Mr. D McManaman)
Logic/A Brief Introduction
The Categorical Syllogism
The following is a very brief introduction to intentional logic. Out
of all that is treated in logic (both intentional and symbolic), I have
decided to focus simply on the categorical syllogism.
Terms:
Logic: the study of how to reason well. Reasoning, recall, is the
third act of the intellect. Validity: Valid thinking is thinking in
conformity with the rules. If the premises are true and the
reasoning is valid, then the conclusion will be necessarily true.
Non-sequitur: (it does not follow). This means that the proposed
conclusion cannot be deduced with certitude from the given
premises. The categorical proposition: A complete sentence,
with one subject and one predicate, that is either true or false.
The Subject: that about which something is said.
All giraffes are animals. (giraffes = subject)
The Predicate: that which is said about something.
All giraffes are animals. (animals = predicate)
The copula: connects together or separates the S and the P.
All giraffes are animals. (is/is not)
Standard Propositional Codes.
These codes come from the Latin words "Affirmo" and "Nego".
Affirmo: I affirm. Note the A and the I Nego: I deny. Note the E
and the O
Hence, we have A statements, E statements, I statements, and O
statements. A - universal affirmative: All S is P
E - universal negative: No S is P.
I - particular affirmative: Some S is P
The parts of a categorical syllogism:
O - particular negative: Some S is
a. The two premises.
• All A is B (first premise)
• Some B is C (second premise)
• Therefore, Some C is A
b. The Conclusion.
In the above syllogism, Therefore, Some C is A
• The major term: this term is always the P (predicate) of the
conclusion. In the example directly above, A is the major
term.
• The minor term: this term is always the S (subject) of the
conclusion. In the example directly above, C is the minor
term.
• The middle term: this term is never in the conclusion but
appears twice in the premises. (The function of the middle
term is to connect together or keep apart the S and P in the
conclusion).
Distribution: This is a very important term in logic. A distributed
term covers 100% of the things referred to by the term. An
undistributed term covers less than 100% of the things referred to
by the term (few, many, almost all). For instance, All men are
mortal. In this statement, "men" is distributed; for it covers 100%
of the things referred by the term "men". In Some men are Italian,
"men" is undistributed; for the term covers less than 100% of the
things referred to by the term "men".
• Universal Affirmative statements (A statements): the subject is
distributed, the predicate is undistributed.
• Universal Negative statements (E statements): both the subject
and the predicate are distributed.
• Particular Affirmative statements (I statements): neither subject
nor predicate is distributed (both are undistributed).
• Particular Negative statements (O statements): the predicate
alone is distributed.
Note the following (bold and underline = distributed):
A = All S is P
E = No S is P
I = Some S is P
O = Some S is not P
Question: Why is the predicate (P) distributed in the E and O
statements?
E = No dogs are reptiles. We do assert something about those
individuals (dogs) designated by the subject term. We also know
that each and every dog is not a reptile.
O = Some men are not Italian. We are saying something about all
things which are Italian (P). Of all the things which are Italian,
those men mentioned in our statement are excluded from all those
designated by Italian. In O statements, the particular individuals
designated by the subject term are said to be excluded from those
individuals designated by the predicate term. Those particular
individuals have to be excluded from all those individuals
designated by P.
Rules of Syllogistic (categorical) reasoning.
The middle term must be distributed at least once. Any term
which is distributed in the conclusion must also be distributed in
the premises. From two negative premises, no conclusion can be
drawn. If a premise is particular, the conclusion must be
particular. If a premise is negative, the conclusion must be
negative.
The Conditional (Hypothetical) Syllogism
There is another type of syllogism besides the categorical. This is
the conditional syllogism, and it has the form
If p, then q p q
• If federal public housing will be undertaken, taxes will rise.
• Federal public housing will be undertaken.
• Therefore, taxes will rise.
The major premise is a conditional proposition. The "if" clause is
called the antecedent, while the concluding clause introduced by
"then" is called the consequent. Now in a true conditional
proposition, the major premise provides a condition upon which
the consequent depends for its truth. The minor premise, in
affirming the antecedent, states that this condition is fulfilled. The
conclusion then affirms the consequent.
But it is possible to deny the consequent, and then conclude by
denying the antecedent. Hence:
• If India is ready for nuclear war, it has a stockpile of nuclear
weapons.
• India does not have a stockpile of nuclear weapons.
• Therefore, India is not ready for nuclear war.
In a valid conditional syllogism, the minor premise either affirms
the antecedent (AA) or denies the consequent (DC). But in an
invalid conditional syllogism, the minor premise denies the
antecedent (DA) and proceeds to conclude by denying the
consequent clause, or it affirms the consequent (AC) and proceeds
to affirm the antecedent. Hence,
• If India is ready for nuclear war, it has a stockpile of nuclear
weapons.
• India is not ready for nuclear war.
• Therefore, India does not have a stockpile of nuclear
weapons.
This is evidently false. India could very well have a stockpile of
nuclear weapons and at the same time be unprepared for nuclear
war for other reasons.
• If India is ready for nuclear war, it has a stockpile of nuclear
weapons.
• India has a stockpile of nuclear weapons.
• Therefore, India is ready for nuclear war.
This too is evidently false. As was said above, India may not be
prepared for nuclear war and yet have a stockpile of nuclear
weapons.
To sum up, it is an invalid procedure to affirm consequent, and it
is an invalid procedure to deny antecedent.
An easy way to remember this is the following.
• Affirming the Consequent = AC = Acne = no one wants acne.
• Denying the Antecedent = DA = Dumb A** = no one wants to
be a dumb a**.
• Denying the Consequent = DC = Washington DC = a nice city
to visit.
• Affirming the Antecedent = AA = Alcoholics Anonymous = a
good program.
