MODULE V: ARGUMENTS
ARGUMENTS
A. Introduction
Arguments. What good are they? Don’t they just drive people apart? Why can’t we all just get along? Well, the reason is because we see things differently… sometimes radically differently. We all know what an argument is, right? It is the heated exchange of words that parents and kids have when the kids want more freedom and the parents want more responsibility or between spouses when they have been hurt, disappointed, or deceived. This is a fine use of the term. It is the most common use of the term. However, in the areas of critical reasoning and logic, this is not how this term is used. In these disciplines, the term refers to the set of sentences one puts forth to establish their claim. An argument is a pretty darned formal thing, then, in critical reason, quite unlike the hair-pulling, name-calling, and voice-raising arguments with which you might be familiar. In critical reasoning, it takes only one person to have an argument. This is the case because an argument is not something people have, it is something a person asserts. And, it is not just anything someone asserts. I can assert that cats ought to be outlawed, but this is not an argument. It is simply an assertion, a claim. To be an argument, I must back up that claim with some sort of reason. If I state that cats should be outlawed because kitty boxes are the most disgustingly smelly things, then I have stated an argument. There is a reason and there is a conclusion. Ta Da. That is it. In simple terms, an Argument Δ= a set of sentences that have a reason / conclusion structure.
Anytime you have at least two sentences that have this structure, you have the presence of an argument, or at least part of one. An argument is sorta like a math problem, though. If I have
3
x __
= 51
then I have a math problem… I have an incomplete structure. By definition, there is something missing here. And it is known as a problem because it requires something to be fixed and someone to fix it. There is something left out… and it is begging for someone to fill in what is missing.
Arguments are exactly like this. They have a formal structure, when laid out. If I had 51 books and wanted to share them equally with my three daughters, then I would have to fill in the missing number. Of course, I just gave you a story problem. In real life that is how things work. We have a problem, we need a number, so we do that math problem. In real life, people use arguments all the time, all day long. Pretty much anytime you use the word because, you are giving an argument, whether you realize it or not. Days are filled with these arguments…you may not recognize them, and they may appear very informally, but they are there nonetheless. Most often, the arguments that are put forth during the day are informal, and incomplete. Just like the math problem above. Something is missing.
Arguments, by definition need a reason and a conclusion. A complete argument needs two reasons and a conclusion. Just like the math problem above, there needs to be two numbers to complete it. Two factors and a product are required to complete the problem.
Arguments, just like math problems, have a certain structure that they have to follow… different math problems, different structure, e.g.:
3 [pic]
x __ (2a+B)*3 = c/2
= 51
Each has a different structure. For each problem, it is absolutely essential that the problem is structured appropriately. Each problem has a correct structure. Arguments are exactly the same. Each argument type has to be structured appropriately. When an argument is structured correctly for that particular argument type, then we say that the argument is valid. Validity Δ= the quality of an argument when it is structured correctly.
Of course, in real life, arguments come in the form of sentences. The teen tells his mom that he should be able to go the party because his friend’s parents will be home. Notice the “because” – this makes it an argument. To understand and analyze arguments, logicians have devised a way to present them….in the form of syllogisms. Syllogism Δ= formally structured argument/discourse, one thing being stated and something other than what is stated follows necessarily from it
e.g. p1 Tom is left-handed (“p” = premise i.e. the reason)
p2 All left-handed people are artistic
C∴ Tom is artistic ( “C” = conclusion)
( ∴ = therefore)
Again, in real life, arguments don’t show up in this form…unless you hang out with some pretty weird friends….or logicians! If you notice the argument above, there are two premises and once conclusion. But, in the example of the teenager, there is only one premise(= reason) and then the conclusion. This is completely normal. This is how people typically think and talk. It is how we argue. These arguments – that have only one reason given – are real arguments…they simply are incomplete as stated. This is how they appear in syllogistic form:
e.g. p1 Tom is left-handed (p1 = stated/explicit premise)
p2 _________________ (p2 = unstated/implicit/hidden premise)
C∴ Tom is artistic (C = conclusion)
This incomplete argument form is known as an enthymeme. Enthymeme Δ= an argument with one implicit (hidden) premise and one explicit (stated) premise and a conclusion. One premise is stated and obvious to the reader/hearer. The other premise is there… it is simply hidden/invisible/silent. Logically, it has to be there. It has to. You see, just like the math problem:
3
x __
= 51
there is something missing… something left out. YOU simply cannot get 51 directly from 3…there needs to be a third element….something that ties them together…. something that links the two numbers. What links the two in the math problem is left out intentionally with the purpose of having the reader figure out what the missing element is. The reader has to use her mind and logic to determine what exact element connects the other two….given the particular form. And please get this….there is only ONE element that works. There is no fudging here….no maybe this or maybe that. In math, it is an exact science….actually… exact logic!
Enthymemes are just like math problems….something is left out / implied (Imply Δ= to leave unstated, to leave out) and therefore you have to fill in / infer (Infer Δ= {literally, to carry in} to fill in what someone has left out (implied) and this can involve inferring either a premise or a conclusion). Let me stress this point: this is not a guessing game not does it involve subjective ideas or beliefs… no different than math problems. Also, just like inferring the missing element in math problems involve steps to get to the solution (and you do remember that you math teachers always wanted NOT just the right answer but also the right steps to the solution?), you need to know how to get to the correct inference. The reason is, that unlike a math problem, he person may NOT want you to figure out the missing element…they might be hiding it from you on purpose!
There are six basis steps to get to the right answer. And guess what? You get to learn them…but before we get there, I would like to give you the reason why it is imperative, as a Critical Reasoner, to do so. Let’s say that someone says that Physician Assisted Suicide should be legal because it ends needless suffering. There is a hidden premise there…and I submit to you two Critical Facts: 1) the hidden premise is always the one that the person is going to have much more difficulty defending… and therefore the one you really need to be attacking, and 2) the hidden premise is often one that the speaker doesn’t even hold to in the first place, so if they are pressed, they will deny that claim…and then, voila, their argument falls apart, because it had to be there to have the argument completed and valid.
The hidden premise to that argument is “All things that end needless suffering should be legal.” I would be hard pressed to find anyone who would make that broad of an assertion…and try to defend it…but the thing is….that is the exact hidden premise that needs to be there to have that argument be valid. If you show them that this is their hidden premise…and they backtrack and say that not ALL things should be legal (such as going to the cancer ward with an AK-57 and taking everyone out who is terminal), then their argument fails, as it is simply not valid with that hidden premise.
Got all that? I hope so, because we need to press on here to the nitty-gritty of argument analysis…
B. Argument Analysis: Categorical Deductive Arguments
The first thing you need to know is that there are two basic types of argument: deductive and inductive. The categories are distinguished by the direction of the argument. Deductive arguments move down. Inductive arguments move up…or sideways.
Deductive arguments start with general premises (Premise Δ= a reason, technically, a reason in a formal argument) and argue to specific conclusions. Technically, deductive arguments start with general premises and move to less general conclusions… that is they move from general to particular, but the particular could still be sorta general. For example:
e.g. p1 All mammals are vertebrates (stated premise)
p2 _______________________ (hidden premise)
C∴ All bears are vertebrates (C = conclusion)
The conclusion is still general, in that it talks about a class of entities, but the class of mammals in the premise is a larger and inclusive of the class of bears. This argument moves from a more general class to a less general class. The following is an example of moving from general to specific:
e.g. p1 (hidden premise)
p2 All bears are mammals (stated premise)
C∴ Smokey the Bear is a mammal (C = conclusion)
This argument has a conclusion that refers to a single entity and a premise that is a class. Still a deductive argument, though. Still moves from more general to less general.
Deductive arguments come in three general subtypes and these types will be looked at later. For now, just know that the most common type is the one in which the general premise is categorical statement (Categorical Sentence Δ= a sentence that assigns a quality to all individuals in a set, e.g. all guys are cool). Surprisingly enough, this subtype is known as the Categorical Deductive argument. This is the default deductive argument, as you will see.
So, someone presents you with an argument, say: Abortion should be allowed because abortion is a woman’s choice. This is not an uncommon argument... and this is an enthymeme…. one premise and one conclusion. Now ‘tis time to learn how to correctly infer the hidden premise.
6 Steps To Correctly Infer A Hidden Premise In A Categorical Deductive Argument
1) Identify the minor term
2) Identify the major term
3) Identify the middle term
4) Double all terms
a) the minor term must be used as the complete subject of one of the premises
b) the major term must be the complete predicate of the OTHER premise
c) the middle term must be used once as a subject of a premise
and once as the predicate of the other premise
5) Make sure the middle term that is in the subject position is distributed*
6) Make sure hidden premise is grammatically correct
* To distribute means to put an adjective in front of a noun extending it to full set of the items (e.g. all, each, every)
Simple enough, right? Let’s do this together. “Abortion should be allowed because abortion is a woman’s choice.” Identify the premise and the conclusion…the conclusion comes right after the “therefore” or before the “because”, and the premise comes right after the ”because”. So here is the argument in syllogistic form:
p1 Abortion is a woman’s choice (stated premise)
p2 ________________________ (hidden premise)
C∴ Abortion should be allowed (C = conclusion)
Now we need to infer the hidden premise.
Ready?
