LOGIC

From Great Issues in Philosophy, by James Fieser

Home: www.utm.edu/staff/jfieser/120

CONTENTS

A. What is an Argument?

Propositions and Non-Propositional Utterances

Premise and Conclusion Indicators

Argument Diagrams

B. Informal Fallacies

Fallacies of Relevance

Other Common Fallacies

C. Propositional Statements

Complex Propositions and Logical Connectives

Nested Logical Connectives

D. Propositional Logic

Valid Argument Forms

Fallacious Argument Forms

Sound and Unsound Arguments

E. Inductive Logic

Inductive vs. Deductive Arguments

Inductive Probability

Inductive Argument Forms

In-Class Exercises

Take Home Study Questions

In ancient Greece, an infamous group of philosophers called "Sophists" could argue for any point, no matter how absurd. One Sophist offered this argument:

(1) Fido is Joe’s dog.

(2) Fido is a mother.

(3) Therefore, Fido is Joe’s mother.

The first two statements are compelling, since we can easily assume that Fido is Joe’s dog, and that Fido has had puppies and is thus a mother. The strange part about this argument, though, is that the third statement, which is absurd, appears to logically follow from the first two. Something has gone wrong here, since Joe, who is a human being, was clearly not given birth by his dog. This argument, then, 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 David Hume was a bachelor, then he was unmarried.

(2) David Hume was a Bachelor.

(3) Therefore, David Hume was unmarried.

The first two statements are true, and, further, the third statement seems to follow with necessity. The goal of logic is to help us understand what makes arguments like the Fido one bad, and others like the Hume one good.

The study of logic is different than the study of other areas of philosophy. Philosophy of mind or ethics, for example, involves an unfolding story of competing theories where one philosophical school attempts to outdo or improve upon the views of rivals. Logic, by contrast, is an analytic skill that requires mastery of abstract argument structures. Learning logic involves looking at examples and doing practice exercises, similar to how we learn other analytic subjects like math or computer programming. This chapter presents basic themes covered in typical introduction to logic text books, with accompanying practice exercises and solutions similar to those in such works.

A. WHAT IS AN ARGUMENT?

Whether an argument is good or bad, there are two components that they all have: premises and conclusions. In both examples above, statements 1 and 2 are the premises, and statement 3 is the conclusion. Here are the definitions of these concepts:

Premise: a statement which is used as evidence for a conclusion.

Conclusion: a statement which is supported by at least one premise.

Argument: at least one premise accompanied with a conclusion.

It helps to think of an argument as a series of factual statements where the first ones provide evidence for the final one.

Propositions and Non-Propositional Utterances

Each line in an argument must be a “proposition”, that is, a true or false statement about the world. Look again the above “Hume” argument. In premise one, it is either true or false that “If David Hume was a bachelor, then he was unmarried.” In this case, it happens to be a true statement. In premise two, it is also either true or false that “David Hume was a Bachelor.” Again, this is also true. Finally, the conclusion “David Hume was unmarried” is either true or false, which, again, is also true.  While all three of these specific statements happen to be true, false statements in arguments also count as propositions, such as “Lincoln was the first US President”. Even strange statements can be propositions so long as they are either true or false, such as the following:

• “Fido is Joe’s mother”

• “Joe is 20 feet tall”

• “Gramps just swallowed his teeth”

With each of these, we can judge whether it is true or false, which is all it takes to qualify as a proposition.

While the concept of a proposition may be clear enough (i.e., an either true or false statement about the world) it is easy to confuse propositions with other kinds of verbal expressions. We must, then, distinguish between these two notions:

Proposition: an either true or false statement about the world, such as, “The door is brown,” “David Hume was unmarried,” or “Fido is Joe’s mother”.

Non-propositional Utterance: a verbal expression that conveys meaning, but is not a true or false statement about the world. Non-propositional utterances include questions (“who am I”), commands (“get that porcupine out of my face!”), expressions of feelings (“three cheers for old glory!”).

With each of these non-propositional utterances, it makes no sense to ask whether it is true or false that “who am I”, or whether it is true or false that “get that porcupine out of my face”. Thus, even though these non-propositional utterances have meaning, they technically do not count as propositions and, in their present form, cannot be used as premises or conclusions in formal logical arguments. However, often non-propositional utterances like these can be rephrased to express some underlying or implied proposition. Take the following:

“Keep your flesh-eating zombie out of my yard!”

While this is in the form of a non-propositional command, its central meaning can nevertheless be rephrased as a proposition such as this:

“You better keep your flesh-eating zombie out of my yard!”

This revised statement is either true or false. More precisely, it is the phrase “you better” which is the true or false component of this statement. That is, it is either true or false that “you better keep your flesh-eating zombie out of my yard.” So, this revised statement can now function as a premise or a conclusion in an argument. Part of the task of logic is to take non-propositional utterances like this that we use in ordinary discourse, and translate their meanings into propositional form so they are suitable for logical argumentation.

Premise and Conclusion Indicators

An initial challenge with understanding arguments can involve simply identifying which sentences are premises and which are conclusions. Sometimes, for example, the conclusion might appear first in an argument, such as the following:

A trumpet is not a stringed instrument since a trumpet doesn’t have any strings.

In this argument, the word “since” is a premise indicator, that is, it is a clue word that tells us that a premise follows it. Presented formally, the argument is this:

premise (1): a trumpet doesn’t have any strings.

concl.    (2): a trumpet is not a stringed instrument.

There are many premise indicators that we use in ordinary discourse, and the more common ones are these:

• Since

• For

• Because

• Given that

• For the reason that

• In view of the fact that

There are also conclusion indicators, which tell us that a conclusion follows the indicator, such as the following:

A guitar has strings, therefore a guitar is a stringed instrument.

In this argument, the word “therefore” is the conclusion indicator, and, more formally, the argument is this:

premise (1):  a guitar has strings.

concl.    (2): a guitar is a stringed instrument.

The word “therefore” is the most common conclusion indicator used in formal logic, but here is a longer list of such indicators that we use in ordinary discourse:

• Therefore

• Thus

• Hence

• So

• Accordingly

• For this reason

• Consequently

• It follows that

The above arguments about trumpets and guitars contain only one premise each, but a typical argument often contains two or more premises. Regardless of the number of premises, though, we still look for premise and conclusion indicators to help identify the argument's structure. Take, for example, this one with two premises:

The pipa is a musical instrument from China, and it has strings; for this reason it is a stringed musical instrument.