Some Exercises
Indicate a) whether the following arguments are valid or invalid by
checking the appropriate square and b) if the syllogism is invalid
indicate why it is invalid (which rule of logic was violated?)
1. All C is P. J is P. Ergo, J is C.
Valid [ ] Invalid [ ] Reason for Invalidity:
2. Some McGivnians are philosophers. Sean Weetom is a
philosopher. Ergo, Sean Weetom is a McGivnian.
Valid [ ] Invalid [ ] Reason:
3. No man is perfect Some men are presidents. Ergo, some
presidents are not perfect.
Valid [ ] Invalid [ ] Reason:
4. All matter obeys wave equations. All waves obey wave
equations. Ergo, all matter is waves.
Valid [ ] Invalid [ ] Reason:
5. All human action is conditioned by circumstances. All
human action involves morality. Ergo, all that involves morality
is conditioned by circumstances (moral relativism).
Valid [ ] Invalid [ ] Reason:
6. All that is good is pleasant. All eating is pleasant. Ergo, all
eating is good.
Valid [ ] Invalid [ ] Reason
7. All patriots are voters. Some citizens are not voters. Ergo,
some citizens are not patriots.
Valid [ ] Invalid [ ] Reason:
8. All potatoes have eyes. Frank's head has eyes. Ergo, Frank
is a potato head.
Valid [ ] Invalid [ ] Reason:
9. All A are B Some C are not B. Ergo, some C are not A.
Valid [ ] Invalid [ ] Reason:
10. All a priori categories are conditions for the possibility of
knowing anything. Some a posteriori imperatives of an a-cosmic
ethics are not conditions for the possibility of knowing anything.
Ergo, some a posteriori imperatives of an a-cosmic ethics are not
a priori categories.
Valid [ ] Invalid [ ] Reason:
11. No oak trees bear fruit. No maple trees bear fruit.
Therefore, no oak trees are maples.
Valid [ ] Invalid [ ] Reason:
12. All Communists favor more social legislation. Some
Congressmen favor more social legislation. Ergo, some
Congressmen are Communists.
Valid [ ] Invalid [ ] Reason:
13. All educated people have worked hard. Some students are
not educated. Ergo, some students have not worked hard.
Valid [ ] Invalid [ ] Reason:
14. Mathematicians know what mathematics is. No
philosopher is a mathematician. Ergo, no philosopher knows
what mathematics is.
Valid [ ] Invalid [ ] Reason:
15. All scientific knowledge is a work of reason. All scientific
knowledge is true. Ergo, all that is true is a work of reason.
Valid [ ] Invalid [ ] Reason:
16. All married people know about marriage problems. No
priests are married people. Ergo, no priests know about marriage
problems.
Valid [ ] Invalid [ ] Reason:
17. All C are B All T are B Ergo, all T are C
Valid [ ] Invalid [ ] Reason:
18. All chickens are born from eggs. All turkeys are born from
eggs. Ergo, all turkeys are chickens.
Valid [ ] Invalid [ ] Reason:
19. Nothing easy is worthwhile. Nothing good is easy. Ergo,
nothing good is worthwhile.
Valid [ ] Invalid [ ] Reason:
20. All contraceptives acts are for avoiding pregnancy. All use
of NFP is for avoiding pregnancy. Ergo, All use of NFP is
contraceptive.
Valid [ ] Invalid [ ] Reason:
21. All Germans despise Judaism. All of the students of the
OAC Philosophy class at McGivney are Germans. Ergo, all the
students of the OAC Philosophy class at McGivney despise
Judaism.
Valid [ ] Invalid [ ] Reason:
22. All parts of a living organism are inside the body. The fetus
is inside the (mother's) body. Therefore, the fetus is a part of the
living organism (mother's body)
Valid [ ] Invalid [ ] Reason:
The Hypothetical Syllogism
23. If one chooses vice over virtue, one will suffer in life. one
is suffering in life. Ergo, one has chosen vice over virtue.
Valid [ ] Invalid [ ] Reason:
24. If government is for the common good, it is not for the good
of only the few. Government is for the common good. Ergo,
government is not for the good of only the few.
Valid [ ] Invalid [ ] Reason:
25. If life exists on Mars, then Mars has an atmosphere. Mars
has an atmosphere. Ergo, life exists on Mars.
Valid [ ] Invalid [ ] Reason:
26. If life involves quantity, it is physical. Life does not
involve quantity. Ergo, life is not physical.
Valid [ ] Invalid [ ] Reason:
27. If atoms are ultimate particles, they are indivisible. Atoms
are not indivisible. Ergo, they are not ultimate.
Valid [ ] Invalid [ ] Reason:
28. If the premises of this argument are true, then the conclusion
of this argument is true (i.e., the argument is valid). The
conclusion of this argument is true. Ergo, the premises of this
argument are true.
Valid [ ] Invalid [ ] Reason:
29. If the premises of this argument are true, then the conclusion
of this argument is true (i.e., the argument is valid). The premises
of this argument are not true. Ergo, the conclusion of this
argument is not true.
Valid [ ] Invalid [ ] Reason:
30. If Henry has moral scruples against drinking, then Henry
never drinks. Henry never drinks. Ergo, Henry has moral
scruples against drinking.
Valid [ ] Invalid [ ] Reason:
31. If the defendant is willing to testify, then he is innocent.
The defenant is not willing to testify. Ergo, the defendant is not
innocent.
Valid [ ] Invalid [ ] Reason:
32. If you expect people to be responsible for their children, then
you have to give them the right to choose whether or not to have an
abortion. Canadian society gives them the right to choose
whether or not to have an abortion. Ergo, Canadian society
expects people to be responsible for their children.
Valid [ ] Invalid [ ] Reason: