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Solutions to Exercises in

JNDAMENTALS OF LOGIC

James D. Carney
Richard K. Scheer

The Macmillan Company, New York

INTRODUCTORY NOTE

This set of solutions to exercises in the text-
book, Fundamentals of Logic, is presented for the
convenience of instructors. It aims simply to dis-
pose of the class preparation chore of solving the
exercises presented to students in the text.

For all except the simplest exercises in Part II,
"Formal Logic," (Chapters 7 - 15), we provide solu-
tions. Limits of space in a booklet to be presented
free of charge prevent including the solution to
every exercise in Part I, "Informal Logic," and Part
III, "The Logical Structure of Science." Some of
these, of course, are so elementary they offer no
problem to instructors; but others require such
lengthy explanation that it is feasible only to give
solutions to representative exercises of their kind.

In Part I (Chapters 1-6), most of the exercises
have more than one defensible answer. Accordingly,
correct answers may be found that do not appear here.

The instructor should notice that the Roman
numerals designating groups of solutions in this
manual correspond to numerals in the textbook that
designate groups of exercises; these numerals do not
refer to section numbers in the text.

One

Two

Three

Four

Five

Six

CONTENTS

Part I: Informal Logic

Logically Appraising Arguments

Traditional Informal Fallacies

Definitions

Use of Language

Analogy

Dilemmas and Paradoxes

Page

1

2

3

10

13

15

Part II: Formal Logic

Seven Validity

Eight Statement Connectives

Nine Truth Tables

Ten Elementary Inferences

Eleven Quantification

Twelve Aristotelian Logic

Thirteen Inferences Involving Quantifiers

Fourteen Axiom Systems

Fifteen Classes

18
18
20
21
33
34
35
44
48

Part III: The Logical Structure of Science

Sixteen Science and Hypotheses 54
Seventeen Crucial Experiments and Inductive

Techniques 55

Eighteen Patterns of Scientific Explanations 56

Nineteen Some Logical Features of Science 57

CHAPTER ONE -- LOGICALLY APPRAISING ARGUMENTS

I 1. Premiss: Advertising stimulates the economy by

inducing the public to buy what they
do not essentially need.
Premiss: Advertising creates mass production,
employment, and greater physical well
being by informing people of the avail-
ability of new or improved products.
Conclusion: Advertising is something we should
have in America.

6. Hume's argument:

Premiss: It is more probable that the witnesses
to the truth of a miracle are mistaken,
than that the miracle should happen.
Premiss: Witnesses are the only evidence for

miracles.
Conclusion: We do not have conclusive evidence
for miracles.

Boswell's argument:

Premiss: It is reasonable to think that God

employs miracles in order to benefit
mankind.
Premiss: Men who attest to miracles have no

interest in deceiving us, and so on.
Premiss: Prophecies have been fulfilled.
Conclusion: We have as strong evidence for

miracles as it is possible to have
(supposing that miracles are
possible) .

9. Premiss: The invasion would not solve the

problem in Latin America but would
intensify it.

Premiss: If successful, the leaders of the in-
vasion would establish a government
which would fail or partially fail on
the social and political front. In
either case the U.S. would be blamed.

Premiss: If the invasion failed, the two

possible reactions of people to the
U.S. would be detrimental to the U.S.

Premiss: Cuba, if it is not a Soviet military

base, can be tolerated.
Premiss: The U.S. should address itself to the

economic needs of Latin America.
Conclusion: The planned invasion by Cuban

refugees (March, 1961) should be
abandoned.

II Five, six, seven, and nine are clearly correct. See
answers below (Chapter Two, III).

Ill One, three, and four are correct.

CHAPTER TWO -- TRADITIONAL INFORMAL FALLACIES

I 1. Converse accident 2. Ad populum 3. Ad
baculum 4. Ad hominem 5. Converse accident
6. Ad hominem 7. Ad ignorantiam 8. Igno-
ratio elenchi 9. Ignoratio elenchi 10. Petitio
principii 11. Ad ignorantiam 12. Ad hominem
13. No fallacy 14. Ad verecundiam or ad hominem
15. Ad hominem, Ad populum 16. Ad populum,
Genetic fallacy 17. A complex question but no
fallacy. 18. Ad hominem 19. Genetic fallacy
20. Ad hominem 21. Ad hominem 22. Petitio
principii

II 1. Hasty generalization 2. Post hoc 3. Post
hoc 4. Special pleading 5. Opposition

6. Post hoc 7. Opposition 8. Hasty general-
ization 9. Special pleading 10. Opposition
or post hoc

III 1. Division 2. Composition 3. Composition

4. Division 5. Equivocation 6. Equivocation

7. Equivocation 8. Equivocation 9. Equiv-
ocation 10. Equivocation 11. Equivocation
12. Quoting out of context

IV 1. Complex question 2. Tu quoque 3. Post
hoc 4. Post hoc 5. Ad populum 6. Equiv-

ocation or accident 7. False cause but not post
hoc. 8. No fallacy--at least no ad verecundiam
9. Satire, but, if regarded as serious, special
pleading 10. Ad populum, Post hoc, Equivocation,
Ad hominem 11. Petitio principii 12. Tu
quoque, Composition 13. Complex question
14. Ad verecundiam 15. Composition 16. Igno-
ratio elenchi 17. Special pleading 18. Igno-
ratio elenchi 19. Accident, Equivocation, Ad
populum 20. Ad verecundiam, Hasty generalization
21. Equivocation 22. Ad hominem, Ignoratio
elenchi, Post hoc, Special pleading, Ad populum
23. Ignoratio elenchi 24. Ignoratio elenchi
25. Ignoratio elenchi 26. Ignoratio elenchi

V (1) 1. Special pleading (?), Post hoc 2. Special
pleading (?) 3. Equivocation 4. Equiv-
ocation 5. 6. 7.

8. Each side gives only the reasons for, thus
special pleading. 9.

(II) 1. 2. (poor dilemma) 3.

4. Glendower commits a post hoc. 5. Ad
populum, Ad ignorantiam, Special pleading, Ad
hominem 6. Equivocation, Ad populum

7. Equivocation, Ad populum, (poor analogy)

(III) (Supposing circumstances which would make each
incorrect) 1. Petitio principii 2. Ad
baculum 3. Ad populum 4. Division

5. Equivocation 6. Ad ignorantiam

7. Ad ignorantiam 8. False cause (not post

hoc) 9. Ad hominem 10. (poor

analogy)

CHAPTER THREE -- DEFINITIONS

I A shoe is a covering for the foot which does not
reach above the ankle, which is worn for everyday

activities, and which is normally made of leather
and has a more or less stiff sole.

II 1. In Chaucer's Miller ' s Tale there is a character
whose mouth itched for a whole day ("My mouth
hath icched al this longe day").
5. The animals called dachshunds at dog shows.
10. Statements which make up the Bill of Rights.

Ill 1. "Itch": (Cl) The character showed — by facial
expressions—that he had an irritation in his
mouth. (C2) He tried in various ways to re-
lieve the irritation. (C3) He said, "My mouth
hath icched al this longe day."
5. "Dachshund": (Cl) A small dog (animal), (C2)
with short legs, (C3) long body, (C4) with long
dropping ears, and (C5) a short sleek coat.
10. "Laws": (Cl) A body of rules, (C2) formally
enacted by a governing body, (C3) and recog-
nized as binding on the citizens of the country,

IV 1. "Conservative": (l) a member of the Conserv-
ative party of a country (2) one who wishes
to keep intact and unchanged the existing
political institutions (3) one who is dis-
posed to maintain existing views and habits
(4) a preservative agent.
8. "Work": (l) someone's occupation (2) a

literary or musical composition (3) to perform
or do something related to one's occupation.
17. "Satellite": (l) a secondary planet which
revolves round a larger one (2) a country
which is subservient to another (3) a man made
object put in orbit round the earth.

V If X has these characteristics: (Cl) weekly pub-
lication, (C2) paper cover, (C3) articles by
various authors, then, in ordinary circumstances,
we call it a magazine.

If X has these characteristics: (Cl) monthly
publication, (C2) hard cover, (C3) articles by
various authors, then, in ordinary circumstances,
we call it a magazine (e.g. Horizon) .

VI "Moses" (Old Testament figure--as the word is now
used)

"Work" (as used in connection with what someone
does)

VII 1. none of these 8. d

2. none of these 9. f

3. e 10. d

4. c or, perhaps, b 11. a thus c

5. none of these 12. c or, perhaps, b

6. b (in some contexts 13. d
a, thus c) 14. d

7. d

VIII 5. "War": The conflict beginning around 1914

between the Allies and Germans is a clear-cut

example of a war. The conflict in 1962-1963

between the U.S. and France over trade and

defense is a clear-cut example of something we

would not hesitate to say is not a war between

the U.S. and France.

Should the relations between the U.S. and the

Soviet Union following the Second World War be

called a war?

Is the activity of the U.N. army in Africa

against certain forces to be called a war?

IX 1. Vagueness, sense two, does not imply that there
is not a sharp distinction between clear-cut
cases of things which are X and those which are
not X.

2. If "adequate" means "successful," then there
are many adequate ways. If "adequate" means
"the only way open so as to prevent misunder-
standings," then exact definitions, in some
circumstances, could be "inadequate." If
"adequate" means "giving a description of the
necessary and sufficient conditions," then the
first sentence is a tautology and it becomes
logically impossible to give an adequate def-
inition for most of our ordinary words.

3. "Aggression" is a class b or d word, thus there
is no exact definition to be found or there is
no common meaning.

4. Words are picked up by children without being
given verbal definitions. We also give
ostensive definitions which children and adults
properly understand (ordinarily). There is no
infinite regress.

5. A reportive definition can be spoken of as true
or false, since such a definition is a state-
ment of how a word is used (at some time by
some group of persons).

6. If such rules were set down no one would follow
them, so such an activity would be idle and
silly. If such rules were followed, certain
undesirable consequences would follow. For
example, new metaphors would be ruled out.
There would also still be the possibility of
misunderstanding since ostensive definitions
can be misunderstood.

7. If this means that things could turn up which
we would hesitate to call X or not X, then this
statement is true.

8. See answer to 3.

9. The fact that two things are called by the same
name does not insure that they share a common
set of characteristics.

10. For most ordinary words we have more or less
the same paradigm examples in mind. In some
contexts two people can use words in accord-
ance with analytic or exact definitions.

11. There are certain contexts where stipulative
definitions are appropriate. Humpty Dumpty's
use of them, as is clear, creates misunder-
standing and confusion and hence negates the
value they have in appropriate circumstances.

1. This appears to be a pseudo-dispute. That is,
each party means something different by the
word "conservative." A means someone who
opposes the aims of the working class and prob-
ably, one who champions the aims of the wealthy.
B means someone who wants to maintain the status
quo. C does not seem to be serious here, but,
perhaps, he means by "conservative" what A means,

2. Pseudo-dispute. A and B have different criteria
for their use of "progress." A means by "prog-

ress" decreasing human sin. B means techno-
logical improvements.

3. Factual dispute. What did the writers of the
First Amendment intend the phrase "freedom of
speech and the press" to cover?

4. Factual dispute, though it looks like a pseudo-
dispute. A and B mean the same by "negotiate."
B thinks negotiation will lead to an agreement
in which each side gains more than it gives up
(he calls this "true negotiation"). A does not
believe that there is anything that either side
can give up which the other side wants. This
dispute could be looked at as a definition
dispute. B, for some reason which is not clear,
thinks we ought to use the word "negotiation" to
cover "interchange which leads to an agreement
in which each side gains more than it gives up."
A, it seems, would want the word to retain its
common meaning—discussion aimed at settling
differences between two parties.

