If..philosophy.kslinker.com/If-then1.pdf · 1. Introduction to Part One...

If. . .thenAn Introduction to Logic

Kent Slinker

February 5, 2014

Part I.

Informal Logic

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Contents

I. Informal Logic 2

1. Introduction to Part One 4

2. Relations between statements 62.1. Entailment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2. Relevance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3. Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3. Arguments 133.1. Combining propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2. The Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3. Validity, Inference, and Invalidity . . . . . . . . . . . . . . . . . . . . . . . 18

4. More properties of arguments 254.1. Truth versus Inference, Soundness, Strength and Weakness . . . . . . . . 254.2. Valid Argument Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3. Pseudo-valid argument forms. . . . . . . . . . . . . . . . . . . . . . . . . . 324.4. The Counter Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5. Induction 405.1. Evaluating Inductive Arguments . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.1. Exercises Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Answers to Selected Problems 47

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1. Introduction to Part One

Logic, like Mathematics, Psychology, History, Physics, and many other academic fieldsis a broad subject, with many specialized areas and sub-disciplines. The aim of the firstpart of this text is to introduce you to the main ideas behind what is sometimes calledPhilosophical Logic in an informal way. In general, Philosophical logic, at the basiclevel we will deal with in Part One, is a study of the principles of correct reasoning. Ofcourse, to judge something as being correct is to assume that there is an incorrect formas well, and so it is with logic - so to state matters at once as clearly as possible: In Part1 we will be primarily concerned with an examination of the process of reasoning, anddeveloping tools and procedures, concepts and definitions, indispensable for the processof distinguishing correct from incorrect reasoning.To start us down that road, consider the following question:

Suppose I roll two dice and adding together the numbers of each face I get asum of 12. What number must be on the face of each die?1

Before you answer, think carefully about what the question is asking and what infor-mation is given. What are the relevant parts of the question? What words cause you tobe careful with your response, and given a possible answer, what examples convince youthat your answer is correct? What examples or considerations may cause you to retractyour answer? Consider your answer carefully before looking at the answer given in thefootnote below.

Was your answer correct? If not, do you understand why? The process of lookingat information, and drawing conclusions from that information invites us to examinerelations between statements, and our primary question for the next few chapters willbe to learn how to determine whether the truth of one statement affects the truth ofanother statement, and when it does, what accounts for that relationship. In going downthis road, we will build foundational conceptual tools, which we will master by doing

1It is impossible to know for certain which number is on the face of each dice. Since there is nothingabout the word “dice” that insists on it being a standard dice - like the ones used in a game of crapsor monopoly - it leaves open the possibility that one dice might have all zeros and the other dice all12s, or one dice might have only 6 or 5 on each face, while the other dice might have 6 or 7 (givingtwo possibilities 6,6 or 5,7) - this is just a few of the many many possibilities that make certainty inthis case impossible. However, if it were stipulated that the dice were standard dice, then the answerwould be 6 on each face. This tells us to be careful about assuming too much from a given statement,or at least to state any assumptions in our conclusion, such as, “If the dice are standard, then theanswer is 12, otherwise no certain answer can be given . . ."

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1. Introduction to Part One

exercises, and building on these foundational concepts we will eventually reach a pointwhere a more formal treatment of our subject seems both desirable and necessary. Thisformal treatment will be given in Part Two of this text.

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2. Relations between statements

Consider the following two statements:

1. I have a dime, a quarter, and 11 pennies in my pocket.

2. I have some change in my pocket.

For simplicity’s sake, let’s call the first statement p and the second statement q. Now wepose our first question related to logical analysis, “Does the truth of p have anything todo with the truth of q?” - By this, we simply mean, in the broadest of terms, whetherthe truth of the statement, “I have a dime, a quarter and 11 pennies in my pocket”lends credence to the statement, “I have some change in my pocket”. Clearly the answeris yes, for if the first is true, then certainly the second must be true. Why is this so?The answer to this question clearly has to do with the connection between the terms,“dime”, quarter” and “pennies” and the word “change” when used in this context. Nowlets turn the question around and ask if the truth of q has anything to do with the truthof p, or more plainly, does the truth of, “I have some change in my pocket” affect in anyway the truth of, “I have a dime, a quarter, and 11 pennies in my pocket”. To make thisas clear as possible, what we are asking is whether the truth of q gives us any reasonwhatsoever to suspect that p might be true. Again, the answer is yes, as one possibleway to have change in one’s pocket is indeed to have a dime, a quarter and 11 pennies,but as this is just one of many combinations of coins which makes the statement true,the truth of q does not guarantee the truth of p.Let’s consider another example:

1. Today is October 1.

2. The word, “ostentatious” has 12 letters.

As above we will refer to the first sentence as p and the second as q, as ask the samequestions: Does the truth of p have anything to do with the truth of q, and similarly doesthe truth of q have anything to do with the truth of p? In both cases, we are inclined tosay no. The fact that today is the first of October is unrelated to the fact that the word“ostentatious” has 12 letters, and the fact that the word “ostentatious” has 12 lettersas nothing to do with today’s date. As above, the reason is related to the connectionbetween the meanings of the terms, but for now we will put aside that connection andsimply record our observations. Given two statements, p and q, then we have at leastthe following three cases:

1. The truth of one statement guarantees the truth of another statement.

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2. The truth of one statement gives some support to the truth of another statement,but does not guarantee it.

3. The truth of one statement has nothing to do with the truth of another statement.

Let us now give informal definitions to these types of relationships between statements,as they will be used again and again in our informal analysis of Logic.

2.1. EntailmentDefinition 1. If the truth of a statement p guarantees that another statement q mustbe true, then we say that p entails q, or that q is entailed by p.

Examples help us understand definitions, which will play a major part in our study oflogic. Before we give some examples, it might be helpful to point out that our use of theletters p and q is arbitrary. We could just as well use k, m, #, © or any other mark, aslong as it is understood what these symbols stand for. In our case, our symbols are justgoing to stand for sentences which have the property of being either true or false, wherefalse is understood to mean the same thing as “not true". Sentences with the property ofbeing either true or false are called propositions in Logic. Prudence suggests that thesymbols we choose cause as little confusion as possible, and in general it is a good ideato realize that p and q are “dummy” variables, choices which are arbitrary and hencedefinitions which use them do not depend on our choice of symbol. For the most part,we will stick with tradition in this text, and use letters like p, q, r, s, and so forth tostand for propositions. However, for reasons which will become clear when we turn toour formal analysis of Logic, using the letters t, f , or v to stand for propositions mightcause confusion, so we will not use them in this textbook.Notice also the key term, “must be true” in our above definition of entailment.

“Must be” is another way of stating that something is necessary, something for whichno other option or possibility exists. “Must be” encodes one of the most fundamentalconcepts of Logic, the concept of logical necessity. Here are some more examples ofstatements which entail other statements:

Example. Let p be, “Mary knows all the capitals of the United States”, and let q be,“Mary knows the capital of Kentucky”, then p entails q.

Example. Let q be, “Every one in the race ran the mile in under 5 minutes”, and let pbe, “John, a runner in the race, ran the mile in less than 5 minutes”, then q entails p.

Example. Let k be, “The questionnaire had a total of 20 questions, and Mary answeredonly 13”, and let n be the statement, “The questionnaire answered by Mary had 7unanswered questions”, then n is entailed by k.

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Example. Let j be, “The winning ticket starts with 3 7 9”, and let c be the statement,“Mary’s ticket starts with 3 7 9 so Mary’s ticket is the winning ticket”. In this case c isnot entailed by j.

If the relationship of entailment seems conceptually familiar, there is probably a goodreason for this. We say that true conditional statements, or true implications are ex-amples of entailment. Hence any conditional statement, (e.g. a statement in the form“if . . .. then”) which is true, is an example of entailment. In Mathematics, the morefamiliar term is “implication”.The exercises at the end of this chapter will give you more practice with the concept

of entailment, which is central to our study of Logic. Let us now turn to the concept ofrelevance by providing an informal definition.

2.2. RelevanceDefinition 2. Let p and q be two propositions, then p is relevant to q if the truth ofp counts for the truth of q but does not guarantee it.

This is an informal definition indeed, as seemingly all we have done is replace the wordrelevance with the phrase, “counts for the truth of ”. However providing a more detaileddefinition would take away from the basic concepts we seek to cultivate at this juncture.One important point to note about this definition is that entailment is not a specific formof relevance. Other texts may define this term differently, so if you consult another text,make sure you check how they define relevance. This underscores the need to stipulateprecise definitions and allow others to know the intended use of specific terms.