The conclusion indicator here is “for this reason”. The word "and" in the middle of the sentence helps us recognize that there are two premises that lead to the conclusion. Laid out formally, the argument is this:

premise (1): the pipa is an object that has strings.

premise (2): the pipa is musical instrument from China.

concl.    (3): it is a stringed musical instrument.

Argument Diagrams

Once we identify the premises and conclusions in an argument, the next step is to see how the premises lead to the conclusions, and there are different ways that they can do that. It is sometimes helpful to diagram argument structures using arrows and plus signs to reveal an argument’s structure. Take again the above argument:

premise (1): the pipa is an object that has strings.

premise (2): the pipa is musical instrument from China.

concl.    (3): it is a stringed musical instrument.

The argument diagram of this is as follows:

1+2 |→ 3

This tells us that both premise 1 and 2 must be taken together to produce the conclusion; each premise independently will not do that. We call this a joint inference. Premise one by itself only tells us that some strange thing called a “pipa” has strings, and for all we know it may just be a clothesline. Premise two tells us that the pipa is in fact a musical instrument, but by itself it says nothing about it having strings. Both pieces of information in the two premises are needed to lead to the conclusion.

Other times each premise in an argument might lead independently to the conclusion, without the assistance of the other premise, as in the following:

premise (1): a typical trumpet is made of brass.

premise (2): the Harvard Dictionary of Music classifies the trumpet as a brass instrument.

concl.    (3): the trumpet is a brass instrument.

Using an arrow diagram, the structure of the argument is this:

1 |→ 3 and 2 |→ 3

This tells us that premise 1 by itself leads to the conclusion, and that premise 2 by itself also leads to the same conclusion. We call this an independent inference insofar as each premise leads to the conclusion independently of the other premise. Here we actually have two distinct arguments for the same conclusion, each of which stands independently of the other. The benefit of independent arguments like this is that, when debating with someone, it is often more effective to offer separate arguments for the same conclusion, just in case your opponent does not find one of your arguments compelling. Throughout this chapter, though, our focus will be on arguments like the earlier pipa one, where premises are taken jointly to produce a conclusion.

B. INFORMAL FALLACIES

Recall again our opening argument by a Sophist:

(1) Fido is Joe’s dog.

(2) Fido is a mother.

(3) Therefore, Fido is Joe’s mother.

There is some logical trickery going on here, and it contains what logicians call a fallacy, that is, an error of reasoning that makes an argument flawed. To help expose the logical tricks used by the Sophists, the ancient Greek philosopher Aristotle wrote a book titled Fallacies of the Sophists, which catalogs around a dozen fallacious patterns of reasoning. Today we call these informal fallacies since they occur within the context of ordinary discourse, and do not require a formal or abstract analysis of argument patterns. Although the list of informal fallacies has changed since Aristotle, acquaintance with the fallacies is still an effective way of detecting bad argumentation. Some discussions of informal fallacies list as many as 300 different types. We will look at thirteen of the more important ones. For at least most of these fallacies, there will be exceptions where use of that argument pattern will be valid and, accordingly, we should see the informal fallacies below mainly as rough guidelines for analyzing arguments, and not absolute rules. Context is everything when determining whether or not a line of criticism counts as an actual commission of an informal fallacy.

Fallacies of Relevance

A first group of fallacies are called fallacies of relevance, where the premises of the argument are irrelevant for establishing the conclusion. What counts as relevant is a matter of degree: some premises are completely irrelevant to the conclusion, and others only mildly so. Many of the fallacies have Latin names that were introduced by medieval logicians, and these are given in parentheses.

Argument against the Person (argumentum ad hominem): attacking a person’s character instead of the content of that person’s argument. Arguments against the person typically reflect a bias within the listener that prevents her from appreciating the speaker's argument as it is in itself. For example, “Bob is an alcoholic, so don’t take his investment advice too seriously.” “Heidegger was a poor philosopher since he was a member of the Nazi party.” “Of course Jones would argue for gun control, after all, Jones is a Democrat.” In these examples, the specific features of someone’s private life may not have any bearing on whether he is a good philosopher, or has sound investment advice, or has a good argument for gun control. Again, though, this fallacy is only a rough guideline. For example, if Heidegger was primarily a moral philosopher, we might be justified in looking at his personal behavior as a clarification of what his ethical theory permitted. His personal affiliation with Nazism might reveal a weakness within his moral theory that we might have otherwise overlooked.

Argument from Ignorance (argumentum ad ignorantiam): concluding that something is true since you can’t prove it is false. For example, “Zeus must exist, since no one can demonstrate that he does not exist.” The fallacy here is that it is unreasonable to insist that someone be able to prove that Zeus does not exist since it would require virtually complete knowledge about the cosmos. This is reflected in the common expression that “You can’t prove a negative.” Another infamous 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 [Central Intelligence] Agency that there is nothing in the files to disprove his communist connections.” In this example, lack of disproof of about a person’s communist connections does not by itself constitute a proof of anything.

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 Bob not guilty of assault, since his personal 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.” In some situations, we may feel true compassion for the person, and in fact do what we can to help them. But in these cases, the unfortunate situation is irrelevant to the factual issue of whether the student has earned a failing grade or whether the person broke the law.

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, so I should too.” “Everyone I know says zombies exist, so I guess they do exist.” In many cases, there is no fallacy to accept popular opinion, such as with the commonly held view that eating three meals a day is a good thing. However, with the gun and zombie examples, mere popular opinion is not a good indicator of what we should do or believe.

Appeal to Authority (argumentum ad verecundiam): accepting the view of 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.” While it may be proper to cite Einstein as an authority in physics, questions of religion are outside his area of expertise. Bart Simpson, if he counts as an authority at all, is only an expert in causing mischief, not in culinary matters. The commission of this fallacy comes down to whether the person cited is indeed a genuine specialist on the issue in question, or just a popular figure who is speaking without adequate knowledge on the subject.

Irrelevant Conclusion (non sequitur): drawing a conclusion which does not follow from the evidence. Strictly speaking, all the above fallacies of relevance involve the drawing of an irrelevant conclusion. This specific fallacy of “irrelevant conclusion,” though, is a general one, and applies when an argument does not fit any of the more specific patterns of irrelevance above. For example, “My business went under last year, hence the U.S. president should be impeached.” “My shoestring broke; I guess that means it’s time to buy a new car.” In these examples, there is little or no connection between the premise and conclusion. If your business failed, the most reasonable conclusion would be that the fault rests with you, not the president. If your shoestring broke, the most reasonable conclusion would be that you should buy a new shoestring, not a new car.

