CHAPTER 5 Complete evaluation of reasoning
SECTION 5.1 JUDGMENTS ON STATEMENTS AND EPISTEMIC SOUNDNESS

	Critical Life Decisions are ones which require a lot of our resources and will affect our lives in major ways.  It will be useful at this time to go over an eight step method for complete evaluation of a piece of reasoning. More details on how to perform some of the steps will be found in later chapters. For now, the method will be summarized and the instructor or student herself can decide which subjects to pursue further. Some of these  topics are also taught in informal fallacies and formal logic courses.
	
	Earlier we defined a sound argument as one in which a) all of the reasons are true, and b) the reasoning is valid. The purposes of reasoning are: to extend our own knowledge, to persuade ourselves or others to believe a conclusion to be true or to explain why it is true. To make a complete  evaluation of a piece of reasoning, then, we must judge not only the validity of the reasoning but also the truth of the reasons. For this purpose we will use the following categories. A statement is either 1) definitely true (so far as we know), or 2) probably true (so far as we know), or 3) we don’t know whether it is true (we are uncertain) or 4) probably false or 5) definitely false (so far as we know.) For now, these judgments may vary greatly  among students. Part Three will give background helpful to make these judgments on statements more uniform. 

	Two observations are necessary. First, for now at least, “I don’t know” or “I am uncertain” are not bad answers. They can reflect intellectual honesty. They reflect critical self-assessment. They are the sort of answer Socrates often gave. In particular, if a statement refers to people, places or things unknown to the student, he or she should judge the  the statement by responding “I don’t know” or “I am uncertain.” On the other hand, there is certain basic information which everyone in a  culture or country above a certain age is supposed to have. The student is expected to use that fund of knowledge to judge statements and not rely on “I don’t know” when the matter should be known to everyone. Also, if students do not know the meanings of words in statements, it is expected that they look the  words up  in a dictionary before making a judgment.  If after doing and showing this work and trying to write a paraphrase, they still do not understand the statement, then it is acceptable to answer “I do not understand” or “I do not know.” 	
	Second, we can now define a concept of “epistemic soundness.” 

Epistemic soundness is soundness with respect to adding to our knowledge. It is that level of confidence we can have in some arguments which allows us to accept their conclusions. After background knowledge, the student will be expected to judge statements as definitely true or false, or probably true or false, or still uncertain.  However,  for the purpose of making judgments of epistemic soundness, we can lump together “don’t know,” “uncertain,” “probably false” and “definitely false” as all cases of unwarranted statements. Definitely true and probably true statements are “warranted.” An argument is epistemically sound if and only if all of its reasons are warranted and  every step of reasoning in it is at least strong. 
	
Section 5.2 ambiguity and vagueness

	 There are  other problems with evaluating reasoning than the ones  already discussed.  Some of them arise because of difficulties with language. These difficulties affect our judgment as to the truth or falsehood of the reasons. They can also affect the extent to which reasons support a conclusion.  An example is ambiguity.  A word or phrase is ambiguous when it has two or more different meanings.  A statement which can be true in one sense of an ambiguous phrase, can be false in another. Or in one sense it can be very supportive of a conclusion, while in another it can give much less support. Stephen Thomas gives an example like this. How much support does the  reason give to its conclusion? 

Professor Cogan gives objective tests.

Professor Cogan gives fair tests.


	“Objective,” in one sense, means unbiased. In another sense here, it could also mean “multiple choice.” However, multiple choice tests are not necessarily fair. Suppose Professor Cogan gave multiple choice  tests on material he did not cover or even assign students to read. The reason could be true in the sense of “objective,” meaning “multiple choice,” and the conclusion could be false.  

	FIRST - DECIDE WHETHER EACH OF THE BASIC REASONS IS definitely or probably tRUE OR unwarranted AND WRITE "T" for “true” OR "”U” for “unwarranted” BESIDE EACH OF THEM. Where this is not possible because of 

ambiguity, go to step 2. Where it is  not known whether a basic reason is true or false,  judge it to be UNWARRANTED.  (To do this in detail, one will need to read Part Three, Evaluation of Unsupported Claims.)

	SECOND - Use SEMANTIC CLARIFICATION, where necessary ,to determine truth or warrant of ambiguous statements. Write "T" for “definitely or probably true” or "Not-W" for “unwarranted” beside each clarified statement. Semantic clarification  is the process of fixing the meaning of vague or ambiguous words. This is done by either drawing a distinction  between the meanings of word or giving a definition.  A distinction is a sentence in which someone says that a word or phrase has two or more uses, or that a general class has two or more subclasses called species. A definition  is a sentence which states the meaning of a word. 
	
	There are several kinds of definitions which can be important in reasoning. The most common sort is  called a “lexical” definition. This is the sort of definition which attempts to state the most frequent use of a word. The reasoning "Professor Cogan gives objective tests, so his tests are fair"  has no degree of support at all  until we have drawn the distinction between the meanings of “objective’ and substituted one of them for the ambiguous word itself. If this ambiguity of the words "objective test" is not apparent, it is possible to  accept the reason by interpreting it as meaning multiple choice tests, and then fail to notice that this is no evidence at all that his tests are fair!

SECTION 5.3 CRITICAL THINKING AND DEFINITION

	Definition is  important in a few of the most controversial, “life and death” issues. For example, in the issues of abortion and mercy killing the definition of “human life” is criticallly important. Another example would be the issue of “Holocaust Revisionism.”  A Holocaust Revisionist claims that “The so-called ‘Holocaust’ never occurred.” Others become enraged. A critical thinker’s reaction is to ask both sides “What do you mean by ‘The Holocaust?’ ” Shermer (1993, p. 33-34)  defines “The Holocaust” as the intentional or functional near-destruction of an entire people based primarily on race.” He gives a definition of ‘holocaust’ with a small “h” with the same words except adding that the destruction may be based on race, religion, ethnicity, land, and/or property and wealth. By his definitions only what happened to European Jews constitutes “the Holocaust.” What whites did to Indians  and Africans in colonization and slavery, killing up to 1 and 10 million,  religious war in Bosnia, and the  Hutu murders of 500,000 Tutsi’s in Ruwanda are  small “h”  holocausts.

 Only with definitions like these are we able to ask the right questions, such as “How many Jews were there in Europe before WWII?” “How many were alive after it?” “How did the non living die?” “Were gas chambers and crematoria used to engage in mass extermination?” “Is there good evidence that it was Hitler’s intention to exterminate, rather than deport, the Jews?” “Did the Nazi government carry out a coordinated plan to exterminate the Jews?” (This author’s acquaintence with convergent evidence suggests the answer to the last three questions is “yes.”)

	In reading, always look up  unknown words in a dictionary! Words express ideas. A person of few words is one of few ideas, one who can not think interesting or powerful thoughts. A definition should state important, universal features of something, not less important ones. Wrong: “Man is a featherless biped.” Better: “Man is a rational animal.” One way of defining a common noun is to state a genus (animal), such that every member of the species (man) is a member of it, and also a difference (rational). which marks off men from the other animals. Such a definition should be neither too narrow nor too broad. Too narrow: “A chair is a 4 legged wooden piece of furniture for sitting on.” Too broad: “ A chair is anything that can be sat upon.”  A definition should not be vague or use analogies or wording more unfamiliar than the word being defined.  Avoid overly general words like “some-” or “any-” thing, body, one, and obscure words, if possible.

	Another way of defining is to give a synonym. Be careful, however, to find an exact synonym or add words to avoid excess broadness or narrowness.  A definition should be a complete sentence. It should reflect the word’s part of speech.  Do not use a form for verbs when the word is a noun or adjective. Wrong: “A rock is to....” “Rock” is a common noun, not “to” do anything. To define is to explain the meaning of a word. This can not be done using the word or a related word in the definition itself. That is called “circular definition.” A person who does not know what Socratic Method is gets no information out of being told “Socratic Method is the method used by Socrates.” A definition must be relevant to the field in which it is requested. For example: In philosophy “argument” means a group of statements made to persuade. It would not probably not be acceptable to give as an answer, on a philosophy test, a dictionary definition synonym that “argument” means a fight, dispute or disagreement.

	 A disagreement can sometimes be settled by either giving a lexical 


definition or by parties stipulating (agreeing to) a meaning for a word. But in the philosophical sense of argument, any argument in which there was a disagreement (inconsistency) between any or the reasons would automatically be an unsound argument. Most arguments are certainly not unsound for this reason.

	Consider the following illustration of the substitution process to clarify the degree of support, using “gives objective tests,” as short for the reason, and “gives fair tests,” as short for the conclusion. Note that  the ambiguous word is underlined and its substitute meanings are italicized.

Gives objective tests. 	Gives unbiased  tests.	Gives multiple choice  tests.							
Gives fair tests.		Gives fair tests.			Gives fair tests.

	A word or phrase is vague  when it has no clear boundaries of meaning. When it is not certain whether word applies to some object, then the word is vague. For example, color words are vague. Trying to match a very dark piece of cloth for sewing can demonstrate this point. It may not be clear whether the dark cloth is "black" or "dark blue" or a color in between called "navy". There are no precise standards for the use of such 
words. But vagueness is widespread and can be important.

	In a Pennsylvania town, a paralyzed man in a motorized wheel chair could not go on the sidewalk under a bridge. There was a pile of debris blocking his way. So he went out into the middle lane of a three lane road. On the other side of the bridge, a police officer gave him a ticket.  The man in the wheelchair had to go to court. It had not yet been decided in local law whether a motorized wheelchair was a "motor vehicle." The vagueness of "motor vehicle" was important to him in this case. Suppose motorized wheelchairs were made much heavier to protect their users.  They  might now and then run over and hurt people walking. For someone who might get struck by one, this vagueness might become important  in determining an insurance settlement. A precise definition is needed to clear up a matter like this.