Example. Let p be, “Jane knows the capital of Arizona”and let q be, “Jane knows allthe capitals of the United States” , then the truth of p is relevant to the truth of q.

Here q could be made true by the conjunction of several cases; Jane knows the capitalof California, and Jane knows the capital of Nevada, and Jane knows the capital of. . ., and clearly Jane knowing the capital of Arizona is one such case. In this case therelevance results in the fact that p is one of the many ways which together make q true.Note that on one hand p is relevant to q, but on the other hand q entails p. The order ofterms (p first, then q, or q first then p) makes a difference. We will visit this importantcharacteristic of order of terms in more detail in Part 2 of the text.

Example. Let p be, “John ate uncooked meat and later came down with a case of e-coliinfection” and let q be, “Eating uncooked meat is a cause of e-coli infection”, then p isrelevant to q.

Clearly p alone does not establish q, but the fact that John ate uncooked meat andlater came down with a case of e-coli is evidence which supports the truth of q.

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Example. Let p be, “Mr. Smith is a millionaire” and let q be, “The Smiths live in amansion”, then p is relevant to q.

In all of the above cases, we have two statements, p and q, where the truth of p isrelevant to the truth of q. In other words, knowing that p is true lends some credence tothe truth of q, but unlike entailment, the truth of one statement does not guarantee thetruth of the other. We might think of relevance in terms of degrees of evidence. Whenwe want to establish that a certain statement is true, many times we offer evidence fromvarious sources. Each single piece of evidence is related to the truth of the statementwe want to establish (is relevant to it), but in many cases no single piece of evidence byitself firmly establishes it.

2.3. IndependenceAfter becoming acquainted with the concept of relevance, it is natural to ask about theconcept of irrelevance. If q is irrelevant to p, then the truth of q does not support thetruth or falsity of p. Instead of using the term irrelevance for this case, it is customaryto use the term independent instead. Our last relation between statements is that oftruth independence, which we now define.

Definition 3. Let p and q be two statements. We say p and q are logically indepen-dent if the truth of one does not effect the the truth of the other.

The notion of truth independence will play a central role in formal logic, but for nowwe will just examine some examples to get a feel for what it means for two propositionsto be be independent.

Example. Let p be, “Tom’s first child was born October 9, 2010”, and let q be, “Pianoshave 88 keys”, then the truth of q is independent of the truth p.

Example. Let p be, “At normal pressure water freezes at 0 degrees C”, and let q be,“February has 28 days except for leap years”, then q is logically independent of p.

Example. Let p be, “Columbus sailed for the Americas in 1492”, and let q be, “Theconcert ended at 10:15pm ”, then q is logically independent of p.

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Sometimes it is difficult to say whether two statements are logically independent orrelevant, as relevance allows for the truth of two propositions to be connected even inremote ways. As we will soon see, this poses no problem for the Logician, who is primarilyinterested in hypothetical cases where the question, “If. . .then” is of more interest manytimes than the question of “is actually”. As a result, when questions concerning relevanceversus independence arise, the Logician can just consider what follows if p is relevantto q, and similarly what follows if q is independent of p. A similar problem arisesbetween the relations of entailment versus relevance. Exploring this connection hasbeen one of the most fruitful chapters of Logic, and in a sense has defined the directionof logical investigation for years. In particular the question of entailment versus verystrong relevance characterizes what is known as the problem of induction, which we willencounter in some detail in Chapter 5.

Exercises Section 2

Preliminaries:a) Without looking (but with having actually read the text) write out the definition

of Entailment, Relevance and Independence. Compare your answers to the actual defi-nitions given in the text. Are your definitions equivalent? If not, highlight areas in yourdefinition which are different than the original definition.b) Write out the actual definitions several times, until you get them correct. State

that you have done so.c) Find five examples of statements p and q, such that p entails q, five examples where

p is relevant to q, and 5 examples where p and q are independent.d) Give reasons why these distinctions between relations between statements are im-

portant to the process of reasoning,

Main Exercises:State the relation between p and q. Write E if p entails q, R if p is relevant to q, and

I if q is logically independent of p. Recall the order of p and q in the question mattersin the cases of Relevance and Entailment.

1.* Let p be, “The capital of Arizona is Phoenix” and let q be “January has 31days”.

2. Let p be, “The Mississippi river is free from ice” and let q be “Today is May8”.

3.* Let p be, “Mary is a grandmother of 10” and let q be “At least one of Mary’schildren has children”.

4. Let p be, “Tom’s favorite color is green” and let q be “John owns a greentruck”.

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5.* Let p be, “The shortest driving distance from Tucson to Phoenix is 100 miles,and John drove from Tucson to Phoenix” and let q be “John drove at least100 miles on his trip from Tucson to Phoenix”.

6. Let p be, “The book has a total of 237 pages” and let q be “The book weighsmore than 6 ounces”.

7.* Let p be, “Mary knows the capital of French Guyana” and let q be “Maryknows every world Capital”.

8. Let p be, “27 out of 30 students graduated in May” and let q be “The tallyof monthly traffic accidents increased in October by 3%”.

9.* Let p be, “If John gets a raise, then he will buy a new car, and he did get araise” and let q be “John bought a new car”.

10. Let p be, “John bought a new car” and let q be “If John gets a raise, thenhe will buy a new car, and he did get a raise”.

Answer True or False. Justify your answer by appealing to definitions or examples orother reasons.

Example. If p entails q, then q can be false even if p is true.FALSE: The definition of entailment states that if p entails q, then if p is true, q must

be true.

Example. If p entails q, then q can be false.TRUE: Let p be, "Yesterday was Wednesday", and let q be "Today is Thursday",

so p entails q (by definition of weekdays), but if today is some other day rather thanThursday, we have that p entails q, where q is false. This example highlights the “If”part in the definition of entailment, since If p is true, then q must be true, is not thesame as, “p is actually true, so q is also true”.

11.* If p entails q, then q entails p.

12. If p is relevant to q, then q is relevant to p.

13. If q is logically independent of p, then p is logically independent of q.

14. p entails q if it is impossible for q to be false and p to be true.

15.* Since our choice of letters to represent propositions is arbitrary, we can letp stand for one proposition, and a different proposition at the same time.(hint: what would happen to our definitions if this were allowed?)

16. Not all sentences in English are propositions.

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17.* Propositions are sentences which have the property of being true or false.

18. Given three propositions p, q, and r, it may be the case that neither p orq alone entail r, but taken together they do. In other words, it may be thecase that p does not entail r, and q does not entail r, but p and q togetherentail r.

19.* If two statements are logically independent, then both p and q can be false.

20. If p entails q, then both p and q can both be false.

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3. ArgumentsIf Philosophical Logic has a single most important object of study, that object is certainlythe argument. In this chapter we will learn what an argument is in the Logical sense, anduse the relations of entailment, relevance and independence to understand the logicalconcept of inference. This will allow us to classify all arguments into two basic types,valid and invalid. Take care, many of the terms that are central to Chapter 3 will havea very different meaning in the context of Logic than elsewhere. It will be importantto keep this in mind to avoid errors and to master the definitions and concepts in thischapter.

