ;7. (M ∧ F) ∧C = ;8. (K ∧ B)∧~G = ;9. (~C∧L)∧~E= .

A 3,5B 8, -3C 7, -1,6,-9D 4, -1E 6, -9F 2,7, -5G 4,6, -8H 4, -6J -1,-5K 2,8L 9, -2M 3,7

http://durendal.org/lcsl/

Dodgson and the syllogism• (a) All babies are illogical.• (b) Nobody is despised who can manage a crocodile.• (c) Illogical persons are despised.

• “universe”: set of all people. • “it” in this context refers to a general person. • rewrite each statement in the puzzle as an implication in

symbols

• B : it is a baby• L : it is logical• M : it can manage a crocodile• D : it is despised ,

• In symbols:• (a) B → ~L : If it is a baby then it is not logical.• (b) M → ~D : If it can manage a crocodile then it is not

despised.• (c) ~L → D : If it is not logical then it is despised.

• Aim: stringing together chain of implications, apply transitive reasoning:

• A → B ^ V → C → A → C .• We have: B → ~L , and ~L → D so we conclude B → D• Contra positive of (b): D → ~M so…• B → ~L → D → ~M .

• Our ultimate conclusion is the statement:

• B → ~M : If it is a baby then it cannot manage a crocodile .• Vernacular: Babies cannot manage crocodiles

• Puzzle # 2

• (a) None of the unnoticed things, met with at sea, are mermaids.

• (b) Things entered in the log, as met with at sea, are sure to be worth remembering.

• (c) I have never met with anything worth remembering, when on a voyage.

• (d) Things met with at sea, that are noticed, are sure to be recorded in the log.

• Universe: “things met with at the sea”

• N : it is noticed• M : it is a mermaid• L : it is entered in the log• R : it is worth remembering• I : I have met with it at sea .

(a) ~N → ~M (M → N)(b) L → R (~R → ~L)(c) I → ~R (R → ~I)(d) N → L (~L → ~N )_____________________________________Strung together:

I → ~ R → ~L → ~N → ~M .I → ~M : If I have met with it at sea then it is not

a mermaid .“I have never met with a mermaid at sea.”

• Puzzle # 3

(a) No interesting poems are unpopular among people of real taste.(b) No modern poetry is free from affectation.(c) All your poems are on the subject of soap-bubbles.(d) No affected poetry is popular among people of real taste.(e) No ancient poem is on the subject of soap-bubbles.

Universe: “poems”, I : it is interestingP : it is popular among people of real tasteM : it is modernA : it is affectedY : it is your poemS : it is on the subject of soap bubbles .

(a) I → P (~P → ~I)(b) M → A (~A → ~M)(c) Y → S (~S → ~Y)(d) A → ~P (P → ~A)(e) ~M → ~S (S → M).

(a) I → P (~P → ~I)(b) M → A (~A → ~M)(c) Y → S (~S → ~Y)(d) A → ~P (P → ~A)(e) ~M → ~S (S → M).__________________________________________I → P → ~A → ~M → ~S → ~Y ,“contra-positive chain” Y → S → M → A → ~P → ~I .Solution: I → ~Y, or Y → ~I. “Your poetry is not interesting.”

For Quiz

• Every one who is sane can do Logic.• No lunatics are fit to serve on a jury.• None of your sons can do logic.

User your clicker to express the right conclusion

Every one who is sane can do Logic.No lunatics are fit to serve on a jury.None of your sons can do logic. A.) Jurors can do logicB.) Jurors are saneC.) Your sons are not saneD.) Your sons are not fit to serve on a jury

Solution

Every one who is sane can do Logic.No lunatics are fit to serve on a jury.None of your sons can do logic. Sa->DLFJ->SaDL->~SoAnyone fit to serve on a jury is not your son

(your sons are not fit to serve on a jury)

For group work/quiz1. The only animals in this house are cats.2. Every animal is suitable for a pet, that loves to gaze at the

moon.3. When I detest an animal, I avoid it.4. No animals are carnivorous, unless they prowl at night.5. No cat fails to kill mice.6. No animals ever take to me, except what are in this house.7. Kangaroos are not suitable for pets.8. None but carnivora kill mice.9. I detest animals that do not take to me.10. Animals, that prowl at night, always love to gaze at the

moon.