	


	Another example of how to fix vagueness in an  argument could be: 

David had 6 bookcases at home.

      He must keep some books there.

This is a pretty strong looking argument.  However, both the words "some" and "there" are vague. Remember that one way of judging the degree of support reasons give to a conclusion is to consider how willing the reader or listener would be to bet something of great value on the truth of the conclusion, given the truth of the reasons. 
	
	Suppose there was someone, Ms. Confirmation, who had a lot of money and was not interested in trying to trick anybody. She goes around telling people that certain statements are true.  Then  she tells them another statement, a possible conclusion.  Then she offers them a chance to place an "even money" bet, given the truth of the earlier statements, on the truth of the conclusion. If she told Gary that it is true that David has 6 bookcases at home and Gary had a total of $100,000 life savings, would he bet $10,000 on the truth of "David must keep some books there?" How about the whole bundle, all $100,000? Before Gary bets, it would be wise to come to  agreements with her on three things. 

	 First, Gary would want her to agree that “must” is only an inessential conclusion signal. That is, that it does not mean that David is required  in some way to keep books there. This fixes an ambiguity in “must.” Second, how much is "some?” Third, what, exactly, is included in "there?" If "some" meant "two or more," then Gary would lose if David had only one book. If it meant "at least one," then Gary would win even if David had only one book.  If "there" meant "on his bookshelves," then Gary would lose if David had books in the house but not on the bookshelves. Gary would rather that "there" meant "anywhere in the house." In this specific meaning, David would win even if he had no books on any of his 6 bookcases.

	In both ambiguity and vagueness, what is fundamentally wrong is that the meaning of the statement is not clear. Therefore it is not clear exactly what the statement claims. Therefore it is not clear how much support it gives to a conclusion. It may also not be clear whether it is true or false. Vague and ambiguous statements will be classified as 

unwarranted until clarified. More information on this is to be found in Part Three.
THE STUDENT SHOULD NOW DO EXERCISE 10 ON SEMANTIC CLARIFICATION.    

Section 5.4 DRAWING CONCLUSIONS AND FINDING MISSING ASSUMPTIONS

	THIRD -  ON THE PRINCIPLE OF ChaRITY, CONSIDER WHETHER THERE IS ANY definitely or probably TRUE STATEMENT THE MAKER OF THE ARGUMENT DID NOT STATE WHICH WOULD RAISE THE DEGREE OF SUPPORT THE REASONS GIVE TO THE CONCLUSION. IF SO, ADD THESE OMITTED TRUTHS. We are not talking any longer about just evaluating somebody else’s argument. We are talking about using argument to get at the truth for ourselves  and others. This is much more what fair minded critical thinking is all about (Paul, 1996, p. 32.)

	Sometimes people leave out statements which seem to them so obvious that they don't need to say them. Sometimes they seem obvious because they are parts of patterns of valid reasoning which are particularly simple and clear to most people. These omitted statements are either conclusions or reasons. In case they are conclusions, filling in missing statements is called "drawing one’s own conclusions." In case they are reasons, it is called "bringing out the unstated assumptions" of the argument. Everyone has a natural sense, more or less developed, of patterns of valid reasoning. This sense can be aided by being taught certain valid patterns.

	Here is how to draw conclusions: Ask, "If these reasons are true, then what conclusion must be true?" Say aloud the reasons of the argument in question. Then say "SO.....", in a leading way, and then go on to fill in the missing statement. Try it with the following examples. They were picked from advertisements by Thomas (1993), probably because most readers would be familiar with the conclusions they imply. Try to be specific and get the exact conclusion. 

EXAMPLE 1. "The more carefully brewed a beer, the better it tastes; and Budweiser is the most carefully brewed beer in the world. SO.. <........>."

Did the reader remember or understand the conclusion the advertisers wanted us to get from this commercial? It is that Budweiser is the best tasting beer in the world. Here is another example from commercials. 

EXAMPLE 2. "The bigger the burger, the better the burger; and Burger King's burgers are bigger, SO... <........>." 

Did the reader draw the conclusion that Burger King's burgers are better?  That is what the advertisers wanted us to understand. The argument to this conclusion is valid. When doing this, stick to THE WORDING OF THE REASONS.  Arguments to conclusions less similarly worded to the way the reasons are stated are weaker.

	Remember, also, the two common types of linked reasoning from Part One: syllogistic reasoning and conditional reasoning. Try to do this syllogistic example before reading the answer. 

EXAMPLE 3: "All horses are mammals and no mammals are cold-blooded. SO...<.............>." 

Did the reader reason out the conclusion that no horses are cold-blooded? Consider the following example of conditional reasoning. 

EXAMPLE 4. "If this object is red, then it has some color. But it has no color. SO...<.............>" Did you correctly deduce that "it is not red?"

	Here is how to deal with missing reasons. Say aloud the reasons given. Then say "so" and the conclusion. Then say “BECAUSE “ in a leading way, and go on and complete the sentence with the missing reason.

EXAMPLE 5: "All men are mortal, so Socrates is mortal BECAUSE...<..............>" Did you get the missing assumption that Socrates is a man? 
 
EXAMPLE 6: "If you go, then we will go.  So we will go" BECAUSE...<.............>. Here the missing reason is "You go."
 
	Step Four: MAKE AN ARGUMENT AS STRONG AS IS POSSIBLE BEFORE EVALUATING IT.  This is one of the things that the Principle of Charity says we should do. However, there are limits to how far we should carry that. Any inductive argument can be made into a deductively valid one by adding certain reasons. But sometimes to do this, we would need to add reasons which are obviously false. For example, we could make our strong inductive argument, "All horses ever observed have hearts, so all horses have hearts", valid by adding to the reason the claim: "We have observed all horses." But this reason is definitely false. 

	Suppose we are trying to make an argument as strong as possible. We find that the only way we can make it deductively valid is to add some assumption like this which "blows up in our face." When we uncover it, it 

is an assumption we immediately realize to be false. Such an assumption is called a “land mine assumption.” In that case, the argument can not be deductively valid.

	On the other side of the coin, Fifth, we SHOULD TAKE ACCOUNT OF TOTAL AVAILABLE EVIDENCE. To be sure, to reach a conclusion reliably, we should add  any true or warranted statements which LOWER  the degree of support. Suppose, for example, a woman is considering whether to marry someone. She may think "He's good looking. He's fun to be with. I’m attracted to him. He has a good job. He's generous with his money. He loves me. So I should marry him." Suppose it is also true that he has slapped her around a couple of times. Also, he has been arrested for assault. She had better add this information in reasoning about the conclusion unless she wants to risk ending up a battered or murdered woman. By writing the reasoning out, we could just write these statements down with dotted line arrows to the conclusion, or with the words “reasons against” beside the arrow. The dotted line arrows could represent reasons against a conclusion. These reasons would greatly lower the degree of support the woman  has for the conclusion that she ought to marry him.

	An argument justifies its conclusion if and only if all its basic reasons are true, or at least warranted, and every step of inference is at least strong, if not deductively valid. Here is a possible picture of the whole reasoning on marrying the man.

He's good looking. + fun + sexy + good job + generous + loves me.
			 He slapped me around. + He was arrested for assault.
									
                  	 I should marry him.

	SIXTH, REJUDGE THE DEGREE OF SUPPORT FOR EACH STEP OF INFERENCE AND WRITE  ONE OF THE DEGREE OF SUPPORT PHRASES BESIDE EACH ARROW. 

	When it comes to complete evaluation of reasoning, we NEED TO use  a third princple: THE WEAKEST LINK OF THE CHAIN PRINCIPLE. A chain is only as strong as its weakest link. Similarly, aN ARGUMENT IS ONLY AS STRONG AS ITS MOST DOUBTFUL NECESSARY BASIC REASON AND ITS WEAKEST STEP OF INFERENCE. If presented with an argument with seven necessary basic reasons, and six are definitely or probably true while  one is unwarranted,  the whole set must be evaluated as unwarranted. If there are five necessary steps of 

inference and four are deductively valid while one is only moderate, evaluate the whole set of inferences as only moderate. This principle is illustrated in the diagrams below. 

	Suppose we have an argument in which statement 1 is claimed to support 2 while 2 is claimed to support 3 and 3 and 4 are linked and claimed to support 5. In such a case we always have 2 questions of evaluation to deal with: A. How strong are the inferences? and B. Are all the basic reasons true? Suppose further that when we make our preliminary evaluation, we judge the following.


			1	        	 	 2	 			 3 + 4
								   
			 2			3				    5

	Here is an example of a preliminary diagram of a piece of reasoning before clarifying and adding missing statements, along with what it might look like after clarifying and adding assumptions which make it as strong as possible. The ideal is to be able, as in this example, to find sufficient unstated but true assumptions which will make the every step of inference deductively valid. In the diagrams  notice what effect this has on the questions of evaluation. However, it is often the case that we can not find such statements and this does not prevent an argument from being a strong inductive one.

					BEFORE

	 1 		 		2 Questions of Evaluation:
         	  												 2				1. Are all basic reasons true?	
			
	3 + 4				2.  How strong are the inferences?	
           
    	   5

AFTER

	 1 + [A]			One Question of Evaluation left:
	    
	 2 + [B]			1. Are all the basic reasons true?
             