Step 1) Identify the minor term. Minor Term Δ= the complete subject of the conclusion in a categorical deductive argument. The complete subject in the conclusion above is “abortion”. To identify the complete subject, simply find the verb, and it is typically all the words before it.
Minor Term = Abortion
Step 2) Identify the major term. Major Term Δ= the complete predicate of the conclusion of a categorical deductive argument. To identify the complete predicate, find the verb, and that word along with all the words after it, are the complete predicate.
Major Term = should be allowed
Step 3) Identify the middle term. Middle Term Δ= the term that appears in the stated / explicit premise that does not appear in the conclusion; the predicate of the minor premise; the subject of the major premise. To most easily identify the middle term, simply find the word or phrase that appears in the explicit premise but does not appear in the conclusion. Make sure you get the whole phrase!
Middle Term = is a woman’s choice
Step 4) Double the terms. In a categorical deductive argument, each of the three terms must appear twice. Minor terms MUST be subjects. Major terms MUST be predicates. Middle terms are in the subjects in one premise and are in the predicate in the other premise. Middle terms go both ways.
Start the analysis with the conclusion. Start with the minor term in the conclusion and ask, is it used once as the subject of the conclusion as also as the subject in one of the premises? (it doesn’t matter which one – actually it doesn’t matter which order the premises are in at all) If it is used twice, cool, you can go the major term. If it is not used twice, then write it in as the subject of your hidden premise as the subject. As you can see, in our argument, the term “abortion” is already doubled, so nothing needs to be done with the minor term.
Then go to the major term. Is the major term used once as the predicate of the conclusion and then once as the predicate of one of the premises? If so, you are good to go with the major term. If not, write in the major term in the hidden premise as the predicate.
Very Important Note: the minor and major terms will never appear in the same premises. The minor term will be the subject of one premise and the major term will be the predicate of the other premise. So, this is what we are left with at this point:
p1 Abortion is a woman’s choice (stated premise)
p2 is a woman’s choice should be allowed (inferred premise)
C∴ Abortion should be allowed (C = conclusion)
Unless the enthymeme was of a particular construction, you will have an inferred premise that sounds a bit strange here. But take heart, we are not finished.
A Not So Important Note: The premise that contains the minor term is referred to as the Minor Premise (p1 above). The premise with the major term in it is known as the Major Premise (p2 above). The conclusion is known as the conclusion. ( Let’s move on.
Step 5) Distribute the middle term that is in the subject position. Distribute Δ= to add an adjective to a term so as to extend its number to the entire set of those things, e.g. all, every, each, any, no, none, etc. So, find the premise that has the Middle Term in the subject position (the major premise) and put the word “all” in front of the middle term. Just do it. It doesn’t have to sound correct. Now what do we have? A mess, right?
p1 Abortion is a woman’s choice (stated premise)
p2 All is a woman’s choice should be allowed (inferred premise)
C∴ Abortion should be allowed (C = conclusion)
Yuppers, p2 is kind of a mess right now. Sorta reminds you of your own mind right about this time, right? Be patient, it will all become very clear… very soon.
Step 6) Make sure hidden premise is grammatically correct. This involves two substeps.
A) To start with, if the middle term begins with a verb, you need to put a noun in there. The particular noun will be determined by the middle term itself. There are three ways to do this. Two easy ways and the difficult way. The difficult way to do it is to ask yourself what class does the term fit into? It is an action, a person, a process, a procedure, an object, etc? Whatever class you determine it is in, you put that word there after “all”. Easy Way 1 involves just putting the word “things” (if there is no verb in the Middle Term) or the phrase “things that” or “things that involve” (if there is a verb in the Middle Term). Easy Way 2 involves simply removing the verb….but this is only allowed if it is a verb of being, like “is” or “are”.
Examples: Difficult Way - p2 All actions that is a woman’s choice should be allowed
Easy Way 1 - p2 All things that is a woman’s choice should be allowed
Easy Way 2 - p2 All woman’s choice should be allowed
B) Make the sentence fit together. This involves matching the subject and verb numbers. We need subject / verb agreement. This would end up with the following:
Examples: Difficult Way - p2 All actions that are women’s choices should be allowed
Easy Way 1 - p2 All things that are women’s choices should be allowed
Easy Way 2 - p2 All women’s choices should be allowed
I can see that look in your eyes. Either you are high on something, have not slept for 36 hours, or you are feeling a bit overwhelmed here. Please trust me. This gets much, much easier after you have done it a few times.
Trust me. So, now we have completed the 6 Steps… let’s see what we have.
p1 Abortion is a woman’s choice (stated premise)
p2 All actions that are women’s choices should be allowed (inferred premise)
C∴Abortion should be allowed (C = conclusion)
Or
p1 Abortion is a woman’s choice (stated premise)
p2 All things that are women’s choices should be allowed (inferred premise)
C∴Abortion should be allowed (C = conclusion)
Or
p1 Abortion is a woman’s choice (stated premise)
p2 All women’s choices should be allowed (inferred premise)
C∴Abortion should be allowed (C = conclusion)
Phew….we got it.
As you should be able to tell, it really does not make any difference how the hidden premise is worded. Each of these renderings really mean pretty much the same thing.
Again, let me remind you that it makes no difference what order the premises appear. Logically,
p1 All women’s choices should be allowed (hidden premise)
p2 Abortion is a woman’s choice (stated premise)
C∴Abortion should be allowed (conclusion)
Is equivalent to
p1 Abortion is a woman’s choice (stated premise)
p2 All women’s choices should be allowed (hidden premise)
C∴Abortion should be allowed (conclusion)
And this, my friends is the complete syllogism that is a result of the analysis of, “Abortion should be allowed because abortion is a woman’s choice.” We have inferred the hidden premise. Ta Da. Let’s take a moment, however, and look at the substance of this argument, not just the form. Most people, I suspect, would not take issue with the first premise…Premise 1. Some hard core Pro-Life people would want to, I guess, but not if they put aside the Abortion Issue and just looked at the sentence as a sentence. As a sentence, as it is saying is that abortion is something that is a option that a woman has amongst a range of options and that the woman has free will in determining which she chooses. Leaving aside the morality of either having a range of choices or a woman exercising her free will in these choices, all the sentence is saying is that abortion is something involving a woman choosing to do something. It is really no different from saying, “Going to night school is a woman’s choice,” or “Wearing black shoes is a woman’s choice, or “Making meatloaf is a woman’s choice, or “Running for President is a woman’s choice.” Understand? Given this, as I said above, most would agree with Premise 1.
However, I think MOST would also NOT agree with Premise 2. I have yet to have a single student tell me that they hold to Premise 2….except in jest. Sure, some individual women jokingly wish that they could do whatever they wanted…but they sure wouldn’t want every other woman to have the same freedom…just themselves! No sane person I can conceive of actually would believe that every woman on the planet should be free to do whatever she wants. If that we the case, we would have a lot of dead or maimed husband around. But, we would also have a lot of dead or maimed paramours, neighbors, mothers-in-law, judges, check out clerks, and appliance manufacturers.
Let’s say then, for argument’s sake, that the person says to you, “Abortion should be allowed because abortion is a woman’s choice.” What you do is, immediately in your head analyze the argument and infer the correct hidden premise. Then you ask the person, do you believe that all women’s choices should be allowed? If they say, “Yes,” then you should run the other way if it is a woman… if they say no, then you realize that they do not have a valid argument… right? Right? Did I lose you somewhere? Let me explain. If they do not believe that all women’s choices should be allowed, they probably believe that some women’s choices should be allowed…and that abortion is in that group. But there’s the rub. You see, if you change the word “all” to “some”, then the argument no longer follows the correct structure of a categorical deductive argument….and is invalid (unless they restructure the argument (which we will talk about below). You see? If you change “all” to “some” the argument collapses. It is not valid… the conclusion does not follow from the premises. It is a waste of words. And I will show you why…and how to prove it!
I promise.
At the end of this section, there are so practice pages. This might be a good time to stop and go work through the first page. Print it out and then lay it if front of you covered with another sheet of paper. Don’t peek. Now, slide the top paper over so can see just the enthymemes on the left side, but not the answers.
Try to do the first one without any reference to the notes. Good luck there, eh? Actually some people can do it…to some people this stuff is as easy as breathing…yes, I agree, they must be emotionally unbalanced. Anyway give it a shot. On another (or the top piece) of paper, write down the answers in this order: 1) identify the minor term, 2) identify the major term, 3) identify the middle term, 4) infer the correct hidden premise.
If you did not get them all correct, for the next problem, you may refer to the list of the 6 Steps:
1) Identify the minor term
2) Identify the major term
3) Identify the middle term
4) Double all terms
a) the minor term must be used as the complete subject of one of the premises
b) the major term must be the complete predicate of the OTHER premise
c) the middle term must be used once as a subject of a premise
and once as the predicate of the other premise
5) Distribute the middle term that is in the subject position
6) Make sure hidden premise is grammatically correct
If you were not able to do the next one correctly even with this list, then figure out exactly what you did wrong (e.g. misidentified the middle term, or skipped Step 5) and then read through the explanation above concerning that Step and redo the one you missed.
Then, move on to the next problem. Each time, if you miss one, just go back and figure out where you missed something in the steps, because that is where the error comes from – either mis-indentification from not knowing the definitions or mis-stepping from not understanding the step or jumping over it. This is the hard part. It is just like elementary school and you have to do this step by step.