Other Common Fallacies

Fallacies of relevance are just one type of erroneous argumentation. Other informal fallacies are often classed together under various headings, such as "semantic" fallacies or "inductive" fallacies, and the ones below are among the more common of these.

False Cause (post hoc ergo procter hoc): inferring a causal connection based on mere correlation. For example, “crime rate is directly proportional to ice cream sales; thus, ice cream causes crime.” “Successful people have expensive clothing; hence the best way to become a success is to buy expensive clothing.” Both examples present genuine correlations, but they wrongly assume that one correlated event causes the other. In the ice cream example, there is a third factor, hot weather, which causes people to want ice cream more, and also causes people to be out of their homes and in public more. Thus, while quantities of ice cream sales and crime are correlated, the two are not causally connected. In the second example, while success and expensive clothing are correlated, if there is a causal connection it would likely be in the reverse direction where a person’s success puts her in a position where she can buy expensive clothing.

Circular Reasoning: implicitly using your conclusion as a premise. For example, “It is impossible to talk without using words, since words are necessary for talking.” In this case, the premise and conclusion mean exactly the same thing, just using different words, and each implies the other. Here is a common example where the circularity can be seen more clearly when laid out:

(1) The Bible says that God exists.

(2) The Bible is true because God wrote it.

(3) Therefore, God must exist.

In premise 2, by stating that “the Bible is true because God wrote it”, the arguer is already assuming that God exists, which is the very point that her argument aims to prove in the conclusion. The argument, then, amounts to simply this: "God exists, therefore God exists".

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.” The word being equivocated upon here is “rare”, where the first part of the sentence takes it to mean “unusual” and the second part “lightly cooked”. “Jones is a poor man, and he loses whenever he plays poker; therefore, Jones is a poor loser.” The equivocated word in this case is “poor” which initially means “financially destitute” and later means “bad”. “You don’t find cars like yours in these parts, so don’t let your car out of your sight.” Here the equivocated word in question is “find” which initially means “have” and later means “locate”. This is the fallacy committed in our opening example about Fido, where the word “mother” at one time implicitly means “canine mother” and later implicitly means “human mother”.

Composition: assuming that the whole must have the properties of its parts. For example, “Each part of this machine is light, therefore the whole machine is light.” “Each person in this corporation is important, therefore the entire corporation is important.” The fallacy takes place when “composing” the whole from the parts and transferring over one of the properties of the parts. In these examples, it is clear that the properties of the parts do not necessarily transfer to the whole.

Division: assuming that the parts of a whole must have the properties of the whole. For example, “This whole machine is heavy, therefore each part of this machine is heavy.” “This corporation is important, hence each worker in this corporation is important.” This is the reverse of the previous fallacy, and in this case the fallacy takes place when “dividing” the whole into parts and carrying over one of its properties of the whole. Again, in these examples, the properties of the whole do not necessarily transfer to the parts.

Red Herring: introducing an irrelevant or secondary subject and thereby diverting attention from the main subject. The argument gets its name from the sport of fox hunting, where a hunter would place a strongly scented fish on a trail to distract his hounds from the fox he is pursuing. In this way, a red herring is a distraction from the primary object. For example, “seat belts in cars do not really increase safety, and, besides, it’s my business, not the government’s, how I choose to sit in my car.” In this example, the real issue up for discussion is seatbelt safety, and the diversion is government intrusion on liberty. “Women should have the freedom to choose to have an abortion, since restricting this freedom is just another instance of male oppression of women.” In this example, the real issue up for discussion is freedom of choice, and the diversion is male oppression of women.

Straw Man: distorting an opposing view so that it is easy to refute. The argument name originates from knights practicing jousting using a straw man, which is an easy target. For example, “vote against gun control, since gun control advocates believe that no one should own any type of firearm.” This misrepresents the gun control view which typically seeks to restrict only specific types of firearms, not all of them. The misrepresented view is thus easy to attack. So too with the argument that “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.” The pro-life position is here misrepresented since it typically does permit abortion when the mother’s life is at risk.

C. PROPOSITIONAL STATEMENTS

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, lays out specific rules for constructing arguments which fit valid argument forms. Consider again our earlier example:

(1) If David Hume was a bachelor, then he was unmarried.

(2) David Hume was a bachelor.

(3) Therefore, David Hume was unmarried.

The logical structure of this argument is this:

(1) if P then Q

(2) P

(3) therefore, Q

The arguments of propositional logic have a special grammar to them, and they often consist of splicing together simple propositions into longer ones. Consider premise one above: “If David Hume was a bachelor, then he was unmarried”. This contains two simple propositions that we have abbreviated with the letters P and Q. Here, P stands for the simple proposition that “David Hume was a bachelor”, and Q stands for the simple proposition that “He was unmarried.” P and Q each express a single, simple idea that is either true or false. However, the entirety of premise one expresses a complex idea that is formed by combining these two simple ideas. In propositional logic, the letters P, Q, R, etc., are commonly used to designate simple propositions. However, there is nothing important about this choice, we could just as easily use Greek letters or geometrical shapes as abbreviations. What matters is the consistent use of some symbol to represent every occurrence of a simple idea.

Complex Propositions and Logical Connectives

In the above “Hume” example, premise one is a complex proposition that splices together the simple propositions P and Q in an “if-then” statement. Within the system of propositional logic, there are four and only four basic logical connectives that are used to construct complex propositions from simple ones, and here they are in their standard forms:

• P and Q

• P or Q

if P then Q

not P

Each of the above four logical connectives in their standard forms have special names and have specific meanings, which we will next examine.

Conjunction: “P and Q”. An example of a conjunction is “Bob is rich and Joe 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 “Joe 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. For example, the statement “Bob is rich and Joe is poor” means the same thing as “Joe is poor and Bob is rich.” In ordinary discourse we regularly use conjunctions, but often in a disguised form using a synonym for “and”. When working within the system of propositional logic, though, the word “and” is the only acceptable term for a conjunction, and all synonyms must be translated into this. Thus, the standard form for a conjunction is “P and Q”. Here is a list of some typical synonyms that require translation:

• P, but Q

• P, although Q

• P; Q

• P, besides Q

• P, however Q

• P, whereas Q

Disjunction: “P or Q”. An example of a disjunction is, “Beth went to London or Beth went to Berlin.” This can be abbreviated as “P or Q” where “P” stands for the simple proposition “Beth went to London,” and “Q” stands for the simple proposition, “Beth went to Berlin.” 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 discourse the word “or” can be used in two distinct ways. First, the word "or" is used inclusively in the above example since Beth could have gone to London, or to Berlin, or both of these. Second, in ordinary discourse, the word “or” can also 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 discourse, in logic, however, it is used only inclusively. There is a more complex way of logically expressing the notion of “or” in an exclusive sense, which we will look at further down.