	3 + 4 + [C]			  
	    
	      5

Section 5.5: Soundness Verdicts, Final Evaluation of Conclusions
 
	The seventh step of our method is to judge the epistemic soundness of the argument in question. Based on the definitions above, we will judge arguments to be either “epistemically sound”, for which we will use the letter “S”, “probably epistemically sound”, symbolized by “PS” or “unsound,” indicated by a “U.” Any argument which is logically sound is epistemically sound. This means that it is one on the basis of which we may include the conclusion in our beliefs. The eighth step  is  to make the final judgment on  whether or not to accept the conclusion and add it to our stock of beliefs. If the argument is epistemically sound, then judge the conclusion as definitely or probably true; believe it; rely on it. If it is unsound, judge the conclusion as unwarranted, unproven by this argument. Do not yet add it to your stock of beliefs. Do not rely on it.

	There will be  times in  life  when it will be  helpful to arrive at a final verdict on the soundness of an argument and then use that verdict to accept a conclusion or continue to suspend judgment on it. These times are  when critical life decisions must be made. Critical life decisions are situations in which one has has to make a choice which will involve  a lot of one’s resources and affect one’s life and happiness for a long time.  Critical thinkers make such decisions by completely evaluating arguments for the choices they have. Critical life decisions include  whether to go to war or to college, what to study,  whether to leave one job and home for another, whom to marry or partner with, whether to have a child, and what course of treatment to undertake for a serious illness. Everybody has to make  these decisons. So critical thinking is useful for everybody.
	
	In order to judge an argument to be sound, judge the degree of support the reasons give to the conclusion at every step of inference. Judge also whether each basic reason is true, warranted, unwarranted or false. The distinction among true, warranted,  unwarranted and false statements will be explained in Part Three.   But it is desirable here to  illustrate the final steps of evaluating argument.
 
	The purpose of logic and critical thinking is to seek reliable patterns of reasoning. These are ones which enable us to be certain, or as near to certain of our conclusions as possible. So we are conservative in our evaluations. We apply the Weakest Link of the Chain Principle  to statements as well as to steps of inference. The group of all the basic reasons has, as a whole, the value of the least warranted statement which is necessary to argue for the conclusion. Again, if there are seven necessary basic reasons, and six are known to you to be   definitely true, but the seventh is probably untrue, then they are probably untrue as a whole. This does not prove that the conclusion is false! All this shows is that the present reasoning for it is unsound. There may be other true or warranted basic reasons which could be relied upon to prove the truth of the conclusion. The Principle of Charity suggests that we look for them.

	 When evaluating long, complex pieces of reasoning, write degree of support judgments beside each arrow. After clarification, if necessary, write judgments of definite or probable truth or  unwarrantedness beside each statement. Next,  write the label "Final Verdict" and either "sound" or "probably sound" or "unsound" for the argument as a whole. If the argument is sound, then we know we can have a very high level of confidence in the (definite) truth of the conclusion. If it is probably sound, then we will say we should treat the conclusion as probably true. If it is unsound, then  treat the conclusion as unwarranted, at least by this argument. Here is a table of final verdicts on reasoning. It  is a simplification  because “don’t know”, “probably false” and “definitely false” have all been collapsed in meaning into “unwarranted”. An argument with any necessary unwarranted basic reasons is unsound. Its conclusion, like bad meat, is unfit for (epistemic) consumption.






	When a person goes through the eight step process  with many long, complex decision problems like the career change example mentioned above, he or she will find that at first there are one or more unwarranted reasons. So  the reasoning appears unsound. This should motivate the individual to do sufficient research so that a decision can be made strictly on warranted or true reasons, and at least strong, if not deductively valid, reasoning. Warrantedness is not exactly an hereditary property of statements like truth is under deductive validity. However, if a statement is originally unwarranted to the reader and  she comes up with a sound or probably sound argument for it, then she should upgrade it in her thinking to warranted.  

	Here are a few examples showing how to make final judgments of soundness and warrant of conclusions.
.
                                            TRUE                       TRUE
  All men are mortal.  +   Socrates was a man.
						    
     Socrates was mortal.

Final Verdict:  This argument is sound, so  the conclusion is true.


                                                 TRUE                    FALSE
  All gods are immortal. +   Socrates was a god.
		
  Socrates is immortal.

Final Verdict:  This argument is unsound due to  a false reason,  so it’s conclusion is unwarranted.
                                              TRUE                           TRUE
All clams live in water. + All mollusks live in water.

        All clams are mollusks.

Final Verdict:  This argument is  unsound due to weak reasoning,  so  it’s conclusion  is unwarranted (by this argument.)

	Now let's look at a complex case of a sort young students may frequently face.  A young woman comes to the guidance counselor and says the followingÖ "I'm doing poorly in most of my courses, because I watch too much TV, the daytime 'soaps.' Then there's my job: I work 20 hours a week at MacDonald's. That takes time from school work. So does my sorority. We have at least one major drinking party every weekend, and I'm 
wrecked the next day. I don't want to flunk out, so I've got to do something! Joe, my boyfriend, wants to get married. I could drop out and get married. This would mean that I'd end up a homemaker dependent on my husband for an acceptable income. That's a bad idea in this day and age, because 1 in 2 marriages end in divorce and lots of men don't pay child support, leaving the woman stuck with kids and poverty. So I shouldn't drop out and get married. T.V. is addictive to me, so I probably can't cut back there. I could quit my job, because I don't really need the money for school expenses. But I do really need it for the nice clothes girls in the sorority are expected to wear. I could stop going to sorority parties, but that would mean that they would probably drop me from the pledge class." Let's bracket, number and diagram this passage, picking out her alternatives as we go.

 1   because  2  3  

4   5 Note that "So" is not functioning as an indicator here, but like "also". 6   and 7   8  so 9  10  ALTERNATIVE: 11  11  would entail that 12  Number 11, dropping out and getting married, is referred to by the pronoun "this," so the pronoun alone is bracketed and gets the same number as the statement it refers to. 13  because 14  and 15  So 16  17   so  ALTERNATIVE: 18    ALTERNATIVE: 19    because 20  21  ALTERNATIVE: 22  but that would mean that  23 

	The discourse  has the statements about TV, the job, and the sorority as converging reasons which explain doing poorly. Doing poorly, linked with not wanting to flunk out, support the conclusion of having to do something. Joe wanting to get married and her being able to do this support the conclusion that she'd end up a dependent homemaker. My judgment is that they provide strong support for this conclusion. The statistics on divorce and child support provide strong support for the conclusion that dropping out is a bad idea, which supports the conclusion that she shouldn't do it. But the statement that TV is addictive is only warranted, not definitely true. Also, the step to "so it can't be cut back" is, at best, only moderate and so that inference is unsound. 
	
	The passage also illustrates how deliberation may include reasoning which weakens other reasoning. The reasoning from needing the money for nice clothes for the sorority  weakens  the argument that she doesn't really need the money for school expenses, so she could quit her job. But it is doubtful that it makes it less than a strong argument. Finally, the argument that her stopping going to the sorority parties would mean that they would drop her is less than strong. It is at best moderate. So it looks like TV could be controlled somewhat. She could quit or cut back on her job hours, and might even explain that she needed to stop going to the parties for a while, so that she could stay in school. The reasoning  would then look like this, with all judgments of warrant, degree of support and final soundness judgments in place.

We will use “Warr.” as an abbreviation for “warranted” here.

TV         job       sorority      ALL TRUE
2        3 + 4      5 + 6 + 7
        	      
			 JUDGMENT: This  is PROBABLY SOUND, SO 1 IS WARRANTED.

1  I’m doing poorly -WARRANTED  +   8  Doesn’t want to flunk out.  -TRUE
						
  		                        9 Must do something

JUDGMENT: This argument is PROBABLY SOUND, SO 9 IS WARRANTED.


10 Joe wants to  marry -WARR.  + 11  I can drop out and marry  -WARR.
              					
    12 I’d become dependent homemaker.

judgment:   This argument is probABLY  sound, so 12 is warranted.

 14 1/2  end divorce. -WARR.*.15  Many men don’t support kids. - TRUE
                            			
13  Being dependent is bad idea.									
* 14 is commonly believed, but may be a significant exaggeration.
 	
JUDGMENT:  This argument is PROBABLY SOUND, SO 13 IS WARRANTED.

 



 12  I’d be dependent -WARR. +  13  Dependency is a bad idea. -WARR.
					
                     16 I  shouldn't drop and marry.

JUDGMENT: THIS ARGUMENT IS PROBABLY SOUND, SO 16 is WARRANTED.
.
         17   TV is addictive. -WARR.
			
         18   I  can't cut TV.

JUDGMENT:    THIS ARGUMENT IS UNSOUND, so 18 is UNWARRANTED.

20  Don't need school $ -WARR.    21  Need clothes $ -WARR.
				
                                   19  Could quit job

JUDGMENT: The argument 20 ---> 19 is PROBably SOUND despite 21, so 19 is WARRANTED.

  22  I could stop going to parties. -WARRANTED.
               
 23  The sorority will probably drop me.

JUDGMENT: This argument is UNSOUND, so 23 is UNWARRANTED.

                                      
More examples of such decision problems can be found in more advanced textbooks of direct natural logic, such as  Thomas (1993.)




CHAPTER 6:  SYLLOGISTIC LOGIC
SECTION  6.1: CATEGORICAL STATEMENTS

	A syllogism is any reasoning having exactly two reasons and one conclusion. A categorical syllogism  is a syllogism having two categorical reasons and one categorical conclusion. A categorical statement is a statement which affirms or denies that the members of one class are included in another class, in whole or in part. A categorical statement has a certain quantity  and a quality. Its quantity is how many of the subject class it talks about. This is usually expressed by a quantifer like “all” or “some.” Its quality is either affirmative or negative.  Classes are expressed by terms.