5. Pseudo-dispute. Goldwater, it appears, consid-
ers being able to dispose of one's profits in
the way one wants to a necessary characteristic
of being free as he uses the word "freedom."
The disputer does not regard this characteristic
as necessary in order to speak of someone as
"free," though he does think having some prop-
erty is a necessary characteristic in order to
speak of someone as free. It is quite possible
that there are some characteristics which both
parties include in "being free." For example,
belonging to the political party of one's choice
and bringing up one's children as one chooses.
If the disputer argued that the possession of
these characteristics is incompatible, in our
society as it is today, with one's being free

to dispose of profits as one likes, and if
Goldwater opposed this, then this would be in
all likelihood a factual dispute.

6. Word-extension dispute. A thinks that the ab-
sence of this characteristic: freedom from
military control, is sufficient to prevent our
speaking of what was done in the 19th century
in the South as "ratification." B thinks all

that is necessary was present, so that it is
proper to speak of the action in question as
"ratification. "

7. Word-extension dispute. Goldwater believes
that the characteristics, few though they may
be, found in common between federal matching
funds and clear-cut cases of blackmail and brib-
ery are sufficient to use these latter notions
to cover federal matching funds.
'8. Factual dispute. A appears to have misunder-
stood the Louisiana statute. The statute does
not cover things which "might well have led to
disturbing and alarming the public."

9. Pseudo-dispute. The lumberman and passer-by

have, in these circumstances, different criteria
for their use of the phrase "the same axe." The
passer-by counts "being of the same material"
among his criteria for saying X is the same axe.
The lumberman employs "having been used over
such-and-such a period of time" as his criteria
for saying X is the same axe.

10. Not a verbal dispute of the kind described.
Also it is significantly different from clear-
cut cases of factual disputes.

11. Could be construed as a definition dispute (of
the second kind discussed). Each has a theory
of perception which he believes is true and is
imposing on the situation in question.

12. Word extension dispute. An extraordinary case.
Commonly when we speak of "same person" such
characteristics as same appearance, same char-
acter, same body, and so on are present. In
the Jekyll and Hyde case some are present and
some are missing. Should we extend "same per-
son" to cover Jekyll and Hyde?

13. Certainly in this case the man had some of the
characteristics commonly present in those sit-
uations where we say of a person that he is
dead — but, of course, his heart started beating
again. There seems to be no basis for saying
that one party wants to extend the word "dead"
in this extraordinary case. But B seems to
want this. He also, for some reason, is anx-
ious to find an instance of a "resurrection."

8

If B does call this situation an instance of a
resurrection, and A opposes this, then this
would, it seems, be a pseudo-dispute.

14. A verbal dispute. Jesus makes the point that
only in the context of earthly life can we
speak of "being married" and "not being married."
Depending on how the Sadducees react to this
point, it could develop into a word extension
dispute.

15. Factual dispute. It is false advertising. B
is making a joke or is attempting to defend a
false position by making it appear to be a ver-
bal dispute of the pseudo type — i.e. in one
sense of "economy" it is the economy size.

16. A good case can be made for this being a word
extension dispute. The case in question has
several of the characteristics of clear-cut
cases of false advertising which come to mind.
For example, most were deceived by what the
salesmen said. The salesmen intended to de-
ceive. They said just those things which in
these circumstances would deceive. The story
goes, however, that this case came to court,
and the judge said that this was not a case of
false advertising.

17. On the whole this can be regarded as a word
extension dispute. The episode in question has
some (and thus lacks some) of the character-
istics of a "military expedition or enterprise
from the U.S."

XI Since the distinction between real and nominal
definition is not clear, we omit this distinc-
tion in the answer, though it is worthwhile raising
this issue in class. A great deal of writing would
be necessary to justify any answer we might here
provide. For one thing, further stipulations are
needed to employ the distinction in these exercises.

1. Definition by analysis. Persuasive definition.

2. Definition by analysis. Persuasive definition.

3. Definition by analysis. Stipulative definition.

4. Definition by analysis. He intends that this
be a lexical definition, thus we have a false
lexical definition.

5. Persuasive definition of "civilized man".
Definition by analysis.

6. Cites of a few characteristics which are
present in those who are correctly called
"cynics" and "hypocrites." There appears to
be no attempt at a definition.

7. Persuasive definition.

8. Definition by, example. Lexical definition.

9. Definition by analysis. Theoretical defini-
tion (or real definition).

10. Definition by analysis. Stipulative definition,

11. Definition by analysis. Theoretical definition,

12. Definition by analysis. All are intended to be
lexical definitions (or, perhaps, real defini-
tions) .

13. No definition. "Everyone who commits sin is a
slave" appears to be a factual claim.

14. The Century Dictionary definition gives a
lexical definition for "philosophy" as the
word was and is used in certain contexts.
Definition by analysis. James is expressing
dismay or some other emotion and is not giving
a definition.

Wittgenstein is not giving a definition. Per-
haps this can be regarded as a stipulative
definition.

15. Lexical definition for "philosophy" as it is
used in certain contexts. Definition by
analysis.

16. Lexical definition. Definition by analysis.

17. Stipulative definition for the symbol "«".

18. Definition by example. Stipulative definition
for "sense-data."

CHAPTER FOUR -- USE OF LANGUAGE

Uses of Language

I The answers to the elementary examples are obvious
The others involve an analysis which is prohibi-
tively long for this manual.

10

II Though it is easy enough to imagine or report

examples the analysis of the examples is involved,
so no attempt will be made to do these exercises.

Ill 1. Confuses making a promise with making a pre-
diction.

2. Confuses contingent statements with logical
truths.

3. Confuses statements which express acts of
consciousness with statements which report
physical acts.

4. Confuses expressions of intention with pre-
dictions.

5. Confuses statements of intention with statements
which assert a similarity between two things.

6. Confuses dream reports with reports of past
experiences.

7. Confuses contingent statements with logical
truths.

8. Confuses logical truths (or grammatical state-
ments) with statements which are justified by
observation.

9. Confuses expressions of wishes with predictions.

10. Confuses the ceremonial use of language with the
informative use.

11. Confuses the directive use of language with the
informative use.

12. Confuses statements in fiction with the ordinary
non-fictional informative use of language.

13. Confuses aesthetic judgments with the informa-
tive use of language.

14. Confuses statements which express how things
appear with those which express how things are.

Nonsense

I 1. 1 2. 2 or none of these 3. 2, 10, or

none of these 4. 10 5. 10 6. 10

7. 10 8. 10 9. 10 10. 1 or none of

these

11. 1

12. 6 and 9

13. none

II When we say "context-mixing" below, this does not
necessarily exclude the possibility that the item
might also involve what we are calling category-
mixing (and vice-versa). We give what seems to us

11

to come in mind most readily.

1. Category-mixing. Confuses "nobody" with a
proper name.

2. Category-mixing. Context-mixing. An infinite
series is not a finite series. Uses the notion
of "not being able" as it is used in contexts,
say, of counting to the end of a finite series.

3. Context-mixing. Uses "looks like X" as it is
used in context where "X" already has a meaning.
In this context "horse" so far has not been
given a meaning.

4. Context-mixing. It would only make sense to
speak of "stingy right hands" in the context of
a person's relationship with other persons.

5. Context-mixing. Uses notions of "ownership" as
it is used in those contexts where there is
agreement and established laws.

6. What X says is misleading and would be context
mixing if he uses "walk such-and-such number of
miles an hour" as it is used in ordinary con-
texts (such as walking down a road).

7. Perhaps heaven differs sufficiently from what
is around us so that it would be context-mixing
to speak of it as a "place."

8. Category-mixing? (Regards God as like a person
in ways in which he is not.)

9. Context-mixing. Uses "hang downward" (and so
on) as it is used in ordinary contexts. (Bats
hang downward with their feet higher than their
heads. )

10. Category-mixing. Regards colors and shapes as
more like tables and cigars than they are.

11. Category-mixing. Regards the word "good" as a
word which names something as do, for example,
proper names.

12. Category-mixing. Confuses names of real things
with names of fictitious things.

13. Category-mixing. Regards "thing" as a name
like "table," "stone," etc.

14. Open invitation to context-mixing, it seems.
Some concepts curdle into (context-mixing)
nonsense when put together.

15. Context-mixing. Uses notions which need human
contexts to have the sense they have.

12

CHAPTER FIVE -- ANALOGY

1. Explanatory analogy. It would seem that this is a
poor explanatory analogy. It gives the impression
that institutional authority is a bit more arbi-
trary, independent of popular support, and so on,
than it is generally. The analogy can thus be
criticized by calling into question the supposed
similarities.

2. Whether or not this is an argumentative or explan-
atory analogy it is poor. If explanatory, then
the supposed similar elements are not similar. If
it is an argumentative analogy, then one has dis-
similar elements.

3. Argumentative analogy. A poor argumentative
analogy. There are a great number of relevant
dissimilarities (in the context of Western democ-
racies). In fact, what are the similarities?

4. Three successive arguments by logical analogy.
All are good since what Alice says is not true and
they bring this out.

5. Argumentative analogy. Actually two analogies are
used to support the conclusion — the first state-
ment: Equal armaments on both sides will not
prevent war, since in the past it has not; and new
weapons will not prevent war, since the develop-
ment of new weapons did not prevent war in the
past. A case can be made for these both being
poor. There are several relevant differences. In
the first argument, it can be argued that more or
less equal armaments can be maintained--!. e. , arma-
ments necessary for deterrence. In the second
argument it can be maintained that the old weapons
were not able to destroy whole hemispheres.

6. Russell is giving two examples of prohibition.
Prohibition in America and the prohibition against
eating laurel leaves. Apparently the force of this
is: what is true in these cases--if a person(s) is
not prohibited from doing something, then he will
not do it or will not do it in the degree he would
do it if he were not prohibited — is true of pro-
hibiting pornography. If this is correct, this is
an argumentative analogy. What makes the laurel
leaves analogy poor is the fact that some persons

13

are interested in pornography even in those
circumstances where it is available and nothing
prevents them from indulging in it. The Pro-
hibition argument looks considerably stronger.

7. A good argument by logical analogy.

8. This is an explanatory analogy. Given what Jesus
had in mind, it is a good one.

9. A good explanatory analogy.

10. An explanatory analogy. If there exists the
possibility of either capitalism or communism gain-
ing the world, and one can make a good case for
this being true, then, obviously, this argumenta-
tive analogy is poor. The element said to be
similar would not be similar, and this element must
be similar for the argument to stand up.

11. Explanatory analogy. A good explanatory analogy,
though the candidate might be offended.

12. No analogy. Don B. is making a distinction between
official Christians and real Christians. The force
of the distinction shows that Don G. is correct if
he has official Christians in mind, whereas Don
B.'s first statement (lesser evils are not valid in
a religious society) is true if real Christians are
kept in mind.

13. A good argumentative analogy.

14. A good refutation by logical analogy of "medical
materialism." The argument of medical materialism
implies not only that religious opinions are mis-
taken but all beliefs are mistaken.

15. A poor analogical argument. The conditions which
made the few communists a danger in the other
countries (supposing this is true) were missing in
the U.S. when this was written.

16. With a little effort one can find a fairly good
argument by logical analogy in this item.

17. An implicit argumentative analogy. We would not
think highly of individuals who act in such-and-
such a way, so, similarly, we should not think
highly of nations when they behave in similar ways.
One point against this analogy is that ordinarily
an individual would know that he was acting in
this way if he did act in this way, whereas
individuals which make up countries generally feel
they are doing what is right and are ignorant that

14

they are ignoring all interests except their own.
They also often do not have the motives that
Russell's individuals have (thinking they are
morally and intellectually superior). These two
differences certainly tend to lessen our ill feel-
ings towards the actions of the people of a
country.

