ELEMENTARY LOGIC
James Fieser
9/9/2008
CONTENTS
1. What is an Argument?
2. Informal fallacies
3. Propositional logic
4. Propositions
5. Complex Propositions and Logical Connectives
6. Nested Logical Connectives.
7. Valid argument forms
8. Soundness
9. In Class Exercises
10. Take Home Exercises
1. WHAT IS AN ARGUMENT?
In ancient Greece, a group of traveling teachers called Sophists had the reputation of being able to argue for any point, no matter how absurd. One Sophist offered the following argument:
(1) Fido is my dog.
(2) Fido is a mother.
(3) Therefore, Fido is my mother.
As bizarre as this argument is, at least some parts of it seem compelling, specifically the first two statements. The strange part about this argument is how the third statement appears to follow from the first two. The goal of logic is to construct good arguments, and to recognize when people use bad ones. Clearly, the above argument is a bad one, even though we might not yet be able to identify exactly where the problem lies. Here, by contrast, is an example of a good argument:
(1) If Gandhi was a sadist, then he would have hurt people.
(2) It is not the case that Gandhi hurt people.
(3) Therefore, it is not the case that Gandhi was a sadist.
Whether an argument is good or bad, there are a few basic ingredients that all arguments have, namely that they consist of premises and conclusions. A premise is a statement which is used as evidence for a conclusion; in both of the examples above, statements 1 and 2 are the premises. A conclusion is a statement which is supported by at least one premise; statement 3 in both of the examples is the conclusion. An argument, then, is at least one premise accompanied with a conclusion.
2. INFORMAL FALLACIES.
To help expose the logical tricks used by the Sophists, Aristotle wrote a book titled Fallacies of the Sophists, which catalogs dozens of illogical patterns of reasoning, or informal fallacies as we now call them. Although the list of common informal fallacies has changed since Aristotle, acquaintance with the fallacies is still an effective way of detecting bad argumentation. Some of the more important fallacies will be presented here. For each of the following fallacies, there will be exceptions where use of that argument pattern will be valid. Thus, the informal fallacies should be seen as rough guidelines, and not absolute rules.
Argument against the Person (argumentum ad hominem): attacking a person’s character instead of the content of that person’s argument. For example, “Heidegger was a poor philosopher since he was a member of the Nazi party.” “Bob is an alcoholic, so don’t take his investment advice too seriously.” “Of course Jones would argue for gun control, after all, Jones is a democrat.”
Argument from Ignorance (argumentum ad ignorantiam): concluding that something is true since you can’t prove it is false. For example, “God must exist, since no one can demonstrate that he does not exist.” The classic example of this fallacy is from Senator Joseph McCarthy who accused a certain person of communist connections with the following argument: “I do not have much information on this except the general statement of the agency that there is nothing in the files to disprove his Communist connections.”
Appeal to Pity (argumentum ad misericordiam): appealing to a person’s unfortunate circumstance as a way of getting someone to accept a conclusion. For example, “you need to pass me in this course, since I’ll lose my scholarship if you don’t.” “I implore you to find Mrs. Bobbit not guilty of mutilating her husband, since her home life was so traumatic.” “Please don’t arrest me, I have a wife and kids to support.” “Yes, I murdered my parents, but take pity on me for now I’m an orphan.”
Appeal to the Masses (argumentum ad populum): going along with the crowd in support of a conclusion. For example, “Gee mom, all the guys in school carry guns.” “Don’t be so prudish, Lulu, I’m sure all of your dorm mates are sleeping around.”
Appeal to Authority (argumentum ad verecundiam): appealing to a popular figure who is not an authority in that area. For example, “Einstein believed in God, so God must exist.” “Bart Simpson likes Butterfinger candy bars, so they must be good.”
Irrelevant Conclusion (non sequitur): drawing a conclusion which does not follow from the evidence. For example, “My business went under last year, hence Bush is a bad president.” “My shoe string broke; I guess that means it’s time to buy a new car.”
