Chapter 2

Determine the logical form of the following arguments. Use symbols to express that form and determine whether the form is valid or invalid. If the form is invalid, determine the type of error made. Comment on the soundness of the argument as well, in particular, determine whether any of the premises are questionable. (a) All who are guilty are in prison. George is not in prison. Therefore, George is not guilty. (b) If one eats oranges one will have high levels of vitamin C. You do have high levels of vitamin \(\mathrm{C}\). Therefore, you must eat oranges. (c) All fish live in water. The mackerel is a fish. Therefore, the mackerel lives in water. (d) If you're lazy, don't take math courses. Everyone is lazy. Therefore, no one should take math courses. (e) All fish live in water. The octopus lives in water. Therefore, the octopus is a fish. (f) If a person goes into politics, they are a scoundrel. Harold has gone into politics. Therefore, Harold is a scoundrel.

In the movie "Monty Python and the Holy Grail" we encounter a medieval villager who (with a bit of prompting) makes the following argument. If she weighs the same as a duck, then she's made of wood. If she's made of wood then she's a witch. Therefore, if she weighs the same as a duck, she's a witch. Which rule of inference is he using?

There is a common variant of the existential quantifier, \(\exists !,\) if you write \(\exists ! x, P(x)\) you are asserting that there is a unique element in the universe that makes \(P(x)\) true. Determine how to negate the sentence \(\exists ! x, P(x)\)

There are 3 operations used in basic algebra (addition, multiplication and exponentiation) and thus there are potentially 6 different distributive laws. State all 6 "laws" and determine which 2 are actually valid. (As an example, the distributive law of addition over multiplication would look like \(x+(y \cdot z)=(x+y) \cdot(x+z)\), this isn't one of the true ones.)

Design a digital logic circuit (using and, or \& not gates) that implements an exclusive or.

Below is a rule of inference that we call extended elimination. $$ \begin{aligned} &(A \vee B) \vee C \\ & \neg A \\ & \neg B \\ \hline \therefore C & \end{aligned} $$ Use a truth table to verify that this rule is valid.

In constructive dilemma, the antecedent of the conditional sentences are usually chosen to represent opposite alternatives. This allows us to introduce their disjunction as a tautology. Consider the following proof that there is never any reason to worry (found on the walls of an Irish pub). Either you are sick or you are well. If you are well there's nothing to worry about. If you are sick there are just two possibilities: Either you will get better or you will die. If you are going to get better there's nothing to worry about. If you are going to die there are just two possibilities: Either you will go to Heaven or to Hell. If you go to Heaven there is nothing to worry about. If you go to Hell, you'll be so busy shaking hands with all your friends there won't be time to worry \(\ldots\) Identify the three tautologies that are introduced in this "proof."

The order in which quantifiers appear is important. Let \(L(x, y)\) be the open sentence " \(x\) is in love with \(y\) " Discuss the meanings of the following quantified statements and find their negations. (a) \(\forall x \exists y L(x, y)\) (b) \(\exists x \forall y L(x, y)\). (c) \(\forall x \forall y L(x, y)\) (d) \(\exists x \exists y L(x, y)\).

Complete truth tables for the compound sentences \(A \Longrightarrow B\) and \(\neg A \vee B\).

Determine a useful denial of: \(\forall \epsilon>0 \exists \delta>0 \forall x(|x-c|<\delta) \Longrightarrow(|f(x)-l|<\epsilon)\) The denial above gives a criterion for saying \(\lim _{x \rightarrow c} f(x) \neq l\).

Consider the sentence "This sentence is false." Is this sentence a statement?

Identify the rule of inference being used. (a) The Buley Library is very tall. Therefore, either the Buley Library is very tall or it has many levels underground. Identify the rule of inference being used. (a) The Buley Library is very tall. Therefore, either the Buley Library is very tall or it has many levels underground.

A Sophie Germain prime is a prime number \(p\) such that the corre- sponding odd number \(2 p+1\) is also a prime. For example 11 is a Sophie Germain prime since \(23=2 \cdot 11+1\) is also prime. Almost all Sophie Germain primes are congruent to \(5(\bmod 6),\) nevertheless, there are exceptions - so the statement "There are Sophie Germain primes that are not \(5 \mathrm{mod} 6 . "\) is true. Verify this.

Find the negation of each of the following and simplify as much as possible. (a) \((A \vee B) \Longleftrightarrow C\) (b) \((A \vee B) \Longrightarrow(A \wedge B)\)

Write two-column proofs that verify each of the following logical equivalences. \(\neg(A \vee \neg B) \vee(\neg A \wedge \neg B) \cong \neg A\)

Determine a sentence using the and connector \((\wedge)\) that gives the negation of \(A \Longrightarrow B\).