18. An explanatory analogy. What makes this poor is
that the statement in the analogy: "the flame is
not a distinct entity," seems to involve some kind
of equivocation. There are also grounds for argu-
ing that even if we assume that the flame is not a
"distinct entity," the elements thought to be
similar are not similar.

19. An explanatory analogy. It is seriously doubted
by many that explanations, aims, and methods in
history are the same as those in the physical
sciences, as Hume says. This makes the analogy
poor. Perhaps there was a closer connection be-
tween the two in Hume's time than there is today.

20. Analogy used to suggest hypotheses.

21. An explanatory analogy. Criticism of this analogy
would most likely be directed to Descartes' belief
that our beliefs are just a matter of opinion, as
is suggested by this analogy.

22. A good argument by logical analogy.

CHAPTER SIX -- DILEMMAS AND PARADOXES

Dilemmas

1. Take the dilemma by the horns. The second if-
then premiss is weak.

2. Much depends on what is meant by "fated." In one
sense of the word the argument is invalid--the
conclusion does not follow from the premisses.

3. This does not seem to be a dilemma ^ though with
great effort it might be reworked into a dilemma.
On the face of it the argument is an instance of
modus tollens :

If God desires to prevent helpless human beings

15

from suffering (is benevolent) and has the
power (is_ omnipotent), then no helpless human
being would suffer.
Helpless human beings do suffer.
Therefore either God does not desire to pre-
vent helpless human beings from suffering or
has not the power.
Understood in this way, the first premiss is open
to criticism. One can desire something and be able
to do it and still not do it (because of other
considerations) .

4. Slip between the horns.

5. Take the dilemma by the horns. Is it true that if
one does not know a subject, then he cannot inquire
about it?

6. In the circumstances this appears to be a realistic
d i 1 emma .

7. (Perhaps) take it by the horns. There is a problem
about what is meant by "pushing too far" and
"carried to its fullness." On an interpretation
which does not make the 'if --then1 premisses
tautologies,.it would seem that a good case can be
made for doubting both.

8. Either we have the will to use nuclear weapons
or we do not.

If we do not, then there is no deterrent.
If we do, then "having the will" implies "being
willing to exercise it," and thus destroying
what we are trying to save by having the
deterrent.
This seems realistic (supposing that this formu-
lation is correct).

9. One can, it seems, question whether the only way to
stop Caesar's "potential tyranny" is by murdering
him. Depending on how the dilemma is explicity
formulated, one would slip between the horns or
take it by the horns.

10. Either we should or should not pay taxes.

If we should pay taxes, then we should support

or go along with a tyrant and with unjust

government.

If we should not pay taxes to Caesar, then we

should disobey the existing laws and ruling

body.

16

The dilemma appears to be realistic (considering
the circumstances). Jesus' reply stresses, it
seems, that for the Christian everything is "due
to God." What would a Christian conscience
demand here?

11. Realistic dilemma.

12. Hume is confronting certain theologians with this
dilemma. Hume himself does support all aspects
of it, but is starting from what he believes
certain theologians hold. From these assumptions,
he generates what seems to be a realistic dilemma.

Paradoxes

As was mentioned in the text, there is no commonly
accepted solution or resolution of any of these para-
doxes. The instructor may find suggested solutions
for some in

G. Ryle; Dilemmas.

Whitehead and Russell; Principia Mathematica.

W. V. Quine, "Paradox," Scientific American,
April, 1962.
and in the various journals, especially Mind.

For a discussion of 5 see Martin Gardner, "Mathe-
matical Games," Scientific American, March, 1963.

17

CHAPTER SEVEN — VALIDITY

I 1. (C • Ci) 3>Ci? valid

2. Not deductive

3. ((E v I) '*o 1)3^ Ej invalid

4. ((PDE) • ^E) "3^P; valid

5. Not deductive

6. ((M v^M)"vM)DTj invalid

7. ro(ruR • H) ID Rj invalid

8. ((N Z)B) • (BDS)) ID (~S 3^N); valid

9. (((C v B)D^T)-^C)D^T; invalid

10. (((P"vR)D'vB)-B)D(Rv^P)i invalid

II 1. C (supposing the father was not adopted)

2. LT (supposing the father was not adopted)

3. CT (if true)

4. LT

5. CT (if true)

6. CT (if true)

7. CT

8. LT

9. CT
10. LT

III In normal circumstances, allowing for the usual

meanings of the words, none of these is necessarily
contradictory.

CHAPTER EIGHT -- STATEMENT CONNECTIVES

1.

TF

2.

N

3.

TF

4.

TF

5.

GC

6.

N

7.

N

8.

LT

9.

TF

(counterfactual)

(count erf actual)

10. LT

11.

N

12.

N

13.

N

14.

N

15.

GC,

16.

N

17.

N

18.

TC

19.

N

20.

TF

(counterfactual)
LT

(counterfactual)

18

1.

b

2.

a

3.

b

4.

b

II 1. b 5. b 8. b

6. b 9. b

7. a 10. b

III 1. NP

2. NP

3. S 73T

4. M • F

5. H DB

6. H DB

7. (generalized conditional) NP

8. B vrwH

9. L • ^B

10. NP

11. r^F • J

12. S w>S

13. YDI

14. Ja • Ji

15. B-i • B2 or just B (ambiguous)

IV 1. F

2. F

3. O(BvJ) or TD((B v J)tv(B-J)); T

4. F

5. F

6. (T • BOf\j (J v H); T

7. T

8. F

9. T ■ rvj (B v J); T

10. F

11. ((T • J) • B) 3(H • D); T

12. T

13. T

14. (J-H).(TD (Ja • Ha)); F

V Assuming, where no parentheses exist, that '-"^' is

the weakest connective.

1. <\J, •

2. 3, ro

3. • (2nd), r\j (1st), * , wand ID.

4. • (1st) , 73 and'vj (2nd), ^ and • .

5. ID = fXJ .

6. ID(lst), <\j(3rd), 3 (2nd), cu (1st, 3rd),
= , v and * , <v .

19

7. • , r\j (1st), D(lst), rvi (2nd), v and Z) , rvi

8. • (2nd), rv> (1st), s , ' , v, ru .

9. v (1st), v (2nd), *>J (1st), • (1st), • , v, <v .
10. = , «v» (2nd), • (2nd), * , nj

VI

1.
2.
3.

4.

F

T
F
T

5.
6.
7.

F
T
F

8. F

9. F
10. F

CHAPTER NINE -- TRUTH TABLES

I 2, 3, 4, 6, 7, 12, 13, 15, 16, 17 are equivalent,

II

1.
2.
3.
4.
5.
6.
7.

T
N
N
N
C
T
N

8.
9.

10.
11.
12.
13.
14.

T
T
C
T
N
N
T

15.
16.
17.
18.
19.
20.

N
T
T
C
C
C

III

1.

2.

3.

4.

5.

6.

7.

8.

9.
10.
11.
12.
13.
14.
15.

16.
17.

rv»(G • S) • S)~3<v>G Valid

(R v N)"\JN) DR Valid

(AvP)-ruA) Z)rvjp Invalid

(BZDT)-rvjB) -J)r\JT Invalid

<vi (C-^S)-r>jS) "3 C Invalid

(L v B)-r\jB) DL Valid

(C ZJfvP) ' (WPDM))D(CDM)
(T v l)-rv(T • I)) Z) (T 3<v»l)
P 3((S v E)-ojS)) DE
(D v S)-<\JS) "3D
(HZDWJ-ojhJID^W
(HDC) • (W T3H)-rvJC)3'V)W

Valid

Valid

Invalid

Valid

Invalid

Valid

LD(M v (P • S)))-oj (M v (P • S)) IDrviL Valid

(ND(D v G)) • (ojN- <VJD)) Z) G
(rvJFi Z)rv»Fr) • (rvJT "3^Fi)) ~3((T

(f>j(M v E) Z)oJA) Z>((<v>E • A) 73 M)
Invalid

Invalid
v Fr) TDFi)
Valid
Valid

20

18. Invalid

19. Valid

20. Valid

IV 1, 2, 3, 5, 6, 7, 9, 11, 12, 14, 15, 16, 18, 20
are valid.

CHAPTER TEN -- ELEMENTARY INFERENCES

I A,

B,

1.

3 (

[1,2) HS

2.

4 (

,1,2) Conj

5 (

[3,4) CD

3.

2 (

[1) Add

4.

3 1

[1,2) HS

5.

3 1

[1,2) DS

6.

3 (

[2) DM

4

[3) DN

5 (

[4) DN

6

[1,5) MP

7.

5

[1,2) Conj

6

[3,5) CD

7

[4,6) DS

8

[7) Add

1.

1

G D(SDU)

2

G

3

S DU

4

ojU ~3rv»S

5

rviU

6

OJS

2.

1

NDM

2

M73D

3

MIDP

4

OJP

5

M v N

6

r\jp 730JM

7

OJM

8

N73D

9

N

10

D

8.

9.

10.

2 (

1) Add

3 (

2) Add

4 (

3) Add

5 (

'4) Add

3 (

1,2) MP

4 (

,3) Simp

5 (

,4) Simp

6 (

[5) Add

2 (

,1) Simp

4 (

,2,3) DS

6 (

[4,5) DS

p
p

(1,2) MP
(3) Trans
P

(4.5) MP
P

P
P
P
P
(3) Trans

(4.6) MP
(1,2) HS

(5.7) DS
(8,9) MP

21

3. 1 BDJ P

2 HDD P

3 oj(r\Jj vnJD) DU P

4 rOU P

5 fv>U ZD^v)^(^ J vrviD) (3) Trans

6 ojoj(ojj yfvJD) (4,5) MP

7 r\Jj vrviD (6) DN

8 J D^D (7) Imp

9 B D<^D (1,8) HS

10 WD D^->H (2) Trans

11 B D^H (9,10) HS

12 ojb vfVJH (11) Imp

4. 1 P P

2 (P v R) D D P

3 P v R (1) Add

4 D (2,3) MP

5 P • D (1,4) Conj

5. 1 oj ((A • A) v D) 3 Z P

2 ^Z P

3 »>J Z Z)oj D P

4 ^JZ 3ojoj((A -A) v D) (1) Trans

5 oJoj((A -A) v D) (2,4) MP

6 (A • A) v D (5) DN

7 oJD (2,3) MP

8 A • A (6,7) DS

9 A (8) Taut

6. 1 EDF P

2 (FDD) • (FDC) P

3 ojD vf>JC P

4 FDD (2) Simp

5 rviDD^F (4) Trans

6 FDC (2) Simp

7 ojc D^F (6) Trans

8 (^DD^F) • (oJCD^F) (5,7) Conj

9 ojf vOJF (3,8) CD

10 r\J (F • F) (9) DM

11 <v»F (10) Taut

12 ojf D^E (1) Trans

13 oje (11,12) MP

7. 1 TD(C-O) P

2 T • B P

3 (F • F) vf\->(<v>W . b) P

4 W D)w(C v D) P

5 T (2) Simp

22

II A,

1.

2.

3.

4.

5.

B.

6

C • 0

(1,

5) MP

7

C

(6]

Simp

8

B

(2)

Simp

9

(rv»W • B) 73 (F • F)

(3;

Imp

10

(oJW B) DF

(9)

Taut

11

rv>oj(C V D) 3^ W

(4)

Trans

12

(C v D) DrvtW

(11) DN

13

(C v D)

(7)

Add

14

r\JW

(12

>,13) MP

15

f\JW • B

(8,

14) Conj

16

F

(ic

),15) MP

2

;i) DN 6.