Negation: “not P”. An example of negation is, “it is not the case that Fido just chased a raccoon into Walmart.” This can be abbreviated “not P” where “P” stands for the simple proposition “Fido just chased a raccoon into Walmart.” 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 Bill was not really a communist.” 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 (sometimes also called the “sufficient condition”), and the Q part is called the consequent (sometimes also called the “necessary condition”). An important feature about conditionals is that if the P’s and Q’s are switched around, the meaning of the sentence changes. This is precisely why the P and Q components of conditionals have separate names, unlike with conjunctions and disjunctions whose parts can be switched and retain the same meaning. For example, compare the above example to this: “if you die, then you will have eaten of the forbidden fruit.” Clearly, the two sentences do not 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 who dies in a skydiving accident. In ordinary discourse we have many phrases for expressing conditional propositions, and some common ones are these:

• If P, it follows that Q

• P implies Q

• P entails Q

• Whenever P, Q

• P, therefore Q

• Q follows from P

• Q, since P

As before, when working within the system of propositional logic, “if-then” is the only acceptable term for a conditional, and all synonyms in ordinary discourse must be translated into this.

Nested Logical Connectives

Complex propositions can contain several logical connectives nested within each other. Consider the following: “I will not cry and roll around on the floor if you simply give me that piece of candy.” This proposition contains a negation, conjunction, and a conditional. Translated into standard form the proposition reads,

If you will simply give me that piece of candy, then it is not the case that (I will cry and I will roll around on the floor).

Abbreviated this would be as follows:

if P then not (Q and R)

where,

P = you simply give me that piece of candy

Q = I will cry

R = I will roll around on the floor

The benefit of nesting logical connectives is that it is possible to put complicated propositions into standard form. For example, we saw earlier that an “or” (that is, disjunction) in logic is inclusive. Yet, by nesting several logical connectives, it is possible to convey an exclusive meaning of “or” without breaking any rules of the logical connectives. 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, which has an inclusive meaning. To express this following the rules of propositional logic, we translate this into, the following:

(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,

Q = Mary is alive

This is just one example of a nested proposition, and there are literally an infinite number that we might construct. When a nested proposition is constructed properly, it is called “well-formed”. For example, the following statements are not well-formed:

P not (if and)

(P Q) not R

or P then Q if

P if Q and R or S not

In each of these statements, the sub-components are randomly strung together, and each whole statement is gibberish. By contrast, the following propositions are well-formed:

(P and Q) or R

If P then (Q and R)

not (P or R)

(P and Q) or (R and S)

In these cases, (a) all of the sub-components within the parentheses fit the basic pattern of one of the four logical connectives, and (b) the whole proposition also fits the basic pattern of one of the four logical connectives. For example, in the first proposition, “P and Q” within parentheses fits the pattern of a conjunction, and the whole proposition fits the pattern of a disjunction.

D. PROPOSITIONAL LOGIC

So far, we have seen that well-formed propositional statements may be simple, complex, and nested:

• Simple:

P

Q

R, etc.

• Complex:

P and Q

P or Q

If P then Q

not P

• Nested:

If (P or Q) then (not R or S)

(If R then S) and (P or not Q), etc.

Any of these propositions may be used as elements of an argument. Propositional logic is a system of logic that builds arguments from such propositional statements.

Valid Argument Forms

As noted earlier, many arguments are bad ones, and when constructing logical arguments in propositional logic, our goal is to make good ones. The first step in forming a good argument is to follow a valid argument form, 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 below).

There are an infinite number of valid argument forms, but we will be interested in just four commonly used ones:

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 bomb will explode.

(2) He pushed the button.

(3) Therefore, a nuclear bomb will explode.

Modus Tollens

premise (1) if P then Q

premise (2) not Q

concl.    (3) therefore, not P

(1) If Bob climbed up the police radio tower then he would have been arrested.

(2) It is not the case that Bob was arrested.

(3) Therefore, it is not the case that Bob climbed up the police radio tower.

Disjunctive Syllogism

premise (1) P or Q

premise (2) not P

concl.    (3) therefore, Q

(1) Joe will eat an apple or Joe will eat a banana.

(2) It is not the case that Joe will eat an apple.

(3) Therefore, Joe will eat a banana.

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.

Fallacious Argument Forms

All four of these valid argument forms have a strong intuitive appeal. However, it is easy to make a tiny mistake when constructing these arguments, which can radically alter its validity. This happens so often that, based on such mistakes, logicians have introduced a group of fallacious argument forms, or formal fallacies (in contrast to the informal fallacies discussed earlier). These arguments are so similar to genuine valid argument forms, that they can be frequently mistaken for the real thing. We will consider three of these.

Fallacious Modus Ponens: fallacy of affirming the consequent

premise (1) if P then Q

premise (2) Q

concl.    (3) therefore, P

(1) If the president pushes the button, then a nuclear bomb with explode.

(2) A nuclear bomb exploded.

(3) Therefore, the president pushed the button.

Intuitively, we can see that this argument fails since, even if a nuclear bomb did explode, it wouldn’t necessarily be because the president pushed the button. The Whitehouse janitor, for example, might have accidentally pushed it. For comparison, here is the genuine version of modus ponens:

premise (1) if P then Q

premise (2) P

concl.    (3) therefore, Q

By comparing the two, we can see that the error with the fallacious form occurs in premise two by affirming the consequent, rather than by affirming the antecedent as is required in the genuine version of modus ponens.

Fallacious Modus Tollens: fallacy of denying the antecedent

premise (1) if P then Q

premise (2) not P

concl.    (3) therefore, not Q

(1) If Bob climbed up the police radio tower then he would have been arrested.

(2) It is not the case that Bob climbed up the police radio tower.

(3) Therefore, it is not the case that Bob was arrested.

This argument fails since Bob might be arrested for any number of reasons, such as tap dancing on top of a patrol car. Climbing the police radio tower is just one reason for someone to be arrested. Here again is the genuine version of modus tollens:

premise (1) if P then Q

premise (2) not Q

concl.    (3) therefore, not P

The mistake with the fallacy occurs in premise two by denying the antecedent, rather than by denying the consequent as is required in the genuine version of modus tollens.

Fallacious Disjunctive Syllogism: fallacy of asserting an alternative

premise (1) P or Q

premise (2) P

concl.    (3) therefore, not Q

(1) Joe will eat an apple or Joe will eat a banana.

(2) Joe will eat an apple.