False Cause (post hoc ergo procter hoc): inferring a causal connection based on mere correlation. For example, “the number of stroke victims in hospitals is directly proportional to the number of tar bubbles on the road; thus, tar bubbles cause strokes.” “Successful people have expensive clothing; hence the best way to become a success is to buy expensive clothing.”
Circular Reasoning: implicitly using your conclusion as a premise. For example, “God must exist since the Bible says that God exists, and the Bible is true because God wrote it.” “It is impossible to talk without using words, since words are necessary for talking.”
Equivocation: an argument which is based on two definitions of one word. For example, “Good steaks are rare these days, so you shouldn’t order yours well done.” “Jones is a poor man, and he loses whenever he plays poker; therefore, Jones is a poor loser.” “You don’t find cars like yours in these parts, so don’t let your car out of your sight.”
Composition: assuming that the whole must have the properties of its parts. “Each part of this machine is light, therefore the whole machine is light.” “A bus uses more gas than a car, therefore all busses combined use more gas than all cars combined.”
Division: assuming that the parts of a whole must have the properties of the whole. For example, “This corporation is important, hence each worker in this corporation is important.” “Students study Math and English, therefore each student studies Math and English.”
Red Herring: introducing an irrelevant or secondary subject and thereby diverting attention from the main subject. For example, “side impact air bags in cars do not really increase safety, and, besides, most cars with side impact air bags are Japanese imports.” “Women should have the freedom to choose to have an abortion; restricting this freedom is just another instance of male oppression of women.”
Straw Man: distorting an opposing view so that it is easy to refute. For example, “vote against gun control, since gun control advocates believe that no one should own any type of fire arm.” “The pro-life position on abortion is wrong since pro lifers believe a woman would have to bring her fetus to term even when her life is in danger.”
3. PROPOSITIONAL LOGIC.
Learning the informal fallacies may help you detect specific types of logical errors, but this will not guarantee that you can construct error-free arguments of your own. A different approach to argumentation, called propositional logic (also sentential logic), assists in constructing arguments which fit valid argument forms. Consider again the following example:
(1) If Gandhi was a sadist, then he would have hurt people.
(2) It is not the case that Gandhi hurt people.
(3) Therefore, it is not the case that Gandhi was a sadist.
The logical structure of this argument is this:
(1) if P then Q
(2) not Q
(3) therefore, not P
To understand logical arguments, such as the one above, it is best to begin with elementary concepts and build upon these. Similar to the way that geometry is founded on the basic notions of points, lines, and intersections, logic is founded on the concept of a proposition.
4. PROPOSITIONS.
It is important to distinguish between three related concepts:
Utterance: The most general form of verbal expression. Utterances include nonsense expressions, such as “ob la di ob la da,” as well as statements (see next definition).
Statement: An utterance which conveys meaning. Statements include questions (“who am I?”), commands (“get that gun out of my face!”), expressions of feelings (“three cheers for old glory!”), and propositions (see next definition).
Proposition: An either true or false statement about the world, such as, “Bob was bitten by a rattlesnake,” “Pastor Jack is in Jail,” or “Aunt Edith recovered from cancer.”
The distinction between the above three notions is important since every statement in an argument must be a proposition (i.e. an either true or false statement about the world). For example, each statement in the above Gandhi argument is either true or false. It is also important to note, though, that a statement does not need to be true to qualify as a proposition: false statements also count as propositions. Consider the following:
“Joe is 20 feet tall.”
Since people don’t grow to be 20 feet tall, then this statement is obviously false. However, given that the sole requirement of a proposition is that it be either a true or false statement, then the above statement qualifies as a proposition. Also, we are not required to know all the details of a statement to determine whether it is a proposition. Consider the following:
“Gramps just swallowed his teeth.”
Even if we don’t know who Gramps is, we can still see that it is either true that Gramps just swallowed his teeth or it is false that he just swallowed his teeth. So, this too is a proposition. Consider one more:
“Keep your boy away from my daughter!”