3

(1,

2) Conj

3

\2) DM

4

(3)

Imp

4

[3) DM

5

(4)

DN

2

[1) Assoc

6

(5)

Dist

3

[2) Comm 7.

2

(1)

Equiv

4

|3) Assoc

3

(2)

Trans (twice)

3 (

'2) Trans

4

(3)

DN

4 (

3) DM

5

(4)

DM

5 (

4) Imp

6

(5)

Imp

6 (

1,5) MP

7

(6)

Trans

4 (

2) Exp

8

h)

DM

5 (

3) Comm 8.

2

(1)

Equiv

6 (

5) Comm

3

(2)

DM

7 (

4,6) MP

4

(3)

Comm

8 (

7) DN

5

(4)

Imp

9 (

8) Imp

6

(5)

Comm

10 (

1,9) MP

7

(6)

Dist

2 (

1) Add

3 (

2) DM

4 (

3) Equiv

1

M 73 (oj R 73 U )

P

2

M • OJR

P

3

M

(2)

Simp

4

f>JR73U

(1,

3) MP

5

OJR

(2)

Simp

6

U

(4,

5) MP

1

W =F

P

2

OJ (W v D)

P

3

(W73F) • (F73W)

(1)

Equiv

4

W73F

(3)

Simp

5

oJW • oJ D

(2)

DM

23

6 fVD

7 ojd vojF

3. 1 CD(KDW)

2 K- rvJ W

3 ^(KDW) ID^C

4 roru(K« rJW)

5 nj (r\j K voJojw)

6 ^ (r>J K v W)

7 fvJ(KDW)

8 <^C

4. 1 GD(TDU)

2 G- OJU

3 G

4 TDU

5 oJUZ^fviT

6 oju

7 OJT

5. 1 G v (L • T)

2 G 3^T

3 T

4 r\jr\JT 730JG

5 T I>vJG

6 ^rvJG v (L • T)

7 ^GD(L-T)

8 T73 (L • T)

9 L • T
10 L

6. 1 (FDM) - (E DF)

2 E

3 E 73F

4 F

5 F73M

6 M

7. 1 P v D

2 S

3 SDC

4 C

8. 1 PDD

2 DZ)U

3 O I

4 PDU

5 PD I

9. 1 A
2 N

(5) Simp

(6) Add
P

P

(1) Trans

(2) DN

(4) DM

(5) DN

(6) Imp
(3,7) MP
P

P

(2) Simp
(1,3) MP
(4) Trans
(2) Simp

(5.6) MP
P

P

P

(2) Trans

(4) DN

(1) DN

(6) Imp

(5.7) HS

(3.8) MP
(9) Simp
P

P

(1) Simp

(2,3) MP

(1) Simp

(4,5) MP

P (unnecessary)

P

P

(2.3) MP
P

P
P
(1,2) HS

(3.4) HS
P

24

10.

11

12.

13.

14.

3

(A • N • F)3^ S

P

4

(Fr DF) • Fr

P

5

Fr IDF

(4) Simp

6

Fr

(4) Simp

7

F

(5,6) MP

8

A • N • F

(1,2,7) Conj
(twice)

9

r^S

(3,8) MP

1

PZ)C

P

2

C 3)f\J (F • A)

P

3

F

P

4

P Z)rv» (F • A)

(1,2) HS

5

rwrvj(F • A) Z)rv» P

(4) Trans

6

(F • A) Z)<^> P

(5) DN

7

F 3(A ID^P)

(6) Exp

8

A D^P

(3,7) MP

9

rvJrxj p 730JA

(8) Trans

10

P ID^A

(9) DN

1

CD(HvD)

P

2

roH

P

3

C ~3(oj^H v D)

(2) DN

4

CD(^HDD)

(3) Imp

5

(C- roH) Z)D

(4) Exp

6

^H 3(CZDD)

(5) Exp

7

CIDD

(2,6) MP

1

((R • D) v W) • ru((R- D)

• W) P

2

W

P

3

ojrv) w

(2) DN

4

^((R • D) • W)

(1) Simp

5

oJ (R • D) vOJW

(4) DM

6

^(R • D)

(3,5) DS

1

DDT

P

2

rvJDTDH

P

3

H~~JF

P

4

(DZ>T) • (HDF)

(1,3) Conj

5

^^J D v H

(2) Imp

6

D v H

(5) DN

7

T v F

(4,6) CD

8

r\Jr\JT v F

(7) DN

9

CKJJ "3F

(8) Imp

1

(D • oj(A v B))Z)C

P

2

rv>(C V A)

P

3

D • (C vfvJB)

P

4

D

(3) Simp

25

Ill 1.

2.

3.

4.

5 D Z> (OJ (/

l v B) DC)

(1) Exp

6 fv»(A v B)

oc

(4,5) MP

7 <XJC IDojoj (A v B)

(6) Trans

8 ^C3(A

v B)

(7) DN

9 ^C 73(^j^>A v B)

(8) DN

10 roc 3(oJ

ADB)

(9) Imp

11 (oj C • ro /

0 DB

(10) Exp

12 ^v>C • ^A

(2) DM

13 B

.

(11,12) MP

15.

1 (D • (M v

F)) v (D • (G \

r A)) P

2 D • ((M v

F) v (G v A))

(1) Dist

3 D

(2) Simp

1

A73B

P

2

A

Hyp. RCP

3

B

(1,2) MP

4

A • B

(2,3) Conj

5

AD(A'B)

(2,4) RCP

1

LDF

P

2

LD(FDB)

P

3

FD (BDS)

P

4

L

Hyp. RCP

5

F

(1,4) MP

6

FDB

(2,4) MP

7

B

(5,6) MP

8

BDS

(3,5) MP

9

S

(7,8) MP

10

LIDS

(4,9) RCP

1

<\JF D(RDL)

P

2

F v H

P

3

R

P

4

ojp

Hyp. RCP

5

RDL

(1,4) MP

6

L

(3,5) MP

7

<^F~3 L

(4,6) RCP

1

AD((B-C) v

E)

P

2

(B • C) ~Jr\JA

P

3

D ID>r\JE

P

4

A

Hyp. RCP

5

(B • C) v E

(1,4) MP

6

rUoJA ZDrvi (B •

■c)

(2) Trans

7

A Z)<\J(B • C)

(6) DN

8

r>J (B • C)

(4,7) MP

9

E

(5,8) DS

26

5.

6.

IV 1.

2.

10

'VrNJEID'VD

(3) Trans

11

E~3^D

(10) DN

12

rv»D

(9,11) MP

13

AD^D

(4,12) RCP

1

(PvQ)DR

P

2

(S v T) Z) ((A v B)Z)P)

P

3

S

Hyp. RCP

4

A

Hyp. RCP

5

S v T

(3) Add

6

(A v B)DP

(2,5) MP

7

A v B

(4) Add

8

P

(6,7) MP

9

P v Q

(8) Add

10

R

(1,9) MP

11

ADR

(4,10) RCP

12

S D(ADR)

(3,11) RCP

1

(SZDW) • (EDF)

P

2

S • E

Hyp. RCP

3

SDW

(1) Simp

4

S

(2) Simp

5

w

(3,4) MP

6

EDF

(1) Simp

7

E

(2) Simp

8

F

(6,7) MP

9

W • F

(5,8) Conj

10

(S-E)Z) (W -F)

(2,9) RCP

1

B v<vC

P

2

C

P

3

«v»b

Hyp. RAA

4

rv»C

(1,3) DS

5

C • 'viC

(2,4) Conj

6

B

(3,5) RAA

1

BDA

P

2

rvi(A • rv»C) 73 B

P

3

rvJA

Hyp. RAA

4

rvJA Z)r\J B

(l) Trans

5

fVJB

(3,4) MP

6

(rviA v^uajc) Z) B

(2) DM

7

(rvA v C) 3 B

(6) DN

8

rvJA v C

(3) Add

9

B

(7,8) MP

10

B- fVl B

(5,9) Conj

11

A

(3,10) RAA

27

3.

4.

5.

1

^B73E

2

D 73^E

3

r\j (r\j D • f\J B)

4

ajB

5

E

6

fviruE Z) (V D

7

EZD^j d

8

ruD

9

r\jrv» D v^JfVB

10

D v B

11

B

12

B • rvB

13

B

1

B73W

2

WZDrxjp

3

ojb v P

4

°J (B3^ W)

5

OJ (r\j B v<v»W)

6

B • W

7

B

8

B~3^P

9

OJP

10

AjrviB

11

P

12

p • c\jp

13

BD^W

1

F v W

2

FIDO

3

W73K

4

^(D v K)

5

^D • ^K

6

rvJD

7

rvJK

8

rvJK73<^ W

9

rvjW

10

fNjoj F v W

11

^F~3 W

12

•^W 3r\joj p

13

rvjWJ3F

14

^DU ruF

15

F

16

oj f

17

F • (NJF

18

D v K

(twice)
DS

Conj
RAA

P

P

P

Hyp. RAA

(1,4) MP

(2) Trans
(6) DN

(5.7) MP

(3) DM

(9) DN
(8,10)
(4,11)
(4,12)
P

P
P
Hyp. RAA

(4) Imp

(5) DM

(6) Simp
(1,2) HS

(7.8) MP

(7) DN

(3.10) DS

(9.11) Conj

(4.12) RAA
P

P
P
Hyp. RAA

(4) DM

(5) Simp
(5) Simp
(3) Trans
(7,8) MP

(1) DN

(10) Imp

(11) DN

(12) DN

(2) Trans

(9.13) MP

(6.14) MP
(15,16) Conj
(4,17) RAA

28

V

VI

1.

Inconsistent: P - F,

Q - T

2.

Consistent

3.

Inconsistent: A - F,

B - T

4.

Consistent

5.

Consistent

6.

Consistent

7.

Inconsistent: P - F,

Q - T,

R - F

8.

Consistent

1.

1

w(PvQ) v R

P

2

P • S

P

3

(PvQ)DR

(1) Imp

4

P

(2) Simp

5

P v Q

(4) Add

6

R

(3,5) MP

7

P • R

(4,6) Conj

2.

1

(SDQ)DR

P

2

(P-S)DQ

P

3

PD(SDQ)

(2) Exp

4

P IDR

(1,3) HS

3.

1
2
3

S73P

P "3rvj (v • N)

oj VID^ P

4

ru p • N

5

P

Invalid

4.

1

r\J (P v M) v (S • R)

P

2

rxJS

P

3

(PvM)D(S- R)

(1) Imp

4

rvj (S • R) D^J (P v

M)

(3) Trans

5

(fNJS v^JR) 73 ^ (P

v M)

(4) DM

6

rvJS v^R

(2) Add

7

rv> (P v M)

(5,6) MP

8

r\) p . r\j m

(7) DM

9

ixJM

(8) Simp

5.

Inconsistent Premisses

6.