3.1. Combining propositionsIn our analysis in the previous section, we allowed ourselves to refer to specific propo-sitions with single letters, such as p, q, and r. We will continue to use this conventionthroughout this book, indeed, when we start our formal analysis of arguments and logicalconcepts, such single letters will dominate our discussion, together with other symbolswhich do not stand for propositions but instead words like, “and”, “or”, “if. . .then” andothers. To anticipate such usage, let us informally stipulate that two propositions p andq are equivalent only if p and q are true (or false) under the exact same conditions orcircumstances. For example, if p stands for, “Exactly 24 hours have passed since weput the petri dish in the incubator”, and if q stands for, “Exactly one day has passedsince we put the petri dish in this incubator”, then p is equivalent to q. Keeping thisin mind, suppose we have two propositions, p and q, which are not equivalent, can weform another proposition whose truth is connected to both p and q but not equivalentto either? The answer is yes, as a matter of fact there are many ways this can be done,but for the moment we will just focus on one way (and look at others when we turn toour formal analysis of Logic).To illustrate how this is done, consider two specific propositions p and q and let p be,

“John is a student at Pima Community College” and let q be, “Mary is a student atthe University of Arizona”. Supposing that John and Mary are not connected in anyway, we can stipulate that p and q are independent propositions. We want to createanother proposition, which we will call r, whose truth is connected to both p and q.The natural way to do this is to join p and q together with the conjunction, “and”. Inother words, let r stand for, “John is a student at Pima Community College and Maryis a student at the University of Arizona” . Symbolically we might say that r =“p andq”. Clearly r is not equivalent to p or to q alone, but the truth of r entails the truthof both p and q individually! The process of joining two propositions with the word“and” is called conjunction. In the above example, we restricted the use of conjunction

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3. Arguments

to two propositions which are not logically equivalent. This restriction assured us thatthe resulting proposition was not equivalent to either of the original two. In general,we can ignore this restriction and can use conjunction freely between any number ofpropositions to create another proposition whose truth entails the truth of each of theindividual propositions conjoined with the word, “and”.As it turns out, we can repeat the above process and replace the word “and” with the

word “or”, hence r would stand for, “John is a student at Pima Community College orMary is a student at the University of Arizona” . When we use the word “or” we saythe resulting proposition r is the disjunction of p and q, but in this case the truth of ronly entails the truth of at least one of individual disjunctions p or q.To summarize, we can take any number of propositions, join them all together with

multiple uses of the word “and” or “or” to form another proposition which is the con-junction or disjucntion of all of the individual ones. We will learn more about thisfundamental process when we study formally the truth table definitions of both and andor later on.

3.2. The ArgumentSuppose someone wants to assert a claim that something is true. Following conventionwe will call that claim q (recall that our letters stand for propositions, sentences whichare either true or false). Now if q is not obviously true, it seems appropriate for theperson asserting the claim to support it with some type of evidence, or at the very leastprovide reasons for claiming q is true. Since evidence comes in the form of propositionswhich are either true or false, then we can also denote each piece of evidence as p, r,s, etc. When we do this, we are saying that because of p, r, s, etc. it is reasonable toaccept the claim q as true. One of the major goals of Philosophical Logic is classifyingarguments and discovering whether an argument really gives good reasons for acceptingits main claim. This process of reflection and analysis is called reasoning, and as amatter of fact, it is of such importance that many textbooks define Philosophical Logicas the process of distinguishing between good and bad reasoning! Let us now informallydefine the argument, our central object of study in Part 1.

Definition 4. An argument consists of a claim, which is called a conclusion, togetherwith at least one proposition, called a premise, that is given to support the truth of theconclusion.

It is important to note that an argument consists of two parts, a conclusion and atleast one premise given to support the conclusion. Informally, this is saying somethingsimilar to: “because p is true, then it is reasonable to conclude q is true”. In this case,p is the premise, and q is the conclusion. In real life, arguments are far more complexthan this simple but abstract example, and we will examine many in this chapter. Inparticular, most arguments have more than one premise, and many times more thanone conclusion. When this happens, we can try to identify the main conclusion, andconsider the other conclusions as acting as premises (in context of the entire argument)

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3. Arguments

which are supposed to support the main conclusion. Let’s illustrate this process withsome examples.

Example. Because of heavy rush-hour traffic, Tom missed his flight.

In this case, our conclusion (claim) is, “Tom missed his flight”, and the reason givenfor this claim is the single premise, “Because of rush hour traffic”. In this simple case, itis easy to see which part of the argument is the conclusion and which is the premise.

Example. If Henry is the person who damaged the rental car, then he must have beenin San Diego during Spring Break. If Henry were in San Diego during Spring Break,then he could not have been in Tucson at the same time. But we know Henry was inTucson during Spring Break, so Henry is not the person who damaged the rental car.

In this example, the conclusion is that Henry is not the person who damaged therental car. The premise that Henry was in Tucson during Spring Break supports theclaim that Henry was not in San Diego, which in turn leads directly to the conclusionthat Henry was not responsible for the damaged rental car, since after all, the very firstpremise asserts that, “If Henry is the person who damaged the rental car, then he musthave been in San Diego during Spring Break.”This is an example of a chain of reasoning, where one premise leads to another con-

clusion which together with another premise leads to the final conclusion. Chains ofreasoning of a special type will be studied in detail in Chapter X.

Example. It is a good idea to make sure you have working fire alarms in your house.Just look at the family whose house burned down on Christmas. They lost everything intheir house, and almost lost their daughter who nearly died of smoke inhalation. Theirhouse did not have working fire alarms, and for that reason, the fire itself went unnoticedwhile the family slept. They were saved only by the chance occurrence of a neighbor’steenage son arriving home late from a Christmas-eve date who noticed the fire and wokethe family up.

This argument is a bit more complicated than the previous examples, but it is morelike arguments we encounter in our daily lives. In order to start our analysis of theargument, we have to break it down into its parts, which means finding its premisesand the principle conclusion. To find the main conclusion, consider each sentence inturn, and ask, “Is this the main claim all the other sentences taken together wish toestablish?” Asking ourselves this question with respect to the above argument, we seethat the main conclusion is stated at the very start of the argument, namely, “Its a goodidea to make sure you have working fire alarms in your house.” The remainder of thesentences basically give reasons that support this conclusion.Lets turn to those reasons now and examine how they support this claim. In doing

so, we may have to identify some unwritten assumptions that the author has made, andre-write some sentences so that they take the form of propositions (sentences which are

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3. Arguments

either true or false). To help with this process, lets enumerate each sentence of theargument and consider each in turn:

1. It is a good idea to make sure your have working fire alarms in your house.

2. Just look at the family whose house burned down on Christmas.

3. They lost everything in their house, and almost lost their daughter who nearlydied of smoke inhalation.

4. Their house did not have working fire alarms, and for that reason, the fire itselfwent unnoticed while the family slept.

5. They were saved only by the chance occurrence of a neighbor’s teenage soon arriv-ing home late from a Christmas-eve date who noticed the fire and woke the familyup.

Considering each sentence in turn, we discover that 2 - 5 support 1, which is just sayingthat 1 is the main conclusion and 2 - 5 are premises that support that conclusion.However, the really attentive student might point out that, according to our definitionof an argument, sentence number 2 does not seem to be part of the argument! Why?According to the definition of an argument, premises are propositions, and propositionsare statements which are either true or false. In this case, sentence number 2 is acommand, not a proposition. Commands are neither true or false (whether or not youobey the command is a proposition, but the command itself is not). According to ourdefinition, 2 can not form part of our argument. But surely its truth is connected in someway to the to the argument as a whole and should not be ignored. How do we resolvethe problem? Of course, we could just simply re-define the term “argument” to includecases like these, or just throw out sentence 2, but there is a less drastic option whichLogicians employ constantly, we simply re-state the essence of the sentence in such aform as to keep its connection to the main conclusion, while turning it into a proposition.Such a restatement of 2 may be: “A family’s house burned down on Christmas.” Nowclearly, this statement is either true or false, and is important to the conclusion and theremaining sentences, and this re-writing of 2 fully preserves the original’s connection tothe whole, hence Logicians consider this an acceptable change. After we have made thischange, we see the remaining sentences are indeed propositions and relevant to the mainconclusion, hence they are indeed premises.But what makes them relevant to the main conclusion? To answer this question,

the concept of unstated assumptions, or unstated premises, made by the argument ishelpful.1What are some of these assumptions which make the premises relevant? Let’s start

with our modified premise 2, “A family’s house burned down on Christmas”. What is itabout this premise that makes it relevant to the claim that, “It is a good idea to makesure your have working fire alarms in your house”? There are several possibilities, but

1Unstated premises are technically called enthymemes, but we will just call them unstated premisesor assumptions in this text.

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let’s explore the obvious by first asking, Would it matter if the premise stated that thehouse burned down on another day rather than Christmas? If not, then the fact thatit burned down on Christmas is not as important as another fact, that being that itburned down (to see this, just change 2 to state, “A family’s house did not burn down. . .”, and ask if that changes its connection to the conclusion). So, the fact that ahouse has burned down is at least relevant to having working fire alarms, but we needto pursue the issue more. Suppose the family did have working fire alarms, would thatfact result in the house not burning down? This one is harder to answer, as it requiresmore information than we are given. Whether or not having a working fire alarm wouldprevent the house from burning down, the argument gives us another line of reasoning -premise 3 suggests that losing everything, including the life of a loved one is undesirable,and premise 4 argues that without working fire alarms, one might sleep through a fire,and as a result lose one’s life. Finally premise 5 suggests that unless one wants to leaveone’s alert system to chance, then one should have a working fire alarm. What has beenassumed in all of this? At least the following (and probably more):

• It is possible that houses burn down.