Technically, this statement is not a proposition since it is a command and commands are neither true nor false statements. However, commands may often be rewritten to express propositions. The above, for example, may rewritten as follows:
“You better keep your boy away from my daughter!”
Here it is either true or false that X better keep his boy away from Y’s daughter. The propositions examined so far have been simple propositions in the sense that they express a single simple idea which is either true or false. For example, there is only one true or false idea expressed in the proposition that “Joe is 20 feet tall.” As such, this simple proposition may be abbreviated with the single letter “P”. Similarly, we may use “Q” to abbreviate the simple proposition that “Gramps just swallowed his teeth.” The use of letters such as P, Q, R, etc. is only a convention adopted by logicians. We can just as easily use Greek letters or geometrical shapes as abbreviations for simple propositions.
5. COMPLEX PROPOSITIONS AND LOGICAL CONNECTIVES.
Simple propositions are only one type of proposition used in logic. Others may take a more complex form, where two or more simple propositions are combined. For simplicity, logicians have arrived at four basic logical connectives which are used in constructing complex propositions from simple ones: (1) P and Q, (2) P or Q, (3) if P then Q, (4) not P. Again, the letters “P” and “Q” represent simple propositions. For example, the complex proposition, “Smith is a nerd or Jones is a geek” may be abbreviated, “P or Q”, where P stands for “Smith is a nerd” and Q stands for “Jones is a geek.” Each of the four logical connectives have special names and have specific meanings, which we will now examine.
Conjunction: “P and Q” An example of a conjunction is “Bob is rich and Sam is poor.” This can be abbreviated “P and Q” where “P” stands for the simple proposition “Bob is rich” and “Q” stands for the simple proposition “Sam is poor.” The “P” and “Q” elements of a conjunction are referred to as conjuncts. The P’s and Q’s of conjunctions can be switched around and mean the same thing; clearly, the statement “Bob is rich and Sam is poor” means the same thing as “Sam is poor and Bob is rich.” In ordinary conversation we use conjunctions with great regularly, but often in a disguised form. It is important to see through these disguises and translate concealed conjunctions into standard propositional form (standard form being “P and Q”). Here is a list of some typical disguises:
P, but Q
P, although Q
P; Q
P, besides Q
P, however Q
P, whereas Q
In addition to these, sentences such as “Sherman and Xavier are computer software pirates” could be translated into “Sherman is a computer software pirate, and Xavier is a computer software pirate.”
Disjunction: “P or Q”. An example of a disjunction is, “mom pawned her wedding ring or mom sold blood.” This can be abbreviated as “P or Q” where “P” stands for the simple proposition “mom pawned her wedding ring,” and “Q” stands for the simple proposition, “mom sold blood.” The “P” and “Q” elements of a disjunction are each referred to as disjuncts. Like conjunctions, the two disjuncts in a disjunction may also be switched around and mean the same thing. Disjunctions are more complicated than they first appear since in ordinary conversation the word “or” can be used in two distinct ways. First, the word or is used inclusively in the above example since mom could have pawned her ring, or sold blood, or both of these. Second, the word “or” can be used exclusively as in the statement, “Mary is dead or Mary is alive,” where Mary cannot be both dead and alive at the same time. Although the word “or” can be either inclusive or exclusive in ordinary language, in logic, however, it is used only inclusively.
Negation: “not P”. An example of negation is, “it is not the case that Fido just left his territorial mark.” This can be abbreviated “not P” where “P” stands for the simple proposition “Fido just left his territorial mark.” A sentence which has a negative word in it, such as “not,” “never,” or “none,” may often (but not always) be translated into a negated proposition. For example, consider the sentence, “I knew that Jurgan was not really a Nazi.” Since this sentence is an assertion about my knowledge, it does not translate into a negation.