1

(A v B) 3 (C • D)

P

2

(D v E)DF

P

3

A

P

4

A v B

(3) Add

5

C • D

(1,3) MP

6

D

(5) Simp

7

D v E

(6) Add

8

F

(2,7) MP

29

7. 1

(S v W) 73 (E

1 • T)

P

2

(T v H) DB

P

3

S

P

4

S v W

(3) Add

5

B • T

(1,4) MP

6

T

(5) Simp

7

T v H

(6) Add

8

B

(2,7) MP

8. 1

PDR

P

2

(^PvR)D(SDQ)

P

3

(P-P)DR

(1) Taut

4

P D(PDR)

(3) Exp

5

(PDR) D (S

DQ)

(2) Imp

6

P D(SDQ)

(4,5) HS

9. Invalid Argumen

t

10. 1

(Q • (R v 5))D^JP

P

2

P

P

3

S

P

4

PD^vi (Q • (R

v S))

(1) Trans,

5

r\J (Q • (R v S

))

(2,4) MP

6

~ ((Q • R) v

(Q-s))

(5) Dist

7

f>J (Q • R) • *\J

(Q-S)

(6) DM

8

^(Q-S)

(7) Simp

9

^Q v<^JS

(8) DM

10

SD'xjQ

(9) Imp

11

<VJQ

(3,10) MP

11. 1

'v AID'v B

P

2

ADC

P

3

B v D

P

4

DDE

P

5

BDA

(l) Trans,
(twice)

6

BD C

(2,5) HS

7

rv»nj B V D

(3) DN

8

^BDD

(7) Imp

9

^BDE

(4,8) HS

10

^EDB

(9) Trans,

11

^EDC

(6,10) HS

12

njojE v c

(11) Imp

13

E v C

DN (13)

L2. 1

^A v^B

P

2

(ADC) • ((A

•C)DB)

P

3

ADC

(2) Simp

4

A D^ B

(1) Imp

DN

DN

DN

30

5

(A-C)DB

(2) Simp

6

B ZD'vM

(4) Trans,

7

(A • C) 3>oJ A

(5,7) HS

13.

1

PD(Qv (R • S))

P

2

^R v\JS

P

3

rvJQ

P

4

f>J (R • S)

(2) DM

5

^ (Q v (R • S))73^ P

(1) Trans

6

(r\JQ • nj (R . S))ZD°-» P

(5) DM

7

'viQ • 'vi (R • S)

(3,4) Conj

8

r\J p

(6,7) MP

14.

Invalid argument

15.

Invalid argument

16.

1

(PDQ) ■ (RDS)

P

2

HJ(QSR)

P

3

(rvjp73R) v (Q= R)

P

4

'NJP 73R

(2,3) DS

5

r\Jr\j p v R

(4) Imp

6

P v Q

(5) DN

7

Q v S

(1,6) CD

17.

Invalid

18.

1

C = J

P

2

r^J

P

3

(CDJ) • (JDC)

(1) Equiv

4

C 3J

(3) Simp

5

rvJJ73^C

(4) Trans

6

r\JC

(2,5) MP

7

wC v (E v\JK)

(6) Add

8

CD(E vojK)

(7) Imp

19.

Invalid argument

20.

Inconsistent premisses

21.

Invalid

22.

Not truth - functional

23.

Inconsistent premisses

24.

1

(JvR)D(D- V)

P

2

J

Hyp. RCP

3

J v R

(2) Add

4

D • V

(1,3) MP

5

D

(4) Simp

6

J3D

(2,5) RCP

7

^J v D

(6) Imp

25.

1

U Z)(V v W)

P

2

(W-X)DY

P

3

^Z D (X- *vJY)

P

DN

31

4 U Hyp. RCP

5 V v W (1,4) MP

6 O (XDY) (2) Exp

7 ^(X-^Y)DZ (6) Trans, DN

8 ^X vvj'vtYlDZ (7) DM

9 ^XvYDZ (8) DN

10 (OY)DZ (9) Imp

11 OZ (6,10) HS

12 'VJ'VJ V v W (5) DN

13 ^VDW (12) Imp

14 'VJVIDZ (11,13) HS

15 UD K>VZ>Z) (4,14) RCP

16 U 3(<^^V v Z) (15) Imp

17 V Z>(V v Z) (16) DN

26. 1 HD(L-R) P

2 (LvW)DP P

3 W v H p

4 'vi'vjW v H (3) DN

5 ^OH (4) Imp

6 ^WD (L- R) (1,5) HS

7 ^roW v (L • R) (6) Imp

8 'vj (<v>ojnj Woj(L • R)) (7) DM, DN (twice)

9 ^(ojWto(l • R)) (8) DN

10 oj (oj W • (r\J L V\J R ) ) (9) DM

11 ^v» ((rvJW • rviL) v (ojW"\jR)) (10) Dist

12 *>J (oj W • rv»L) • r\J (rxj W • oj R ) ( 1 1 ) DM

13 oj(ojw.ojl) (12) Simp

14 W v L (13) DM, DN (twice)

15 P (2,14) MP

27. Invalid

28. Invalid

29. Not truth - functional

30. 1 (R • F) v D P

2 ojd v F P

3 ^F Hyp. RAA

4 DDF (2) Imp

5 ^FD^ D (4) Trans

6 ^D (3,5) MP

7 rvj (oj (r. p) • ojd) (l) DM, DN (twice)

8 ^((f^R v^F)'^D) (7) DM

9 'V ((f\JD» ojR) v (f\JD»oJF)) (8) Dist

10 ^> (f>JD • <\JR) • rvj (r\j D • f\J F ) (9) DM

11 *\J (ojd* ^v»F) (10) Simp

12 D v F (11) DM, DN (twice)

32

13 rwF v D

14 'XJFDD

15 D

16 D • ojD

17 F

(12) DN

(13) Imp

(3.14) MP

(6.15) Conj

(3.16) RAA

CHAPTER ELEVEN -- QUANTIFICATION

II

III

1.

2.

3.

4.

5.

6.

7.

8.

9.
10.
11.

1.
2.
3.

4.
5.

6.
7.
8.
9.

10.

1.
2.
3.
4.
5.
6.
7.

(]x)(Px-Nx) 12.

( 3 x)(Bx"\J Sx) 13.

( 3 x)(Tx- rsj Fx) 14.

(3 x)(Bx • Mx) 15.

(3x)(Px-Sx) 16.

( 3 x)(Bx"\J Yx) 17.
( 3 x)(Px • Fx)

( 3x)(Sx • Fx) 18.

(3x)(Px-Sx) 19.

( 3x)(Bx • Wx) 20.
( 3x)(Vx • Ax)

(x)(PxDNx) 11.

(x)(Px3Nx) 12.

<v» (x)(PxDw Nx) 13.

or ( 3*)(Px * Nx) 14.

(x)(NxZ)Px) 15.

rvj (x)(PxDNx) 16.
or ( 3*)(Px * Nx)

(x)(Px3ru Nx) 17.
( 3x)(Fx • Ox)

(x)(CxDOx) 18.

(x)(BxDFx) 19.

(x)(DxDr\JBx) 20.

3 x) (Lx • f\J Ax)
3x)(Fx • Lx)
3x)(Ex • Sx)
3x)(Ex ' Fx)
3x)(Bx ' Fx)
3x)(Hx * Sx) or/and
3x)(Hx- r\J Sx)
3x)(Fx- r\j Hx)
3x)(Sx • Dx)
3 x) (Px • <\j Ax) or/and
3x)(Px • Ax)

x)(WxZ) Vx)

3x)(Hx • ^jTx)
x)(HxD^J Tx)

3x)(BxLO Lx)

x)(Px ZDckJ Lx)

3x)(Px*fvRx) or/and
3x)(Px * Rx)
x)(GxD Lx)
Lx = nothing can save x,

x)(GxD Lx)
x)(LxDGx)
x)(GxD Lx)

(x)(PxD( 3y)(Cy • Exy))
( 3x)(Dx ' (y)(Py_OFxy))
( 3x)(Ax ' ( 3y)(Py • Bxy))
( 3x)(Sx ' ( 3y)(Ty * Exy))
(x)(PxID(y)(CyZ)Hxy))
(x)(Px ZD (y)(Ly ZD Loves xy))
(x)(LxD (y)(Py 3wyx))

33

IV

8,
9.

10,

11.

12.
13.
14.
15.
16.

17.
18.
19.
20.

1.
3.
5.

(x)(CxD(

(x)(Cx~3(

(3x)(Nx •

Oy = y is

Ox)(Px •

(x)(y)(Nx '

(x)(RxZ)(

(x)(CxD(

(x)(Px3(

(x)(Mx "3(

(x)(Px3(

(3x)(Bx-

(3x)(Sx-

(x)(NxD(

y)(Py 3^Cyx))

3y)(Py -rvjVyx))

(y)(OyZ)Lxy))

a number other than 0,

(y)(Cy3rvjVxy))

' Ny Z) (Lxy Z)njExy))

y)(FyIDLxy))

3y)(Hy • Wxy))

y)(Ty 3^Lxy))

3y)(Wy • Fyx)
3y)(Syojjxy))

(3y)(Py.rvj Iyx))
(y)(Ey3)Wxy))

3y)(Ny Gyx))

y free; x bound 2. x,y free; x,y bound

y free; x,y,z bound 4. x,y,z bound

x free; x,y bound 6. x free; x,y,z bound

CHAPTER TWELVE — ARISTOTELIAN LOGIC

1.
2.
3.
4.
5.
6.

Not a

Not a

Valid

Rl

Valid

R2

syllogism
syllogism

7. R2

8. R3,4

9. R6

10. R5,6

11. R6

II 1. Premisses 1 and 3 yield "All B are D" which,
with premiss 2, yields "No B are M".

2. Premisses 1 and 3 yield "No CT are W" which,
with premiss 2, yields "No CT are C".

3. Premisses 1 and 4 yield "All S are C". Premisses
3 and 5 yield "All SH are T". "All SH are T"
and 2 yield "All SH are S", which, with "All S
are C" yield "All SH are C", i.e. "Shakespeare
was clever".

4. Premisses 1 and 3 yield "All T are R", which,
with premiss 2, yields "No T are H", i.e. "No
Hedgehogs take in the Times".

34

5. Premisses 1 and 4 yield "All L are RO", which
with 2, yields "No L are S", which, with 5,
yields "No HA are S", which, with 3, yields
"No EE are S", i.e. "These Sorites are not
easy examples.

Ill 1. All Athenians are men.

2. All rare people should be honored.

3. All human beings make mistakes.

4. You are English.

5. Whatever the critics say is best is best.

6. Some major nations are in Europe.

7. Whenever Russia increases hers, we need to
increase ours.

8.

CHAPTER THIRTEEN -- INFERENCES INVOLVING QUANTIFIERS

I A. 1

2.

3.

1

(x)(CxD Yx)

P

2

(3x)(Ox •

Cx)

P

3

Oa • Ca

(2)

El

4

Ca3 Ya

(1)

UG

5

Oa

(3)

Simp

6

Ca

(3)

Simp

7

Ya

(4,

6) MP

8

Oa • Ya

(5,

7) Conj

9

Ox) (Ox •

Yx)

(8)

EG

1

(x)(CxD^J Ux)

P

2

Ox) (Ax '

Cx)

P

3

Aa • Ca

(2)

El

4

Ca 3r\j Ua

(1)

Ul

5

Aa

(3)

Simp

6

Ca

(3)

Simp

7

rv>Ua

(4,

6) MP

8

Aa • ru Ua

(5,

7) Conj

9

(}x)(Ax-

r\JUx)

(8)

EG

1

(x)(AxDBx)

P

2

(x)(Px3ru Bx)

P

3

Aa3 Ba

(1)

Ul

4

Pa Z)ro Ba

(2)

Ul

35

4.

5.

6.

8.