If you have proceeded as I have instructed, by the time you get to the last few problems, you should be able to do them without using the notes.
There are other exercises, but if you have followed instructions, you should be semi-proficient with this process by now and we will do more of them later.
You should try to become very aware of arguments in daily life. Even simple ones like, “I have to go home because I don’t feel well,” and Bob is a dork because he is from Millard.” Tune into these enough, and it will become second nature for you to hear, recognize, and instantly analyze people’s arguments. My goal is that you do this the same way you check the rearview mirror while driving. When you first learned to drive you did it consciously and with intent all the time as a part of the array of things you were supposed to do when you drove. Now, checking the mirrors is done by your brain (subconsciously, for those of you who categorize that way) automatically without you even thinking about it. As a matter of fact, I bet that entire array of things you were once so nervous about doing as you drive (like keeping your hands at 10:00 and 2:00) never even hardly cross your mind anymore. Nowadays, you have Tom Petty cranked up and are banging you hand on the dash as you sing, “She’s an American girl,” or you are talking on the phone to your mom asking for money…everything BUT thinking about all those things. Don’t get me wrong. I bet you still do most of them…it is just, because of so much practice, your brain does it automatically, leaving your mind to ponder the deep verities of life….or just rock out.
If you keep conscious simply of hearing the word “because” and “therefore”, you can get your brain to do this analysis for you too….leaving your mind to do the higher order functions of identifying fallacies and formulating killer responses to the arguments of all the novices!!!
K… on to the next part… verifying your work!
C. Venn Diagrams
So… did you do the math problem above? What times 3 equals 51? I suspect you could do that in your head. How about 413 divided by 7? Not quite as easy. So, if you do the math…work out the problem using the following structure:
59
7)413
35
63
63
0
Simple enough, yes…once you know how to do long division, it is. When you set up the real world problem, of say, having a $416.00 football pot and needs to be split by 7 people in the office, you take the problem and translate it into this structure:
_
7) 416 and then do the math. There is nothing magical or even “right” about doing division this way….it simply is a way to help the mind reduce the process to a set of logical steps toward the answer. This is no different at all from what you learned above when you inferred the hidden premises.
But….here is the rub….how do you know you did it right? How do you know you got the right answer? If you do all the steps, one by one and do them correctly, of course you should end up with the correct answer. But how do you know for sure? With division, the answer is to check your results….and you check the results in division by doing a multiplication of the divisor/quotient back and see if you end up with the original dividend:
59
x 7
= 9 x 7 = 63; so write the 3, carry the 6.
5 x 7 = 35; add the carried 6 and you get 41
41- 6 => 416 so voila, you have done the problem correctly!
You have a tool with which you can check your work, your math, your logic.
We have a tool available to us to check whether or not we have done our inference correctly, also. This tool is known as the Venn Diagram. Venn Diagram Δ= a graphic representation of a statement and/or an argument, translating the premises from sentences to sets, attributed originally to a mathematician named Venn. The process is the same, conceptually as above. We have a real world problem, we reduce it to a syllogistic form, we take the enthymeme (as real world arguments are almost always enthymemes) and apply the six steps to it and come up with our inference. Let try this one. People are sitting around in the lunch room discussing the current congressional debate over raising the minimum wage. After a while, Ezra says, raising the minimum wage should be done because raising the minimum wage will pull workers out of poverty. You, of course, having instantly did the six steps in your head, know immediately that Ezra’s unspoken premise is “all things that pull workers out of poverty should be done.” Hmmm… you ask yourself… does Ezra really believe that? Can it honestly and reasonably be defended that all things that pull workers out of poverty should be done? Probably not. So you ask Ezra that question, and he responds by saying, “no,” not all things…he is not in favor of taking 50% of everyone’s paycheck and depositing in the accounts of people who make below the minimum wage. Ok, cool. So, then, you say that there must me some things that are ok to do and some things that are not ok. Ezra agrees. Now is when the testing actually becomes meaningful. Sit back and watch as I show you how this works.
As defined above, a Venn Diagram is simply a graphic representation of a statement and an argument. Lets say we have an assertion that, “All pigs suck slime.” This verbal assertion can be translated to the following sets:
[pic]
Are you with me so far? Good….cuz now it starts to get fun…and personal!
So, say, Ezra comes back and says that not all things that pull people out of poverty should be done…just some things. You see…this is a HUGE problem…the problem is that the argument is not valid anymore. You no longer have a distributed middle term in the subject position, therefore you no longer have the correct form. And this is key: the correct form is not just a rule or somebody’s ideas. It is the logically necessary form.
Lets look at why. Lets say that someone asserts that, “Porky is a Pig, and therefore Porky sucks slime.” We have an enthymeme, correct? And the correct hidden premise is, “All pigs suck slime, “ right? Let’s see how this Venn Diagram looks:
[pic]
A valid categorical deductive argument produces a necessary conclusion (more on that later). The conclusion necessarily follows from the premises if the argument is structured correctly. If Porky is a pig, and if all pigs suck slime, then Porky MUST suck slime. Got it?
But…here is the rub. Let’s change the premise to read, “Some pigs suck slime.” We have, in essence, un-distributed our subject-positioned middle term. So what, you ask? Lets us see so what.
[pic]
[pic]
But, we could also put Porky in the unshaded portion of the set of “All Pigs”, and that would satisfy the assertion that Porky is a pig. Either one works, does it not? Do you see the problem? If either position works, then we cannot definitively infer a relationship between Porky and Slime Suckers. Given the “some”, Porky might be a Slime Sucker or might not be. Hence, there is no way to infer a relationship between Porky and Slime Suckers. Maketh this sense unto thee?
To put it in technical terms, the presence of an undistributed subject-positioned middle term in a categorical deductive argument renders the argument invalid.
So… it this what you want to say in response to Ezra’s argument? I suspect not. I suspect that you realize that the only thing you would get back from Ezra is a vacant look. No, you need to deal with people on their level. So what do you say back? Ezra, you flippin’ idiot! Probably not helpful either. How about this? How, about you say, “Ezra, since you are claiming that not all things that pull people out of poverty should be done, but that some things should be done, then for me to understand why it follows that the minimum wage should be raised, I need to know why you put ‘raising the minimum wage’ in the ‘some things that should be done’ category. Why there and not the other category? What is it about this solution that puts it into the should category and not into the should not category, where we find the ‘taking 50% of everyone’s paycheck and depositing in the accounts of people who make below the minimum wage’ solution?”
If you are an astute thinker, you can see what is going on here, logically. What Ezra is going to have to do is come up with a new category into which the ‘raising the minimum wage’ solution fits… and then that category will have be distributed. The middle category HAS to be distributive.
A brief aside: can you see now why they call the terms minor, middle and major? Let’s use the following argument:
e.g. p1 All Cretans are liars
p2 Alex is a Cretan___________
C∴ Alex is a liar
[pic]
Remember the definitions for our terms? The Minor term is the complete subject of the conclusion….the Major term is the complete predicate of the conclusion… and the Middle term is the term that shows up in the premise but not in the conclusion….
So…
the minor term is Alex
the major term is Liars
the middle term is Cretans
Rearranged…
Major = Liars
Middle = Cretans
Minor = Alex
Does this arrangement remind you of anything in the Venn Diagram above?
Aha…..do you see now? You see, gang, they picked these names for these terms for a reason. The major term is the most general term. The minor term is the most specific, and the middle term is the one that is placed between them, and each of them after they are each related to the middle term, are then related to each other. There is a method in all this madness, I assure you. And the rules that govern arguments exist for a reason too. They are not arbitrary like speed limit laws, or how late a 17 year-old can stay out on a week night. The rules governing logic are intrinsic and inherent to the process.
I submit to you that the most difficult thing to do in constructing a good argument is to find the perfectly sized middle term. Just like Baby Bear’s porridge… if it is just right, it will render your argument more palatable to your hearers.
D. Argument Analysis: Other Deductive Arguments
Time now to press on the other two deductive arguments: Hypothetical Deductive and Disjunctive Deductive. Since they are deductive in form, they follow the General to Specific structure. The three deductive arguments are distinguished by their General Statement. Categorical Deductive arguments have a categorical statement (an ALL type statement) as their General Premise (in this case, the Major Premise). Hypothetical Deductive Arguments have Hypothetical statements and Disjunctive, of course have Disjunctive statements as their General Premises. They are all deductive arguments, but because of the differing General statements, they have different structures and different analysis. You with me? K… let’s get to it.
Hypothetical Deductive Arguments
A hypothetical statement is one that infers an if/then condition. You must make sure, though that this is what you are dealing with. A true hypothetical statement is making an assertion about a future event in time. The statement, if a figure had three sides, then it is a triangle, is not a true hypothetical statement. It should also be noted that all hypothetical statements can logically be resolved into categorical statements; e.g. If it rains tomorrow the game will be postponed is logically saying the same thing as, All situations where it is raining are situations where the game will be postponed.
But, let’s go ahead and see how to infer the correct hidden premise for a Hypothetical Deductive argument.
First off, you need to see that a hypothetical statement has two parts to it, the “if” portion and the “then” portion (the word “if” will almost always be stated, but the word “then” is most often understood but not stated – implied, if you will). The “if” part is known as the Antecendent and the “then” part is known as the Consequent.