• It is better that family members live than die in house fires.

• One can die from house fires which go unnoticed.

• Many things go unnoticed while sleeping.

• A working fire detector can sound the alarm, which, at the very least will alerteven sleeping people to a fire which might otherwise go unnoticed.

• A fire detector can monitor one’s house at all times, especially late at night whenneighbors sleep.

• It is better not to leave the fire alarm alert to chance.

With these assumptions in mind, we can re-read Example 2 and see that each sentencein our argument supports each of the above assumptions, and the natural conclusionis that it is better to have a working fire alarm than none at all. It is, of course, alegitimate question as to whether the person making the above argument should justdo so by using the above assumptions for premises, but the point of example 2 was toexamine an argument that is more like one encountered in our daily lives.Notice that we have not stated anything about the strength of the above argument,

or just how relevant the premises are to the conclusion, whether any single premiseor conjunction of premises entails the conclusion nor have we said anything about thelikelihood of the conclusion being true, or in general, whether a given argument presentsa good case for its specific conclusion(s). These concepts will be the detailed subjectof later investigations, but to get us started down that road, we now turn to the veryimportant concept of inference.

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3. Arguments

3.3. Validity, Inference, and InvalidityLet us begin our investigation of inference by considering three examples:

Example 1If John makes the free throw, then the U of A will win the game.John made the free throw, so the U of A won the game.Example 2If John makes the free throw, then the U of A will win the game.John did not make the free throw, so the U of A lost the game.Example 3If John makes the free throw, then the U of A will win the game.John did not make the free throw, so yesterday my neighbor ate oatmeal for breakfast.Let’s sort out the premises and conclusion to each of the above examples and use our

method of connecting premises together with the word “and” as we studied in Section 3.1with the goal of determining whether the conjunction of all of the premises is relevant,entails, or is independent of the conclusion.To simplify our analysis, let p be, “If John makes the free throw, then the U of A will

win the game”, and let q be, “John made the free throw”, and let r be, “The U of Awon the game”. This allows us to re-write Example 1 as, “p and q, therefore r.”2Now we pose the following question, When is “p and q” true? If you are tempted to

consider specifics about the rules of basketball, game conditions, and players, then STOP.These indeed are important questions, but surprisingly they are of minor importance toour immediate goal. Let’s assume we know all of these answers (even though we do not)and ask the question again, when is “p and q” true? Hopefully we can all agree thatthere is no hope for the conjoined statement “p and q” being true unless both p and qare individually true. This important point applies in general, not just when p and qstand for propositions related to free throws and basketball games, but in every case!As a matter of fact, the conjunction of any number of propositions is true only whenevery single individual proposition in the conjunction is true, as we discussed at the endof Section 3.1.With this in mind, let us now consider the argument given in Example 1, and ask,

using the vocabulary we have learned, whether the conjunction “p and q” entail r, isrelevant to r or is independent to r? In order to answer this question, it is a good ideato just assume (pretend) every single premise is true, which means in this particularcase, we assume that the statement, “If John makes the free throw, then the U of Awill win the game” is indeed true, and the statement, “John made the free throw” isalso true. Given these assumptions it is natural to infer that the U of A will win the

2Notice that we have replaced the word, “so” with the word “therefore” to highlight the conclusionto the argument. It turns out that there are many words which help us decide which sentences inan argument are conclusions. If one sees (or can add unwritten) words and phrases such as; “so”,“then”, “therefore”, “hence”, “as a result”, “it follows that”, “because of”, then what follows is usuallya conclusion (major or minor) of the argument. On the other hand words and phrases like, “because”,“since”, “for the reason that”, mark premises in an argument.

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3. Arguments

game. Is it possible for the U of A to lose, if our assumptions are true? Clearly not,for it that were possible, then at least one of our premises must be false. Make sure youclearly understand this point before you go on. To repeat; if we assume the followingtwo premises are true, “If John makes the free throw, then the U of A will win the game”and “John makes the free throw”, then it must be the case that the U of A wins thegame.Of course to assume something is true is not the same as to assert that it really is true.

However, these types of assumptions are key to understanding one of the major tools inthe Logician’s tool box. Clearly if we know that if all of the premises to an argumenthappen to be true, then the conclusion to that argument must be true, such knowledgeis important if one wants to analyze arguments! This brings us to a very importantdefinition, which is of such importance that the student should spend as much time asneeded to fully understand the definition, which requires not only knowing what thedefinition says, but what it means.

Definition 5. An argument which has the property that the conjunction of all of itspremises entail its conclusion is said to be a valid argument. Equivalently an argumentis valid if it has the property that it is impossible for the conclusion to be false and allof the premises to be true. If an argument is valid, we also say that the inference fromthe premises to the conclusion is valid.

Note the introduction of the term “inference” above. In general, an inference is theact of drawing a conclusion from a set of premises. We will say that an argument is validor that the inference of an argument is valid interchangeably.

Before we examine Examples 2 and 3, let’s discuss just what this important definitionmeans, by considering the most common mistakes students make with respect to thisnew concept. All of the statements below are incorrect or incomplete:

i) A valid argument must have a true conclusion.

ii) If all premises of a valid argument are false, then the conclusion must befalse.

iii) A valid argument is an argument with all true premises and a true conclusion.

iv) The conjunction of all the premises in a valid argument must be true.

The first statement omits a very key part of that definition, that part that says, “If allpremises are true”. Missing or forgetting key parts of definitions is perhaps the numberone reason for student errors.The second statement is not the definition of a valid argument or a statement entailed

by that definition. The definition of a valid argument is silent about what, if anything,must be the case if all the premises of a valid argument are false, hence no assumptionshould be made on the part of the student concerning that possibility. In Part 2 of thistextbook, we will prove this statement to be incorrect formally. For now, don’t makethe error and consider this statement as a proper characterization of valid arguments.

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3. Arguments

The third statement omits a very important part of our definition for a valid argument.Why this is an error is left as an exercise for the student.The fourth statement is a bit more difficult. So we will demonstrate is falsity by exam-

ple. Suppose I state the following argument: The Cullinan diamond weighs more than100 grams, and was worn by Queen Victoria when she was crowned in 1964, thereforethe Cullinan diamond weighs more than a 2 gram feather. Clearly, this argument isvalid (it is impossible to have a false conclusion with all true premises), but since QueenVictoria was not crowned in 1964, the conjunction of the premises can’t be true (recallthis is another way of saying at least one of the premises is not true). The argument isvalid because the truth of premise one alone entails the truth of the conclusion, whichmeans it is impossible to have a false conclusion and all true premises.

We are now ready to examine Examples 2 and 3. Repeating the procedure as above,we assume that the conjunction of the premises are true (recall, this is just another wayof saying that all of the premises are individually true). Assuming that both, “If Johnmakes the free throw, then the U of A will win the game.” and “John did not make thefree throw” are true, is it possible for “the U of A lost the game” to be false? Carefulthinking shows that indeed this is a possibility. To see this, simply suppose that Mikemakes the winning free throw rather than John. Then both premises are true, but sinceMike made the winning free throw, the U of A won the game (so the conclusion statingthat the U of A lost the game is false).A similar analysis will show that it is also possible for all of the premises to be true

but the conclusion to be false in example 3. The possibility that all of the premisesto an argument can be true, but the conclusion can still be false leads us to our finaldefinition.

Definition 6. If it is possible for an argument to have all true premises but still have afalse conclusion, then the argument is said to be invalid. Equivalently, if the premisesof an argument do not entail the conclusion, the inference from the premises to theconclusion is called invalid.

Again, be careful to not add anything to the definition of an invalid argument thatis not stated in the definition. As a rule of thumb, any argument where the premisesare relevant (but do not entail) the conclusion is an invalid argument, as well as anyargument whose conclusion is logically independent of the truth of the premises - thereis one important case which prevents us from saying this is must always be the case,which we will encounter when we examine arguments formally in Part 2. Invalid doesnot mean “false” or even “bad reasoning” - it means only that the possibility is open (nomatter how remote) for the conclusion to be false, even if all of the premises are true.What is the case is that all arguments are either valid or invalid, simply because invalidmeans “not valid”. Hence, any argument can be classified as either “valid” or “invalid”,and determining which is the case will be an important first step in our analysis ofarguments. However, like any subject of scope and depth, the first step is not the last- and the same is true in our analysis of arguments. Knowing whether an argument

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is valid or invalid is important, but we will need to examine and answer many morequestions before we can decide whether an argument gives good reasons for supportingits conclusion!