Conditional: “If P then Q”. An example of a conditional proposition is, “if you eat of the forbidden fruit, then you will surely die.” This can be abbreviated “if P then Q” where P stands for the simple proposition “you eat of the forbidden fruit,” and Q stands for “you will surely die.” The P part of a conditional is referred to as the antecedent, and the Q part is called the consequent. An important feature about conditionals is that if the P’s and Q’s are switched around, the meaning of the sentence changes. Compare the above example to this: “if you die, then you will have eaten of the forbidden fruit.” Clearly, the two sentences don’t mean the same thing. Assume that everyone who eats the forbidden fruit subsequently dies; still, not everyone who dies will have eaten of the forbidden fruit, such as someone why dies in a skydiving accident. Some typical disguised conditionals are,
If P, it follows that Q
P implies Q
P entails Q
P only if Q
Whenever P, Q
P, therefore Q
P is a sufficient condition for Q
Q follows from P
Q is a necessary condition for P
Q, since P
Sometimes it is easy to determine which kind of logical connective is being used in a complex proposition. Take, for example, the ordinary statement “America: love it or leave it.” Technically this is a command, but when translated the implied proposition is, “you must love America or you must leave America.” Clearly this is a disjunction, given the prominence of the “or”. Other complex propositions are more difficult to recognize. Take, for example,
“Bestowing gifts of gold, frankincense and myrrh upon the teacher is a necessary condition for passing the course.”
There are two ways to figure out what kind of logical connective is used here. First, you may recognize that the phrase “necessary condition” is one of the clue phrases given above for a conditional (if-then) proposition. Alternatively, you may recognize that there is a causal (if-then) connection involved between bestowing gifts and passing the course. Translated into standard form this conditional proposition reads,
“If you pass the course, then you will have bestowed gifts of gold, frankincense and myrrh upon the teacher.”
Abbreviated this would read,
if P then Q
where, P stands for “you pass the course,” and Q stands for “you will have bestowed gifts of gold, frankincense and myrrh upon the teacher.”
6. NESTED LOGICAL CONNECTIVES.
Complex propositions often contain a number of logical connectives nested within each other. Consider the following: “I will not hurt you and your old lady if you simply hand over your wallet.” This proposition contains a negation, conjunction, and a conditional. Translated into standard form the proposition reads, “If you will simply hand over your wallet, then it is not the case that (I will hurt you and I will hurt your old lady). Abbreviated this would be as follows:
if P then not (Q and R)
where,
P = you will simply hand over your wallet
Q = I will hurt you
R = I will hurt your old lady
The benefit of nesting logical connectives is that it is possible to put even quite complicated propositions into standard form. As noted above, an or (that is, disjunction) in logic is inclusive. Yet, by nesting a number of logical connectives, it is possible to put exclusive or’s into standard form without violating any rules. Take again the example “Mary is dead or Mary is alive.” The intent of this proposition is that Mary cannot be both dead and alive, hence the “or” here is used exclusively. To be in proper logical form, this sentence must be translated into, “(Mary is dead or Mary is alive) and it is not the case that (Mary is dead and Mary is alive). Abbreviated this says,
(P or Q) and not (P and Q),
where,
P = Mary is dead
Q = Mary is alive
It is important to recognize when a nested proposition is formed properly. For example, the statement “or P then Q if” is clearly not properly formed. The basic rule is that well formed nested propositions contain only complete sub-propositions. For example, “(P and Q) or R” is well formed since it consists of “P and Q” as well as “() or R,” both of which are complete sub-propositions.
7. VALID ARGUMENT FORMS.
So far we have seen that propositions may be simple, complex, and nested. Any of these may be used as elements of an argument. As noted earlier, many arguments are bad ones, and when constructing logical arguments our goal is to make good ones. The first element of good argumentation is validity, and for our purposes we will define a valid argument as follows:
Valid Argument: an argument which fits a valid argument form (such as modus ponens).
The first step in forming a good argument is to follow a valid argument form. There are an infinite number of valid argument forms. We will be interested in just four:
Modus Ponens
premise (1) If P then Q
premise (2) P
concl. (3) Therefore, Q
(1) If the president pushes the button, then a nuclear holocaust will commence.
