A Brief Introductory Guide to Formal Logic
PART 1: THE BASICS
How syllogistic logic works and how to apply it to formal arguments
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
For easy reference the four combinations above have been given the names, A
, 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
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 PNote: 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.)
(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)
(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
(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
(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