Valid argument/inference, three equivalent definitions:

An argument is valid iff:

1. The truth of the premises entail the truth of the conclusion.

2. If all the premises are true, the conclusion must be true.

3. It is impossible for the conclusion to be false and all the premises to be true.

Invalid argument/inference, two equivalent definitions.

An argument is invalid iff:

1. The truth of the premises do not entail the truth of the conclusion.

2. The mere possibility for the conclusion to be false with all true premises makes anargument invalid (no matter how remote that possibility).

As always, examples help us to grasp new concepts.

Example. If the washes are full of water in Tucson, then it rained.

This argument as simple as it gets, with one premise (the washes are full of water)and one conclusion (it rained). To determine whether this argument is valid or invalid,we ask, Is it possible that it rained and the washes are not full of water? It does nottake much imagination to conceive of this possibility, (think about it raining for just afew seconds) - hence this argument is invalid.

Example. The lights are not working because the circuit breaker has tripped or theelectrical storm knocked out the power. The circuit breaker is tripped therefore theelectrical storm did not knock out the power.

This argument may not be as obviously invalid as the previous example, in orderto analyze it, we must know what the conclusion is and what the premises are. Theconclusion (the claim the argument wishes to establish) is that the electrical storm didnot knock out the power (use the clue that conclusions follow words such as “then” and“therefore”). The premises are: “the lights are not working because the circuit breakerhas tripped or because the electrical storm knocked out the power” and “the circuit

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breaker is tripped”. Given this information, we pose the question, Is it possible for theconclusion to this argument to be false, even if all the premises are true? If the answeris yes, then according to our definition the argument is invalid. In this case, we canconsider the possibility that both the circuit breaker is tripped and the electrical stormknocked out the power. While perhaps a rare occurrence, this is certainly a possibility,and a possibility which clearly makes both premises true but the conclusion false. Hence,the argument is invalid.

Exercises Section 3

Preliminaries:a) Without looking (but with having actually read the text) write out the definitions

of: Argument, Valid argument, and Invalid argument. Compare your answers to theactual definitions given in the text. Are your definitions equivalent? If not, highlightareas in your definition which are different than the original definition.b) Write out the actual definitions several times, until you get them correct. State

that you have done so.c) Make up (create your own) 3 valid arguments and 3 invalid arguments. Clearly

state whether each argument is valid or invalid and why (hint, for the why part, appealto the definitions in each case).d) How are the terms valid and invalid used in Logic differently than elsewhere?

Provide an example of the term “valid” and “invalid” which clearly do not mean thesame as our definitions in class (you may use the internet to aid in your search).

Main Exercises:Identify the premises and main conclusion to the following arguments. Use this in-

formation to state whether the argument is valid or invalid. If the argument is invalid,modify the premises to make it valid. Remember to know and use the definitions of validand invalid in your reasoning process (memorize them or write them down so you havethem handy when doing the exercises)

Example. Malaria is caused by a parasite that lives in mosquitoes that inhabit tropicalregions throughout the world. John has malaria, so he must have recently been in atropical country.

Answer: The premises are: 1) Malaria is caused by a parasite that lives in mosquitoesthat inhabit tropical regions throughout the world. 2) John has malaria. The conclusionis that John must have recently been in a tropical country. The argument is invalid,as John may have only been in a tropical country 3 years ago, this fact alone wouldmake the conclusion false even if all the premises were true. Another possibility (amongothers) is that the parasite that causes malaria is also found in other non-tropical regionsof the world. This possibility is not excluded by that fact that the parasite lives alsoin tropical areas, but this fact allows the conclusion to be false (as John may havegotten malaria by a mosquito in a non-tropical region), even if all the premises are true.One way the argument could be made valid would be the modification of premise one

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to state: “Malaria is caused by a parasite that only lives in mosquitoes that inhabittropical regions throughout the world and is only caught by individuals that reside intropical regions”

Example. If John gets a new job, then we will take a vacation in the Summer to Italy.If we take a vacation to Italy, we will have to obtain US passports, which require a birthcertificate. In order to get my birth certificate, I will need to look through Mom’s boxes,which will require that I visit my parents. John got a new job, so I will have to visit myparents.

Answer: The premises are: 1) If John gets a new job, then we will take a vacationin the Summer to Italy, 2). If we take a vacation to Italy, we will have to obtain USpassports, which require a birth certificate, 3) n order to get my birth certificate, I willneed to look through Mom’s boxes, which will require that I visit my parents, 4) Johngot a new job. The conclusion is that I will have to visit my parents. The argument isvalid. In order to see this, assume the conclusion is false. This means that I did not visitmy parents, which is turn means I did not get my birth certificate, which means I didnot get a new passport, which means I did not get a Summer vacation it Italy, whichmeans John did not get a new job, which means premise 4 would be false, thus makingit impossible to have a false conclusion and all true premises.

1.* If it rains in Tucson, then the washes will be full of water. The washes arenot full of water, therefore it did not rain in Tucson.

2. Every time I have taken an algebra class in the past, I fail. So if I take anAlgebra class this semester, I will just fail again.

3. The suspect in the homicide investigation wore a large hat. The 3rd man tothe left in the police line-up has a large head, therefore that man is probablythe murderer.

4. The battery is bad, or the alternator is not working. The battery is not bad,so the alternator is not working.

5.* The Power ball has reached a near-record jackpot of $210 million dollars.Almost anyone would like that kind of money, and one thing is for sure, ifyou don’t play, you can’t win, so everyone should buy a power ball ticket.

6. Many people who have had a bad case of the flu and took zinc supplementsreport their flu did not last as long as it would have otherwise. Therefore,zinc supplements help alleviate the flu.

7. It is a good idea to make sure you have working fire alarms in your house.Just look at the family whose house burned down on Christmas. They losteverything in their house, and almost lost their daughter who nearly died ofsmoke inhalation. Their house did not have working fire alarms, and for thatreason, the fire itself went unnoticed while the family slept. They were saved

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only by the chance occurrence of a neighbor’s teenage soon arriving homelate from a Christmas-eve date who noticed the fire and woke the family up.

8.* If Descartes thinks, then he must exist. Descartes is not thinking, thereforehe does not exist.

9. Kaiser Permanente, one of the nation’s largest managed care organizations,has ordered its pharmacies to stop dispensing Bextra, an arthritis and paindrug made by Pfizer that some tests have indicated may increase the risk ofheart attacks and strokes Therefore, one should not take Bextra.

10. If water is boiled at 212 degrees, then all harmful organisms will be killed.We boiled this water at 212 degrees, therefore there will be no living harmfulorganisms in the water.

Answer True or False and give a brief justification for your answers (hint: appeal to thedefinitions of valid and invalid)

11. A valid argument must have a true conclusion

12. An invalid argument always has a false conclusion.

13. Every argument can be classified as being invalid or valid.

14. Arguments always have at least one premise and at least one conclusion.

15. Invalid arguments are false arguments

16. Invalid arguments are always instances of poor reasoning.

17. The claims of any argument that is not valid should be dismissed.

18. An invalid argument can have a true conclusion.

19. It is possible to have a false conclusion and all true premises if an argumentis valid.

20. Suppose there exists a truth relationship between the premises of an argu-ment such that it is impossible for all the premises to be true. The resultingargument is therefore invalid.