	  Term  means a noun phrase which stands for a class. Each class has an opposite class within a universe of discourse  (such a universe is also called a “parameter.”) A universe of discourse is a larger class into which all terms in a statement or syllogism fall. Each term has a complement which refers to  members of the opposite class. If A is a term, we express its complement as “non A.” For example, the complement of “dogs” is “non dogs.” Between them, they exhaust the universe of discourse “animals.” People frequently use prefixes “un-,” “in-,” and “dis” in ways that we can express as complements without doing violence to an argument. For example, “effective” and “ineffective” form a term and its complement with the parameter “acts.” The complementary terms are “effective acts” and “ineffective acts.” Other examples would be “sympathetic” and “unsympathetic,” “honest” and ‘dishonest” in the universe of discourse “people.”  

	There is a standard form for a categorical statement. It begins with a quantifier.  It then has a subject term, the verb "to be,” and a predicate term. It may have the word "not" between the verb and the predicate term. Here is the standard form of a categorical statement.

	 QUANTIFIERs	subject	TO BE		NOT (maybe)	predicate.
 	All, No or Some 	     F		is/are		not	   	      G.					    
	The words "affirms,” "denies,” "in whole,” and "in part" set up  types of categorical statements. The letters "F,” "G,” and "H" will be used as variables to stand for different classes.  The quantifiers  listed in the form above are called the classical universal  quantifiers: “all,” and “no;”  and the particular  quantifier “some.”  They are not the only quantifiers. 
 											                           
SECTION 6.2: Marking  TERMS AND THEIR POSITIONS 

	Affirmative categorical statements say that one or more of a subject class is included  in the predicate class. Negative ones say that one or more of the subject class is excluded  from the predicate class. Many statements which do not appear to be categorical can easily be put into categorical form. A verb ordinarily marks the break between the subject and the predicate of a statement. The verb answers the question “What action or state of being is described here?” The subject answers the question “Who or what is doing the action or is in the state of being?” Always begin   to put a statement into categorical form by finding this subject - verb break.  The verb is often not “to be.” Second, if the verb is not “to be”, then  fix this by inserting between the subject and verb “are,” a  plural noun for the type of things being talked about by both the subject and predicate, and relative pronoun like “that,’ “who,” or “which.”  The verb “to be” is called the copula (KOH puh la).  The plural noun for the type of thing being talked about is called a “parameter.” Examples of typical parameters are “people,”  and “things.” For example, “All young colts love to run” must be put into this form: “All young colts are animals that love to run.” Do not throw a parameter into a subject which is already a plural noun. Wrong: “All animals that are young colts are animals that love to run.”
     
	Categorical statements can have short subjects and predicates or  long ones like this: "All members of the Catholic church who are preparing to go into the priesthood of the Catholic church are persons who should be prepared for the frustrations of a celibate life." Third, automatically put parentheses around the whole subject noun phrase after the quantifier and up to the verb “are.” Put another set around the whole predicate term after “are” or "not" to the end of the sentence. All (members of the Catholic Church who are preparing to go into the priesthood...) are (persons who should be prepared for the frustrations of a celibate life.) These are the subject and predicate terms of this statement. Failure to do this correctly can prevent one from evaluating a syllogism correctly.
	
	Syllogisms all have a linking  middle term  which appears once in each reason. It links, or appears to link, the subject of the conclusion and the predicate of the conclusion. The subject and predicate of the conclusion  are called the end terms.  The subject of the conclusion is called the minor term.  The reason containing it is called the minor premise. The predicate of the conclusion is called the major term. The reason containing this term is called the major premise. One reason ties 

the subject to the middle term. The other ties the middle to the predicate. The conclusion contains the subject and the predicate with the middle term left out. The major premise is always listed first, then the minor premise and finally the conclusion. Each statement has a single letter abbreviation (such as A, E, I or O.) The mood  of a syllogism is represented by 3 of these letters  for the major premise, minor premise and conclusion respectively, e.g. EIO. E is the major premise, O the conclusion.

	The letters "S,” "M,” and "P" will stand for the Subject of the conclusion, the Middle term of the syllogism and the Predicate of the conclusion. There are four different possible ways the subject, predicate and middle terms can appear in syllogisms. A row of letters "S,” "M" and "P" will stand for each statement. Here are the four possible positions of the middle term. The word figure  is used to denote these four possible positions of the middle term. They  make up what are called “the four figures of the syllogism.”
          
                 1st. Fig    2nd. Fig    3rd. Fig     4th. Fig
 
Reason 1      M - P        P - M        M - P         P - M  
Reason  2     S - M       S - M         M - S         M - S
		  Conclusion   S - P        S - P        S - P          S - P


        Each statement in a standard form syllogism can be either an A, E, I or O, so there are at least 4 x 4 x 4 types of syllogisms.  But the middle term could be in any one of the 4 positions. So there  are 4 x 4 x 4 x 4  or 256 possible types of syllogisms. Under Aristotelian rules of validity, 24 forms are valid. Under  Boolean rules, only 15 are valid. 
 
SECTION 6.3 DISTRIBUTION

	The concept of distribution  deals with the number of items in a class which the quantifier refers to in the subject or predicate. Universal   quantifiers distribute their meaning maximally, i.e.,  to all members of a subject class.  Particular  quantifiers distribute minimally to at least one of the subject members. Affirmative statements distribute their import minimally to some of the predicate class. Negative statements distribute maximally to all of the predicate.  We will use subscript distribution index numbers, “5” for maximal distribution and “1” for minimal distributION under the term letter to indicate how much the term is distributed.  This gives us the following table.

 Syllogistic  Distribution Table
                           			Quality
	 	 Affirmative   				NEGATIVE

		A:  All F 5 are G1				E:  No F5 are G5

QUANTITY

		I: Some F1 are G 1       			O:  Some F1 are not G5.
                                                  
This pattern of distribution can be summarized in a handy little rule abbreviated as USNP. Universal statements (A and E) maximally distribute their Subject terms; Negative ones (E,O) maximally distribute their Predicates, and all others are minimally distributed.

Section 6.4:  A COMPARISON OF ARISTOTELIAN AND BOOLEAN RULES

	A statement has “existential import” provided that it implies the existence of some objects. Aristotelian rules assume that all four types of statements have existential import; Boolean ones that only particular statements do. Boolean rules were written to avoid inferences  to the existence of things we do not think to exist. For example, when we say “All trespassers will be shot,” we do not imply that there are trespassers. Boole’s interpretation said this means: “If  anything is a trespasser, then it will be shot.” 

	Aristotelian Rules are the following: A standard form CATEGORICAL SYLLOGISM IS VALID IF, AND ONLY If: RULE 1: THE MIDDLE TERM must be maximally DISTRIBUTED  at least ONCE. RULE 2: If an end term is maximally distributed in the conclusion, then it must be maximally distributed in the premises. RULE 3: The NUMBER OF NEGATIVE REASONS EQUALS THE NUMBER OF negative  conclusions.
	
	Boolean Rules are the following:  RULE 1: THE MIDDLE TERM must be maximally DISTRIBUTED EXACTLY ONCE.  RULE 2: NO END TERM IS maximally DISTRIBUTED ONLY ONCE. RULE 3: The NUMBER OF NEGATIVE REASONS EQUALS THE NUMBER OF negative  conclusions.

	Boolean Rule #1 is narrower than Aristotelian Rule #1. AR#1 allows syllogisms with M distributed twice to be valid; BR#1 does not. Similarly, BR#2  is narrower than AR #2. AR #2 allows an end term to be distributed in a reason and not in the conclusion. BR# 2 does not allow this. The result 

is that 9 of the 24 syllogistic forms which are valid by Aristotelian rules are not valid by Boolean Rules. The totality of all valid forms by Aristotelian rules are shown below. Those not valid by Boolean rules are marked with asterisks.

FIRST FIGURE	SECOND FIGURE	THIRD FIGURE	FOURTH FIGURE

AAA				AEE			AAI*			AAI*
AAI*				AEO*			AII			AEE
AII				AOO			EAO*			AEO*
EAE				EAE			EIO			EAO*
EAO*				EAO*			IAI			EIO
EIO				EIO			OAO			IAI

	Boole’s interpretation removes existential import from universal statements but leaves it for particular statements. A valid argument can have no information in the conclusion which was not already contained in the reasons. So those syllogisms valid on  Aristotelian rules, but not on Boolean ones, are those which involve trying to draw a particular conclusion with existential import from two universal premises (the ones with asterisks.)

	An extended syllogistic like that in Thompson (1992) has 5 levels of quantifiers: Universal (all-no), Predominant (almost all - few), Majority (most - most are not), Common (many - many are not) and Particular (some-some are not.) In such a system, we  assign subscripts for intermediate quantities of distribution. If universal quan- tification is represented by  5, predominant would be 4, majority 3, common 2, and particular 1. Predicates of affirmatives would get distribution 1 and of negatives 5. HERE IS A DISTRIBUTION TABLE for this.
					QUALITY
		AFFIRMATIVE			NEGATIVE

		A:  All F 5 are G 1 		 E:  No F5 are G5

		P:  Almost all F4 are G1 	 B:  Few F4 are G5 

		T:  Most F3 are G1		 D:  Most F3 are not G5

		K:  Many F2 are G1 		 G:  Many F2 are not G5

		I: Some F1 are G1	 		 O:  Some F1 are not G5.
							