So, let’s start with an assertion:
e.g. p1 If Bob eats road kill, he will puke.
p2 ___________________________
C∴
In a categorical deductive argument, there is one and only one possible hidden premise… just like in the math problem. In Hypothetical Deductive arguments, there are tow possible hidden premises. There is a several step process to correctly infer the rest of the argument, just as there was with the Categorical Deductive, and the process is completely different.
Steps To Correctly Infer The Two Possible Hidden Premises Of A Hypothetical
Argument –
1) Identify the antecedent (the if portion of the hypothetical sentence)
2) Identify the consequent (the then portion of the hypothetical sentence)
3) For the hidden premise, either:
a. Affirm the antecedent (state the if portion as if it has already
occurred, or
b. Deny the consequent (negate the verb of the consequent)
4) For the conclusion, either:
a. Affirm the consequent (if the antecedent was affirmed in p2), or
b. Deny the antecendent (if the consequent was denied in p2)
Let’s see how this looks:
If we affirm the antecedent in p2, then we affirm the consequent in the
conclusion-
e.g. p1 If Bob eats road kill, he will puke.
p2 Bob ate road kill______________
C∴ He will puke (note NOT that he puked, past tense)
If we deny the consequent in p2, then we deny the antecedent in the conclusion-
e.g. p1 If Bob eats road kill, he will puke.
p2 He will not puke (past tense also works here
C∴ Bob did not eat road kill
See? Two possible p2’s and two possible conclusions.
Note the errors of denying the antecedent and affirming the consequent:
e.g. Error 1 p1 If Bob eats road kill, he will puke.
p2 Bob did not eat road kill___________________________
C∴ He will not puke
Can you see why this is invalid, why the conclusion does not then follow from the premises? You see, eating road kill is not given as the only cause for Bob’s puking. It is given as a sufficient cause, but not a necessary cause (for those of you who have taken philosophy, that should be very clear). You see, Bob could have puked for various other reasons: getting the flu, drinking ipecac, looking at your sister, and etc. The only way you could have that p2 is if p1 said, “If and only if Bob eats road kill, he will puke.” This then gives a necessary condition…only under this specific condition will the consequent occur.
Try the other way:
e.g. Error 2 p1 If Bob eats road kill, he will puke.
p2 He puked__________________________
C∴ Bob ate road kill
Same problem as above. This only works if the Hypothetical Premise sets forth a necessary condition. If the situation is that if and only if Bob eats road kill, and he did in fact puke, then it has to follow that he ate road kill.
So….what does this mean in the real world? These errors are seen quite often in argumentation. You need to become acutely aware of Hypothetical Arguments and once you recognize them, make sure that they are valid. If you do not do both of these things, you can quite easily be swayed by an invalid argument. Make sure you “hear” if/then statements. Make sure you have the other person clarify whether the causal relationship between the two elements is sufficient or necessary.
Disjunctive Deductive Arguments
A disjunctive statement is one that infers an either/or condition. You must make sure, though that this is what you are dealing with. A true disjunctive statement is making an assertion about contradictory elements. As you, I am sure, recall from the earlier class on statements, contradictory elements are elements that are logically negatives of each other; e.g. Harry is a magician - Harry is not a magician. With contradictory assertions, one of them HAS to be true and the other HAS to be false. It should also be noted that, just like hypothetical assertions, all disjunctive statements can logically be resolved into categorical statements; e.g. It is the case the Harry is either a magician or that Harry is not a magician. This is, of course, sort of a useless assertion as it is saying nothing much more than giving an example of contradiction. It is no different, really than saying, “It is the case the laundry is either a fun activity or that laundry is not a fun activity.” You are really doing nothing more than saying that things have contradictories.
But, let’s go ahead and see how to infer the correct hidden premise for a Disjunctive Deductive argument.
First off, you need to see that a disjunctive statement has two parts to it, the “Option A” portion and the “Option B” portion (the word “either” will almost always be stated, and the word “or” is most often stated also). The “if” part is known as the Alternative A and the other part is known as the Alternative B (although I would prefer to call it Alternative Not-A).
Categorical Deductive arguments had one and only one possible second premise.
Hypothetical Deductive arguments have two possible second premises.
Disjunctive Deductive arguments have four possible second premises.
Let’s figure out how to correctly infer them, ok?
Steps To Correctly Infer The Four Possible Hidden Premises Of a Disjunctive
Argument-
1) Identify the two alternatives
2) Affirm one and deny the other
or
3) Deny one and affirm the other
Disjunctive statement: Either Q or X
Alternative A: Q
Alternative B: X
Remember, the ~ symbol means not.
e.g. p1 Either Q or X
p2 X____ _____(affirm Aternative A)____
C∴ ~Q (deny Alternative B)
e.g. p1 Either X or Q
p2 ~ X ____ __(deny Aternative A)____
C∴ Q (affirm Alternative B)
e.g. p1 EitherX or Q
p2 ~Q____ ____(deny Aternative B)____
C∴ X (affirm Alternative A)
e.g. p1 Either X or Q
p2 Q__ (affirm Aternative B)____
C∴ ~X (deny Alternative A)
Was this too abstract for you? It isn’t for mathematical types, but that might not be you. Hence I will give you another example… something a little more close to real life.
Disjunctive statement: Either Alice goes to college or she will not get a decent job
Alternative A: Alice goes to college
Alternative B: Alice does not get a decent job
e.g. p1 Either Alice goes to college or she will not get a decent job
p2 Alice went to college____ _____(affirm Aternative A)____
C∴ Alice will get a decent job (deny Alternative B)
e.g. p1 Either Alice goes to college or she will not get a decent job
p2 Alice didn’t go to college____ __(deny Aternative A)____
C∴ Alice will not get a decent job (affirm Alternative B)
e.g. p1 Either Alice goes to college or she will not get a decent job
p2 Alice got a decent job____ ____(deny Aternative B)____
C∴ Alice went to college (affirm Alternative A)
e.g. p1 Either Alice goes to college or she will not get a decent job
p2 Alice does not have get a decent job__(affirm Aternative B)____
C∴ Alice did not go to college (deny Alternative A)
This is pretty simple straightforward logic, based on the nature of contraction. It can get tricky, as seen above when one of the original options is stated as a negative assertion, because then you have to negate a negative assertion.
The biggest thing you have to do here is to make sure that they are true contradictories. In the above example, the disjunctive statement does NOT contain contradictory elements, only contary. Now, you have to realize, that Alice’s mom might believe that they are contradictory, or more likely, that she thinks that it is generally true, and she is asserting it this way to Alice in order to convince her to go to college. And Alice’s best defense for staying home and sitting all day and watching tv is that her mom’s argument fails because she has not asserted a true contradictory condition. In fact, her mom has uses a fallacy, but we will get to those later!
One last point before we proceed to inductive arguments. Note that the conclusions necessary follow from valid deductive arguments… but that does not mean that the conclusions are true. Validity has nothing to do with the truth value of any of the premises or the conclusion. Valid arguments can have completely false premises and conclusions. If you do have an argument that is both valid and has true premises and a true conclusion, then you have what is known as a Sound Argument.
Now…on to Inductive Arguments!
(Note: there is a practice page for these arguments at the end of the section)
E. Argument Analysis: Inductive Arguments
Next we move to inductive arguments. These are just the opposite in structure from deductive arguments; they move from specific to general (or sometimes specific to specific, although I prefer not to refer to these types as inductive but transductive, e.g. Nebraska is north of Mexico; Mexico is north of Peru; therefore Nebraska is north of Peru – this argument does not move in the direction of specifics or generalities, but simply “across” the terms ).
Inductive arguments move from specific to general. They move from examples or test cases to generalities inferred from common elements or characteristics found within the individual examples or test cases.
e.g. p1 Apples are sweet
p2 Pears are sweet
C∴ All fruit is sweet
You need to know that there are a couple of differences between deductive and inductive arguments. First off, deductive arguments can only have two premises, while inductive can have more than two.
e.g. p1 Apples are sweet
p2 Pears are sweet
p3 Bananas are sweet
p4 Oranges are sweet
C∴ All fruit is sweet
Secondly, while there is a necessary structure in inductive arguments, because the consequences of validity are so different in deductive and inductive arguments, many do not use the term validity as applicable to inductive arguments. I think they do so incorrectly, as the simple denotation of validity is that is refers to the quality of an argument as simply fitting the specifically defined structure for that type of argument.
With deductive arguments, validity results in conclusions that necessarily follow from the premises….that is, if the deductive argument is structured correctly, then the conclusion has to follow. With inductive arguments, even if they are structured correctly, the conclusion never has to follow from the premises. More on this later.
So, let’s look at exactly what is going on in the inductive argument above. You have four individual items. These would correspond to multiple minor terms in the deductive arguments (here, the complete subject of the individual premises) What is done in the conclusion is to 1) take the mentioned common element (the common complete predicate in each of the premises (serving like the major term in deductive arguments), 2) devise some common category into which all the individuals fit, )corresponding to the middle term in the deductive argument, and then 3) distribute that category by asserting that all other non-mentioned individuals that might also fit in that category also possess that same. Does this make any sense at all?
The individual items are given in the premises (apples, pear, bananas, oranges). The common quality that all of the individuals share (sweetness) is also given. What the conclusion does, then, is simply devise some category (fruit) within which all of the individual items would fit and then assert that then entire category is populated by (both named and non-named individuals) items that have that same quality. Logically, it does not matter what the category is that you devise. As long as all the named individuals fit into the category, the argument is valid.