5

rv>Ba 7Dr\j Aa

(3) Trans

6

Pa ZD^kj Aa

(4,5) HS

7

(x)(Px ~3rv>Ax)

(6) UG

1

( 3 x)(Px • Bx)

P

2

(x)(Px DHx)

P

3

Pa • Ba

(1) El

4

Pa 3 Ha

(2) Ul

5

Pa

(3) Simp

6

Ba

(3) Simp

7

Ha

(4,5) MP

8

Ha • Ba

(6,7) Conj

9

(3 x)(Hx • Bx)

(8) EG

1

(x)(Rx 73 Px)

P

2

( Bx)(Rx • r%jDx)

P

3

Ra • oj Da

(2) El

4

Ra 73 Pa

(1) Ui

5

Ra

(3) Simp

6

^Da

(3) Simp

7

Pa

(4,5) MP

8

Pa • nj Da

(6,7) Conj

9

( 3 x)(Px • r\JDx)

(8) EG

1

(x)(WxDAx)

P

2

(x)(Ax IDajGx)

P

3

Wa 77>Aa

(1) Ul

4

Aa Z}^ Ga

(2) Ul

5

Wa ZDr^jQa

(3,4) HS

6

^rsJGaZDr^J Wa

(5) Trans

7

Ga 77>r\j Wa

(6) DN

8

(xHGxZD'vj Wx)

(7) UG

1

(xj(WxDAx)

P

2

(x)(AxD^Gx)

P

3

( ]x)Gx

P

4

Ga

(3) El

5

Wa 73 Aa

(1) Ul

6

Aa Z)r\J Ga

(2) Ul

7

Ua 77>r\JGa

(5,6) HS

8

^^JGa 7Dr\j Wa

(7) Trans

9

Ga TD^Wa

(8) DN

10

<^JWa

(4,9) MP

11

Ga • r\j Wa

(4,10) Conj

12

(3 x)(Gx • ouWx)

(11) EG

1

( 3 x ) ( Px • rsj Lx )

P

2

(x)(Px73Gx)

P

3

Pa • r\j La

(1) El

36

2.

3.

4.

5.

4

Pa 3 Ga

(2) Ul

5

Pa

(3) Simp

6

^v»La

(3) Simp

7

Ga

(4,5) MP

8

Ga • oj La

(6,7) Conj

9

(3 x)(Gx ' r\jLx)

(8) EG

1

(x)(NxDr\jGx)

P

2

(3x)(Nx • Bx)

P

3

Na • Ba

(2) El

4

Na Z3 r\j Ga

(1) Ul

5

Na

(3) Simp

6

Ba

(3) Simp

7

^JGa

(4,5) MP

8

Na * Ba • ro Ga

(5,6,7) Conj
(twice)

9

( 3 x)(Nx • Bx • ruGx)

EG (8)

1

(x)(Gx DBx)

P

2

»v>B(e)

P

3

G(e) DB(e)

(1) Ul

4

rvJB(e) 73r\jG(e)

(3) Trans

5

^G(e)

(2,4) MP

1

(x)((f\JMx* rviCx) 73 Dx)

P

2

^C(e)

P

3

^M(e)

P

4

(f\JM(e) • oj C(e)) DD(e)

(1) Ul

5

^M(e) • <v» C(e)

(2,3) Conj

6

D(e)

(4,5) MP

1

( 3*)(Sx • Ix • ojCx)

P (unnecessary)

2

(x)(SxDHx)

P

3

(3 x)(Sx • rsj Ix)

P

4

(x)(Gx Z)Ix)

P

5

Sa • ru la

(3) El

6

Sa ID Ha

(2) Ul

7

Ga Z> la

(4) Ul

8

Sa

(5) Simp

9

Ha

(6,8) MP

10

<v IaZDru Ga

(7) Trans

11

fvi la

(5) Simp

12

fvJGa

(10,11) MP

13

Ha • <^JGa

(9,12) Conj

14

(3 x)(Hx • ^Gx)

(13) EG

1

(x)(Mx 3 (Sx v Ax))

P

2

(3 x) (Mx ' ^ Ax)

P

37

6.

8.

9.

3

Ma • ru Aa

(2) El

4

Ma 73 (Sa v Aa)

(1) Ul

5

Ma

(3) Simp

6

Sa v Aa

(4,5) MP

7

'vi'viAa v Sa

(6) DN

8

^Aa 73 Sa

(7) Imp

9

<^Aa

(3) Simp

10

Sa

(8,9) MP

11

Ma • Sa

(5,10) Conj

12

( ]x)(Mx • Sx)

(11) EG

1

(x)(Bx = (Fx v Lx))

P

2

Bs

P

3

Bs = (Fs v Ls)

(1) Ul

4

(Bs 73 (Fs v Ls)) • ((Fs

v Ls) 73 Bs)
(3) Equiv

5

Bs 73 (Fs v Ls)

(4) Simp

6

Fs v Ls

(2,5) MP

1

(x)((Dx • Ix)77>rvj Cx)

P

2

(x)(Tx73Dx)

P

3

(x)K>Cx 77) Ax)

P

4

(Da • la) 73rNj Ca

(1) Ul

5

Ta 73 Da

(2) Ul

6

^Ca 73 Aa

(3) Ul

7

Da 73 (la 73r\j Ca)

(4) Exp

8

Ta 73 (la 73^ Ca)

(5,7) HS

9

(Ta • la) 73 Aa

(8) Exp

10

(x)((Tx • lx) 73 Ax)

UG (9)

1

(x)(Ax 73^v)Rx)

P

2

(x)(Ax73Gx)

P

3

(3x)Ax

P

4

Aa

(3) El

5

Aa 73<^j Ra

(1) Ul

6

Aa 73 Ga

(2) Ul

7

<-\JRa

(4,5) MP

8

Ga

(4,6) MP

9

Ga • ru Ra

(7,8) Conj

10

( ^ x)(Gx • ^\jRx)

(9) EG

1

(x)(Bx 73 (Sx • Fx))

P

2

(x)((Bx • Cx) 73 Jx)

P

3

(x)((Fx- r\jRx) 73 Cx)

P

4

( 3 x)(Bx • fviRx)

P

5

Ba • aj Ra

(4) El

6

Ba

(5) Simp

7

Ba 73 (Sa • Fa)

(1) Ul

38

II 1.

2.

3.

1
2
3
4
5
6
7
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2

(6,7) MP

(2) Ul

(3) Ul
(10) Exp
(8) Simp
(11,12) MP
(5) Simp
(13,14) MP
(6,12,15) Conj
(twice)
(16) EG

follows :

8 Sa • Fa

9 (Ba • Ca) 3 Ja

10 (Fa • rvjRa) 73 Ca

11 FaD(^RaDCa)

12 Fa

13 ^RaDCa

14 ^Ra

15 Ca

16 Ba • Fa • Ca

17 ( 3 x)(Bx • Fx • Cx)
10. The syllogisms are as

i. (i), (5) .-. (n)

2. (8), (11) ;. (12)

3. (4), (12) /. (13)

4. (6), (13) .\ (14)

5. (10), (14) .\ (15)

6. (2), (15) /. (16)

7. (9), (16) /. (17)

8. (7), (17) ;. (18)

9. (3), (18) /. (19)

(x) (y) ( ] z) ( (Px • Py • Lxy) 73 Nxyz)

Ps • Pa • Lsa

(y) ( 3 z) ( (Ps ' Py " Lsy) 73 Nsyz

( 3 z)(Ps • Pa • Lsa) 73 Nsaz

(Ps • Pa • Lsa) 77) Nsab

Nsab

( 3 z)Nsaz

(x)(y)((FxDGx) 73 (Ay 73 By) )

(3x)^\jFx

(3y)Ay

'vJFa
Ab

(y)((Fa73Ga) 73 (Ay 73 By))
(Fa73Ga) 73 (Ab 73 Bb)
((Fa73Ga) • Ab) 73 Bb
rv Fa v Ga
Fa 73 Ga
(Fa 73 Ga) • Ab
Bb

(]x)Bx

(x)(y)((Fx • By) 73 Exy)
(x)(y) ((Fx • '■vJFry) 73^ Exy)

P
P

(1)

(3)
(4)

Ul
Ul
El

(2,5) MP
(6) EG

P

P

P

(2)

(3)

(1)
(6)
(7)
(4)
(9)

El

El

Ul

Ul

Exp

Add

Imp

(5.10) Conj

(8.11) MP
(12) EG

P
P

39

4.

5.

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

1

2

3

4

5

6

7

8

(3x)Fx

(3x)Bx

Fa

Bb

(y)((Fa • By) 3 Eay)

(Fa • Bb) 3 Eab

Fa 3 (Bb3Eab)

(y) ((Fa • ^viFry) 3^ Eay)

(Fa • *JFrb) 3^ Eab

Fa 3 (ojFrb 3^ Eab)

Bb 3 Eab

Eab

^JFrb Z)f\j Eab

Eab3Frb

Frb

(B*)(Frx)

(x)(y)(Bx3(Gy3rxjFxy))
(3x)(3y)(rxj(Bx-Gy)-3Hxy)
(x)(y)(Hxy3Ix)
(x)(y)Fxy

(3y)(oj(Ba * Gy) 3 Hay)
^ (Ba ' Gb) 3 Hab
(y) (Hay 3 la)
Hab 3 la

(y)(Ba 3 (Gy 3r\J Fay) )
Ba 3 (Gb 3^ Fab)
(y)Fay
Fab

(Ba • Gb)3^ Fab
^^Fab 3^vJ (Ba '
Fab3<"\J (Ba ' Gb)
rvJ (Ba ' Gb)
Hab
la

(3x)Ix

(y)(x)((Gy ' Dx) I
(x)(y)((Gy -Dx) 3

(3x)(3y)(Dx • gy *Cxy)

( jy) (Da • Gy * Cay)

Da • Gb • Cab

(x)((Gb • Dx) 3 (Fxb3Yb))

(Gb • Da) 3 (Fab 3Yb)

Gb 3 (Da • (Fab 3Yb))

Gb)

(Fxy3Yy))
(Cxy 3 Fxy) )

p

p

(3)

El

(4)

El

(1)

Ul

(7)

Ul

(8)

Exp

(2)

Ul

(10) Ul

(11) Exp

(5,«

?) MP

(6,

13) MP

(5,

12) MP

(15

) Trans,

DN

(twice)

(14,16) MP

(17

) EG

P

P

P

P

(2)

El

(5)

El

(3)

Ul

(7)

Ul

(1)

Ul

(9)

Ul

(4)

Ul

(11)

Ul

(10)

Exp

(13)

Trans

(14)

DN

(12,15) MP

(6,16) MP

(8,17) MP

(18)

EG

P

P

P

(3)

El

(4)

El

(1)

Ul

(6)

Ul

(7)

Exp

40

6.

7.