Read the following proof that the sum of two odd numbers is even. Discuss the rules of inference used. Proof: Let \(x\) and \(y\) be odd numbers. Then \(x=2 k+1\) and \(y=2 j+1\) for some integers \(j\) and \(k\). By algebra, $$ x+y=2 k+1+2 j+1=2(k+j+1) $$ Note that \(k+j+1\) is an integer because \(k\) and \(j\) are integers. Hence \(x+y\) is even. Q.E.D.

Rewrite the sentence "Fix the toilet or I won't pay the rent!" as a conditional.

Sometimes in constructing a proof we find it necessary to "weaken" an inequality. For example, we might have already deduced that \(x

The so-called "ethic of reciprocity" is an idea that has come up in many of the world's religions and philosophies. Below are statements of the ethic from several sources. Discuss their logical meanings and determine which (if any) are logically equivalent. (a) "One should not behave towards others in a way which is disagree- able to oneself." Mencius Vii.A.4 (Hinduism) (b) "None of you [truly] believes until he wishes for his brother what he wishes for himself." Number 13 of Imam "Al-Nawawi's Forty Hadiths." (Islam) (c) "And as ye would that men should do to you, do ye also to them likewise." Luke 6:31, King James Version. (Christianity) (d) "What is hateful to you, do not to your fellow man. This is the law: all the rest is commentary." Talmud, Shabbat 31a. (Judaism) (e) "An it harm no one, do what thou wilt" (Wicca) (f) "What you would avoid suffering yourself, seek not to impose on others." (the Greek philosopher Epictetus - first century A.D.) (g) "Do not do unto others as you expect they should do unto you. Their tastes may not be the same." (the Irish playwright George Bernard Shaw - 20th century A.D.)

The famous logician Raymond Smullyan devised a family of logical puzzles around a fictitious place he called "the Island of Knights and Knaves." The inhabitants of the island are either knaves, who always make false statements, or knights, who always make truthful statements. In the most famous knight/knave puzzle, you are in a room which has only two exits. One leads to certain death and the other to freedom. There are two individuals in the room, and you know that one of them is a knight and the other is a knave, but you don't know which. Your challenge is to determine the door which leads to freedom by asking a single question.

You encounter two natives of the land of knights and knaves. Fill in an explanation for each line of the proofs of their identities. (a) Natasha says, "Boris is a knave." Boris says, "Natasha and I are knights." Claim: Natasha is a knight, and Boris is a knave. Proof: If Natasha is a knave, then Boris is a knight. If Boris is a knight, then Natasha is a knight. Therefore, if Natasha is a knave, then Natasha is a knight. Hence Natasha is a knight. Therefore, Boris is a knave. (b) Bonaparte says "I am a knight and Wellington is a knave." Wellington says "I would tell you that \(\mathrm{B}\) is a knight." Claim: Bonaparte is a knight and Wellington is a knave. Proof: Either Wellington is a knave or Wellington is a knight. If Wellington is a knight it follows that Bonaparte is a knight. If Bonaparte is a knight then Wellington is a knave. So, if Wellington is a knight then Wellington is a knave (which is impossible!) Thus, Wellington is a knave. Since Wellington is a knave, his statement "I would tell you that Bonaparte is a knight" is false. So Wellington would in fact tell us that Bonaparte is a knave. Since Wellington is a knave we conclude that Bonaparte is a knight. Thus Bonaparte is a knight and Wellington is a knave (as claimed).

Express the statement \(A \Longrightarrow B\) using the Peirce arrow and \(/\) or the Scheffer stroke.

Find the contrapositives of the following sentences. (a) If you can't do the time, don't do the crime. (b) If you do well in school, you'll get a good job. (c) If you wish others to treat you in a certain way, you must treat others in that fashion. (d) If it's raining, there must be clouds. (e) If \(a_{n} \leq b_{n},\) for all \(n\) and \(\sum_{n=0}^{\infty} b_{n}\) is a convergent series, then \(\sum_{n=0}^{\infty} a_{n}\) is a convergent series.

The integral test in Calculus is used to determine whether an infinite series converges or diverges: Suppose that \(f(x)\) is a positive, decreasing, real- valued function with \(\lim _{x \longrightarrow \infty} f(x)=0,\) if the improper integral \(\int_{0}^{\infty} f(x)\) has a finite value, then the infinite series \(\sum_{n=1}^{\infty} f(n)\) converges. The integral test should be envisioned by letting the series correspond to a right-hand Riemann sum for the integral, since the function is decreasing, a right-hand Riemann sum is an underestimate for the value of the integral, thus $$ \sum_{n=1}^{\infty} f(n)<\int_{0}^{\infty} f(x) $$ Discuss the meanings of and (where possible) provide justifications for the inverse, converse and contrapositive of the conditional statement in the integral test.