(2) He pushed the button.
(3) Therefore, a nuclear holocaust will commence.
Modus Tollens
premise (1) If P then Q
premise (2) Not Q
concl. (3) Therefore, not P
(1) If Bob desecrated the Bible then he would have been struck down.
(2) It is not the case that she was struck down.
(3) Therefore, it is not the case that Bob desecrated the Bible.
Disjunctive Syllogism (two versions)
premise (1) P or Q (1) P or Q
premise (2) not P (2) not Q
concl. (3) Therefore, Q (3) Therefore, P
(1) Either Smith bites the dust or Smith bites Jones.
(2) It is not the case that Smith bites the dust.
(3) Therefore, Smith bites Jones.
Hypothetical Syllogism
premise (1) if P then Q
premise (2) if Q then R
concl. (3) Therefore, if P then R
(1) If you bribe the officer then he will tear up the ticket.
(2) If he tears up the ticket then you won’t pay a fine.
(3) Therefore, if you bribe the officer then you won’t pay a fine.
All four of these valid argument forms have a strong intuitive appeal. Unfortunately, it is all too easy to accidentally switch a P and Q. This happens so often that it is customary for logicians to introduce a group of fallacious argument forms. These arguments bear a strong resemblance to genuine valid argument forms and consequently are frequently mistaken for the real thing. We will be interested in three of these.
Fallacious Modus Ponens: fallacy of affirming the consequent
premise (1) if P then Q
premise (2) Q
concl. (3) therefore, P
Fallacious Modus Tollens: fallacy of denying the antecedent
premise (1) if P then Q
premise (2) not P
concl. (3) therefore, not Q
Fallacious Disjunctive Syllogism: fallacy of asserting an alternative
premise (1) P or Q (1) P or Q
premise (2) P (2) Q
concl. (3) therefore, not Q (3) therefore, not P
8. SOUND AND UNSOUND ARGUMENTS.
As noted, the first step in forming a good argument is that it must be valid. However, more is needed. A good argument must be valid and have all true premises. This is called soundness:
Sound Argument: an argument which (a) follows a valid argument form, and (b) has only true premises.
The Gandhi argument presented at the outset is an example of a sound argument since it is both valid and has only true premises. A few examples will illustrate why a sound argument must be both valid and have all true premises. The following argument follows a valid form but does not have true premises:
(1) Either Uncle Bob is a tennis shoe, or Uncle Bob is a golfing shoe.
(2) It is not the case that Uncle Bob is a golfing shoe.
(3) Therefore, Uncle Bob is a tennis shoe.
The above argument is valid since it follows the form of disjunctive syllogism. But, premise one is obviously false, so the argument is unsound. There are also some arguments which have true premises, but do not follow a valid argument form, and are likewise unsound. For example:
(1) If the President was the pilot of Air Force One, then he could fly in the presidential plane.
(2) The President can fly in the presidential plane.
(3) Therefore, the President is the pilot of Air Force One.
Premises (1) and (2) in this argument are true, but the argument follows the fallacious argument form of fallacious modus ponens, so this argument is also unsound. In short, since soundness entails both validity and true premises, there are two ways that an argument can be unsound: (1) it will be invalid, or (2) it will have at least one false premise.
9. IN CLASS EXERCISES.
Identify the informal fallacy in each of the following:
(1) “The Dead Milkmen” is a rock band. Most people who were once milkmen in the U.S. are now dead. Yikes! That’s one big rock band!
(2) Hey, forget about Lulu, she’s nothing special. Is there anything special about her kidneys, tonsils, or small intestine? She’s only a collection of those things.
(3) Of course the Major thinks that the Army offers good career opportunities. He’s an Army man himself.
(4) I think Lulu will go out with you. I haven’t heard anything which suggests that she wouldn’t.