9

(y

)((Da -Gy) 73 (Cay D Fay))

(2) Ul

10

(Da • Gb) 73 (Cab 3 Fab)

(9) Ul

11

Da

(5) Simp

12

Gb

(5) Simp

13

Da • Gb

(11,12) Conj

14

Cab73 Fab

(10,13) MP

15

Cab

(5) Simp

16

Fab

(14,15) MP

17

Da • (Fab73Yb)

(8,12) MP

18

Fab73 Yb

(17) Simp

19

Yb

(16,18) MP

20

Gb • Yb

(12,19) Conj

21

(]

x)(Gx • Yx)

(20) EG

1

C-

x)Rx

P

2

(x)(Rx3(3y)(By • Vxy))

P

3

(x)(Rx 73(y)(Ly~3'>J Vxy))

P

4

Ra

(1) El

5

Ra 3(3y)(By • Vay)

(2) Ul

6

(By) (By -Vay)

(4,5) MP

7

Bb • Vab

(6) El

8

Ra 73 (y)(Ly 73^ Vay)

(3) Ul

9

(y)(Ly 73^Vay)

(4,8) MP

10

Lb 73 'vj Vab

(9) Ul

11

Vab

(7) Simp

12

'vJ'vj Vab ZD'vJ Lb

(10) Trans

13

Vab 73^ Lb

(12) DN

14

^JLb

(11,13) MP

15

Bb

(7) Simp

16

Bb • ^ Lb

(14,15) Conj

17

(^ x)(Bx • 'vjLx)

(16) EG

1

(x)((Px • Cx) 73 Sx)

P (unnecessary)

2

( 3 x)((Px • Cx) • (y)(Vy~3'v>Tyx

)) P

3

Hx)Vx

P

4

(x)(y)((Px • Vy)

73 (^TxyD^J Wx))

P

5

Va

(3) El

6

(Pb • Cb) • (y)(VyZ)^J Tyb)

(2) El

7

(y)(((Pb • Vb) 73^ Tby^'V Wb)

(4) Ul

8

(Pb • Va) 73 (^Tba 73^ Wb)

(7) Ul

9

( ( Pb • Va ) • oj Tba ) 73^ Wb

(8) Exp

10

(Pb • Cb) 73 Sb

(1) Ul
(unnecessary)

11

(y)(Vy73^Tyb)

(6) Simp

12

Va 73^ Tab

(11) Ul

41

13

rviTab

14

(x)(y)(<>JTxy 73rv>Tyx)

15

(y ) (r\J Tay 73^ Tya )

16

^Tab ZD^JTba

17

^JTba

18

Va • <-\j Tba

19

Pb '

20

Pb • Va • ro Tba

21

rvlWb

22

Cb

23

(Pb • Cb) • ^Wb

24

(3 x)((Px • Cx) • ^JWx)

1

(3 x)(y)(rvJDx 73 (rxJGy • Dxy))

2

(y) (ru Da 73 (oj Gy • Day) )

3

'VJ Da 73 (rv) Gb • Dab)

4

Gb

5

rvJ (rvJGb • Dab)Z)fv)'vi Da

6

^ (rv»Gb • Dab) 3 Da

7

('vi'viGb v^ Dab) 73 Da

8

(Gb VVJ Dab) 73 Da

9

Gb WJ Dab

10

Da

11

Gb73Da

12

(y)(Gy73Da)

13

(3x)(y)(Gy73Dx)

(5,12) MP
P

(14) Ul

(15) Ul
(13,16) MP
(5,17) Conj

(6) Simp, Assoc.

(18,19) Conj

(9,20) MP

(6) Simp, Assoc,

(19,21,22)

Conj (twice)

(23) EG
8. 1 1 x)(y)K>Dx 73 (^oGy • Dxy) P

(1) El

(2) Ul
Hyp. RCP

(3) Trans

(5) DN

(6) DM

(7) DN

(4) Add

(8.9) MP

(4.10) RCP

(11) UG

(12) EG

III 1. In a universe of one individual
Sa T
Na T
.*. Sa • f\JNa F

2. Valid

3. In a universe of one individual

Aa 73 Ba
Ca 73 Aa
.'. Ca • Ba Take Ca, Ba, and Aa as false

4. In a universe of one individual

Ca 73 (Pa v Ta)
Ha 73 Pa
Ca * Ha
.'. Ca • Ta Take Ta as false and the

rest true.

42

5. In a universe of two individuals
(Ba • ojCa) v (Bb • r\JCb)
(Ca • Da) v (Cb ; Db)
(Ba • oj Da) v (Bb • roDb)
Take Ba, Bb, Da, Cb, Db as true — the rest false.

IV 1. 1 (x)(BxDHex) P

2 Bh P

3 BhIDHeh (l) Ul

4 Heh (2,3) MP

2. 1 H v E P

2 G 3rv) H P

3 ^E Hyp. RCP

4 ^vJ^JE v H (1) DN

5 ^EDH 4 Imp

6 H (3,5) MP

7 ^^HD^G (2) Trans

8 HZ}r\JG (7) DN

9 ^G (6,8) MP
10 rvJED^G (3,9) RCP

3. 1 (x)(FxZ)Hx) P

2 (3 x)(Nx • Fx) P

3 Na • Fa (2) El

4 FalDHa (l) Ul

5 Fa (3) Simp

6 Ha (4,5) MP

7 Na (3) Simp

8 Na • Ha (6,7) Conj

9 (^ x)(Nx • Hx) (8) EG

4. Invalid: "( (T ID (N Z) C) ) • (oJ N ID E) • oj E) 7Jr\J T"
is not a tautology

5. 1 (x)(Fx Z> (Hx vPx)) P

2 Fh P

3 Fh ID(Hh v Ph) (1) Ul

4 Hh v Ph (2,3) MP

5 ^J^JHh v Ph (4) DN

6 ^HhZ>Ph (5) Imp

6. Invalid in a universe of two individuals.

(Ba ZD Ta) • (Bb ID Tb)
(Ba • ^JCa) v (Bb ; ojCb)
/. (Ta ~Or\jCa) . (Tb ZDro Cb)
Take Ba, Ta, Tb, Ca as true and the others false,

7. 1 PID^T P
2 H v P P

43

3 H 73N P

4 ojn P

5 fviND^H (3) Trans

6 f\JH (4,5) MP

7 pJ^jh v P (2) DN

8 ^JHDP (V) Imp

9 P (6,8) MP
10 ^JT (1,9) MP

8. Invalid in a universe with one individual

Fa 73 Na

Ba 73 (Fa vojNa)
Fa

,\ oj Na Take Fa, Na as true.

9. 1 (xM'vjFx 3 (*v> Wx • oj Bx)) P

2 (x)(^TxD^Fx) P

3 oj Fa ID (oj Wa • oj Ba ) (l) Ul

4 ojja ZD^Fa (2) Ul

5 ^TaD(^Wa'OJBa) (3,4) HS

6 fv (oj Wa • oj Ba) ZD^J^J Ta (5) Trans

7 oj (oj Wa • oj Ba ) 73 Ta (6) DN

8 (Wa v Ba) 73 Ta (7) DM

9 (x)((Wx v Bx) 73 Tx) (8) UG
10. 1 (x)(Tx73Vx) P

2 ( 3x)(Rx • (y)(Vy 73^Lxy)) P

3 Ra • (y)(Vy 73^ Lay) (2) El

4 (y)(Vy 73rvJLay) (3) Simp

5 Vb 73^ Lab (4) Ul

6 Tb 73 Vb U) Ul

7 Ra (3) Simp

8 Tb 73ojLab (5,6) HS

9 (y)(Ty73ojLay) (8) UG

10 Ra • (y)(Ty 73oj Lay) (7,9)Conj

11 (3x)(Rx • (y)(Ty 73^Lxy)) (10) EG

CHAPTER FOURTEEN — AXIOM SYSTEMS

I 1. Replacing 'p' by 'ojp', we get 'oj p 73 (q v'vJ'vjp) '
or "ojojp v (q v'vojp) • or 'p v (q v p)'. The
value of 'p73(q v^p) ' or 'oj p v (q Wp)' is
the same as that of 'p v (q v p) ' .

44

II 3. For 'v', use the table

V

0

1

2

3

0

0

0

0

0

1

0

1

2

3

2

0

2

2

0

3

0

3

0

3

III 1. pD(pDq)

rv> p v (fVp v q); not a tautology

2. p =^v> p

(pI3r\j p) . (ojp3p)

(oj p v*Vp) • (p v p) ; not a tautology

3. (p Z) q) * f\J q ZD1^ p
(ro p v q) • <\J q ZD'vJ p
^((rUp v q) • ru q) vvip

fv>ru (rv) (nj p v q) v^J^q) v<>Jp

((p'^Jq) v q) v^p

((q v p) • (q V>Jq)) WJp

Kip v (qvp))'(^p v (q WJq))

("\J p v q v p) • (fvJ p v q v^Jq); a tautology

4. (pDq) • (qDr) 3(pDr)

(f\J p v q ) • (*\J q v r)D (f\J p v r )

f\J ( (ro p v q) • (f\J q v r ) ) v K p v r)

(r\j (oj p v q ) v^J K q v r ) ) v (rv)p v r )

((p»*vi'q) v (q"\->r)) v (^p v r)

((p*rvjq) v q) • ((p"Vq) v\Jr) v (fv> p v r)

((q v p) • (q v\Jq)) • ( (^ r v p)

(r\j p v r v q v p ) • (f\J p v r
(rvj p v r v^r v p) • (<"\-i p

(*\j r v'Vq) )
v (^o p v r )
v q v^q) •
v r v^r v^Jq) ;
a tautology
p • q — r^> (ru p v<>J q )

( (p • q) ZW K> p V\Jq)) • (rvj (fv» p vviq) Z) (p • q))
(ro'NJ (f^p v<^->q ) v*\J ("VJ p v^q ) ) •

(rvJOJ (r\j p v^vjq) v (p • q) )
( (<\J p WJq) VVJ (<^> p vvq) ) • (('vip v\jq) v (p • q) )
((rv»p y\Jq) v (p«q))«((r\jpvr\jq) v (p • q))
(^\J p v^q v p) • ("VJ p v<^q v q) • (<\J p w>q v p) •
(<v» p vf\jq v q); a tautology

IV Because the transformations used in obtaining the
CNF from 'P' are themselves equivalences. The CNF

45

of 'P' is equivalent to 'P' and is therefore deriv-
able if 'P' is.

V If each conjunct does contain an individual

variable and its negation then it is of the form
'(...v p v.-.w-ip v...)' which is clearly a tau-
tology. Thus each conjunct of a derivable CNF must
contain a variable and its negation.

VI 1. Take 'e' as '1' if 'o' is multiplication and
'x-1' as the inverse of 'x'. Take 'e' as '0'
if 'o' is addition and 'x-!' as '-x'.

2. Take 'e' as '1' and 'x-1' as the inverse of 'x1.

3. Take the variables to be integers; for example,
'x-1' as '-x' and 'e' as '0'.

4. No

VII T6 (pvq)D((rv q) v p)

Proof 1 q 3 r v q A2 p/r

2 (qDr v q) A4 r/r v q

D(P v qDp v (r v q))

3 p v q Dp v (r v q) (1,2) Rl

4 pv(rvq)Z)(rvq)vp A3 q/r v q

5 pvqT3(rvq)vp (3,4) DT2

T7 rv(pvq)~3rv(pv(qvs))

Proof 1 qDq vp A2, A3 DT2

2 qDq v s (l) p/s

3 (q Z)q v s) A4 r/q v s

Z) (p v q Z)p v (q v s))

4 pvq3pv(qvs) (2,3) Rl

5 (pvqDp v (q vs)) A4 p/r, q/p v q,

Z)(r v (p v q) r/p v (q v s)

3r v (p v (q v s)))

6 r v (p v q) (4,5) Rl

73 r v (p v (q v s) )

T8 (pvq)v(qvr)IDpv(qvr)
Proof 1 (pvq)v(qvr) T6 p/p v q,
Z)(p v (q v r)) v (p v q) q/q v r, r/p
2 (p v (q v r)) v (p v q) T7 r/p v (q v r),
D ( p v ( q v r ) ) s/r

v (p v (q v r))

46

3 (p v (q v r)) Al p/p v (q v r)
v (p v (q v r))

Dp v (q v r)

4 (p v q) v (q v r) (1,2) DT2

Z)(p v (q v r))
v (p v (q v r ) )

5 (p v q) v (q v r) (3,4) DT2
lb p v (q v r)

T9 (pvq)vrZ)pv(qvr)
Proof Use T3, T8, DT 2

T10 _

Til

T12 PD(q3p'q)
Proof 1 (wp v^Jq)

vrNJ (r\J p yro q )