Let’s try another…maybe that will be helpful in your understanding.
e.g. p1 People from Nebraska are hicks
p2 People from Arkansas are hicks
p3 People from Kansas are hicks
So, you have given to you the individual items (the several minor terms), and the quality that is common to all of them (the major term). What is required, then, is to infer/fill-in some category into which all of the stated items would fit. You could use “midwestern states” as your middle term and conclude that “all people from Midwestern states are hicks,” and your argument would be valid. But, you could also use, “states whose names have the letter “k” in them and conclude that, “all people from states whose names have the letter “k” are hicks… and this also would result in a valid argument. One might also use a larger middle term like “states in the U.S. and this would be fine. Heck, you could use the middle term “states in the western hemisphere, and this would also fit the structural requirement for an inductive argument. Again, it is all about the structure… not about the truth value of the premises or conclusion.
This sounds silly doesn’t it? How can you conclude that all people in states in the western hemisphere are hicks after having named only three of fifty in the U.S. and many more in the other countries in the western hemisphere? Well… you just can. The thing is, as mentioned above, it is about structure and not content, per se.
As also mentioned above, valid deductive arguments produce conclusions that have to follow from the premises…they produce necessary conclusions…it cannot be any other way. Inductive arguments, however, do not produce necessary conclusions… ever. Valid inductive arguments produce possible and ranges of probable conclusions. Look again at the last argument above. Given those three premises, it is necessary that all people from Midwestern states are hick? Sure it is. Given those premises, is it not also possible that all people from states in the western hemisphere are hicks? Yessiree, Bob, it is. It is less likely, to be sure, but it is possible. Valid inductive arguments always produce possible conclusions… and depending on the relative size of the number of individuals mentioned and the size of the middle term, also a range of how possible. The smaller the size of the middle term and / or the larger the number of individuals named, the greater the possibility.
Let’s go back to our old friends the Venn Diagrams. I think this will help you to understand what I am saying. Take following premises:
e.g p1 Refrigerators are expensive
p2 Washing machines are expensive
p3 Dishwashers are expensive
So, we have three categories of items mentioned, let is make a symbol for each-
[pic]
Now, let’s put them all into the major category of expensive things:
[pic]
Now, we take the givens….the things stated…and we imply a new category – appliances.
[pic]
So, we end up with:
e.g p1 Refrigerators are expensive
p2 Washing machines are expensive
p3 Dishwashers are expensive
C∴ All appliances are expensive
Now we have a valid argument… it is constructed correctly. You can see however, that in the real world, that the conclusion is not “true”. Again, remember that validity has nothing to do with truth. Nothing. The argument is valid or not depending on the structure alone, without any consideration of the truth value of the premises or conclusion.
Now, let’s look at the question of probability. Given p1-3, is it possible that all appliances are expensive? Sure. Is it probable? Well, what does “probable” mean? I define the word Probable Δ= having more than a 50% chance of being the case. Is it probable that all appliances are expensive, given the premises? Not so much, no. And why not? Because the set of all possible appliances is pretty darned big…and there have been only three types given… there is a huge disparity between the number of items given and the size of the category. At this point, the most we have is a possible conclusion that all appliances are expensive. We have to options if we want to improve the odds. We can 1) increase the number of items in the set (add Freezers, Microwaves, and etc.) or we can 2) decrease the size of the set (have she set be “All appliances that weigh over 20 pounds.”) The smaller the ratio, the greater the probability.
In logic, the possibility of the conclusion following from the premises in an inductive argument is known as Safety of Inference. The greater the ratio of number of items named/number of possible items, the higher the Safety of inference.
Let’s try another example. Let’s say we have 26 students in the Critical Reasoning (CR) classroom in a college of 2000 students. And let’s give them all letter designations (i.e. Student A, Student B, and etc. – that is how we instructors look at you guys anyway…) So, we start on the left side of the room and starting taking a poll, asking one student at a time which is their dominant hand. Student A answers and says she is right handed. I then ask the student behind her, who answers that he is also right handed.
At this point, it would be perfectly fine, from a structural perspective, to have the following argument:
e.g p1 Student A is right handed
p2 Student B is right handed________________
C∴ All students in the college are right handed
This is a valid inductive argument. However, it is not a safe inference because of the small ratio of students polled and actual students in the set. Only 2 of all the students in the college have been polled….a small percentage. Is it possible that all students are right handed after only determining that with 2 of them? Possible, yes. Probably, no. the ration of 2/2000 is too small to be probable. So, to increase the Safety of Inference, we can either put more individuals in the set or decrease the size of the set… or both. So, we poll the next three rows of students and we find that A, B, C, D, E, F, G, H, I, J, K, and L are all right handed. Has this increased the safety of our inference? Sure… but not by much. It is still 12/2000, a pretty small ratio. It is increasing, yes…so the safety of inference has increased, but not significantly. The most significant thing that has happened is that we have not found any variances… they all have been right handed. That is significant. You see, if just one of the students…student G for example, was left handed, then the entire argument fails. A single contradictory example destroys the validity in an inductive argument. Even if 11 were right handed, the existence of a left handed student makes it not logical to conclude that “All students are right handed”… it would be an assertion contradictory to the premises.
An increase in the number of non-variant individuals increases the Safety of Inference. Also, however, decreasing the size of the set would have the same result. If we decreased the set to, say, students in the CR classroom, then our ratio skyrockets from our current 11/2000 to 11/26. At that ratio, we have a greatly increased Safety of Inference over what we had before. Our possibility that all Students in the set are right handed has increased dramatically because we have greatly reduced the size of the set = the population of the set.
We have a greater possibility that the conclusion is true with the increase of Safety of Inference. Once our sample number reaches and surpasses 50% of the population, then we have moved from a possibility to probability. At the current status, we have 12/26 students confirmed as right handed, so it is still only possible that all students in the CR classroom are right handed. If we poll two more students and they are non-variant, then we are above the 50% mark and now we can say that it is probable that all students are right handed. From this point on, the more non-variant individuals we get, the greater our level of probability.
Let’s say we have polled all the students except one….student Z, and we have found no variants. Now our safety of inference is very strong… we have a very high probability that all students in the CR classroom are right handed. Can we say for sure that they are all right handed? No, of course not. But it is pretty darned likely, yes? Or is it? Well, I am not going to enter into a discussion of probability theory with you, but I will say this…that it is still a 50% chance that student Z is left handed or right handed, so I think that there is still a 50% chance that all students in the classroom are right handed, but I think that this really is a question of semantics. Given that there are no non-variants, one could argue that there is something odd going on here…or something not taken into account (e.g. psychologists have determined that people interested in and therefore predisposed to taking a Critical Reasoning class tend to be heavily left brained…and people who tend to be strongly left brained statistically tend to be right handed) [by the way…I am making that up]. But that is all for another class.
Right now, what I want you to understand is that where there is even only ONE individual in the population that is not known to be non-variant, then the most we can have is strong probability as the level of Safety of Inference. What then happens after we poll student Z and find her to be right handed? Have we achieved a level of 100% Safety of Inference? Drum roll please……..Ta Da…..NO!!! We have achieved a level of 100% statistically that all students in the CR classroom is right handed….but not 100% Safety of Inference….and why not, you ask…why the heck not? Because there is not more inference. By definition, inference is bringing in/filling in what is missing. There is nothing missing anymore, so there is nothing more to infer.
Do you know what we have now? Now we simply have a categorical descriptive assertion that describes the situation. There is no inductive reasoning going on anymore. We have the general statement as a given, not as an inferred conclusion. We have the beginnings of a deductive argument now, not the end of an inductive.
What can we say then about inductive arguments? They can NEVER give us necessary conclusions. They can never give us absolute assertions. They can, be definition, ONLY give us possibilities and probabilities. Got that?
A question that has puzzled many of us Critical Reasoning types is, how can so many people put so much stock in inductive arguments? You see it when you have known one or two people from a particular ethnic background act a certain way, and observers take that and make generalizations about the entire group. You see it when a woman gets burned by a couple of guys and concludes that all men are jerks. You also see it when you get one bad meal or bad service at a restaurant and then you don’t go back because you think that they will always have bad food or poor service. All of these scenarios involve making generalizations (which is what inductive arguments do) on a very small sample of the entire set. We have done things like this haven’t we? Do you know why? My take is that when it all comes down to it, it is mostly just intellectual laziness. It is so much easier to put things in categories… and not take the time to investigate, scrutinize, and distinguish between the new individuals and the old ones. This is the heart and soul of Critical Reasoning… making distinctions where they ought to be made…and non-Critical Reasoners don’t want to take the time and effort to do so.
But there is another troubling aspect to this. You see, by definition, valid deductive arguments produce necessary conclusions. The conclusions have to follow… there is not guesswork or chance. Valid deductive arguments give us certainty and solidity. By contrast, inductive arguments, by definition only give us possibilities and probabilities. Nothing certain. Nothing sure. But here comes the rub. You see, in my experience, most people have inductive arguments as the basis for the things they hold to be the most sure, the things that they would be least likely to change their mind about. Do you fit this category? I suspect that you do. What things do you hold as most certain, as most undeniable? Unless I miss my guess, it involves things that have been proven scientifically. Do you doubt that the chair you are sitting in is made of atoms? Do you doubt that water boils at 100C? I bet not. And yet, science, by definition (aren’t you getting tired of that phrase?) uses inductive arguments in its work. Science takes individual experiments and then makes generalizations about the world from them. It’s all inductive. Of course, it them takes those generalizations and later applies them to life…but the “truth” value of science is founded 100% upon inductive arguments…which, by definition, cannot give us truth!