	Notice the regularity of the statements and distributions of the table. It should be fairly easy to memorize quantifiers in order. The standard quantifiers on affirmative statements are “all,” “almost all,” “most,” “many,” and  “some.” The ones on negative statements are ”no,” “few,” “most - are not, “many - are not,” and “some - are not.” One could memorize the letters which stand for the statements with phrases like “All Poisons Taken  Kill Internally” and  “Every Boy Deserves Good Oranges.” One could also just memorize a nonsense pronounciation of the letters. It is better in applying the third rule of validity to learn a phrase or pronunciation for EBDGO. Say “EHBD,” as in “the tide ebbed,” and  “GO”  (The water went away.)  For APTKI one could say “ahpt” as in “He was apt,” meaning good at doing something, “Kuh” for “K,” and “EYE” for the last letter “I.” Then all one needs to remember is that the distribution numbers for subject and predicate start with 51 for A, and go down ten for each affirmative statement and 55 for E and down 10 for each statement. 

		A:  All  51	 			 E:  No 55
		P:  Almost all 41			 B:  Few 45
		T:  Most 31				 D:  Most-are not 35
		K:  Many F 21 			 G:  Many-are not 25
		I: Some 11				 O:  Some-are not  15

This yields 4,000 syllogistic forms, of which 105 are valid by Thompson’s rules. Thompson’s rules for validity are these: An extended form syllogism is valid if, and only if, 

Rule 1. The value of the middle terms added together is more than 5.  The values under the two “M’s” must add up to 6 or more. 

Rule 2. No end term may have a greater value in the conclusion than in a reason. The value of “S” in the conclusion must be less than or equal to its value in the reason. The value of “P” in the conclusion must be less than or equal to its value in the REASON.

Rule 3. The number of negative reasons equals the number of negative conclusions.  If there is no negative conclusion, there can be no negative reason. If there is a negative conclusion, there must be exactly one negative reason. No syllogism is valid with two negative reasons. A syllogism with no negative reasons and no negative conclusion satisfies the rule. Zero negative reasons equals zero negative conclusions.

	Why are these rules of validity?  In a valid categorical syllogism all of the information presented in the conclusion must already have been contained in the reasons. Distribution, making reference	to members of classes, is information. Rule 3 is a rule of validity because any  negative information of  exclusion in a conclusion could only be supported by some negative information of exclusion in a reason, but two reasons of exclusion do not contain enough information to support any claim about the relations of the end terms. Rule 2 is a  rule of validity because if an end term was more distributed in the conclusion than in a reason, this would mean that the conclusion presented information not contained in the reasons. Rule 1 is a rule of validity because, if the information in the reasons does not refer to some M’s twice, it would be possible that “S’s” are related to some “M’s” and “P’s” are related to other “M’s” so that the reasons contain no information about the relation of “S’s” and “P’s.”

	To be a valid syllogism, an argument must have two reasons and one conclusion containing exactly three terms total. The S and P end terms each occur twice, once in a reason and once in the conclusion. The M term occurs once in each reason and not in the conclusion.  A term can be expressed by different words such as synonyms. For example, if one statement had “attorneys” in it and another had “lawyers,” the student would be expected to count them as one term, pick one of them and substitute it for the other to reduce the number of terms to exactly three.

	Students are also expected to recognize “equivocation,”  using the same word in two or more different meanings. For example,  “Love is blind. God is love. So God is blind.” Each reason seems true. However, the conclusion is nonsense. The reason is that  “love” is used differently in the two statements. The two uses of “love” are not really the same middle term. The student is expected to be able to understand that in such a case there are really four (4) terms. The argument can not be a valid syllogism because it does not have exactly three terms. Look out for occasional equivocation. In that case write: “This syllogism cannot be valid. The term ‘T’ has two meanings.”  The student should be able to state the  different meanings.

	Here are our three examples from Chapter Four with quantifiers, what are called “parameters” and standard plural copulas put in to clearly show their form. “All vegetables are plants and  no dogs are plants, so  no dogs are vegetables.” “ (All) (pieces of) music are combinations of sounds pleasing to the human ear, and (No) (pieces of) Heavy Metal (music) are (combinations of sounds) pleasing to the human ear, so No (pieces of)

 Heavy Metal (music) are (pieces of) music.” and “All responsible drivers (are) (people who) get lower insurance rates.  No teenagers (are) (people who) get lower insurance rates, so no teenagers are responsible drivers.” These all have the following form and distributions.

 All P5 are M1 +   No S5 are M5
   
No S5 are P5.

This form obeys Rule 1, because the “5” under one M and the “1” under the other add up to more than 5. It obeys Rule 2, because the “5” under each of the “S” and “P” in the conclusion is not greater than the “5” under the corresponding “S” and “P” in the reasons. It obeys Rule 3 because 1 negative reason equals 1 negative conclusion. So all of these arguments are deductively valid. 

	This example points out something else. One may not be able to prove that  Bigfoot, flying saucers or ESP do not exist. However, one can “prove a negative,” in some forms of syllogistic argument. 

SECTION 6.5:  DETERMINING VALIDITY OF SYLLOGISMS

								
Step 8: Use the distribution table to write subcript 5’s, 4’s, 3’s, 2’s or 1’s under the letters S, M and P. Put 5 under the S, M or P in the subject position of an A or E, 4 under a P or B, 3 for a T or D, 2 for a K or G, 1 for an I or O. If the statement is an APTK or I, put 1 under its predicate. If it is an EBDG or O, put a 5 under its predicate.
  

A SYLLOGISM IS VALID IF AND ONLY IF ALL  RULES ARE SATISFIED.
To say that a rule “is satisfied” is to say that the cross hatch contains marks which conform to the rule.

FOR ARISTOTELIAN SYLLOGISTIC RULES IT IS VALID IF AND ONLY IF
	Rule 1: There is a “5” under at least one “M.”
	Rule 2: If there is a “5” under “S” or “P” in the conclusion, then there must be a “5” under it in the reason.
	Rule 3: If the conclusion is “E” or “O”, then there is exactly one “E” or “O”  reason. Also, no syllogism with EE, EO, OE, or OO reasons is valid.

FOR BOOLEAN SYLLOGISTIC RULES IT IS VALID IF AND ONLY IF
	Rule 1: There is a “5” under one “M” and not the other.
	Rule 2: If there is a “5” under either “S” or “P” then there must be a “5” under the other “S” or “P” respectively.
	Rule 3: Same as R3 above.


FOR THOMPSON’S EXTENDED RULES IT IS VALID IF AN ONLY IF
	Rule 1: The sum of the values under the  “M”s  is greater than 5.
	Rule 2: The value  under “S” and “P” in the conclusion must be equal 			to or less than that in the reason.
 	Rule 3: If the conclusion is an E, B, D, G or O, then there is exactly 			one E, B, D, G or O reason. Also, no syllogism with any combination of  		E’s, B’s, D’s, G’s or O’s is valid.

Here is an example from the Extended System:


Section 6.6: Ways of expressing quantification
	
	We have five levels of quantification. Their names, together with the standard quantifiers which express them are: Universal (“all” or “no,” meaning “absolutely every one”); Predominant (“almost all,” “few,”) meaning “more than a majority, but less than all”; Majority (“most,” “most are not,” meaning “more than half”); Common “(many,” “many are not,” meaning “more than one” ) and Particular (“some,” “some are not,” meaining “at least one”). Each can be expressed in many ways. I am grateful to Bruce Thompson (1992) for having gathered them most systematically. Among these ways are the use of synonyms, adverbial forms, numbers, and alterations of other quantifiers. Also, a statement can express a quantificational attitude (universal, etc.) without any explicit quantifier on it.

	Universal quantification: First note some synonyms for “all.” The universal affirmative words  “any,” “any one of the,” “every,” “every one of the,” “each,” and “each one of the” F’s are G’s and “the only F’s are G’s.” should be translated into “All F’s are G’s.” The same is true of compounds like “everybody” or “anybody”  or “everybody” or “everything.” This is also true for words like “wherever” and “whenever.” When these words are found we can add a parameter  like “places” or “times” to  a quantifier to  make a noun phrase. “Wherever” means something like “all places.” “Whenever” means “all times.”

	Universal quantification can also be expressed this way: F’s are always G’s (or never G’s.) Notice that “universal” is not equivalent to affirmative in meaning!  “F’s are never G’s” means “No F’s are G’s.” In Chapter 4 we dealt with the syllogism “Music is a combination of sounds pleasing to the human ear and Heavy Metal is not pleasing to the human ear, so Heavy Metal is not music.” Such statements are probably intended to be universal. But while the first means “All music...” the second and third are properly translated into universal negatives: “No  Heavy Metal is a combination of sounds pleasing to the human ear” and “No  Heavy Metal is music.” 

	Note also that “All S is not P” is not a standard form. Never translate any statement into this form!  To put the  music syllogism into standard form we have to do something unusual, add a pluralizing parameter to the subject  to make it plural as well as to the predicate. Music comes in pieces. So we can translate it as “All pieces of  music are combinations of sounds pleasing to the human ear and no pieces of  Heavy Metal music are combinations of sounds pleasing to the human ear, so no pieces of  Heavy Metal music are pieces of  music.”
	
	With numbers, universal quantification is expressed when one says both  of the F’s are G’s or each or all of the n F’s are G’s, where “n” is a number. That is, “Each of the 17 F’s is a G” should be translated into “All F’s are G’s.” Numbers are relative to context.  “Each  of the 37 billion inhabitants of the planet Crude-O watched the televised executions” simply means “all inhabitants watched them.”

	There are also “negative exceptive quantifiers” which should be translated as universal.  In “Only F’s are G’s,” the word “only” functions to reverse the subject and predicate. To assure recollection of this, consider 1. “Only women are nuns” and 2. “Only nuns are women.” 1 is true and 2 is false. 3, “All women are nuns” is false, while 4, “All nuns are women” is true. For one statement to be a correct translation of another, it must have the same truth value, true or false, as the other. Since 1 has the same truth value as 4, not 3, 4 is the correct translation of 1.  This is always the case. “Only F’s are G’s” equals “All G’s are F’s.” Never translate it as “All F’s are G’s,” even if it sounds false when translated. We do not employ a Principle of Charity when translating fixed meaning words like “only.” “Just F’s are G’s,” “None but F’s are G’s,” and “None except F’s are G’s” are also translated as “All G’s are F’s.”