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4. More properties of arguments

4.1. Truth versus Inference, Soundness, Strength andWeakness

The concepts of validity and invalidity are so central to the study of Logic, that it isworthwhile discussing them more at some length, as long experience has shown that theuse of these terms outside of logic, where the meanings are different, linger in the mindsof students causing continued confusion and wrong inferences based on these differentmeanings. Perhaps the most persistent confusion concerning the concepts of validity andinvalidity is the tendency to think they are synonymous with the concepts of truth andfalsehood, namely that valid is equivalent to true, and invalid is equivalent to false. Thisis not the case, and great effort should be taken by the student to learn the differencebetween these concepts. Truth and falsity are properties of propositions, whereas validityand invalidity are properties of inference (or, as we often say, arguments). The inferenceof an argument is either valid or invalid, whereas the individual components of theargument - its premise(s) and conclusion(s) are either true or false. The non equivalenceof these terms leads to the possibility that all of the propositions which make up anargument (the premises and conclusion) might be false, whereas the inference made bythe argument is valid. For example consider the following argument:

Today is Wednesday and the day that follows Wednesday is Thursday.Therefore tomorrow is Thursday

If you are reading this on any day other than Wednesday, both the premise and con-clusion are false, but the inference is valid - since if the conclusion were false (tomorrowis not Thursday) it is impossible for the premise, “Today is Wednesday and the day thatfollows Wednesday is Thursday” to be true. If the above example still seems odd, reviewagain the formal definition of validity.Another common misconception often demonstrated by student’s answers on exams is

the incorrect view that invalid arguments are somehow bad arguments, in the sense thateither the conclusions to invalid arguments must be false, or somehow any reasoningwhich is invalid is poor reasoning. This too is incorrect. Consider the following invalidargument:

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I parked my car in the west parking lot this afternoon, as I do every time I drive toschool.My car has always been exactly where I parked it at the end of the school day.Therefore, after classes, my car will be parked in the west parking lot.

In this case, it is possible for the conclusion to be false (your car will be missing),even though the premises are true. What this means is that the argument is invalid,but the reasoning used is good (imagine drawing the opposite conclusion from the samepremises, that your car will not be parked in the west parking lot). Again, this illustratesthe difference between invalidity and poor reasoning. An inference may be invalid, butstill present good reasoning - the difference being only that the conclusion to an invalidargument is not guaranteed to be true even if all of the premises turn out to be true,whereas with a valid argument we have such a guarantee.In order to reenforce this distinction we introduce three more concepts and their

appropriate definitions.

Definition 7. An argument is sound if it is valid and all the premises are actually true.

Notice that for an argument to be sound, two conditions must be meet: 1) the argu-ment must be valid, and 2) the argument must actually have all true premises. Thinkabout what this means for a second, and answer the following question:

What can be said about the conclusion to a sound argument?

Hopefully, your reasoning went something like this: Since the argument is sound, thenit is both valid and actually has all true premises, so the conclusion must be true, bydefinition of validity. The following is an example of a sound argument:

If a number is greater than 7 it is greater than 3.8 is greater than 7.Therefore 8 is greater than 3.

Definition 8. An argument is said to be strong if the inference is invalid but thepremises are judged to give better reasons to accept the conclusion than to reject it.

This definition is not as precise as we would like, and indeed it seems to pass the buck,trading the meaning of strong for the meaning “judged to give better reasons to accept".What is more, what may be judged better reasons for one person may not be for another.However, Logicians find it helpful to have a term for arguments of this type, and, as wewill see later, certain types of arguments, called inductive arguments, lend themselvesnaturally to this type of description. Some textbooks define strong arguments in termsof the likelihood of the conclusion being true. This, of course, poses the question of whatdetermines likelihood, and has naturally lead to the mathematical study of probability,

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which is fascinating in its own light. Since this is an introductory course, we will simplyencourage the interested student to take future courses where the subject of probabilityis central. An example of a strong argument is the following:

The sun has risen every day for all of recorded history.Therefore the sun will rise tomorrow.

The student should recognize this as an invalid argument, but a strong one. Literallyevery day of all of recorded history is given to accept the truth of the conclusion, whereasno single reason is given to suspect it is false. This is not to say none could be given (asthe nature of invalid arguments are such that if their conclusions can be false, even if allthe premises turn out to be true). Contrast this with the following argument:

Our dog had six puppies, the first in the litter being male.Therefore all the puppies in the litter will be male.

In this case, the reason given does not support the conclusion, it is just as reasonable(if not more so) to accept the negation of the conclusion, that at least one puppy in thelitter will not be male. Arguments of this type are called weak arguments.

Definition 9. An argument is considered weak if it is invalid and the premises are notjudged to give better reasons to accept the conclusion than to reject it.

With these concepts in mind, lets us now develop tools which help us determinewhether an argument is valid or invalid and consider some guidelines which give usconfidence about judgments of strength or weakness.

4.2. Valid Argument FormsOne of the oldest discoveries in Logic, made by Aristotle, is that validity is not a propertyof argument content - what the argument is about - but is rather a property of argumentform, by which we mean the way elements of the argument are connected. The follow-ing argument forms are all valid, which means, by replacing the argument’s statementvariables with any particular proposition, the resulting inference is valid - irrespectiveof our choice for the statement variables. This is what we mean by saying that validityis a property of form rather than content. Below are several important argument formswhich are valid. There are many, many more, but as we will see in Part 2 all can becontructed from a small set of rules. Here we give the traditional argument names (andsome modern ones) along with their forms.

Modus Ponens/Affirming the AntecedentIf p then qpTherefore q

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Modus Tollens/Denying the ConsequentIf p then qnot qTherefore not p

Hypothetical Syllogism/Chain of ReasoningIf p then qIf q then rTherefore if p then r

Disjunctive syllogism/Process of Eliminationp or qnot pTherefore q

ConjunctionpqTherefore p and q

The above are common forms of valid arguments and familiarity with these five willsuit our purposes for now. We will study these forms in more detail in Part 2 of thistextbook, but for now we will focus on two facts:

1. Given any valid form, replacement of the variables with any proposition results ina valid argument,

2. Any argument that can be demonstrated to have a valid form must also be valid.

We will make use of both of these facts.As an illustrative exercise, let:

p ="Today is Wednesday”q ="Tomorrow is Thursday”

Use these assignments as replacement values for Modus Ponens/Affirming the An-tecedent, to obtain the following argument whose inference is valid:

If today is Wednesday then tomorrow is ThursdayToday is WednesdayTherefore tomorrow is Thursday

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The resulting argument has a valid inference. Consider now the following propositionalassignment to the variables p and q:

p ="It rains”q ="Streets get wet”

Using these assignments as replacement values for Modus Ponens/Affirming the An-tecedent„ we also obtain the following valid argument:

If it rains then (the) streets get wetIt rain(ed)Therefore the streets are wet.

Notice that there is a change in tense between “it rains” and “it rained” - for now, wewill not worry about such changes in tense, and just consider the argument as valid.1Examples like those above are useful, but obscure Aristotle’s discovery that validity

is a function of form, rather than content. To illustrate this, recall that we stated thatany choice of propositions for the variables produces a valid argument. To illustrate thisconsider:

p ="Today is Wednesday”q ="Tomorrow is Sunday”

Then, using these as our p and q, (and again choosing as our form, Modus Ponens),we obtain:

If today is Wednesday then tomorrow is SundayToday is WednesdayTherefore tomorrow is Sunday

The above argument is indeed valid!It is at this point that a wave of confusion engulfs many students. So it is time to

put into place what we have already learned. Look at the two premises, suspend whatyou know, and pretend both premises are actually true - then ask yourself, Must notthe conclusion also be true? If you agree that the conclusion must be true if both of thepremises were really true, then review the definition of a valid argument and convinceyourself that indeed the argument is valid. If you are still puzzled, remind yourself that“if” is not the same as “actually is”. The definition of validity does not state that all

1For those who are not satisfied with this, replace “If” with “whenever”. For this to work, we mustagree that “If p then q” agrees in meaning with, “whenever p then q”.

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the premises are actually true, only what must be the case if they are. Of course, this isanother example of the distinction between valid inference and truth and falsity. Reviewand fill in the blank:

The above argument is valid but not 2

In Part 2, we will explore why form guarantees validity, (even if the propositions chosenfor the variables are logically independent) - for now we can take it as a yet unprovenfact.As mentioned earlier, given any valid form, replacement of the variables with any

proposition results in a valid argument. It is now time to explore the second case,that any argument that can be demonstrated to have a valid form must also be valid.Consider the following argument:

The batteries are dead or the bulb is shot.The batteries are not deadSo the bulb must be shot.