(3) Therefore, it is not the case that Joe will eat a banana.

The fallacy in this example rests on forgetting that the “or” in logic is inclusive. Even if Joe does eat an apple, he could also eat a banana. Here, by comparison, is the genuine version of disjunctive syllogism:

premise (1) P or Q

premise (2) not P

concl.    (3) therefore, Q

The error with this fallacy is by asserting one of the disjuncts in premise two, rather than denying a disjunct.

Sound and Unsound Arguments

So far, we've seen that the first step in forming a good argument within propositional logic is that it must be valid. However, more is needed. A good argument must be valid and have all true premises. The combined requirements of validity plus truth is called soundness:

Sound Argument: an argument which (a) follows a valid argument form, and (b) has only true premises.

The "Hume" argument presented at the outset is an example of a sound argument since it is valid and has only true premises:

(1) If David Hume was a bachelor, then he was unmarried.

(2) David Hume was a Bachelor.

(3) Therefore, David Hume was unmarried.

First, the argument is valid since it follows a valid modus ponens argument form. Second, both premises are true. Premise 1 is true by definition, since a bachelor is defined as an unmarried man. Premise 2 is true because it is a fact of history: David Hume in fact was a bachelor his entire life. Since this argument is valid and has only true premises, then it is thereby a sound argument.

Two examples will illustrate why a sound argument must be both valid and have all true premises. The argument below follows a valid form but does not have true premises:

(1) Uncle George is a golfing shoe or Uncle George is a tennis shoe.

(2) It is not the case that Uncle George is a golfing shoe.

(3) Therefore, Uncle George is a tennis shoe.

The above argument is valid since it follows the form of disjunctive syllogism. But, premise 1 is obviously false: Uncle George is a human being and thus is not any type of shoe. The argument, then, is unsound because premise 1 is false.

There are also arguments that 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. With premise 1, “Air Force One” is simply the nickname of the president’s personal plane; whoever is the pilot of Air Force One will thereby be flying in the presidential plane. Premise 2 is true since flying in the presidential plane is a privilege of the president’s job. However, even though both premises are true, the argument follows the fallacious argument form of fallacious modus ponens, so this argument is also unsound.

In summary, 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.

E. INDUCTIVE LOGIC

Inductive vs. Deductive Arguments

All four of the above valid argument forms in propositional logic are classified as deductive, as defined here:

Deductive argument: an argument whose conclusion follows necessarily from its basic premises.

Deduction was the central intuition with soundness: if you have a valid argument with all true premises, then the conclusion follows with necessity. In a sense, the conclusion in its full force is already built into the combination of premises, and all that a deductive argument does is extract the conclusion from those premises. To “deduce” means to take some facts and see what necessarily follows from them.

The argument forms of propositional logic are just one type of deductive logical system, and, in fact, it is only within recent times that it has become the dominant deductive system within the field of logic. Prior to that, the main deductive approach was categorical syllogistic logic, a system first created by Aristotle over 2,000 years ago. Here is the classic example of a categorical syllogism:

1. All human beings are mortal things.

2. Socrates is a human being.

3. Therefore, Socrates is a mortal thing.

The categorical syllogism, as the name implies, is about categories of things, and how some categories are contained within other categories. Think of it like a series of boxes that are contained within each other. In this case, there is a large box labeled “Mortal Things”, within it a smaller box labeled “Human Beings” and within it an even smaller box labeled “Socrates”. Using this box metaphor, the above argument is this:

1. The box labeled “Human Beings” is inside the box labeled “Mortal Things”.

2. The box labeled “Socrates” is inside the box labeled “Human Beings”.

3. Therefore, the box labeled “Socrates” is inside the box labeled “Mortal Things.”

The conclusion states the obvious: the box containing “Socrates” is within the box of “Mortal Things”, and this is so because the Socrates box is in the “Human Beings” box, and that box is within the larger box of “Mortal Things”. The conclusion follows from the two premises with necessity.

The larger point here is that both propositional logic and syllogistic logic are deductive systems since their conclusions necessarily follow from their premises. But inductive logic takes a completely different approach: the conclusion is likely to be true, but it doesn’t follow from the premises with absolute necessity. Induction is about probability, not necessity. In contrast with deductive argumentation, the definition of an inductive argument is this:

Inductive Argument: an argument in which the premises provide reasons supporting the probable truth of the conclusion.

Here is an example of a basic inductive argument:

1. Rock 1 falls to the ground when I open my hand.

2. Rock 2 falls to the ground when I open my hand.

3. Therefore, all rocks similar to 1 and 2 will probably fall to the ground when I open my hand.

The first two premises here are observations about how two rocks behave in a similar fashion. Strictly speaking, all that this tells us is how only those two rocks behave. But the conclusion moves well beyond those two rocks and makes the sweeping claim that all other rocks similar to those two will behave similarly. The conclusion here does not follow with necessity from the premises, since, for all we know, the next rock that I pick up that is similar to rocks 1 and 2 will levitate to the sky. That is why the conclusion contains the critical word “probably”, which warns us that there is no guarantee that the next rock that is similar to rocks 1 and 2 will behave the same way. Even if the word "probably" does not appear in the conclusion of an inductive argument, it is always implied.

Induction is often connected with scientific experimentation. Scientists will look at a comparatively small group of cases and conclude that this is how things work in all similar cases. A test group of 100 bald men are given a drug, and they all grow hair. The scientists then conclude that this drug will probably promote hair growth on all bald men. This is no guarantee that it will work on every bald man, but the evidence does support the probable truth of the conclusion.

Inductive Probability

With deductive arguments, validity is the key concept: an argument is either valid or invalid, and there is no in between. With induction, however, the concept of validity makes no sense: induction is all about the probability of a conclusion, not its necessity. Thus, in place of validity, inductive arguments use a different standard, namely, inductive probability, which is the degree to which a conclusion is probable given the truth of the premises. There are degrees of inductive probability based on the relative strength or weakness of the inference. There is no sharp line between strong and weak inductive reasoning, but for convenience we will use the following four degrees of strength:

• Inductively very strong: probability is close to certain.

Example: “Every living person that we know of breathes air; therefore, Joe probably breathes air.”

• Inductively strong: probability is high.

Example: “Regular cigarette smoking reduces a person’s life expectancy by seven years on average; therefore, Joe, who smokes regularly, will probably die seven years earlier than he would otherwise.”

• Inductively weak: probability is low.

Example: “Some people watch reruns of the Andy Griffith Show; therefore, Joe probably watches reruns of the Andy Griffith Show.”

• Inductively very weak: probability is close to non-existent.