And why is this so? I suggest one big reason. It lies not in the sheer number of individual test cases they have done as compared to the whole set. I mean, how many things have they dropped to determine that the rate of gravitational acceleration on planet earth is 9.8 m/s2? I don’t know… but I am guessing that they have dropped thousands… maybe hundreds of thousands of things and measure how fast they fall. But tell me this, how many things are there on planet earth that can fall? Billions and billions. So how can we trust that experiments done on even hundreds of thousands out of the billions in the set…and say that the rest of the items in the set will behave in exactly the same manner? Sheer ratio would argue against it. But there is another factor…another factor that basically makes the small ratio utterly meaningless….and that is the factor of non-variance. Even if they have tested only 500,000 of the 100,000,000,000 items…as long as the 500,000 test cases all produced the same results, then the generalization carries a great deal of weight. If you look back up to the last Venn Diagram, imagine what would happen to the generalization that all people in the CR class room are right handed if student Z was determined to be left handed. The fact the other 25 were consistently right handed is utterly irrelevant once the variant individual in the set is discovered. One counter example can completely blow apart any generalization. It is, as it were, very easy to destroy any generalization – it takes only one instance of variance. So, as long as it is the case….when no counter examples are found, a generalization that has checked out a reasonable number of representative individuals in the set compared to the population of the entire set is considered to be such that one can take it as a “law”.
Phew…. What a long section. Let’s take a short break before we plow through the next stuff, ok?
F. Argument Identification – Discovering The Structure
Now then, we come to the place where the rubber meets the road so to speak, where all of the stuff about argument analysis. As mentioned above, the most important thing about an argument is validity. No matter how well supported the premises are, no matter the truth value of the assertions, if the conclusion does not follow from the premises, it means nothing. Without validity, an argument is as valuable as a Midnight Blue ’68 GTO with white leather interior and a white leather convertible top…..but with no engine and no transmission. It might look good…might give the promise of good times….but it is a mere shell. It gives empty hopes and vain promises.
So, the number one thing you, as a Critical Reasoner have to do when listening to or reading an argument is to make sure that the argument is valid. This involves several steps: 1) Make sure that what is presented is, indeed, an argument, 2) If it is an argument, identify the type of argument, and 3) Having identified the argument, inferring any hidden premises and determining if it is valid.
Alrighty then, the first thing you need to do is figure out if there is an argument presented. Do you remember the definition on an argument? It is a set of sentences that have a reason/conclusion structure. Look at the following set of sentences:
Betty wants to sell her kidney. Her sister is named Ethel. Betty lives in southern Kansas, but grew up in Kentucky. The doctor who examined Betty claims that her kidneys are healthy. Her doctor got his training in Kansas City at one of the best medical schools in the Midwest.
We have a set of sentences…..half the definition of an argument….. but do we have the other half? Do you see any reason/conclusion structure within the sentences. Look hard. Look carefully, look fruitlessly…..because there is none. There are many, many articles written, and many speeches delivered, and many positions passionately presented with no real argument within. When this occurs, there is simply no reason to waste your time trying to pick it apart.
So, how do you know if there is an argument? You find out if there are reasons and find out if there is a conclusion… simple as that. Reasons are all assertions, so you cannot waste your time looking for assertions. They are almost always there…but they do not prove that there is an argument. To show that there is an argument, you have to show that a conclusion is drawn from those assertions. Therefore, what you really need to so keep your eyes peeled for is the presence of a conclusion. So, how do you find that? Well, there are 4 easy steps…and guess what….you get to learn them.
Steps To Identifying The Conclusion In An Argument -
1) Look for signal words* that identify the conclusion.
If none > 2
2) Look for a summary statement at the end of the piece.
If not present > 3
3) Look for a topic sentence at the beginning of the piece.
If not present > 4
4) Look for an Issue*, then find a statement that answers the issue.
If none >assume that there is no argument
*Signal Words Δ= Words that indicate the presence of conclusions and or reasons;
Words that signal conclusions follow:
Therefore (thus)…
In conclusion…
In summary…
It follows that…
So…
Hence…
We may infer (or deduce) that…
Shows (proves) that…
Words that signal a reason follows, and typically the conclusion precedes:
Because…
For…
Since….
Given that….
*Issue Δ= a question that is being addressed and answered in the argument. It is often not stated in the argument, but is implied…and sometimes can be inferred.
Ok then….now let’s assume that you found an argument.(If you made it to the end and did not, then just quit and go get a beer.) Next, you need to determine just what kind of animal you are dealing with. To figure out if the argument is valid, you need to know what kind of argument you have before you. Since every argument has a different structure, you have to identify the type of argument before you can know if it is valid.
How can you identify the type of arguments? Would you believe me if I said that there were several steps?
Steps To Identifying the Type Of Argument -
Ask the following questions:
1) Is there a hypothetical sentence that serves as a reason?
If Yes => the argument is Hypothetical Deductive
If No > Go to Question 2
2) Is there an either/or sentence that serves as a reason?
If Yes => the argument is Disjunctive Deductive
If No > Go to Question 3
3) It there a series of test cases or examples that a generalization is
drawn from?
If Yes => the argument is Inductive
If No => the argument is Deductive Categorical
The irony is that although most arguments that you will likely encounter in life are going to be categorical deductive, they are the ones you look for last. Why is that? Well, ‘tis the nature of the enthymeme. As I said above, most arguments are going to appear as enthymemes with one of the premises left unspoken. In the case of Hypothetical and Disjunctive Deductive arguments, the stated premises will be the hypothetical and the disjunctive premises and you will have to infer the minor premise. In the case of the Categorical Deductive, however, it is the categorical premise… the major premise that is unstated. Therefore, you only have the minor premise stated… and this doesn’t give you much to go on. Let me show you how this works:
e.g p1
p2 Judy went to Westside High School
C∴ Judy will be going to college
Being the experts that you are on correctly inferring hidden premises, you can easily see that the hidden premise here is, “All people who went to Westside High School will be going to college.
e.g p1 All people who went to Westside High School will be going to college.
p2 Judy went to Westside High School____________________________
C∴ Judy will be going to college
But that is if it is a categorical deductive argument.
Could the following be also the case?
e.g p1 If Judy went toWestside High School, she will be going to college.
p2 Judy went to Westside High School____________________________
C∴ Judy will be going to college
How about-
e.g p1 Either Judy did not go to Westside High School or she will be going to
college.
p2 Judy went to Westside High School____________________________
C∴ Judy will be going to college
Do you see the problem? If all you have is the minor premise, the major premise could be either categorical, hypothetical, or disjunctive. As I said a ways back, hypothetical and disjunctive statements can be resolved into categorical statements, so technically, it doesn’t make much difference which statement is there…only for the sake of validity does it matter.
The deal is this…if all you have is a minor premise, you simply cannot know if the major premise is categorical, hypothetical, or disjunctive. And since the hypothetical and disjunctive can be resolved into categorical, you simply end up with the categorical as the default. If the person arguing the position wanted specifically to offer up a hypothetical argument, then the hypothetical statement would be have stated explicitly. The same is true with a disjunctive argument. So, the bottom line is that if you have no reason to think that the major premise is hypothetical or disjunctive, then operate as if it is categorical.
Got this? I know it is a little out there. Just do this. Remember, if the arguer really wants to have the argument be hypothetical deductive, she will clearly state a hypothetical premise. If she really wants the argument to be disjunctive deductive, she will clearly state a disjunctive premise. If neither of these are stated, then you proceed as if it is a categorical deductive. Plus, as mentioned above, in categorical deductive enthymemes, it is almost universal that the major premise – the categorical statement – is the one that is left out… so you simply have to proceed in the dark and assume this structure.
So, the rule is: if the arguer has not made it clear that the deductive argument is either hypothetical or disjunctive, assume that it is categorical.
Of course, in the case of inductive arguments, you are left with merely looking at the argument as a whole and seeing that they are presenting examples and/or test cases and then drawing a generalization from them.
Alrighty then…just one last point on arguments.
(Note: there are practice pages at the end of this section)
G. Meta-structure Of Argumentation
As I try to keep saying, all of this stuff needs to be applied in the real world…it needs to be practiced and used in your day to day lives. That is why I keep pounding home not just theory, but real world application.
Continuing in that practice, I want to finish up this lengthy discussion on arguments and talk about how they show up in real life.
First off, they show up in verbal form….in discussion in the breakroom, over lunch, during drinks after work, or in the living room. Life is full of discussions and these typically contain arguments. You, as a Critical Reasoner, must learn to think on your feet and hear argument structure, distinguish sentence types, probe for source strength, and root out fallacies. And the best place to do that, is to start doing so with written arguments. Practice there, and then it will become easier and, hopefully, second nature.
Secondly, in real life, arguments hardly ever show up in the simple form I have presented above. Not even close. I have, thus far, presented to you what is known as Simple Arguments. And I hate to do this to you…but they are called simple….because they are…there are much more complicated ones ahead! Just what you wanted to hear, right? Well, don’t get discouraged, it won’t really be that tough.