	

	Cases where articles “a,” “an,’‘ “the,” “these,” or “those” are followed by a term should frequently be translated as universal. For example, “a” or “the bat is a mammal”  means “All bats are mammals.” However, take care when there is no quantifer  at all. “Crows are birds” should be translated as “All crows are birds.” “Crows are eating my corn,” however, should be translated as “Some crows are eating my corn.” Again, the point is that the student must ask him or herself: “Is this statement universal,  predominant, majority, common or particular?”  

	Proper names, abstract nouns, descriptions like “the so and so,” ”these so and so’s” should be translated using universal  quantification. Treat  proper names like “Socrates” as standing for a class with only one member. Use a parameter like “people.” Construct a universally quantifed phrase like “All people identical with Socrates” or “No people identical with Socrates.” “Socrates is a philosopher,” would become “All people identical with Socrates are philosopers.” “Socrates is not a philosopher” would be “No people identical with Socrates are philosophers.” “The red roses  are wilting” becomes “All roses identical to the red roses are wilting flowers.”

	 Nouns with articles,  mass nouns, even    abstract nouns in front can and should be translated as  having universal  quantification. This enables us to translate into standard categorical statement form any phrase standing for an individual. We can translate into universal statements not only  any statements about “this car,” or “that lake,” but also about “metal,” or “liquids,”  and even  “justice,” “truth,” and so on. “This car failed to stop” would be “All cars identical with this car are cars which failed to stop.” “That lake has no fish” would be “No lakes identical with that lake are lakes with fish in them.” 

	Abstract singular nouns like  “justice” or “truth”  sometimes need to have a  parameter added in the subject to put the statement into standard form.   We saw this in the music example. Another example is this. “All justice is loved by God” might be translated as “All just acts  are acts loved by God.” “There has never been a good war or a bad peace” means “No times of  war are good times  and no times of  peace are bad times,” or “No periods of  war are good periods ....” To determine the proper parameter for a whole  syllogism, ask “What is it about?” “Terms ‘S,’ ‘M,’ and ‘P,’ are all kinds of ___” (fill in the blank.) Common parameters needed largely for predicates are “people,” “things,” “places,” and “times.” Sometimes a parameter must itself be quite abstract like “cases,” “instances,” “items,” or “principles.”


	Predominant quantification is  frequently expressed by putting a modifier in front of a universal quantifier. So “nearly,” “practically,” “almost,” “just about,” in front of “all,” “every,” “any,” or “the only F’s are G’s,” should all be translated as “Almost all F’s are G’s.” For example, “Nearly any F is a G” = “Almost all F’s are G’s.” Negative predominant quantification is expressed with the same modifiers in front of “no.” “Nearly,” “practically,” “almost,” “just about,” No  F’s are G’s and also “scarcely,” “hardly,” any F’s are G’s  should all be translated into “Few F’s are G’s.” Adverbs following “is” are also to be translated into predominant quantification. “F is almost, or nearly, or practically or just about  always G” should also be translated as  “Almost all F’s are G’s.”  Similarly, “F’s are almost, or nearly, or practically or just about  never G’s” all should be translated as “Few F’s are G’s.” 

	Certain adverbs of time also express negative predominant quantification. “F is seldom, rarely, or not often  G” should be translated  as “Few F’s are G’s.”   “Only” or “Just”  some, a few, or a couple  F’s are G’s also means “Few F’s are G’s.” There are negations which mean the same thing. “Not many, a lot, more than a few, more than a couple  F’s are G’s also mean “Few F’s are G’s.” Finally, quantifiers using many of these adjectives to modify a number should be translated into  “Few F’s are G’s.” Only, just, not more than, not over, fewer than, less than, at most, barely  n F’s are G’s all express the attitude that “Few F’s are G’s.” This can be true even when “n” seems to be a large number. This depends on context. The student needs to understand that when a chemist says, “At most, a billion chlorine atoms are in this pool,” or a member of Congress says, “Only a billion dollars is unaccounted for in the Defense Budget” they mean “Few chlorine atoms” and “Few dollars.”

	Majority quantification  is standardly expressed using “most.” The   majority of, more than half of, over half of   F’s are G’s mean “Most F’s are G’s.” A lot of adverbs also express majority quantification. “F is primarily, mainly, usually, principally, chiefly or generally  G,”  all should be translated as “Most F’s are G’s.” For the most part , and By and large, F’s are G’s also say “Most F’s are G’s.”

	Common quantification has “many” as its standard quantifier. It means a significant number less than half. So “A lot of,”  “Quite a few,” and “Quite a lot of  F’s are G’s” all should be translated as “Many F’s are G’s.” Adverbs after “is” which express common quantification are the following. F is often, quite often, frequently, quite frequently, commonly, 

quite commonly  G. These should all be translated as “Many F’s are G’s.” Alteration on other quantifiers, which make common quantification, include “More than a few “ and “More than a couple  F’s are G’s,” both of which should be rewritten as “Many F’s are G’s.” Negations of “only” or “just” in front of particular quantifications make common quantification. “Not only,” or “Not just  some, a few or a couple F’s are G’s” mean “Many F’s are G’s.” Finally, modifiers on numbers can make for this level of quantification. “More than n,” “Over n,”  “At least n,”  “No “ or “Not fewer than “ or “Not less than n”  F’s are G’s should be translated as “Many F’s are G’s.” 

	Particular quantification is standardly expressed by the word “some.” We take a minimal interpretation of all affirmative quantifiers. That is, “Q F’s are G’s” is interpreted as true if at least  Q F’s are, in fact, G’s. “Some F’s are G’s” is interpreted as true even if only one F is a G. “A few,” “a couple,” “several” F’s are G’s should be translated as “Some F’s are G’s.” F is sometimes, or occasionally  G should also be translated as “Some F’s are G’s.” So should other preceding phrases like “From time to time” and “Once in a while”  F is G. Negations which translate into “Some F’s are G’s” include  “Not all F’s are G’s”  and “Not every F is a G.” Note that the addition of a little word like “a” in front of “few” changes the level of quantification from predominant to particular. “Few F’s are G’s” is a predominant negative. “A few F’s are G’s” means “some F’s are G’s.” The student will have to read very carefully to get the right quantificational attitude.

	Negative exceptive quantifiers  like “Not only” or “Not Just F’s are G’s” are tricky as usual. Since “Only” or “just F’s are G’s” means all G’s are F’s, we have to apply  negation to this statement. A negation of a statement produces its contradictory (see below.)  The contradictory of “all G’s are F’s” is “Some G’s are not F’s.” So “Not only” or “Not Just F’s are G’s”  mean “some G’s are not F’s.” When you have plain numbers unqualified in any way, it is particular quantification. “N F’s are G’s” means “some F’s are G’s.” Numbers out of context are neither small nor large. In a swimming pool there may be quintillions or more atoms. So “1 billion chlorine atoms are in this pool” is  to be translated as “Some chlorine atoms are in this pool.” Finally, the indefinite article “a” should sometimes be translated as “some” rather than “all.” To repeat, in “A crow is a bird” it means “all,” but  “A crow is in the field”  should be translated as “Some crow is in the field.”

The student should now do Exercise 11A on translating into standard form.

SECTION 6.7: THE SQUARE OF OPPOSITION

	A Square of  Opposition is a kind of table which shows logical relations among categorical statements. Logical relations are relations between the truth or falsehood of one statement and that of another. Contradiction, Contrariety (say KON TRA r eye eh tee), sub-contrariety and implication are logical relations between statements.

	Contradictories are pairs of statements which can not both be true and can not both be false. If one of them is true, then the other must be false. Any pair of statements having the same subject and predicate   where one is of the form “All F’s are G’s” and the other is “Some F’s are not G’s”, are contradictories. So are “No F’s are G’s” and “Some F’s are G’s”. The contradictory of a statement is also called its negation or denial. Note that all one needs to say to contradict or deny a universal statement is to claim that there is one F which is or is not a G. Thus A and O, E and I type statements contradict each other.

	Contraries are a pair of statements which can not both be true but can both be false. Statements of the form “All F’s are G’s” and No F’s are G’s” are contraries. They can not both be true. However, they can both be false. This will happen when some F’s are G’s and other F’s are not G’s. For example, “All calculators are complicated to use” and “No calculators are complicated to use” cannot both be true. However, both are false, since some calculators are complicated to use and others are not. A and E statements are contraries.

	Sub-contraries are pairs of statements which can not both be false but can both be true.  “Some F’s are G’s” and “Some F’s are not G’s” are sub-contraries. If “Some whales  are mammals” was false then “Some whales are not mammals” would have to be true. I and O-type statements are sub-contraries.

	Implication: To say that one statement A implies another,  B, is to say if A were true, then B would have to be true. “Implies” here means that B follows with deductive validity from A. It means  “A -> B” is a deductively valid argument. 	“All F’s are G’s” implies every less general affirmative type of statement. If “All F’s are G’s” is true, then  “Almost all F’s are G’s,” “Most F’s are G’s,” “Many F’s are G’s,”  and “Some F’s are G’s” are also true. They are less informative, but still true because we interpret quantifiers below the universal minimally, as meaning “at least Q.” We have plenty of use in real life for conservative generalizations 

when we are not justified in making universal ones. What is true of A implying P, T, K, I (and all top - down combinations like A implying T, P implying K) is also true for the negative statements. “No F’s are G’s”  implies “Few F’s are G’s” and so forth. We  sum up  the relations of implication and contradiction up in a Square of opposition. Arrows show implications,  diagonal lines show contradictions, dashes show contraries, and dots show subcontraries. 