Example: “I once saw a guy balance a chair on top of his head; therefore Joe can probably balance a chair on top of his head.”

Depending on the type of inductive argument used, there are several factors that determine an argument’s inductive strength. But the critical point is whether the conclusions move too far beyond the data in the premises. For example, if every person we know of can balance a chair on top of their heads, then it is reasonable to conclude that a random person named Joe can do this too. But if very few people can balance a chair on top of their heads, then it is improper to conclude that Joe can do this.

Inductive Arguments Forms

There are several types of inductive arguments, but we will consider here just four commonly used forms. For each of these forms there are specific fallacies that often occur, which we will also examine.

Simple Enumerative Induction

The argument form of simple enumerative induction involves drawing a generalized conclusion about an entire class of things based on a few observations about members of that class. The rock example above is a good case in point. Below is the formula for this argument, an example of it, and the fallacy associated with it:

premise (1) Item x has attribute A

premise (2) Item y has attribute A

concl.    (3) Therefore, all items of the same type as x and y probably have attribute A

1. Rock 1 falls to the ground when I open my hand.

2. Rock 2 falls to the ground when I open my hand.

3. Therefore, all rocks similar to 1 and 2 will probably fall to the ground when I open my hand.

Fallacy of hasty generalization: drawing a general conclusion based on one or several atypical instances.

Simple enumerative induction is grounded in our psychological capacity to make generalizations based on a limited number of observations, and it is an essential survival mechanism for both humans and animals. The world is filled with facts, and, to get on with our lives, we need to simplify our experiences and identify common features in similar objects. Eating plants that smell like this will make us sick. Getting too close to animals that look like this will result in death. While our natural tendency to generalize does help us survive, in its raw form it is too inaccurate to use in scientific investigation. Francis Bacon, the father of the scientific method, famously criticized the use of simple enumerative induction on these grounds: “The induction which proceeds by simple enumeration is childish, leads to uncertain conclusions, and is exposed to danger from one contradictory instance, deciding generally from too small a number of facts, and those only the most obvious” (New Organon, 1.105). This is the point behind the fallacy of hasty generalization, that is, drawing a general conclusion based on one or several atypical instances. It can also lead to superstition: “yesterday I saw a black cat and then a tree fell on my garage, today I saw a black cat and then the water line in my house broke, therefore when you see black cats bad things happen”. It can also lead to harmful racial stereotypes: “Joe is Irish and he’s a drunkard, Bob is Irish and he’s a drunkard, therefore all Irish are drunkards.” There are, though, more scientifically rigorous ways of constructing inductive arguments that are based on statistical analysis, and thus attempt to avoid hasty generalization. We turn to these next.

Statistical Induction

The argument form of statistical induction involves drawing a conclusion about a population based on a statistically acceptable sample. Here are the details of this:

premise (1) n percent of a sample has attribute A.

concl.    (2) Therefore, n percent of a population probably has attribute A.

(1) 27% of 1033 randomly surveyed adults believe that God helps decide who wins sporting events.

(2) Therefore, 27% of the population probably believes that God helps decide who wins sporting events.

Fallacy of small sample: a conclusion is too strong to be supported by a small sample number.

Fallacy of biased sample: a conclusion is too strong to be supported by a nonrandom sampling technique.

With statistical induction, there are two distinct fallacies that are often committed, which are more specific than the broader fallacy of hasty generalization noted above. First, with the fallacy of small sample, the issue rests on what an appropriate size sample should be for a study, and this differs based on the field of research. For example, with psychological studies, such as testing the reaction time of elderly people, 30 is an acceptable number. With national surveys, such as the one above about sporting events, 1,000 is acceptable, which will have a margin of error of 3%. As an example of the fallacy of small sample, if in the above illustration we only randomly surveyed 50 people rather than 1033 people, then the margin of error would be too high to reliably support the conclusion. With the second fallacy, the fallacy of biased sample, the issue is not about the number of people sampled, but the nonrandom nature of the sample. As an example of the fallacy of biased sample, suppose in the above illustration the sample group was restricted to 1033 members of an organization called “The Association of Christian Sports Fans.” Here the sample size is large enough, but it would not be a random sample of the whole population, and, so, we could not reliably draw a conclusion about the views of the whole population.

Statistical Syllogism

The argument form of statistical syllogism involves drawing a conclusion about an item based on statistics about the population as a whole. Below are the details:

premise (1) n percentage of a population has attribute A.

premise (2) x is a member of that population.

concl.    (3) Therefore, there is an n probability that x has A.

(1) 36% of Americans ages 18-24 have tattoos.

(2) Joe is an American within that age range.

(3) Therefore, there is a low probability that Joe has a tattoo.

Fallacy of small proportion: a conclusion is too strong to be supported by the small population proportion (or percentage) with the attribute.

Statistical syllogism is like an added step to statistical induction. With the above example, suppose that you establish the truth of premise 1 through statistical induction using a random survey of 1033 adults. A next step might be to conclude the level of probability that a particular American named Joe will have a tattoo. The probability here is low since premise one tells us that only 36% of Americans ages 18-24 have tattoos. As an example of the fallacy of small proportion, suppose we concluded that “There is a very strong probability that Joe has a tattoo.” This new conclusion would be too strong since the population proportion is only 36%. The name “statistical syllogism” is based on its similarity to deductive syllogisms like the Socrates example above, where it moves from a general claim about Americans with tattoos to a specific one about Joe. Statistical induction, however, is inductive rather than deductive since its conclusion is about the probability that Joe has a particular feature (i.e., has a tattoo), rather than, in the Socrates example, the certainty that Socrates has a particular feature (i.e., is mortal).

Argument from Analogy

The argument from analogy involves a conclusion about one item based on its similarities with another item. Here are the details:

premise (1) Items x and y each have attributes A, B and C.

premise (2) Items x has an additional attribute D.

concl.    (3) Therefore, object y probably also has attribute D.

(1) Humans and chimpanzees each have pain receptors, neurological pain pathways within their brains, and natural pain killers.

(2) Humans consciously experience pain.

(3) Therefore, chimpanzees probably also consciously experience pain.

Fallacy of false analogy: comparing two items that have trivial points in common, but differ from each other in more significant ways.

Philosophers often use arguments from analogy like this one, such as the design argument for God's existence from analogy, or the argument for the existence of other minds from analogy. The vulnerability of all arguments from analogy is that the two items being compared may not be sufficiently similar, and thus commits the fallacy of false analogy. Here is an example:

(1) Humans and store manikins both have the human form, stand upright, and wear clothes.

(2) Humans consciously experience pain.