(5) We have a good faculty here at Preppy State University. Therefore, Dr. Wilbur E. Drunkard, who teaches here, is a good faculty member.
Determine whether the following are propositions. If some are not propositions, see if they can be rewritten as propositions:
(6) I have a very refined sense of smell.
(7) Not here, Billy Bob!
(8) I think I’m going to sell little Joey into slavery.
In each of the following identify the logical connective being used and translate the proposition into standard form.
(9) Father O’Brian will marry Lulu only if he first leaves the priesthood.
(10) I was accepted at Yale University, but I’d much rather attend Thrift Community College.
(11) Jones’s name does not appear in the Book of Life.
Determine which of the following are well-formed nested propositions:
(12) if P then (Q or R)
(13) (P and Q) not
(14) not (P or Q)
(15) P and (if Q then R)
Translate the following premises and conclusions into standard form and decide which valid argument form or fallacious argument form is being used:
(16) If the band “Satan’s Pitchfork” performs in town, they will play “Hell, Sweet Hell.” If they perform “Hell, Sweet Hell” then dudes will stage dive. So, if they perform, dudes will stage dive.
(17) Either Bob will go bankrupt, or I will. Bob will go bankrupt. I will not.
(18) If Sam is committed to a sanitarium, then his brother Tom will inherit everything. Sam will not be committed. Tom will not inherit everything.
Make up a valid argument that leads to the conclusion given. Use the rule indicated in parentheses. You will need to invent some simple proposition to make your premises complete.
(19) Polly wants a cracker. (disjunctive syllogism)
(20) If you insult Ed’s mother, you will go to the hospital. (hypothetical syllogism)
(21) Clara won’t win at the crap table. (modus ponens)
(22) Thrift Community College is a good school. (modus tollens)
Are the following arguments valid, invalid, sound, or unsound?
(23) If Fido is a Dalmatian, then Fido would have lots of spots
It is not the case that Fido is a Dalmatian
Therefore, it is not the case that Fido has lots of spots
(24) If Joseph Stalin had U.S. citizenship, then he would have been born in the U.S.
It is not the case that Joseph Stalin was born in the U.S.
Therefore, it is not the case that Joseph Stalin had U.S. citizenship
The following test your knowledge of soundess.
(25) Can a valid argument have a false conclusion?
(26) Can a sound argument have a false conclusion?
Answers to in class exercises
(1) Equivocation. This argument assumes two meanings of the phrase “Dead Milkmen.”
(2) Composition. This argument assumes that a property of each part (i.e., “nothing special”) applies to the whole (i.e., Lulu herself).
(3) Argument against the person. This argument attacks the personal attribute of the Army Major, without examining the content of the Major’s argument.
(4) Argument from ignorance. This argument concludes that something is true (i.e., that Lulu will go out with you) since it isn’t proven false.
(5) Division. This argument assumes that a property of the whole university (i.e., “good faculty”) applies to each member (i.e., Dr. Wilbur E. Drunkard).
(6) This is a proposition as written.
(7) This is not a proposition. It can be rewritten as a proposition as follow: “I request that you do not do that here Billy Bob.” The true/false component of the revised version is “I request that. . . .”
(8) This is a proposition as written. The true or false component is “I think. . . .”
(9) This is a conditional (if-then). The clue term is “only if”, and the proposition translates “IF Father O’Brian marries Lulu THEN he must first leave the priesthood.”
(10) This is a conjunction (and). The clue term is “but”, and the proposition translates “I was accepted at Yale University AND I’d much rather attend Thrift Community College.”
(11) This is a negation. The clue term is “not”, and the proposition translates “It is not the case that Jones’s name appears in the Book of Life.”
(12) This is well formed as written.
(13) This is not well formed as written since (a) a logical connective needs to precede “not” and (b) a proposition needs to follow (not).
(14) This is well formed as written.
(15) This is well formed as written.
(16) This is a valid hypothetical syllogism argument. Here it is translated into proper form:
IF the band “Satan’s Pitchfork” performs in town, THEN they will play “Hell, Sweet Hell.”