 Square of Opposition FOR THOMPSON’S EXTENDED SYLLOGISTIC LOGIC		
A:  All F  are G 							 E:  No F are G


P:  Almost all F are G 						 B:  Few F are G 


T:  Most F are G							 D:  Most F are not G


K:  Many F are G 							G:  Many F are not G


I: Some F are G	 						O:  Some F are not G

	The student should study the diagonal relations of contradiction in particular. They are essential to transforming denials into standard form. Knowing that “Almost all F’s  are G’s ” contradicts “Many F’s  are not G’s”  enables the student to translate “It is not the case that Almost all F’s  are G’s” or “It is false that  Almost all F’s are G’s” into “Many F’s are not G’s.” 	No standard form has a denial at the beginning. All denials must be translated into their contradictories, driving the negation into the statement, to get standard forms.

How to Rewrite Negations according to the Square of Opposition
		Universal: 		Not All S are P  -> Some S are not P 
		      		        	Not No S are P -> Some  S are  P
		Predominant:	Not almost all  S are P ->Many S are not P
		  		         Not few S  are  P-> Many  S are P 
		Common: 		Not many S are P ->Few S  are  P
		   		         Not many S are not P->Almost all S are P 
		Particular:	  	Not Some  S are P -> No S  are P
		           Not Some S are not P  -> All  S are P	  


SECTION 6.8: IMMEDIATE INFERENCES

	The standard form of a categorical syllogism has its major premise listed first or on top, its minor premise second and its conclusion last. The major premise is the one containing the major term, “P,” the predicate of the conclusion. The minor premise is the one containing the minor term, “S,” the subject of the conclusion.

	An immediate inference is a valid argument from just one reason to one conclusion. Implications like “A implies P,” shown in the Square of Opposition, are immediate inferences. Syllogisms and any multi-premise arguments  are mediate inferences. There are three types of immediate inferences which are particularly useful for translating syllogisms into standard form. They are called “Conversion,” Contraposition” and “Obversion.” They are most usefully described as translation operations on statements. They produce equivalent statements, cases where each statement validly implies the other, so one can be rewritten as the other.

	Conversion is the operation of exchanging the subject and predicate. It is valid only for E and I-type statements. We can rewrite “No F’s are “G’s” as “No G’s are F’s” and vice versa.  We can also rewrite “Some F’s are G’s” as “Some G’s are F’s” and vice versa. Conversion leaves a statement of the same type. E-type statements remain E’s, and I’s remain I‘s. Conversion is not valid for any other type of categorical statement. Never convert an A, P, T, K, B, D, G, or O-type statement. In such cases, the reason could be true and the conclusion false. Neither term, nor both terms may validly be turned into their complements. A helpful symbol for remembering conversion is:  For E and I only, [S - P] exchanges with [P - S.] 

	Contraposition replaces both terms with their complements and exchanges these complements. It is valid only for A and O-type statements. We can rewrite “All F’s are G’s” as  “All non G’s are non F’s” and vice versa. We can rewrite “Some F’s are not G’s” as  “Some non G’s are not non F’s” and vice versa. It too leaves statements of the same type. 
Contraposition is valid only for A and O-type statements.  Never use it on
the other types of statements. Both terms must be made into complements. A symbolism for contraposition is: For A and O only, [S-P] exchanges with [non P-non S.]

	Obversion is first changing the quality of a statement and then  replacing the predicate with its complement. It is valid for all 10 forms. 

 For example, if “No  F’s are G’s” is true, then “All  F’s are non  G’s” must be true and vice versa. Note that both the quality (affirmative to negative or negative to affirmative) must be changed, and  the predicate only  must be turned into its complement. Universal and Predominant quantification change quality by changing the quantifier, but Majority, Common and Particular change quality by putting in or taking out the “not” after the copula. “Almost all  F’s are G’s” and “Few  F’s are non G’s” are obverses. However, “Most F’s are G’s” and “Most F’s are not  non G’s” are the obverses on the majority level.  “Not” must  be put in when obverting a T, K or I to a D, G or O, and  dropped when obverting a D, G, or O to a T, K or I. Do not confuse it with “non” or a “not” inside a predicate. A symbolism for obversion is: For all types of statements [(Qual) S-P ] exchanges with [Non (Qual) S - non P]. Some syllogisms are not in standard form because they contain terms and their complements. They thus contain four or more terms.Valid reduction leaves valid arguments valid and invalid ones invalid. These operations are used to reduce the number of terms to three when complements are present so that our rules of validity can be used. 

The student should now do Exercise 11B, C, D on Immediate Inferences.

SECTION 6.9: REDUCING TERMS IN A SYLLOGISM

	Sometimes it looks like there are more than three terms, so an argument is not a syllogism. However, if there are only slight variations of wording, synonyms or complements, it is possible to put the argument into standard form. Here is a strategy  for  reducing to three terms.  First,  count the number of terms.  If it is only three, nothing below needs to be done. If it is 4, 5 or 6, then, second,  if any terms can be made the same by just change of word order, do so. For example, one statement might contain the phrase “all of the members” and the other just “members.” Just cross out “of the.”  If any terms are synonyms pick one of the synonyms and substitute it for  the other. This can be done for 1, 2 or 3 pairs of synonyms, reducing to only 3 terms.  Third, note if any terms have complements in the statements. 

	 Complements appear as  contrary words or prefixes. Words like “high,” “low;” “good,”  “bad,“  can often be treated this way: “low” as “non high;” “bad” as “non good.” Prefixes like “un,”  “in,” or “dis” can be treated similarly. “Unjust,”  “just;” “adequate,”  “inadequate;” “honest” and ‘dishonest” are complements. Put parentheses around each term. Label each with a capital  e.g., the first letter of an important word in the term.  Complements are labeled with “non” in front of the letter. 

	Contraposition allows a person to change 2 terms into their complements, obversion only 1 and conversion none. So if there are 4 or 5 terms and the 4th. or 4th. and 5th. are both predicates, use obversion on the statements containing them to get rid of the complements. If there are 6 terms, then using contraposition on a statement with a 5th. and 6th.  term will reduce to a syllogism containing only 4 terms. Cases where a complement is a subject may require two operations. Usually one is conversion of an E or O to get a term in the predicate so that obversion can then be used on it.  In some cases it may require two operations on one statement and one or even three or four total, but no more. In Aristotelian logic one can always reduce a syllogism containing  up to six terms and complements to three. However, this is not always possible in the extended system. For example, any syllogism with premises such as “Most M are P and Most non M are S” will not be reducible, because conversion and contraposition are not valid forms of inference for these types of statements. (I’m again grateful to Bruce Thompson for this observation.) 

	Let us apply this strategy to examples. First, an apparent 4-term syllogism: “No cult leaders are moral and Herff Applewhite was a cult leader, so Herff Applewhite was immoral.” Recall that singular statements about named people must be treated as universals and that a parameter is needed to make “immoral” into a noun phrase. So first, it is “No cult leaders are moral people  and All people identical with  Herff Applewhite are cult leaders, so all people identical with Herff Applewhite  are immoral people.”  It seems that  “cult leaders,” “moral people,” “people identical with Herff Applewhite,” and “immoral people” are 4 terms.   However, obversion of “no cult leaders are moral people” to “all cult leaders are immoral people” leaves us with just three terms. So would obversion of the conclusion. An example with 5 terms, showing that valid reduction of an invalid argument will leave an invalid 3-term syllogism is this. “All females are non sexists and all high school boys are males, so all high school boys are sexists.” “Females,” “non sexists,” “high school boys,” “males,” and “sexists” look like 5 terms. However, we can force this into standard form by treating “females” and “males” as complements without doing violence to the illogic of the argument. Contraposition on “All females are non sexists” gives us “All sexists are non females” (that is, ‘males’.)

	Finally, a 6-term case suggested by one in Copi (1962)  “No non members are golfers. All non golfers are non club users. So all club users are members.” With the terms counted, it looks like this.


			R1 No non members1 are golfers2.
			R2 All non  golfers3 are non club users4.
			So all club users5 are members.6

It does not matter what number we use for which term, so long as we do not mix them up. So suppose we decide to keep  “club users” and “members” in the conclusion fixed. Taking the contrapositive of  R2, [S-P] to [nonP-nonS], makes it  “All club users are golfers.” This reduces from 6 to 4 terms.  Converting  R1 [S-P]  to [P-S] makes it “No golfers are non members.” Now we can obvert this  [(Qual) S-P] to [Non(Qual) S - nonP]. That is,  “No golfers are non members” becomes “All golfers  are  members.”
All golfers1 are members2.
All club users3 are golfers1.
All club users3 are members2.

	 Do not reduce statements and then go back and use the original, unreduced statements. This leads to wrong cross hatches. Also note that there are cases where a syllogism has more than three terms and  can not be reduced by valid inferences to any valid syllogistic form.  In a case like this, write out attempts to  reduce the syllogism to three terms. If it is possible to do so, this should be done and a cross hatch made. If it is not, the syllogism should be written up as “invalid because it contains more than three terms.”

SECTION 6.10: OTHER TRANSLATION PROBLEMS

	It is not difficult to determine whether a syllogism is valid, if its’  statements are in standard form.  Standard form must  begin with a standard quantifier, its subject must  be a noun phrase, its’ verb must  be “are,” and it must  end with a predicate  noun phrase.  Recall that “not” between “are” and the predicate is an essential part of the form of  D’s, G’s and O’s.  One must  get each statement into one of these forms: 

 Standard    quantifiers		subject noun   TO BE	NOT(in D,G, O) predicate noun.
 All, Almost all, Most, Many, Some	     F’s		  are	   	     	 	G’s.	 
No, Few, Most, Many, Some		     F’s		  are     not	       		G’s.