As usual, we want to know whether the above argument is valid. We could just applythe definition of validity as before, and think it through by brute force. The reasoningmight go something like, If the conclusion is false, is it possible to have all true premises?If not, then the argument is valid, if yes then the argument is invalid. But now we haveanother way - we can check the argument form to see if it matches a known valid form -and if it does, we can declare the argument to be valid. Since our argument forms havevariables rather than specific propositions, we need to change the above argument to anabstract form with variables. We can do this by the following:

Let p = “The batteries are dead”Let q = “the bulb is shot”

Using this assignment of variables we can re-write our argument as:

p or qnot pTherefore q

Notice that our translation goes from specific statements to an abstract form, and ourspecific assignment of variables did not include the word “or” and we used the fact that“so” is equivalent to “therefore”, and that “not q” is the same as “the batteries are notdead”. Much of this will come natural, but the omission of the word “or” usually does

2sound

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not. Words like, “or”, “and”, and phrases like “if . . . then” have a very importantrole in Logic, they are called logical operators. We won’t say much about them now,except that in the process of assigning variables to propositions the logical operators arenot part of the assignment. To illustrate this, we need more examples. Consider thefollowing:

If I work overtime for three months, then I will have enough saved to go to Peru.I don’t have enough saved up to go to Peru.Therefore I did not work overtime for three months.

Recall that in the process of assigning variables to propositions, the logical operatorsare left out. In this particular case, the “if” and “then” will not be part of our assignment.

Let p = “I work overtime”Let q = “I will have enough saved to go to Peru”

This assignment of variables allows us to re-write our argument in the following ab-stract form:

If p then qnot qTherefore not p

With this assignment of variables, we can see the argument is valid, as it has the sameform as Modus Tollens/Denying the Consequent. The process of assigning variables topropositions contained in the original argument in order to determine if an argument isvalid must be done very carefully. Recall that the definition of proposition also allowsthe following assignment of variables:

Let p = “If I work overtime, then I will have enough saved to go to Peru”Let q = “I don’t have enough saved up to go to Peru”Let r = “I did not work overtime for three months”

This assignment of variables gives the following form:

pqTherefore r

The argument form above is not in our list of valid argument forms, but our previousconsiderations show that the argument is indeed valid. What this means is that a neg-ative match to a valid form does not mean an argument is invalid. This example also

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emphasizes the need to take care in our choice of variable assignments, since if a par-ticular assignment of variables does produce a valid argument form, then the argumentmust be valid - no matter what other assignments indicate. One last note, consider thefollowing argument forms:

pIf p then qTherefore q

q (because)If p then q (and)p

Both of the above forms are actually instances of modus ponens/affirming the an-tecedent. The first instance just switches the order of the premises, and the last ex-ample places the conclusion first, the word (because) is given to indicate what followsare premises to the claim just made. The lesson is that order of presentation does notchange the fact that an argument form is valid. The form is slightly obscured, but giventhe fact that premises are distinct from conclusions and the truth of p and q is the sameas the truth of q and p, as discussed in Section 3.1 Combining Propositions, the resultingargument is valid. Keep this in mind, as it is quite common in real life situations forthe conclusion to be stated first, and then premises (reasons) are given to support theconclusion.

4.3. Pseudo-valid argument forms.Our last example shows that an argument which does not match a known valid form(and we only know 5 valid forms at this point) can have a valid inference. In the lastexample, the assignment of the variables included the logical operators “if . . . then” -a procedure we specifically warned against. One might think that if certain rules werealways followed, then there are invalid forms of arguments that are just like valid forms ofan argument - which have the same properties of valid forms, namely that 1) Given anyinvalid form, replacement of the variables with any proposition results in an argumentwhich does not have a valid inference, 2) Any argument that can be demonstrated tohave an invalid form must also be invalid. As it turns out, this is not true in generaland we will have the tools to prove it in Part 2. However, Logicians have noted thatsome forms of argument are often thought to be valid, and used in arguments as if theywere. Many textbooks label these argument forms as invalid deductive forms. We willnot use that terminology here, to avoid possible confusion with the method of deductionwe will use in Part 2, but rather call them pseudo-valid forms. The following are themost common pseudo-valid argument forms.

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Denying the AntecedentIf p then qnot pTherefore not q

Affirming the ConsequentIf p then qqTherefore p

False Dichotomyp or qpTherefore not q

Notice that these forms are very close to valid argument forms, so make sure yousee the difference in, for instance, the valid argument form, modus ponens/affirmingthe antecedent and the pseudo-valid form, denying the antecedent. The following is aninstance of an argument with a pseudo-valid form:

If it rains in east Tucson, then the low level road crossing will be flooded.It did not rain in east TucsonTherefore the low level crossing will not be flooded.

As before we can see the above argument is an instance of denying the antecedent bythe following assignment of variables:

Let p = “it rains in east Tucson”Let q = “the low level road crossing will be flooded”

Verify and convince yourself that the above assignment does indeed produce thepseudo-argument form of denying the antecedent.Given that we can not be sure if an argument which is an instance of an invalid form

is really invalid, how do we determine invalidity? Think about the above argument.Suppose you agree that both premises are true - do you agree that the conclusion mustbe true? If not, then think how the premises can be true and the conclusion false.In the process of doing this, you are formulating what is known in Logic as a counter-argument, which will be one of the most important tools in critical reasoning with respectto invalidity of inferences.

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4.4. The Counter ArgumentSuppose that someone were to argue for a certain conclusion but you are unconvincedby the argument. Implicitly, this means that either you believe the argument’s inferenceis invalid or, if it is valid, you do not accept the truth of at least one of the premises.However, if you are not a Logician, or are familiar with the logical concepts of validand invalid inference, you will probably not express your doubt about the truth of theconclusion by identifying the argument’s inference as being invalid (or, as being validbut having at least one false conclusion), but rather, you will express your doubt byoffering what is called a counter argument. Informally speaking, a counter argumentis an argument offered to give reasons not to accept the conclusion of another argument.There are many types of actual counter arguments encountered in real life, we will focuson two types: those which highlight that the inference to the argument countered isinvalid, and those that cast doubt on the truth of at least one of the premises of theargument being countered. Since our last section was about pseudo-valid arguments, wewill focus on invalid inferences first, as the ability to provide a counter argument is theprimary way to demonstrate that an argument that has a pseudo-valid argument formis actually invalid. Consider again the argument given at the end of the last section:

If it rains in east Tucson, then the low level road crossing will be flooded.It did not rain in east TucsonTherefore the low level crossing will not be flooded.

Now consider your answers to the question posed at the end of the last section, namely,why is the argument invalid? Your reasoning might have gone something like this,Well even if all the premises are true, it is possible that it rained elsewhere in Tucsonand flooded the low level road crossing, so given that possibility, the low level crossingin question could still be flooded. This type of reasoning is an example of a counterargument that highlights the invalid nature of the inference. It proceeds generally alonglines like this, “Even if the reasons for the conclusion are true, given these alternativeconsiderations, it is possible for the conclusion to be false”. Counter arguments thatproceed along these general lines highlight the invalid nature of the argument’s inference.Here is another example:

If Janet gets accepted to the University of Arizona, then she won’t take any more classesat Pima.Janet isn’t taking anymore classes at Pima.Therefore Janet got accepted to the University of Arizona.

The above is an example of the pseudo-valid argument form called affirming the conse-quent. Can you give a counter-argument that shows that the inference is indeed invalid?How about this, “Suppose the premises are indeed true, but Janet did not receive any

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financial aide and as a result she stopped taking classes at Pima but she has yet to beaccepted at the University of Arizona”. This possibility shows that the conclusion couldbe false, even with all true premises.While counter arguments help in showing that an argument is indeed invalid - counter

arguments which begin, “Suppose the premises are true” and end with “then the conclu-sion could be false” are clearly of no help if the argument is valid. To reject claims madeby arguments which are valid, the counter argument can not attack the inference, but itmay attack the truth of one or more of the premises. Consider again the example*:

If today is Wednesday then tomorrow is SundayToday is WednesdayTherefore tomorrow is Sunday

We know this to be a valid form, and indeed one of the premises turns out to befalse. One way one can counter such an argument is to point out the possibility of thatfalsehood: “Suppose today is Wednesday but tomorrow is not Sunday, then clearly todaycould be Wednesday without tomorrow being Sunday”. Of course, in this particular case,we can go even farther and point out the known falsehood of the first premise. But manytimes this is not possible. In these cases, all we can do is point out the possibility thatone or more of the premises are false. Here is another example of a valid argument anda possible counter argument.

If the drought in the Midwestern states continues for the next several years, corn pro-duction will drop by 40%If corn production drops by 40%, then prices for fuel and food will rise accordingly.Therefore if the drought in the Midwestern states continues for the next several years,fuel and food prices will rise accordingly.