IF they perform “Hell, Sweet Hell” THEN dudes will stage dive.
Therefore, IF the band “Satan’s Pitchfork” performs in town, THEN dudes will stage dive.
(17) This is an invalid fallacious disjunctive syllogism argument. Here it is translated into proper form:
Bob will go bankrupt, OR I will go bankrupt.
Bob will go bankrupt.
Therefore, I will not go bankrupt.
(18) This is an invalid fallacious modus tollens argument. Here it is translated into proper form:
IF Sam is committed to a sanitarium, THEN his brother Tom will inherit everything.
IT IS NOT THE CASE THAT Sam will be committed to a sanitarium.
Therefore, IT IS NOT THE CASE THAT his brother Tom will inherit everything.
(19) A possible created proposition is “Polly wants a hamburger.” The argument, then, is this:
Polly wants a hamburger OR Polly wants a cracker.
IT IS NOT THE CASE THAT Polly wants a hamburger.
Polly wants a cracker.
(20) A possible created proposition is “Ed will hit you.” The argument, then, is this:
IF you insult Ed’s mother, THEN Ed will hit you.
IF Ed hits you, THEN you will go to the hospital.
THEREFORE, if you insult Ed’s mother, THEN you will go to the hospital.
(21) A possible created proposition is “Clara partied all night.” The argument, then, is this:
IF Clara partied all night, THEN Clara will fail her exam.
Clara partied all night.
Therefore, Clara will fail her exam.
(22) A possible created proposition is “the faculty members are drunks”. To make sure a negation appears in the conclusion, yet preserves the original meaning of stipulated conclusion, the conclusion should be reworded as follows: “It is not the case that Thrift Community College is a bad school.” The argument, then, is this:
IF the faculty members are drunks, THEN Thrift Community College is a bad school.
IT IS NOT THE CASE THAT the faculty members are drunks.
Therefore, IT IS NOT THE CASE THAT Thrift Community College is a bad school.
(23) This is an invalid fallacious modus tollens argument, since premise two denies the antecedent instead of the consequent. It is, thus, both invalid and unsound.
(24) This is a valid modus tollens argument. However, premise 1 is false since U.S. citizens don’t have to be born in the U.S. (i.e., they can be naturalized). Thus, it is valid, but unsound for having a false premise.
(25) A valid argument can have a false conclusion, such as the earlier argument that concludes “Uncle Bob is a tennis shoe.” Validity only concerns the form of an argument, not the truth of the content.
(26) A sound argument cannot have a false conclusion. Validity is a truth-preserving mechanism; this if a valid argument has true premises (and thus is sound), then the truth carries over into the conclusion.
10. TAKE HOME EXERCISES (“Questions for Review” to put in the blue book; there are no “questions for analysis” for this chapter).
Are the following a proposition? If not, how may they be rephrased as propositions?
(1) I sense that the CIA is spying on me.
(2) Put down your gun and come out with your hands up.
(3) I have contacted the spirits of the dead.
(4) You know where you can put those crystals of yours.
Put the following in standard form propositional form:
(5) x yet y
(6) r if s
(7) x is a necessary condition for y
(8) a although b
(9) r implies s
(10) x but y
Which of the following are not well formed propositions?
(11) not P or Q
(12) (P or Q) not (R and S)
(13) if (P and Q)
(14) (P or Q) if and (R then S)
(15) if [(P and Q) or (R or S)] then T
(16) What is the name of the following argument form or fallacy?
if A then B
not B
therefore, not A
(17) What is the name of the following argument form or fallacy?
X or Y
not X
therefore, Y
(18) Make up a modus tollens which leads to the following conclusion:
Therefore, the CIA is not my friend
(19) Is the following argument sound or unsound? If unsound, why?
If I am at Graceland then I can see Elvis memorabilia
It is not the case that I am at Graceland
Therefore it is not the case that I can see Elvis memorabilia