Assign letters to each of the following statements

• It is an animal that lives in this house• It is a cat• It is suitable to be a pet• It loves to gaze at the moon• I detest it• I avoid it• It is carnivorous• It prowls at night• It kills mice• It takes to me• It is a kangaroo

Translate each of the statements in terms of your symbols, e.g., A→C

1. The only animals in this house are cats.2. Every animal is suitable for a pet, that loves to gaze at the

moon.3. When I detest an animal, I avoid it.4. No animals are carnivorous, unless they prowl at night.5. No cat fails to kill mice.6. No animals ever take to me, except what are in this house.7. Kangaroos are not suitable for pets.8. None but carnivora kill mice.9. I detest animals that do not take to me.10. Animals, that prowl at night, always love to gaze at the

moon.

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Additional exercisesTranslate each sentence into an implication of the form A → B or ˜B

→ ˜A, then determine the intended implication of the following statements:

People who do not sleep enough run red lights.People with too much on their agenda do not cook well. No one gets enough sleep except for artists.People who smell the roses do not run red lights. People who do not smell the roses have too much on their agenda.

All writers, who understand human nature, are clever.

No one is a true poet unless he can stir the hearts of men.

Shakespeare wrote “Hamlet”. No writer, who does not understand human nature

, can stir the hearts of men. None but a true poet could have written

“Hamlet”.

Translate each sentence into an implication of the form A → B or ˜B → ˜A, then determine the intended implication.

People who eat at Chuckie Cheese brag about their exploits.

People who have their own TV shows murder green dogs.

Everyone eats at Chuckie Cheese except people who appreciate math.

People who brag about their exploits have their own TV shows.

Translate each sentence into an implication of the form A → B or ˜B → ˜A, then determine the intended implication of the

following statements:

Good poker players cannot be trusted. People who wear fancy shoes are not lucky in

love. Everyone is good in poker except nerds. You can trust people who wear shabby shoes.

No shark ever doubts that he is well fitted out. A fish, that cannot dance a minuet, is contemptible. No fish is quite certain that it is well fitted out,

unless it has three rows of teeth. All fishes, except sharks, are kind to children. No heavy fish can dance a minuet. A fish with three rows of teeth is not to be despised

Translate each sentence into an implication of the form A → B or ˜B → ˜A, then determine the intended implication.

• No kitten that loves fish is unteachable. • No kitten without a tail will play with a gorilla.

Kittens with whiskers always love fish. • No teachable kitten has green eyes. • No kittens have tails unless they have

whiskers.

Translate each sentence into an implication of the form A → B or ˜B → ˜A, then determine the intended implication.

Animals that do not kick are always unexcitable.Donkeys have no horns. A buffalo can always toss one over a gate. No animals that kick are easy to swallow. No hornless animal can toss one over a gate. All animals are excitable, except buffalo.

Translate each sentence into an implication of the form A → B or ˜B → ˜A, then determine the intended implication.

No Acrobatic feats, that are announced in the bill of Circus, are not attempted there.

No acrobatic feat is possible, if it involves turning in a quadruple somersault.

No impossible acrobatic feat is ever announced in a circus bill.

Translate each sentence into an implication of the form A → B or ˜B → ˜A, then determine the intended implication.

People who have cable get frustrated.Frustrated people have to get an eye patch.No one does not look tough if they wear an eye

patchPeople want to know how tough people who

wear eye patches are.When people want to know how tough

someone is, that person wakes up in a roadside ditch.

Translate each sentence into an implication of the form A → B or ˜B → ˜A, then determine the intended implication.

All who neither dance on tightropes nor eat penny-buns, are old. Pigs that are liable to giddiness are treated with respect. A wise balloonist takes an umbrella with him. No one ought to lunch in public, who looks ridiculous and eats penny-

buns. Young creatures who go up in balloons are liable to giddiness. Fat creatures who look ridiculous may lunch in public, provided they

do not dance on tightropes. No wise creatures dance on tightropes, if liable to giddiness. A pig looks ridiculous carrying an umbrella. All who do not dance on tightropes, and who are treated with respect,

are fat.

Translate each sentence into an implication of the form A → B or ˜B → ˜A, or A∧B→C etc then determine the intended

implication.

Roll your own

Invent universe of “things” (e.g., dogs, people, automobiles,…) and a list of properties A,B,C,D,E that the “things” may or may not possess.

Form the sentences A→B, B → ~C, ~C → D, D→ ~E in such a way that the implication A→ ~E follows from the law of the syllogism.