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Logic is a very important branch of philosophy. In fact, logic is the basic tool of all science; for logic is the study of how to reason validly. Not everyone who reasons does so validly. Some people draw conclusions that simply do not follow from the given premises. In logic, this is called a
**
non-sequitur**. It is very important that we become familiar with the most common logical fallacies so as not to fall victim to the deceptive reasoning of modern sophists, such as journalists and politicians.

Let us jump right into a study of syllogisms. A syllogism is an argument. It involves the deduction of a conclusion from two given premises. The following is an example of a categorical syllogism:

All those on a high fat diet are at risk for colon cancer.

John is on a high fat diet.

Therefore, John is at risk for colon cancer.

Or, a standard and simpler example

All men are mortal

John is a man

Therefore, John is mortal.

Any argument that you might have with somebody is -- if it is a genuine argument -- reducible to a syllogism. Let us begin with the categorical syllogism. This is an argument made up of categorical propositions, such as the two examples above. The categorical proposition is a complete sentence, with one subject and one predicate, that is either true or false. For example,

Allgiraffesareanimals

The subject of a proposition is that
*
about which something is said*. For example, in the proposition above,
*
giraffes
*
is the subject; for something is being said about giraffes.

The predicate of a proposition is
*
that which is said about something*. For example, what is being said about giraffes (the subject)? What is being said is that they are
*
animals*. Thus, animals is the predicate of the above.

The copula joins or separates the subject and the predicate. In our categorical proposition above, "is" joins 'giraffes' and 'animals', thus "is" is the copula. The copula that
*
separates
*
the subject and the predicate will be 'is not', as in
**
Some men
are not
Italian**.

Logicians have devised an easy way to identify categorical statements. These simple codes were derived from two Latin words:
*
Affirm
*
(I affirm) and
*
Nego
*
(I deny). There are only four possible forms of a categorical proposition, two are affirmative statements, and two are negative statements. For example, one can say:

All pigs are smelly

or,

Some pigs are smelly

or,

Some pigs are not smelly

or,

No pigs are smelly

Logicians simply took the first two vowels of
*
Affirmo
*
(I affirm), the A and the I, as well as the first two vowels of
*
Nego*, the E and the O, and employed them as codes for these kinds of statements. Hence, we speak of A statements, I statements, O statements, and E statements.

The A statement is called the universal affirmative, and it takes the form: All S (subject) is P (predicate), for example:
**
All giraffes are animals**. The I statement is called the particular affirmative, and it takes the form: Some S is P, for example,
**
Some men are Italian**. The O statement takes the form: Some S is not P, for example,
**
Some women are not French**. Finally, the E statement, which takes the form: No S is P, for example,
**
No fish is a marathon runner**.

A: All S is P

I: Some S is P

O: Some S is not P

E: No S is P

There are only three terms in a syllogism: the major term, the minor term, and the middle term. The
**
major
**
term is the predicate of the conclusion, the
**
minor
**
term is the subject of the conclusion, and the
**
middle
**
term is never in the conclusion but appears twice in the premises. For example:

All

menare mortal

John is aman

Therefore,Johnismortal.

In the syllogism above, the major term is
**
mortal**, the minor term is
**
John**, and the middle term is
**
man**. The premise that contains the major term is the major premise, and the premise containing the minor term is called the minor premise. Thus, All men are mortal is the major premise; John is a man is the minor premise.

The middle term is
*
the cause
*
of the conclusion. We know something about the nature of man, namely that man is mortal. So,

John is a [Therefore, we can deduce thatman; all men] are mortal.John is mortal.Note the middle term in the middle, causing the conclusion.

Distribution is a very important concept in logic. We speak of terms (such as the major, minor, or middle terms) as being either distributed or undistributed. The word 'distribute' means to spread out. A distributed term is 'spread out'. In other words, a distributed term covers 100% of the things referred to by the term. An undistributed term is not spread out, in other words, it is a term that covers less than 100% of the things referred to by the term. For example, in the categorical proposition:
**
All men are mortal**, 'man' is distributed, because the statement indicates that
*
all men
*
(100%) are mortal.

In the statement:
**
No men have wings**, 'man' is also distributed, because the statement indicates that
*
all men
*
(100%) are lacking wings.

Consider the proposition:
**
Some men are fast runners.
**
In this case, the term 'man' is undistributed, because the statement indicates that only some men are fast runners, not all men. In other words, less than 100% of men are fast runners. In the proposition:
**
Some men are not good cooks**, 'man' is undistributed for the same reason.

Predicates can also be distributed or undistributed. Consider again the proposition:
**
No men have wings**. 'Wings' is distributed, because if
**
no men have wings**, then 100% of winged creatures are not men.

Consider the proposition:
**
All men are animals**. The predicate 'animals' is undistributed, because it is not the case that 100% of animals are men. Rather, only
*
some animals
*
are men, while some are horses, dogs, cats, etc. In the proposition:
**
Some men are fast runners**, the predicate 'fast runners' is undistributed for the same reason. It is not the case that 100% of fast runners are men. Some are men, but other fast runners include cheetahs, deer, cougars, etc.

Finally, consider the proposition:
**
Some men are not American**. The predicate 'American' is distributed. The reason is the following. Imagine we have a group of five men in the room who are not American, and these are the men we refer to when we assert that some men are not American. It is true that 100% of Americans are not these men.

A:

AllS is P

I: Some S is P

O: Some S is notP

E: NoSisP

There are certain basic rules of logic that need to be observed. If any one of these rules are violated, then our reasoning is invalid. This means that the conclusion cannot be deduced from the given premises. In other words, it lacks the force of necessity.

For example, consider the following syllogism:

All

menare mortal (A statement)

Johnis a man (A statement)

Therefore,Johnis mortal (A statement)

Notice that the middle term 'man' is distributed at least once. Note also that we treat a singular as a Universal Affirmative (A) statement. For example, 100% of John is a man. But consider the following argument:

All Conservatives favor privatization. (

A statement)

All members of the CBA favor privatization. (A statement)

Therefore, all members of the CBA are Conservatives. (A statement)

The conclusion simply does not necessarily follow from the given premises (non-sequitur). The problem with this reasoning is that the middle term is, in both, cases undistributed. To make this clearer, we will underline the middle term and highlight all distributed terms in red. Notice that each categorical proposition is an A statement, and in an A statement, the subject is always distributed, while the predicate is always undistributed. Hence,

All

Conservativesfavor privatization. (A statement)

Allmembers of the CBAfavor privatization. (A statement)

Therefore,all members of the CBAare Conservatives. (A statement)

Notice that the middle term remains undistributed in both premises. Hence, it is invalid.

Do not be mislead by the fact that the above statements do not
*
appear
*
to follow the simpler form: All S is P. They actually do, only the copula is hidden. For example, the proposition:
**
All Conservatives favor privatization
**
is really the same as:
**
All Conservatives are people who favor privatization**.

All social workers know about the difficulties of life on the street.

No abstract philosopher is a social worker.

Therefore, no abstract philosopher knows about the difficulties of life on the street.

The conclusion of this argument does not necessarily follow from the given premises. The problem is that there is a distributed term in the conclusion, but that same term is not distributed in the premise in which it first appears. Consider the syllogism with the distributed terms highlighted:

All

social workersknow about the difficulties of life on the street. (A statement)

Noabstract philosopheris asocial worker. (E statement)

Therefore, no abstract philosopher knows about the difficulties of life on the street. (E statement)

The predicate of the conclusion:
**
knows about the difficulties of life on the street**, is distributed, but when it appears in the first premise, it is undistributed, because it is the predicate of an A statement.

Consider the first premise:
**
All
social workers
know about the difficulties of life on the street
(All social workers are people who know about the difficulties of life on the street).
**
The predicate "

No Father Michael McGivney student is Irish (from Ireland).

No Filipino is Irish (from Ireland).

Therefore, No Filipino is a Father Michael McGivney student.

The conclusion, once again, does not follow from the premises. One simply cannot draw any conclusion from the two given premises. Consider the following:

No oak trees bear fruit

No maple trees bear fruit

Therefore, no maple trees are oaks

Even though we know through experience that no maple trees are oak trees, it is not true that we can deduce that no maples are oaks because neither one bears fruit.

No student from this school is musically gifted.

Some of the musically gifted are neurotic.

Therefore, some neurotics are students from this school.

This, of course, is an invalid syllogism. The conclusion is completely unwarranted. Notice that the conclusion is affirmative (*I statement*). But the major premise is negative (*E statement*). But if a premise is negative, the conclusion must be negative. For it is not necessarily the case that some neurotics are from this school, for this school might very well be neurotic free.

No student from this school is musically gifted.

Some of the musically gifted are neurotic.

Therefore, no neurotics are students from this school.

Once again, this is an invalid syllogism. Notice that the conclusion is universal (*E statement: universal negative*), while the minor premise is particular (*I statement: particular affirmative*). But if a premise is particular (i.e., Some S is P or Some S is not P), the conclusion must be particular. Moreover, even though no student from this school is musically gifted, it does not follow that no student from this school is neurotic.

The following syllogism, however, is logically valid:

No student from this school is musically gifted.

Some of the musically gifted are neurotic.

Therefore, some neurotics are not students from this school.

This conclusion necessarily follows from the given premises, and so it has the force of necessity.

Before moving on to the Conditional Syllogism, try to determine the validity of the following categorical syllogisms:

The Categorical Syllogism Exercises

Answers to Exercises

There is another type of syllogism besides the categorical, namely, the conditional syllogism, and it takes the following form:

If p, then q

p

q

For example:

If Johnnie eats cake every day, then he is placing himself at risk for diabetes.

Johnnie eats cake every day.

Therefore, Johnnie is placing himself at risk for diabetes.

The major premise in this kind of syllogism is a conditional proposition: "**If Johnnie eats cake every day, then he is placing himself at risk for diabetes**". There are two parts to the conditional proposition. Notice that one clause begins with "if", another with "then". The "if" clause is called the
**
antecedent**, the "then" clause is called the
**
consequent**.

If Johnnie eats cake every day, then he is placing himself at risk for diabetes.

Johnnie eats cake every day.

Therefore, Johnnie is placing himself at risk for diabetes.

In a true conditional proposition, the major premise provides a condition upon which the consequent depends for its truth. For example, the truth of whether Johnnie is placing himself at risk for diabetes depends upon whether or not he chooses to eat cake every day.

The minor premise is the second proposition in the syllogism. It will do one of four possible things. It will either
**
affirm the antecedent**, or
**
affirm the consequent**, or it will
**
deny the antecedent**, or
**
deny the consequent**. Consider the following:

Affirming the Antecedent

If Johnnie eats cake every day, then he is placing himself at risk for diabetes.

Johnnie eats cake every day. (Affirming the Antecedent)

Therefore, Johnnie is placing himself at risk for diabetes.

Affirming the Consequent

If Johnnie eats cake every day, then he is placing himself at risk for diabetes.

Johnnie is placing himself at risk for diabetes. (Affirming the Consequent)

Therefore, Johnnie eats cake every day.

Denying the Antecedent

If Johnnie eats cake every day, then he is placing himself at risk for diabetes.

Johnnie does not eat cake every day. (Denying the Antecedent)

Therefore, Johnnie is not placing himself at risk for diabetes.

Denying the Consequent

If Johnnie eats cake every day, then he is placing himself at risk for diabetes.

Johnnie is not placing himself at risk for diabetes. (Denying the Consequent)

Therefore, Johnnie is not eating cake every day.

Two of the above syllogisms are valid, while two are invalid. Given that the major premise is true, which ones are valid and which ones are invalid? The syllogism that affirms the antecedent is obviously valid. But is it valid to affirm the consequent and conclude by affirming the antecedent? For example, given that
**
If Johnnie eats cake every day, then he is placing himself at risk for diabetes**, can we conclude that Johnnie eats cake every day because he is placing himself at risk for diabetes? The answer is, no. If Johnnie is placing himself at risk for diabetes, he might be eating cake, but he might not. He might hate cake, but love soda pop instead. Consider the next example, denying the antecedent. Can we conclude that Johnnie is not placing himself at risk for diabetes because he is not eating cake every day? The answer, once again, is no. He might be drinking ten cans of soda pop instead, which places him at risk for diabetes. But can we conclude that Johnnie is not eating cake every day because he is not placing himself at risk for diabetes? Indeed, we can. If he is not placing himself at risk for diabetes, then it follows that he is not eating cake.

Hence,
**
affirming the antecedent
**
(AA) and
**
denying the consequent
**
(DC) are valid forms of reasoning. However,
**
affirming the consequent
**
(AC) and
**
denying the antecedent
**
(DA) are invalid forms of reasoning.

An easy way to remember this is the following.

Affirming the Consequent = AC = Anti-Christ (invalid)

Denying the Antecedent = DA = Dumb Animal = (invalid)

Denying the Consequent = DC = Washington DC = (valid: a nice city to visit).

Affirming the Antecedent = AA = Alcoholics Anonymous = (valid: a good twelve step program)

Try the following exercises:

The Conditional Syllogism Exercises

Answers to Exercises

Next Page:
Chapter 08: An Introduction to Universals

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