(3) Therefore, store manikins probably also consciously experience pain.

While premises 1 and 2 are both true, this argument commits the fallacy of false analogy since the three attributes in premise 1 (human form, standing upright, and wearing clothes) are trivial attributes that are not relevant to the psychological ability to consciously experience pain. Our original argument is much stronger since it itemizes attributes that are relevant to the conscious experience of pain, namely, pain receptors, neurological pain pathways, and natural pain killers.

The full subject of logic within philosophy goes well beyond what we have covered in this chapter, and, at its highest levels, it crosses over into the fields of mathematics and linguistics. The emphasis there is on creating rigorous logical systems, regardless of their application to ordinary discourse. What we have covered here, though, are basic logical tools that are assumed in all philosophical discussion, and which have been used throughout the chapters in this work. Our opening and closing chapters express philosophy at its extremes. The first chapter, on the meaning of life, explores the most concrete and personal issue of philosophical inquiry. This last chapter on logic, though, is about philosophy's most abstract and impartial subject. The bulk of philosophy lies somewhere between these extremes. We examine issues that are of critical importance to us all, such as the human mind, reality and ethics, but do so using the impartial tools of logic. It remains to be seen whether we can ever find definitive answers to the philosophical questions that we ask, even when using these tools. But fans of philosophy are compelled to at least try.

IN-CLASS EXERCISES

Instructions: Determine whether the following are propositions. If some are not propositions, see if they can be rewritten as propositions.

(1) I have superpowers.

(2) Not here, Bob!

(3) I think I’m going to sell little Joey into slavery.

Instructions: Identify the premises and conclusions in the following arguments, and identify any premise and conclusion indicators.

(4) English is the best language since it’s the only one that I speak.

(5) Bob likes to argue all the time, and for that reason he would make a good lawyer.

(6) In view of the fact that Joe cheated on his taxes, we consequently cannot appoint him to the ethics committee.

Instructions: Diagram the following arguments. First number each statement, then use plus signs and arrows to designate the argument structure as either a joint inference or an independent inference.

(7) Joe has no friends since the only people he knows are on social media, and those aren’t real friends.

(8) Bob was voted most popular student in class, and Bob is always seen with lots of people around him. Thus, Bob has many friends.

(9) Joe and Bob aren’t friends because each says that he can’t stand the other, and each angrily insults the other when they pass in the hall.

Instructions: Identify the informal fallacy in each of the following.

(10) “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!

(11) Hey, forget about Beth, she’s nothing special. Is there anything special about her kidneys, tonsils, or small intestine? She’s only a collection of those things.

(12) Of course the Major thinks that the Army offers good career opportunities. He’s an Army man himself.

(13) I think Beth will go out with you. I haven’t heard anything which suggests that she wouldn’t.

(14) We have a good faculty here at Preppy State University. Therefore, Dr. Joseph Drunkard, who teaches here, is a good faculty member.

Instructions: In each of the following identify the logical connective being used and translate the proposition into standard form.

(15) Father Joe’s marriage to Beth implies that he first leaves the priesthood.

(16) I was accepted at Yale University, but I’d much rather attend Thrift Community College.

(17) Bob’s name does not appear on Santa’s “nice” list.

Instructions: Determine which of the following are well-formed nested propositions.

(18) if P then (Q or R)

(19) (P and Q) not

(20) not (P or Q)

(21) P and (if Q then R)

Instructions: Translate the following premises and conclusions into standard form and decide which valid argument form or fallacious argument form is being used.

(22) 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. Therefore, if they perform, dudes will stage dive.

(23) Either Bob will go bankrupt, or I will. Bob will go bankrupt. Therefore, I will not.

(24) If Joe flunks out of college, then his brother Bob will inherit the family business. Joe will not flunk out of college. Therefore, Bob will not inherit the family business.

Instructions: 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.

(25) Joe will fail his exam. (modus ponens)

(26) Polly wants a cracker. (disjunctive syllogism)

(27) If you insult Beth’s mother, you will go to the hospital. (hypothetical syllogism)

(28) Thrift Community College is not a good school. (modus tollens)

Instructions: Are the following arguments valid, invalid, sound, or unsound?

(29) 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.

(30) 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.

Instructions: The following test your understanding of soundness.

(31) Can a valid argument have a false conclusion?

(32) Can a sound argument have a false conclusion?

Instructions: What is the inductive strength of each of the following (that is, very strong, strong, weak, very weak)?

(33) Some notable guitarist have died in their 20s. Joe is a notable guitarist. Therefore, Joe will probably die in his 20s.

(34) College dropouts make \$1 million less during their careers than college graduates. Joe is a college dropout. Therefore, Joe will probably make around \$1 million less during his career than an average college graduate.

(35) 45% of Americans go to church at least once a month. Joe is an American. Therefore Joe will probably go to church this month.

Instructions: For each of the following, indicate the inductive argument form that is followed, and whether it commits any inductive fallacy.

(36) Joe and Bob live in the same town, listen to the same music, and like the same sports teams. Joe is Presbyterian. Therefore, Bob is probably also Presbyterian.

(37) 60% of college students in the U.S. are women. Preppy State University is a U.S. College. Therefore, there is a very high probability that the next student who walks out of Preppy State’s student center will be a woman.

(38) 100% of 20 randomly surveyed adults in the small town of Hornbeak, Tennessee shop at Walmart. Therefore, 100% of Americans shop at Walmart.

(1) This is a proposition as written.

(2) This is not a proposition. It can be rewritten as a proposition as follows: “I request that you do not do that here Bob.” The true/false component of the revised version is “I request that. . . .”

(3) This is a proposition as written. The true or false component is “I think. . . .”

(4) The conclusion is “English is the best language”, the premise is “it’s the only one that I speak,” and “since” is a premise indicator.

(5) The premise is “Bob likes to argue all the time,” the conclusion is “he would make a good lawyer,” and “for that reason” is a premise indicator.

(6) The premise is “Joe cheated on his taxes,” the conclusion is “we cannot appoint him to the ethics committee,” “in view of the fact that” is a premise indicator, and “consequently” is a conclusion indicator.

(7) The numbered argument and diagram are as follows:

1 [Joe has no friends] since 2 [the only people he knows are on social media], and 3[those aren’t real friends].

2 + 3 |→ 1

(8) The numbered argument and diagram are as follows:

1 [Bob was voted most popular student in class], and 2 [Bob is always seen with lots of people around him]. Thus, 3 [Bob has many friends].

