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March 23rd, 10:15 AM   #1
Dr. Hfuhruhurr
Super Moderator
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Join Date: Feb 2007
Posts: 399
Default A Guide to Constructing and Recognizing a Valid Argument

.

A Brief Introductory Guide to Formal Logic

How syllogistic logic works and how to apply it to formal arguments

Dr. Hfuhruhurr
3-23-07






PART 1: THE BASICS


INTRODUCTION

Arguments take many forms. Some are nothing more than personal deductions: The accused couldn't have done it because I was with him the entire time. Some appeal to emotion or authority: He's an expert on recombinant DNA so what he said is credible. Others may be by analogy: Because he looks like a horse, and horses arn't very smart, he must not be very smart. Such arguments are considered informal because they rely solely on their content for their power to convince and may take whatever form their creator desires. Formal arguments derive their power from their construction, the way an argument's terms are arranged and stated. The most common form of formal argument is the syllogism, which is type of symbolic logic because its validity can be expressed in representative symbols. Although a syllogistic argument may be valid it need not be sound. Its soundness also depends on the statements that make it up. Validity only tells us that if its premises are true the argument is sound.

Syllogisms have three parts, and three parts only: Two premises and a conclusion. A syllogistic argument (hereafter simply called, "argument") will typically go something like: Good girls deserve praise. Adrienne is a good girl. Therefore, Adrienne deserves praise. To analyze an argument the three parts are laid out as follows.
Good girls deserve praise (First (major) premise)
Adrienne is a good girl (Second (minor) premise)
_____________________
Adrienne deserves praise (Conclusion)
This happens to be a sound argument: Both of its premises are true and its form is valid. The following is also a valid argument,
All artists are very intelligent
Some artists are men
____________________
Some men are very intelligent
and its conclusion is true, but it is not a sound argument because one of its premises, "all artists are very intelligent" is false. So one has to be careful about assuming an argument is sound because its form is valid and its conclusion happens to be true.



THE CONCLUSION, ITS FORM AND ITS SUBJECT AND PREDICATE

Of course the most important part of any argument is its conclusion; it's its raison d'Ítre. Looking at any conclusion we see it has several parts. Most important are its subject (S), what the conclusion is about, and its predicate (P), what is said about the subject: water is wet, mice eat cheese, fat dogs fart, etc. There are also two other parts: the quantifier (Q), which indicates how many: all or some; and the copula (C), which tells us if the relationship between the subject and its predicate is affirmative or negative: are or are not. The basic arrangement of these four elements is: Q--S--C--P, and can be seen in our conclusion, "Some men are extremely intelligent," (Q: how many) Some, (S: what the conclusion is about) men, (C: the relationship between the Subject and Predicate) are (P:a characteristic of the Subject) extremely intelligent. Often "all" and "are" are left unindicated.

Each of the quantifiers "all" and "some" can be combined with each of the copulas, "are" and "are not" to give us four combinations
All ______ are ______
All ______ are not ______ or more commonly, No ______ are ______
Some ______ are ______
Some ______ are not ______
Filling in the blanks of a conclusion, the subject (S) will always go first, followed by the predicate (P). So regardless of the quantifier and the copula, a conclusion will always take the form S P. Never P S.


For easy reference the four combinations above have been given the names, A, E, I, and O.
AMM All ______ are ______ (All S are P)

EMM No ______ are ______ (No S are P)

IMM Some ______ are ______ (Some S are P)

OMM Some ______ are not ______ (Some S are not P)
To recap: ALL conclusions will take one of these four forms, and in ALL of them the subject (S) will come before the predicate (P).



THE FORM OF THE TWO PREMISES, AND THE MIDDLE TERM, M

Having identified the subject (S) and its predicate (P) in a conclusion we next look to see how they arise in the argument. The argument's two premises, designated Major and Minor, have a set order: The Major premise always comes first, followed by the minor premise. Moreover--and this is very important-- the Subject (S) only occurs in the minor premise and the Predicate (P) only occurs in the Major premise. Tentatively, this gives us.
P somewhere in the Major premise
S somewhere in the Minor premise
___________________________
S P
Of course there's more to an argument than just the subject and its predicate, something has to establish a relationship between them, and this is the middle (M) term. In the argument that Adrienne deserves praise the middle term is "good girl." And using M as its designation we can now construct a syllogism in its symbolic form.
Good girls (M) deserve praise (P)
Adrienne (S) is a good girl(M)
_____________________
Adrienne (S) deserves praise (P)
And, stripped to its essentials
M P
S M
____
S P
Note: the middle term occurs in both premises and never in the conclusion, which holds true for all arguments.



THE FOUR COMBINATIONS of P, S, and M

Because the positions of P and S in their respective Major and Minor premises are not fixed, both they and M can appear in several combinations. There are four, which are called "figures" in logic speak, and when presented with the conclusion are as follows. (Note that quantifiers and copulas are irrelevant to these constructions.)

Figure 1 (The middle term is the subject of the major premise and the predicate of the minor premise)
M P MMM All ducks (M) are birds (P)
S M MMM Daffy (S) is a duck (M)
___
S P MMM Therefore, Daffy (S) is a bird (P)

Figure 2 (The middle term is the predicate of both premises)
P M MMM No drunkards are good mothers
S M MMM Some women are good mothers
___
S P MMM Therefore, Some women are not drunkards

Figure 3 (The middle term is the subject of both premises)
M P MMM All artists starve for their art
M S MMM Some artists are Americans
___
S P MMM Therefore, Some Americans starve for their art

Figure 4 (The middle term is the predicate in the major premise and the subject of the minor premise)
P M MMMAll colored flowers are scented.
M S MMMNo scented flowers are grown indoors
___
S P MMM Therefore, No flowers grown indoors are colored.

Obviously a syllogism need not make sense to be valid.

That's the basics of syllogistic construction. However, just because a syllogism assumes one of these configurations does not mean it's valid. Some figures with certain combinations of quantifiers and copulas are outright invalid.




CONTINUED IN PART 2 BELOW

Last edited by Dr. Hfuhruhurr : April 28th at 06:52 PM.
 
March 23rd, 10:23 AM   #2
Dr. Hfuhruhurr
Super Moderator
Logician
 
 
Join Date: Feb 2007
Posts: 399
Default

'


PART 2: THE VALID AND INVALID FORMS


FOUR RULES

Besides the rules of composition given in part 1, there are four major rules of construction.
1. No valid syllogism can have more than one negative premise. (a premise using, no, not, none are)

2, If either premise is negative the conclusion must be negative. And, if the conclusion is negative one of the premises must be negative

3. The middle term must be distributed at least once

4. Any term distributed in the conclusion must be distributed in the premise.
A distributed middle term (M) establishes a genuine relationship between the major and minor terms, P and S. This means that the premises of an argument have to provide some some kind of information about the entire class represented by the middle term. This probably makes little sense, and because it's a bit complicated to explain and is better explained elsewhere I'm simply going to furnish two links that do just that. Please Click HERE AND HERE.

Now, if you don't want to bother with all these rules just to figure out if an argument is valid or invalid, there's good news. All the valid forms have been figured out and constitute a list of only 15 possibilities. Back in medieval days students of logic assigned each of these possibilities a name, such as Baroco, Disamis, and Camestres--their vowels indicated the type of statements in each. Since then modern logicians have dropped these names and use only the vowels to indicate the three statements.



THE FORMS

The validity of the fifteen is determined in part by the specific combination of the forms of the two premises and the conclusion. These forms are the same as those given above in "THE CONCLUSION, ITS FORM AND THE TERMS S & P".

AMM All ______ are ______

EMM No ______ are ______

IMM Some ______ are ______

OMM Some ______ are not ______
So, for instance, if an argument is so constructed that
the major premise states EMMno____are____
the minor premise states IMM some____are____
the conclusion states.......OMMsome S are not P
the argument is then designated EIO.

Because the S and P terms can be inserted in either position within their respective premises we also designate which of the four figures the argument takes. This is the second determinant of validity. In this case let's say it's
M P
S M
____
S P
which is Figure 1.Therefore, logicians would designate the argument above as EIO-1 or 1-EIO

Now, none of this is particularly important if one simply wants to take an argument and compare it with those listed below, but I think it's helpful to know how the labels were arrived at.



THE FIFTEEN VALID FORMS

1. AAA-1 (Note that this is the only valid form in which the conclusion can begin with "All.")
All M are P
All S are M
_________
All S are P

2. AOO-2
All P are M.
Some S are not M
________________
Some S are not P

3. OAO-3
Some M are not P
All M are S
______________
Some S are not P

4. AEE-4
All P are M
No M are S
_________
No S are P

5. AEE-2
All P are M
No S are M
__________
No S are P

6. EAE-1
No M are P
All S are M
__________
No S are P

7. EAE-2
No P are M
All S are M
__________
No S are P
8. AII-1
All M are P
Some S are M
____________
Some S are P

9. AII-3
All M are P
Some M are S
____________
Some S are P

10. IAI-3
Some M are P
All M are S
____________
Some S are P

11. IAI-4
Some P are M
All M are S
____________
Some S are P

12. EIO-1
No M are P
Some S are M
______________
Some S are not P
13. EIO-2
No P are M
Some S are M
_______________
Some S are not P

14. EIO-4
No P are M
Some M are S
_______________
Some S are not P

15. EIO-3
No M are P
Some M are S
_______________
Some S are not P




SIX INVALID FORMS



AAE-1
Illicit major

All M are P
No S are M
____________
No S are P

All gerbils have fur. No cats are gerbils. Therefore, no cats have fur.

AAA-3
Illicit minor
All M are P
All M are S
_________
All S are P

All elephants are gray. All elephants are big. Therefore, all big things are gray.

AAA-2
Undistributed middle
All P are M
All S are M
____________
All S are P

All college students are over 17. All high school graduates are over 17. Therefore, all high school graduates are college students.

AAI-All
Existential
All M are P
All S are M
__________
Some S are P

All mammals are animals. All humans are mammals. Therefore, some humans are animals. (This is invalid because the conclusion's use of "some" implies that the class of humans has at least one existing member--something not established in either premise.)

AEA-All
Negative premise and affirmative conclusion
All M are P
No S are M
_________
All S are P

All Christmas trees are green. No bears are Christmas trees. Therefore, all bears are green.

EEA-All
Two negative premises
No M are P
No S are M
__________
All S are P

No Martian eats pigs. No Earthling is a Martian. Therefore, all Earthlings eat pigs.



In addition to these invalid syllogisms there are several other invalid forms of argument that deserve mention. These are conditionals that begin with "If."


Affirming the consequent
If p then q
q
________
p

If I go to the store then I can't wash the car. I can't wash the car. Therefore I went to the store.

Denying the antecedent
If p then q
Not p
_____
Not q

If Tweety barks then Tweety is an animal. But Tweety doesn't bark. Therefore Tweety is not an animal.

Negating the antecedent and the consequent
If p then q
_______________
If not p then not q

If my son is ten years old, then I'm over forty. Therefore, if my son is not ten years old I'm not over forty.

Converting the conditional
If p then q
________
If q then p

If you like liver then you're a fool. Therefore, if you're a fool you must like liver.



There is a lot more to formal logic, syllogisms included, but this should give you enough ammunition to construct a decent argument and evaluate those of others.


HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH


If you have any questions or comments about this guide.

Please feel free to post them HERE



All previous replies to this Guide can be found there as well.


Last edited by Dr. Hfuhruhurr : June 20th at 11:03 AM.
 


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