This argument is valid, as it exhibits the valid form hypothetical syllogism/chain ofreasoning. But the premises contain truths about the future, hence we can’t simply saythe premises are factually false - that won’t be known for several years. Our counterargument must take this into consideration by casting doubt on at least one of thepremises. Here is a possible counter that casts doubt on both premises.

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Scientists and weathermen have recorded rainfall in the Midwest for over 150 years, andno drought has lasted for more than 2 years in a row. Also corn production could dropby as much as 50% without affecting fuel prices or food prices, as the primary connectionto corn and fuel price is ethanol, whose production could be replaced by other grassesor crops not affected by the drought, and corn syrup - the connection to corn and foodprices - is already being replaced by cane sugar, grown in areas which are not droughtprone.

The first part of our counter argument casts doubt on whether the drought will actuallylast the next several years by recollecting what has happened in the past. This type ofargument is so important that it will be the subject of our next chapter on induction.The second part of the counter argument casts doubt on the connection between cornproduction and fuel and food prices by offering alternative possibilities. The ability tosee alternatives and use those alternatives in a counter argument designed to cast doubton claims made in another argument is indeed a powerful tool. This should come as nosurprise, for consider again the pseudo-valid argument form:

If p then qnot pTherefore not q

One of the informal reasons we reject this type of inference is because there may bemore than one cause or reason for q besides p. The ability to point out what thosealternatives may be is a skill that grows with practice and is invaluable when reasoningabout the world we live in.

Exercises Section 4

Answer True or False. Support your answer by giving reasons or providing a definitionfrom the text that justifies your answer.

1. A valid argument can have all false premises and a true conclusion.

2. A valid argument can have all false premises and a false conclusion.

3. A true argument is a valid argument with a true conclusion.

4. A false argument is a invalid argument with a false conclusion.

5. All invalid arguments provide poor reasons to accept their conclusions.

6. It is possible for an invalid argument to be sound.

7. According to the definition in our text, it is proper to call a valid argumentstrong.

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8. A weak argument is an invalid argument where the inferences are not judgedto give good reasons to accept the conclusion.

9. Any valid argument form produces a valid argument and any argument whosepremises and conclusion can be assigned variables that result in an validargument form must be valid.

10. It is possible for an argument with a pseudo-valid form to be valid.

Assign variables to the following arguments in order to determine their form. Name theform and indicate whether it is valid or a pseudo-valid form (note: recall that words like,“if”, “then”, “and” and “or” should not be included in your variable assignment). Theproper assignment of variables in the following exercises will always result in one of theforms covered in this Chapter. It is always a good idea to determine which statementsare individual premises and which statement is the conclusion.

Example:The increase in traffic indicates it’s rush hour or a game just let out.It’s not rush hour.So a game just let out.

Let p = (the increase in traffic indicates) it’s rush hour.Let q = (the increase in traffic indicates) a game just let out.

This assignment of variables gives the following argument form:

p or qnot pTherefore qThe above indicates a valid argument form called disjunctive syllogism/process of

elimination.

11. If John gets a raise, then he will take us all out to dinner.John got a raise.Therefore he will take us all out to dinner

12. To get a driver’s license in a foreign country you must have a valid driver’s license in theUnited States or be over 21. John has a valid drivers license in the U.S. Therefore Johncan not get a driver’s license in a foreign country. (Hint: Let p be, “(For anyone) to geta driver’s license in a foreign country they must have a valid driver’s license” and let qbe, “(for anyone) to get a driver’s license in a foreign country they must be over 21”.

13.* If you go to Coffee Exchange, you can buy a large iced-coffee. Maria bought a large icedcoffee, therefore Maria went to Coffee Exchange

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14. If Gravity is equivalent to the curvature of space-time, then light will be bent as it passesby a large gravitational object.Light is not bent as it passes by a large gravitational object.Therefore Gravity is not equivalent to the curvature of space-time.

15. If Congress passes the new transportation bill, then we can expect more jobs in theroad-construction sector.If there are more jobs in the road-construction sector, then John will have a good chanceat getting hired.Therefore if Congress did pass the new transportation bill, then John has a good chanceof getting hired.

16.* If the television is not working, then we will watch the show at Jim’s. We did not watchthe show at Jim’s, therefore the television is working.

17. If there is a security breach at the airport, my flight will be delayed. There was not asecurity breach at the airport. Therefore my flight was not delayed.

18. The cord is unplugged or the fuse is blown. The cord is plugged in, therefore the fuse isblown.

19.* If gas prices continue to rise, then transportation costs will rise also. If transportationcosts rise, then the cost of food will go up too. So if gas prices continue to rise, the costof food will go up.

20. John bought a Toyota or a Ford. He bought a Toyota, so he did not buy a Ford

21. If the contestant picks door number 3, then she will win a new car. The contestantpicked door number 3, therefore the contestant won a new car.

22.* Manatees are mammals.Dolphins are mammals too.Hence manatees and dolphins are both mammals.

23. If John runs a mile a day, he will be in shape in time for the race.John did not run a mile a day.Hence he will not be in shape in time for the race.

24. The class is not officially full (because) when two more people sign up for the class thenthe class will be officially full, but two more people have not signed up.

25. If I get an A in this class, I will get a $500 bonus on my scholarship, because if I get anA in this class, then my GPA will go up by 35 , and if my GPA goes up over

25 , then I

will get a $500 bonus on my scholarship. (Hint: observe that the conclusion is the veryfirst statement. Also notice that 35 >

25)

Provide counter arguments to the following arguments. If the argument is valid, yourcounter argument should cast doubt on the truth of one or more of the premises. Ifthe argument seems to be invalid, then prove it to be invalid by providing a counter

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argument of the form, “even if all the premises were true these alternative possibilitiesindicate the conclusion could be false . . . "

26. The *claims that the earth’s climate is warming compared with past decades is undoubt-edly true, however the claim that this warming is due to human produced green housegases is false. This is simply because the data gathered by scientists can not separatenatural causes of climate warming from anthropogenic (human) causes. Without theability to distinguish between these two causes, no conclusions can be made with respectto the true nature of global warming.

27. If it is impossible to predict accurately the role a person’s behavior will play in his orher health problems, then it is unjust to hold people responsible for the costs of diseasesor disabilities they could not have prevented.Evidence linking lifestyle and disease is based on aggregate statistical methods makingsuch prediction between behavior and health problems impossible.Therefore it is unjust to hold people responsible for the costs of diseases or disabilitiesthey could not have prevented.

28. Men are not more promiscuous than females (because) each act of promiscuity by eachmale requires a female, so the total number of promiscuous men is the same as the totalnumber of promiscuous females.

29. If Einstein is correct, then the universe is expanding in size.(But) if the universe is expanding in size, then it must be expanding into something.But the universe in not expanding into something as the word “universe” means every-thing that exists.Hence, Einstein is not correct.

30. Every cyclist in the Tour de France must test negative for illegal drugs or be disqualified.None were disqualified, so every cyclist in the Tour de France must have tested negativefor illegal drugs.

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5. Induction

The following short essay appeared in a Special Edition of Scientific American in 2012:

The magnitude 9.0 Tohoku-Oki earthquake and tsunami that devastatednortheastern Japan in March 2011 took the seismology community by sur-prise: almost no one thought the responsible fault could release so muchenergy in one event. We can reconstruct the history of seismic activity indi-rectly by inspecting the local geology, but this can never fully substitute fordirect detection. Modern seismographs have been around for only slightlymore than a century, too short a time to give a clear idea of the largest quakesthat might strike a certain area every few centuries or more. If we could letthese instruments run for thousands of years, however, we could map seismicrisk much more accurately - including specifying which regions are capableof magnitude 9.0 even though they have not seen more than magnitude 8.0in recorded history.Multimillennial records would also answer another riddle: Do megaquakes

- by which I mean tremors of magnitude 8.5 or greater - come in worldwideclusters? Records of the past 100 years or so suggest that they might: sixof them occurred in the past decade, for instance, and none in the threepreceding decades. Measurements over a longer period would tell us if thisclustering involves physical interaction or is just a statistical fluke. 1

This essay gives a nice illustration of what is known as the principle of induction. Whenthe author expresses, “If we could let these instruments run for thousands of years,however, we could map seismic risk much more accurately - including specifying whichregions are capable of magnitude 9.0 even though they have not seen more than mag-nitude 8.0 in re

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