A simple argument is one is which there is only one argument present. Consider the following argument that someone states in the office break room:
Capital Punishment should be banned because Capital Punishment is cruel and unusual punishment.
Syllogism:
e.g p1 Capital Punishment is cruel and unusual punishment.
p2 ___________________________________________
C∴ Capital Punishment should be banned.
This is a simple argument, on its face. Nothing to do but infer the hidden premise and then have them support their claims and give authorities for their prescriptions.
But, in real life, either through discussion and expansion or because it was written out that way originally, arguments simply hardly ever show up as Simple Arguments.
Let’s say that in the same discussion the person says:
Capital Punishment should be banned because Capital Punishment is cruel and unusual punishment and because Capital Punishment is prejudicially applied.
There are actually two arguments here.
e.g p1 Capital Punishment is cruel and unusual punishment.
p2 ___________________________________________
C∴ Capital Punishment should be banned.
And
e.g p1 Capital Punishment is prejudicially applied.
p2 ___________________________________________
C∴ Capital Punishment should be banned.
These are, in actuality, two separate and independent arguments here. Each can stand alone. They do, indeed, come to the same conclusion, but they arrive there by separate and independent roads. One could successfully defend one while have the other torn apart…and still end up defending that CP should be banned. When you have two separate and independent simple arguments that end up with the same conclusions, you have a Compound Argument.
When confronted with such, you need to deal with each separately…and you need to work hard to keep clear in your head which you are analyzing and attacking.
Another type of argument is the Complex Argument. This type of argument is one in which there are two arguments, but they are not independent of one another. They do not lead to the same conclusion. The conclusion of one of the arguments literally is inserted into the hidden premise slot in the other enthymeme.
Back to the office break room (PS….don’t spend too much time here.. the boss will get wise to your poor work habits!).
Let’s say that in the same discussion the person says:
Capital Punishment should be banned because Capital Punishment is cruel and unusual punishment and cruel and unusual punishment is against the constitution and should be banned.
There are actually two arguments here.
p1 All things that are cruel and unusual punishment are against the Constitution
p2 All things that are against the Constitution should be banned______________
C∴
And
p1 Capital Punishment is cruel and unusual punishment.
p2 ___________________________________________
C∴ Capital Punishment should be banned.
The conclusion to the first argument is,
“ All things that are cruel and unusual should be banned.”
This is not stated in the argument. It must be inferred. And when you do the correct inferring, you end up with this claim. What happens next is that this inferred claim is inserted into the next argument, completing it in turn. The second argument is dependent upon the first to complete it. It is, as it were, hanging on the waiting inference of the first argument.
Understand? A Complex argument is one in which you have two arguments, one of which produces a conclusion that becomes the inferred hidden premise of the second argument. One is independent; the other is dependent on the first.
These complex arguments sometimes go on for a long time, with several independent arguments are lined up in a row, so to speak. They are like a row of dominos, just waiting for the first one to fall. Once that happens, the rest do the same. In these chain arguments, once the first argument is completed, and the conclusion fills in the next, the chain keeps on going. This form of argumentation – the forming of a chain of linked simple arguments – is known as a Sorites Δ= a chain of linked arguments.
Ok….got this? Good. Check up and down the hall, because we need to take one more trip to the breakroom…
Here we are again… for the last time, I assure you. This time, the guy says the following:
Capital Punishment should be banned because Capital Punishment is cruel and unusual punishment and cruel and unusual punishment is against the constitution and should be banned, also because Capital Punishment is prejudicially applied.
This is, of course, a combination of all that was said before. What we have then is what is called a Complex Compound Argument. And trust me, these really are what you get thrown at you most of time when you are arguing issues. So, get ready for it. How do you do this? Keep reading arguments and taking them apart. Start with simple ones – like from your newspaper’s Public Opinion section – that are just a paragraph or two. Practice, practice, practice. Then graduate up to several paragraph arguments. If you use the steps I gave you above, you really should be able to discover the basic argument structure of any argument of any length within a minute or two…whether it is 3 paragraphs or 9 pages. Learn to look for Signal Words, summary statements, thesis statements and spot issues. Learn to look for hypothetical and disjunctive statements and a setting forth of a series of examples. Learn to “see” these structures in your head….and you will become a formidable force in argumentation.
So, in summary, what you need to learn to do is:
1) Identify the presence of an argument
2) Identify the structure of the argument
3) Identify the meta-structure of the argument
4) Correctly infer all hidden premises and implied conclusions
5) Identify all descriptive claims and check for source weight and truth value
6) Identify all prescriptive claims and check for source acceptance
7) Identify and expose all fallacies (once you have learned them!!!!)
(Note: there are practice pages at the end of this section.)
G. Definitions
Antecedent Δ= the “if” portion of a hypothetical sentence
Argument Δ= a set of sentences that have a reason / conclusion structure
Categorical Sentence Δ= a sentence that assigns a quality to all individuals in a set
Complex Argument Δ= argumentation that uses at least one dependent and one independent simple argument and these are linked together
Compound Argument Δ= argumentation that uses at least two independent simple arguments that come to the same conclusion
Consequent Δ= the “then” portion of a hypothetical sentence
Deductive Argument Δ= an argument that moves from general premises to a specific conclusion
Disjunctive Sentence Δ= a sentence that asserts an either/or relationship between two events/conditions
Distribute Δ= to distribute means to put an adjective in front of a noun extending it to full set of the items (e.g. all, each, every) –> “dog” distributed = “all dogs”
Enthymeme Δ= an argument with one implicit (hidden) premise and one explicit
(stated) premise and a conclusion
Hidden Premise Δ= the premise that is unstated, but that is logically necessary
to have the argument
Hypothetical Sentence Δ= a sentence that asserts an if/then relationship between two events
Imply Δ= to leave unstated, to leave out
Infer Δ= {literally, to carry in} to fill in what someone has left out (implied);
can either be a premise or a conclusion
Inductive Argument Δ= an argument that moves from specific premises to a general conclusion
Issue Δ= a question that is being addressed and answered in the argument
Major Term Δ= the complete predicate of the conclusion in a categorical deductive argument
Middle Term Δ= the term that is stated in the explicit premise but does not show up in the conclusion in a categorical deductive argument
Minor Term Δ= the complete subject of the conclusion in a categorical deductive argument
Necessary Conclusion Δ= a conclusion that has to follow from the premises (ONLY
possible with a deductive arguments, and only then, when they are valid)
Premise Δ= a reason, technically, a reason in a formal argument
Probable Conclusion Δ= a conclusion to an inductive argument whose sample set
comprises more that 50% of the set (the highest degree of certainty attainable with inductive arguments)
Possible Conclusion Δ= a conclusion to an inductive argument whose sample set
comprises 50% or less of the set
Safety Of Inference Δ= (warrant) the criteria of how possible the conclusion follows in an inductive argument; this safety increases as the ratio of the sample set to the whole set increases in size, e.g. moves from 20% to 30% and is accomplished by either increasing the sample set size or decreasing the whole set size
Signal Words Δ= words that indicate the presence of conclusions and/or reasons
Simple Argument Δ= argumentation that uses one single independent argument
Sorites Δ= a chain of linked arguments
Syllogism Δ= formally structured argument with the premise and conclusion specified and set out in a particular format
Transductive Argument Δ= an argument that does not moves from specific to general or general to specific, but simply across the elements
Venn Diagram Δ= a graphic representation of an argument, translating the premises
from sentences to sets, attributed originally to a mathematician named Venn
H. Practice Pages
CATEGORICAL DEDUCTIVE SYLLOGISM EXERCISES I
Cover the answers and work down the exercises one by one
Find: major term, minor term, middle term, hidden premise
P1: All dweebs are nerds
P2: _______________________
C : Paul Simon is a nerd
P1: All wascally wabbits are wacky
P2:_____________________________
C : Bugs Bunny is wacky
P1: The Atlanta Braves are Indians
P2:_____________________________
C : The Atlanta Braves are Redskins
P1: Olive Oil is a sailor's goilfriend
P2:_____________________________
C: Olive Oil is anorexic
P1: Cats should be tortured
P2: ____________________________
C : Sylvester should be tortured
P1: Homophobia is a social disease
P2: __________________________
C: Homophobia should be eradicated
P1:
P2:____Slits are slots
C: Slits are slats
P1: All gosh is be golly
P2: __________________
C: Goshen is be golly
P1:
P2: My baby loves love
C : The Grinch loves love
P1: Salads should be solid
P2:______________________
C: Salads are salubrious
NON-CATEGORICAL DEDUCTIVE SYLLOGISM EXERCISES
Give the possible hidden premises and conclusions that follow:
Cover the answers and work down the exercises one by one
P1: If John goes to college, he will make lots of money
P2: ____________________________________
C:
or
P2: ____________________________________
C:
P1: If I don’t eat, I won’t live
P2: ____________________________________
C:
or
P2: ____________________________________
C:
P1: If Lois dies, Marvin will be sad
P2: ____________________________________
C:
Or
P2: ____________________________________
C:
P1: Either you leave or I’ll call my lawyer
P2: ___________________________________
C:
Or
P2: ____________________________________
C:
Or
P2: ____________________________________
C:
Or
P2: ____________________________________[pic]
C:
P1: Either Tom talks or Bill doesn’t go to prison
P2: ___________________________________
C:
Or
P2: ____________________________________
C:
Or
P2: ____________________________________
C:
Or
P2: ____________________________________
C:
ARGUMENTS FOR ANALYSIS I
For each passage, give:
a. Stated Premises
b. Conclusion
c. Inferred Hidden Premise
d. Argument Type (Deductive –categorical, hypothetical, or disjunctive OR Inductive
e. Meta-structure ( Simple, Compound, Complex, or Compound/Complex)
1. Since Hilda liked dark chocolate and milk chocolate, since she likes caramels and nougats, and since she likes peanut brittle, we may infer that she is fond of most kinds of candy.