	Sophomoric generalizations are ones unquantified at all, such as “Women are...” “Society says...” “Man is...” How many? One can’t tell. Such reasons are vague. In other cases it is difficult to create noun phrases. A noun phrase  answers the question: “Who or what is doing the action or in

the state of being described in the sentence?”  A  phrase which cannot be the subject of a sentence is not a noun phrase. Therefore, always ask “Can this phrase be a subject?” Sometimes abstract singular words like “justice” or “knowledge” used in a categorical statement need help from an unusual parameter. Here are several examples: “Knowledge is intrinsically good” might be translated as “All items of knowledge” or “pieces of knowledge are intrinsically good things.” Notice the need to add the plural-making noun “item” to the subject as well as the parameter “things” to the predicate. “Justice” might need to be made into “principles of  justice” or “just acts.” A word like “disease” might need a parameter like “case” or “instances” or “types of  disease.” 

	statements can be very long and contain unnecessary words like “It is true that,” “It is obvious that,” and so on. They should be  eliminated. The student would do well also to find the break between the subject and the verb. Whenever the verb is not “to be,” it will have to be put in, and the verb presently there will usually become an important part of the predicate term. Here is a general method for  translating non standard syllogisms. 

	A. Identify the conclusion by underlining indicator words and noting premise connectives. B. Bracket and number the statements. C. Inside the brackets cross out unnecessary words (e.g., “it is true that,” but never  negations like “it is false that.”) D. Start either with the conclusion, or the statement closest to being in standard form. Identify the terms: 1) Find the break between the subject and  verb. 2) Put a left parenthesis in front of the the first word of the subject, excluding any quantifier. Put the right parenthesis between the last word of the subject and the verb. Write “S” for subject above it and “P” above the predicate. If exactly the same terms occur elsewhere, mark them “S” and “P” too. Start parentheses for predicates after  “not.” Label the  middle term “M.”  3a) If the verb is “to be,” put a left parenthesis after it unless there is a “not” right after it. In that case, put the left parenthesis after the “not.” Close parenthesis at the end of the thought. Change singular terms to plural and “is” to “are.”  3b) If the verb is  not “to be,” insert “are” and  the left parenthesis to start the predicate term between the last word of the  subject and the verb so that the verb is included in the second term. 
Example of 2 and 3a: “Many a bird is migratory” becomes 
				Many (birds) are (migratory)	
Example of 2 and 3b: “Few men gave blood that day” becomes 
			       Few (men) are (gave blood that day.)
	

	4a) Check if there are exactly three different parenthesized noun phrases. If not, because one or more is not a noun phrase, insert a parameter.  Ask,  “What is the universe of discourse of this statement or syllogism?” “What is it about?” Frequently, the appropriate parameter is used at least once in a syllogism. Use the same parameter wherever necessary to make predicate noun phrases.  
		Examples: Many (birds) are (migratory animals.)
		Few (men) are (people who  gave blood that day.)
	4b) If there are not exactly three terms because of synonyms, then  reduce to three by substitution. 4d) If there are not exactly three terms because of complements, reduce to 3 by obversion, conversion or contraposition. 	5) If there are any pronouns for terms, replace the pronouns with the  terms. 6) Make a key which shows what noun phrase is the subject of the conclusion, the predicate of the conclusion and the middle term as follows: S=  , M=  , P=  .  
	7) Make quantifiers standard.  This includes changing  any negations at the start by writing out the contradictory, not the contrary of a statement!  For example:  “Not every S is P” equals  “Not all S are P” which equals  “Some S are not P.” It does not equal ”No S are P.” While doing this, it is a good idea to write the letter for the type of statement which each categorical statement is above the verb “are.”

	Finally, make a cross hatch, transfer the letters to it, and determine  validity. When in serious doubt about translation, follow the Principle of Charity: If an argument “feels valid,” see if it can be translated into a valid standard form. Here is a simple example without complements. “Most of the older teachers do not believe in ‘social promotion’, so no older teachers are members of the administrative staff, since  almost all of the administrative staff believe in ‘social promotion’.” Here the unnecessary words “of the” and “members of the” are deleted.

Steps  A, B, C: <1Most older teachers do not believe in ‘social promotion’,> so <2 no older teachers are  administrative staff,>  since   <3 almost all administrative staff believe in ‘social promotion.>

				S					M
Step D:  <1Most (older teachers) do not (believe in ‘social promotion’),>
			S					P
so <2 no (older teachers) are (administrative staff),>
					P				M
 since <3 almost all (administrative staff) (believe in ‘social promotion.)>

Step D3b:  there  is a “not”  after “do” in the first statement.  So we must insert  “are” in place of “do,”  “and put the parenthesis after  the “not.”
			S						M
<1Most (older teachers) are  not (believe in ‘social promotion’),>
			S					P
 so <2 no (older teachers) are (administrative staff),>
						P			M
 since <3 almost all (administrative staff) are (believe in ‘social promotion.)>

Step 4a: We need to make the middle term into a noun phrase. This can be done simply by adding “rs” to “believe”: “believers in ‘social promotion.’”
			S					M
<1Most  (older teachers) are not (believers in ‘social promotion’),>
			S					P
 so <2 no (older teachers) are (administrative staff),>
						P				M
 since <3 almost all  (administrative staff) are (believers in ‘social promotion.)>

Step 5: there are no pronouns, so no action is needed. Step 6): The key is S= older teachers, P= administrative staff, and M= believers in ‘social promotion.’ Step 7: No  action is necessary on making quantifers standard. The types of the statements, D, E and P should be put above “are” in each statement.

Step 7: Here is our cross hatch with the major premise on the top.

						



  

The student should now do Exercise 12A on determining validity of syllogisms.


SECTION 6.11: SUPPLYING MISSING REASONS AND CONCLUSIONS

	An “enthymeme” is a valid  argument which is missing a reason or conclusion. In Chapter Four we presented an intuitive way of guessing at a missing statement in  a classical two-quantifier syllogistic system. It is  possible to give a specific recipe for doing this for the extended 5-level system. First, note whether the missing statement is a premise or conclusion. Second,  write down exactly which terms are  missing in alphabetical order. Third, determine what quality (affirmative or negative)  the missing statement must have to make a valid syllogism. (If neither or both of the other statements are negative, then the missing statement must be affirmative. If one is negative,  then it must be negative.) Fourth,  assign distributing index values to the statements given. (This requires remembering A51, P41, T31, K21, I11  and  E55, B45, D35, G25, O15) If the missing statement is a premise, determine the minimum  distribution for each term necessary to make the argument valid. The reason for this is that one wants to assume, under the Principle of Charity,  as little as possible to make the argument valid. (This requires recalling the rules that the sum of the distribution indexes for M must be greater than 5 and for S and P less than or equal to that in the conclusion.)

	Sixth, if the missing statement is the conclusion, determine the maximum  distribution indexes for the terms in the conclusion which satisfy the rule that the value of S and P in the conclusion must be less than or equal to the distribution index in the reason. The reason for this is again, the Principle of Charity: we want make arguments as strong as possible, to learn as much, as highly general a conclusion, as follows validly from the reasons. Seventh, use the quality determination (Step 3) to consult  A51, P41, T31, K21, I11  and E55, B45, D35, G25, O15  to determine which term is the subject and which the predicate.  All pairs of distribution indexes together with qualities and figures have uniquely determined term orders. 


	Here are some examples from condensed forms of syllogisms using  letters standing for actual terms.

Many F are not G.
			Missing term letters F, H. Statement must be affirmative
So some H are not G.

Put in the distribution values:
					Many F2 are not G5.
			    
					So some H1 are not G5.

	A missing reason to make a valid argument must have the other middle term “F,” with at least a distribution of 4 (2+4) needed to be more than 5. It should also have the minimum value which will make the argument valid which is also 4. The term H must have distribution value of at least 1. Since this is also the minimum value we should accept, we assign 1 to H. Since this is an affirmative statement, its predicate must be the term with 1. (All predicates in affirmative statements have distribution 1). Therefore the order of the terms and distributions is F4 - H1. This must, therefore, be a P-type statement. The missing reason is therefore “Almost all F’s are H’s.”

	Second example:

All E are F
	Missing term letters D, F. Statement must be negative.
So some D are not E.

Putting in distribution values give this:
					All E5 are F1
	
					So some D1 are not E5.

	In this case, since the conclusion is negative, the missing reason must be also to satisfy the third rule of validity.  The term “D” must have at least distribution “1” which is also the minimum needed to validly deduce the conclusion. “F,” on the other hand, requires a distribution of 5 to satisfy the middle term rule. The only type of negative statement with this distribution is an O-type of statement with “D” as its subject and “F” as its predicate: “Some D1 are not F5 .” 
			
	Here is an example with a missing conclusion.

	Almost all J are K.
	No K are L. 
	So				Missing terms J, L  Statement is negative.


Putting in distribution values gives us this.

					Almost all J4 are K1.
					No K5 are L5. 
					So			

Since the distribution index of the terms  in the conclusion must be less than or equal to those in the reasons, and those of J and L in the reasons are 4 and 5; they must be 4 and 5 in the conclusion also.  With an negative statement with indexes of 4 and 5, it can only be a B-type statement, Few J4 are L5.

	The student should now do Exercise 12B on  Drawing  Conclusions and Supplying Missing  Reasons.
































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