1 |→ 3 and 2 |→ 3

(9) The numbered argument and diagram are as follows:

1 [Joe and Bob aren’t friends] because 2 [each says that he can’t stand the other], and 3 [each angrily insults the other when they pass in the hall].

2 |→ 1 and 3 |→ 1

(10) Equivocation. This argument assumes two meanings of the phrase “Dead Milkmen.”

(11) Composition. This argument assumes that a property of each part (i.e., “nothing special”) applies to the whole (i.e., Beth herself).

(12) Argument against the person. This argument attacks the personal attribute of the Army Major, without examining the content of the Major’s argument.

(13) Argument from ignorance. This argument concludes that something is true (i.e., that Beth will go out with you) since it isn’t proven false.

(14) Division. This argument assumes that a property of the whole university (i.e., “good faculty”) applies to each member (i.e., Dr. Joseph Drunkard).

(15) This is a conditional (if-then). The clue term is “implies”, and the proposition translates “IF Father Joe marries Beth THEN he must first leave the priesthood.”

(16) 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.”

(17) This is a negation. The clue term is “not”, and the proposition translates “It is not the case that Bob’s name appears on Santa’s ‘nice’ list.”

(18) This is well formed as written.

(19) This is not well formed as written since (a) a logical connective needs to precede “not” and (b) a proposition needs to follow "not".

(20) This is well formed as written.

(21) This is well formed as written.

(22) 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.

(23) 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, IT IS NOT THE CASE THAT I will go bankrupt.

(24) This is an invalid fallacious modus tollens argument. Here it is translated into proper form:

IF Joe flunks out of college, THEN his brother Bob will inherit the family business.

IT IS NOT THE CASE THAT Joe will flunk out of college.

Therefore, IT IS NOT THE CASE THAT his brother Bob will inherit the family business.

(25) A possible created proposition is “Joe parties all night.” The argument, then, is this:

IF Joe parties all night, THEN Joe will fail his exam.

Joe partied all night.

Therefore, Joe will fail his exam.

(26) 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.

Therefore, Polly wants a cracker.

(27) A possible created proposition is “Beth will hit you.” The argument, then, is this:

IF you insult Beth’s mother, THEN Beth will hit you.

IF Beth hits you, THEN you will go to the hospital.

Therefore, IF you insult Beth’s mother, THEN you will go to the hospital.

(28) A possible created proposition is “it will have a football team.” The argument, then, is this:

IF Thrift Community College is a good school, THEN it will have a football team.

IT IS NOT THE CASE THAT it has a football team.

Therefore, IT IS NOT THE CASE THAT Thrift Community College is a good school.

(29) 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.

(30) 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.

(31) A valid argument can have a false conclusion, such as the earlier argument that concludes “Uncle George is a tennis shoe.” Validity only concerns the form of an argument, not the truth of the content.

(32) A sound argument cannot have a false conclusion. Validity is a truth-preserving mechanism; thus, if a valid argument has true premises (and thus is sound), then the truth carries over into the conclusion.

(33) The inductive probability is very weak since the conclusion is much stronger than what the premises indicate. Of the hundreds of notable guitarists, the early death of “some” of them implies almost nothing about the probability of Joe’s early death.

(34) The inductive probability of this argument is strong since the conclusion is consistent with the premises. However, it does not rise to the level of “very strong” since \$1 million is only the average difference, and for specific dropouts it might be much higher or lower than this.

(35) The inductive probability of this argument is weak since the conclusion goes beyond what the premises state. Premise 1 states that monthly church attendance of the average American is 45%, but the percentage would need to be above 50% to support the conclusion that Joe will "probably" attend this month.

(36) This is an argument from analogy that commits the fallacy of false analogy. The attributes of living in the same town, listening to the same music, and having the same parents are trivial and not clearly connected with the attribute of being lazy.

(37) This is an argument from statistical syllogism. It commits the fallacy of small proportion since the clause “very high probability” in the conclusion is much stronger than the 60% indicated in the first premise.

(38) This is an argument from statistical induction that commits both the fallacy of small sample and biased sample. A sample of only 20 people is too small and will produce a high margin of error. And, a sample from only a small town in Tennessee is not a random sample of the whole American population.

TAKE HOME STUDY QUESTIONS (these are the study questions to submit for home work)

Instructions: Determine whether the following are propositions. If some are not propositions, see if they can be rewritten as propositions.

(1) I think that the Feds are watching me.

(2) Put down the porcupine and come out with your hands up.

(3) I have contacted the spirits of the dead.

Instructions: Identify the premises and conclusions in the following arguments, and identify any premise and conclusion indicators.

(4) In view of the fact that I almost never wash my hair, the odds are pretty slim that I’ll get a date.

(5) I’ll do whatever the guy in front of me wants given that he is holding a gun.

(6) There is an ever-increasing population of flesh-eating zombies; it follows that the apocalypse has begun.

Instructions: Diagram the following arguments. First number each statement, then use plus signs and arrows to connect the numbers in the appropriate argument structure.

(7) My life is over since I lost my cell phone, and my life depends upon that device.

(8) I won’t play follow the leader with you since you have no leadership skills, and, even if you did, you’d just lead me right off a cliff.

Instructions: Identify the informal fallacy in each of the following.

(9) Defenders of capital punishment are wrong since they think that even shoplifters should be executed.

(10) John has diabetes, which he probably got from drinking water since he drinks more water than anyone I’ve ever seen.

(11) The sign said “fine for littering”. Since it was fine, I littered.

Instructions: Put the following in standard propositional form by replacing the clue word with the correct logical connective (for example, “d however e” translates into “d and e”).

(12) x follows from y

(13) a although b

(14) r implies s

Instructions: Which of the following are not well-formed propositions?

(15) (P or Q) not (R and S)

(16) if (P and Q)

(17) (P or Q) if and (R then S)

(18) What is the name of the following argument form or fallacy?

if A then B

not B

therefore, not A

(19) What is the name of the following argument form or fallacy?

X or Y

not X

therefore, Y

(20) Make up a modus tollens argument which leads to the following conclusion: “Therefore, the taxman is not my friend.”

(21) 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.

Instructions: What is the inductive strength of the following (that is, very strong, strong, weak, very weak)?

(22) A sixteen-year-old girl was struck by lightning and the next day she won the lottery. Joe was struck by lightning today. Therefore, Joe will probably win the lottery tomorrow.

Instructions: For each of the following, indicate the inductive argument form that is followed, and whether it commits any inductive fallacy.

(23) 21% of Americans believe a UFO crashed at Roswell in 1947. Joe is an American. Therefore, there is a good chance that Joe believes a UFO crashed at Roswell in 1947.