2. Inasmuchas all whales feed milk to their young, it follows that no whales are fish, as not fish feed milk to their young.
3. Because all camels have humps, it follows that some cigarettes have humps, for some cigarettes are camels.
4. That Tristan is Dan’s son is shown by the fact that they have the same last name, they both have red hair, and they live in the same house.
5. Given that the Index of Leading Economic Indicators has been dropping for the last three quarters, we many infer that the economy is falling into recession.
6. The fact that every building on campus was named after a professor is evidence that most of the buildings on campus were named after Professor Hall.
7. We may infer that Lassie is a dog, since all collies are dogs.
8. Humberto attended logic class every day, studied logic for an hour every night, and did well in the course. Megan attended class occasionally, studied logic for an hour every night, and did well in the course. Gwin attended class every day, never studied at night and failed the course. Therefore the key to doing well in logic class is studying nightly.
9. If I get up this morning, I’ll go to work. If I go to work, I’ll get caught in a traffic jam. If I get caught in a traffic jam, I’ll lose my temper. If I lose my temper, I’ll do something rash. If I do something rash, I’ll be arrested. Under those circumstances, would you get up?
10. Learning logic is like learning a foreign language or learning to play the piano. You must master the vocabulary and the grammatical constructions before you can master the more difficult constructions. You must master the scales and finger techniques before you can play Beethoven. Similarly, in logic, you must master the elementary elements of the subject if you are to master the complex elements. Consequently, just as to master a language or the piano you must work with it every day you must study logic daily to master that discipline.
Answers:
ARGUMENTS FOR ANALYSIS II
Argument 1
We’re letting people get away with murder in this country. Somebody offs somebody else, and what to they get? A cot and three hots for the rest of life, plus cable TV and a free college education. What we need is some justice around here. That’s why I think were should have capital punishment restored in this country.
Argument 2
My friend Bob got some bad hamburger at Hinky Dinky a couple of weeks ago. Then, lo and behold, my next door neighbor Phyllis told my wife that she had some green beef from there herself some time back. I guess I won’t buy from their meat department anymore because they’ll fer sure sell me some bad meat.
Argument 3
The honorable Mister Hightower told me himself that the constable showed up more than two hours after he rang the station. I’ll stick by his statement, by Jove, because his reputation is beyond reproach. He a King’s man if ever I saw one.
Argument 4
If God is love then certainly he would do loving things. Since lots of unloving things have happened in this world, I cannot believe in the existence of God.
Argument 5
Either Mr. Clinton is a fool or the slickest man alive. But how could a fool get himself elected Governor of Arkansas, then President of the United States Of America. Therefore, I gotta think he’s worthy of the title Slick Willy.
Extra Credit Argument
Since last Tuesday I’ve had this monstrous headache. That mornin’ I walked into the garage and banged my bloomin’ head on this bike rack contraption my son rigged up. I was standing there cussin’ up a storm when he walked in and I lit into him, “What’s that stupid rack there for? Can’t ya hang it somewhere else?” I wonder if his mother ever had any kids that lived.
Answers:
ARGUMENTS FOR ANALYSIS III
Argument A
As far as I’m concerned either marijuana should be legalized or alcohol should be made illegal. Liquor has shown itself to be far more dangerous than marijuana. It has resulted in the deaths of thousands of people - both directly, through cirrhosis of the liver and indirectly, through drunken driving accidents. And yet it is legal. Since alcohol is more dangerous and yet legal, thank seems logical that marijuana, which is far less dangerous, should also be legal.
Argument B
Many people who are nothing more than vegetables are draining valuable medical resources from our society. This is not good for our society. This is especially tragic when you consider that many of these people actually want to die anyway. Thousands of terminally ill patients do not want to be burdens to society nor their families, and yet they are not allowed to have their doctors assist them with their suicides. We should allow these poor souls to die in peace rather than drag the living down. It as plain as the nose on your face, doctor assisted suicide should be allowed in this country.
Argument C
If abortion continues to be legal in this country, more and more people will be desensitized to human life. Just look at what has happened already: we see doctors and parents allowing their deformed and handicapped newborns to die untreated. What’s to stop this country from healing down the sane road as Nazi Germany? Nothing as far as I know! Abortion devalues human life, therefore abortion should not be allowed in this country!
Argument D
Murder is wrong. Everyone knows that. And when we execute criminals, what are we doing but, as a society, committing murder? If murder is wrong, then murder is wrong. Let’s not placate ourselves by using some fancy euphemisms and convincing ourselves that the execution of convicted murderers is anything less than cold blooded murder itself. Are not these executions cold hearted and planned out far in advance? Why then should we not classify them as premeditated murders? I know of no reason why, not! Since our country does not condone murder, it should not allow executions. Capital punishment should not be allowed in this country.
Answers:
-----------------------
This circle represents a set – the set includes all existing slime suckers.
All Slime Suckers
This next circle represents a set –
the set includes all pigs
All Pigs
To translate the sentence, “all pigs are slime suckers,” all we need to do is insert the set for all pigs inside the set for all slime suckers.
All Slime Suckers
All Pigs
So, the above diagram graphically represents the assertion that all pigs are slime suckers. The diagram and the sentence assert exactly the same thing.
To translate the sentence, “All pigs are slime suckers,” all we need to do is insert the set for all pigs inside the set for all slime suckers. To translate the sentence, “Porky is a pig,” we just need to insert the set for Porky into the set for All Pigs.
All Slime Suckers
All Pigs
So, the above diagram graphically represents the assertion that p1) all pigs are slime suckers, p2) Porky is a pig. And it logically follows, as the diagram shows, that there is a necessary relationship between Porky and All Slime Suckers. Given the first two premises, it HAS to follow that Porky is within the set of Slime Suckers. It HAS to.
Porky
In this diagram, some pigs would be in the shaded portion of the “All Pigs” set and some would be in the unshaded portion. Some pigs are Slime Suckers, and some are not. Got it?
All Slime Suckers
All Pigs
This is how premise 2 would look in diagram form:
All Slime Suckers
All Pigs
You should be able to detect the problem. You see, where do we put Porky? We can put him in the gray-shaded half of the “All Pigs” set and that would satisfy the assertion that Porky is a pig.
Porky
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All Cretans
Alex
Refrigerators
Washing Machines
Dishwashers
Expensive Things
Dishwashersss
Refrigerators
Washing Machines
Expensive Things
Dishwashersss
Refrigerators
Washing Machines
Appliances
Right-Handed People
Students in the College
A
B
Right-Handed People
Students in the CR classroom
G
I
F
K
C
A
H
B
J
D
L
E
minor term: Paul Simon major term: is a nerd
middle term: dweebs P2: Paul Simon is a dweeb
minor term: Bugs Bunny major term: are wacky
middle term: wascally wabbits P2: Bugs Bunnv is a wascallv wabbit
minor term: The Atlanta Braves major term: are Redskins
middle term: Indians Pl: All Indians are Redskins
Minor term: Olive oil Major term: is anorexic
Middle term: sailor’s goilfriend P2: All sailor’s goilfriends are
anorexic
Minor term: Sylvester Major term: should be tortured
Middle term: Cat P2: Sylvester is a cat
Minor term: homophobia Major term: should be eradicated
Middle term: social disease P2:All social diseases should be
eradicated
Minor term: slits major term: slats
Middle term: slots P1: All slots are slats
Minor term: Goshen major term: is be golly
Middle term: gosh P2: Goshen is gosh
Minor term: The Grinch major term: loves love
Middle Term: My baby P1: The Grinch is my baby
Minor term: Salads major term: are salubrious
Middle term: should be solid P2: All things that should be solid are salubrious
P2: John went to college
C: He will make lots of money
P2: He didn’t make lots of money
C: John didn’t go to college
P2: I didn’t eat
C: I won’t live
P2: I will live
C: I ate
P2: Lois died
C: Marvin will be sad
P2: Marvin won’t be sad
C: Lois didn’t die
P2: You left
C: I will not call my lawyer
P2: You didn’t leave
C: I will call my lawyer
P2: I called my lawyer
C: You did not leave
P2: Tom talked
C: Bill does go to prison
P2: Tom did not talk
C: Bill does not go to prison
P2: Bill doesn’t go to prison
C: Tom didn’t talk
P2: Bill does go to prison
C: Tom talked
P2: I didn’t call my lawyer
C: You didn’t leave
P2: Tom talked
C: Bill does go to prison
P2: Tom did not talk
C: Bill does not go to prison
P2: Bill doesn’t go to prison
C: Tom didn’t talk
P2: Bill does go to prison
C: Tom talked
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