T4 p/^p v^q

2 (rvj p v«>Jq ) v p • q ( 1 ) R3

3 (f\J p vviq) v p • q T9 pA-Jp, q/^q,
TJ-Ajp v («\J q v p • q ) r/p • q

4 Mp v (^q v p'q) ( 2 , 3 ) R 1

5 pD (qDp • q) (4) R3 (twice)

T13 (p v q) v r — p v (q v r)
Proof Use T9, T10, DT 12

T14-T17

T18 (pDq)D(^qD^p)

Proof 1 q IDrvj'vi q T15 p/q

2 (qZ3^^q) A4 r/fvcvq,

Z) (fvi p v q ID'VJ p v'V'^q) p/vi p

3 ^p v q 73 oj p v^^q (1)(2) Rl

4 ^p vf\Jf\J q ( 3 )

TJD'vi'Aj q v^vJp

5 'Vp v q ID'NJ'v q v^p (3) (4) DT2

6 (pDq)D(^qD^p) (5) R3 (twice)

T19-T27

T28 p qDp

Proof 1 'VpZD'xjq v\Jp A2 q/^vp, pA-^q

47

2 fVq vvpZD'V) p v'viq

3 «v p Z3<^ p vx^q

4 <v (rv p v<\Jq)ZD<\J'\J p

5 <viaj pDp

6 r\J (*\J p v«v>q) ZD p

7 p • qD p

A3 p/\>q, q/\Jp

(1) (3) DT2

(3) DT18
T16

(4) (5) DT2
(6) R3

CHAPTER FIFTEEN -- CLASSES

I (a) 2, 3, 5, 7, 8, 9, 10 (b) 1 (c) 4, 6

II (a) 1, 4

III 1. 5 2. yes 3. yes; no 4. no; yes

5. no 6. yes

IV (a) 1. \s\ 2. {6,7} 3. { 1,2,3 }

4. null 5. j 1,3.4,5, {3,4,5}}-

)l,2,3,6,7,{l,2,3}} = {4,5, {3,4,5}}
6. {1,2,3,4,5} 7. {6,7 {1,2,3} ,1,2}

8. null 9. {l,2,3,4,5,{{ 6,7, {l,2,3}f}}

10. null
(b) 2, 6, 8 are true.

V 1.

2.

48

3.

4.

4 and 5 are valid
7. 2. There are P's.
6. There are S ' s.

VI

A

(1)

(?)

A U A = A

x€ A U A =

£ A v x€ A

But rvj (x € A ) .

(x6!Avx€ A ) • ^ (x €
And, since both

x€A"3x6:Avx€ A
A D

A ) D x£ A

x 6! A v x
we have
x E A v x
AU A =\/
x€ A U A
And since

e
e

A ^

x€ A

x e a

= x e a v\> (x e a)

x € V and

49

x £ A v\J (x € A)

are both logical truths,

x£Awi(x€A)=x€ V

(14) A = A

x£ As<\j^ (x£ A)

33 r^ (x € A)

(16) AvB^A-^A =?* A vB^A

Since there is an element x such that
x €: A v x £ B, it follows that
either x£ A or x £ B.

(21) A O (A v B) = A

x € A O (A v B) 3= xeA.(xeAvx£B)

as (x € A • x € A) v (x £ A . x e B)
Now

x€A.(x€Avx€B) 3 x € A
and

x£ A ID (x € A • x € A) v (x £ A • x € B)
so
x £ A H (AU B) sss x £ A.

(25) A - A = A

xeA-A== xO"Vi(xeA)
But x € A and x 6 A • r\j (x € A)
are logical contradictions, so
x £ A - A = xeA

(30) (A-B)UB=AvB

x€(A-B)UB== (x £ A • ^ (x€B))vx£B

= (x 6 A v x € B) . (oj (x 6 B) v x £ B)
And, since (*\j (x € B) v x € B) is a
logical truth
x€(A-B)vB = (x € A v x £ B)

50

(11)

VI The proofs are condensed.
(9) A U A = A

1 A U (A H A) = _
(A U A) fl (A U A)
AU A = (A U A) O (AU A)

AUA=(AUA)PiV
A = A U A

U V = V

A u (a n v ) =

(A U A) n (AU V )

a u (a nv ) =

V njA u v )

AUA = \/n(AUV)
A U A = AU V

V = auv
(12) a n A = A

(5)

2
3
4
I
1

3
4
5

(15

(16)

2
3
4

5
I

1
2
3
4
5

1

2
3

4
5

(17)

1
2

A n (a u v ) =

( a n a) u ( A n v )
A n (a u v ) = ( A n a)uA
A n V = ( A n a) uA
A = A n a

A _=

a = b:

AHA
)B = A

A =

A =

A =

B =

B_
B
B
A

= A

A = B 73B
AUB^A DA ^AvB^A
A U B ^A

A = A

A u b 5* A
b * A

AUB^/1 DA =

ADB^A
AUB^A DA ?± A v B ?* A

ahb^Ada^A
a n b ^ A
A = A

#A

3 A n b

4 A 5* A

5 A 5* A

6 a n b^ A

A

(8)

(7)

(1), (2)

(5)

(7)

(2)

(2)

(7)

(6)

(2)

(11)

(1),

(2)

(4)

Hyp.

line

2

(14)

line

3

(1),

(4)

Hyp.

Hyp.

line

1,2

(1)

line

1,2,4

Def.

Hyp.

Hyp.

(for reductio

ad al

Dsurdum proof)

line

1,2

(12)

contradiction

line

2,4

line

1,5

51

d9) An(Bnc) = (a n b) n c

i xe (a n (BO O)

2

3

(24) (AH B) = A U B
1 A U B = In B

x€ A • (x€ B • x€ C)
(x€ A • x€ B) • x€ C

x€ (a n b) n c

(23)

Def.
Taut

2 AUB=AOB

(26)

3
4

1
2

A Q B =

a n b =

A - (A_Q

a n

AW B
A U B
B) = A -

B

(29)

(31)

4
5
6

1
2
3
4
5

1
2
3
4
5

n b

B

(a n b) = a n (A U B)
a n (A U B) =

(a n a) u (a n b)

(AH A) U (A Pi B) =

A u (a n b)

A U (AH B) = A
A O B = A - B
A - (A n B) = A -
(A - B)_ - A_= A_

X.a n b) o_a = A_n (a n b_}_
A_n (aobi= (a n a) n b
(AnA|nB= A hb
A n b = A

(A - B) - A = A
(A U B) - B_= A_- B

(aub)h b_= b n (a y b)

(b n a) u (b n Bj_ = (BO A) U A

£§n a) u A_ = bo a

BOA=AnB=A-B
(AUB) -B=A-B

(14), (15)

(23)
(14)

(24)
(6)

(8)

(1)
Def.

(4)
(19)
(8)
(12)

(4)
(6)
(8)
(1)
(4), Def,

B 1.

a u_(b n c n d) = (a u b) n (a u c) n (A U D)

1 A_U (Bfl CQ D) = (5)
(AUB)n(AU(Cn D))

2 (AUB)H(AU(Cn Di) = (5)
(A U B) f|(AU c)_n (A U D)

2. (A n B) n (AH C) U C U (DPI E) u

(a n (bu o) = v

1 (Left side 2.) = C U (DO E) (18)

u ((a n b) n (An c)
u (a n (bu c))

52

4

5

rs i = cu (d n E) u ((a n B) (6)

n (AH C)) U ((AH B)

U(APlC))

RS 2 = C U (D H E) U (A H B) (23)

U (AO_C) U ((AH B)U(AH C))
RS 3 = C U (DO E) UV (7)

RS_4 = V
3. (AHB)n ((BQ C) U D) = API C

1 LS = ((AQ B) O (BO C)) (6)

U ((A U B) Pl_D)

2 RSU (A_n (B n B)n C) (18)

u ((a n b) n d)

3 RS 2 = (AH C) U ((AHB) n D) (8),(l)

4 RS 3 = (AH C) U (A O B) (5)

n (AH C) U D)

5 RS 4 = ((AH C]_U A) (5)

n ((ah c) u b) n ((ah c) y d)

6 RS 5 = (AH C) H ((AH C) U B) (5), (9)

n ((ah c) u d)

7 rs 6 = (a n c) n ((An C) U D) (21)

8 RS 7 = An C (21)

((An b) n (a u_d)) u ((An b) n (a u d)) = An

1 LS = (((An B) n (AU D)) U (An B)) (5)

n (((a n b) n (a u d)) u (a u d)) _

2 rsi _(_(.An b) u (An b)) n ((a n b) (s)

3

4
5

U (A U D)) n (((A U D)U(AHB))
n ((A UD)U(AU D)))_

rs 2= (a n b) n ((An B) U (A U D))

n ((a u d) y (a n b)) n v
rs 3 = (a n_B) n ( (a n B) U (A U D) )
rs 4 = a n B

5. (A U(BQ C)) U

(p n e) u

= v

c

u (a u b) n (A u C)

1 LS = ((A UB) n (A U C))_

u (a u b) n_(A u c) u c u (d n E)

2 RS 1 = V UCU(DHE)

3 RS 2 = V

(9), (7)
(2), (21)

(5)
(7)

53

PART III

CHAPTER SIXTEEN -- SCIENCE AND HYPOTHESES

2. Galileo's first hypothesis is introduced in

connection with this phenomenon observed on the
7th:

wo ( ) o o

o

and the hypothesis is: All three bodies near
Jupiter are fixed. On the 8th he observed this
phenomenon :

o

« o

By assuming that Jupiter moved east he was able to
retain his hypothesis; however, even with this
assumption the hypothesis does not explain why the
bodies were "nearer one another than before." On
the 10th he observed :

O

O

Now to retain the first hypothesis would be to go
against known astronomical regularities, for if
the hypothesis were true, then Jupiter would have
to move east then west. The explanation now call-
ed for is that the bodies move. On the 11th he
observed :

O

o

All of these phenomena naturally suggest Galileo's
second hypothesis: ". . . there were in the
heaven three stars which revolved round Jupiter."
There are no competing hypotheses since the
second was entertained after the first was aban-
doned. Both hypotheses are empirical, and it is
apparent that further observations would verify
the second hypothesis.

54

CHAPTER SEVENTEEN — CRUCIAL EXPERIMENTS AND

INDUCTIVE TECHNIQUES

I 3. The hypothesis which competed with the phlo-
giston theory—in combustion things combine-
is not mentioned here. The phlogiston theory
is naturally suggested by the appearance of
something being released when, say, a piece of
paper burns. This hypothesis was theoretical-
for as the word "phlogiston" was used in the
theory it was conceptually impossible at the
time to observe phlogiston.

In the passage an experimental result is de-
scribed which, in part, led to the downfall of
the theory. If the phlogiston theory were true,
then it would seem that when metal is calcin-
ated, and thus loses phlogiston, the calx should
weigh less than the metal. The results of cal-
cination, however, were the opposite. This did
not result in the abandonment of the theory,
since defenders supposed that phlogiston had a
"negative weight." Perhaps if this had been the
only way to save the theory, the theory would
have been given up, but there were other ways to
make it compatible with the known P's (see,
e.c[., the reaction to Lavoisier's experiment).

Ill 1. Since he supposed that the nervous paralysis of
the hens had the same cause as the similar paral-
ysis of the prisoners (called "beri-beri")
Eijkman looked for what the hens and prisoners
had in common and discovered that they both fed
almost entirely on polished rice. Here the
method of agreement is employed. It supports
the hypothesis that the disease is caused by the
exclusive diet of polished rice (or by the lack
of something). He then noticed that other pris-
oners who had beri-beri ate polished rice while
those who ate unpolished rice did not have the
disease. Eijkman employed the method of

55

difference here. The results provided addi-
tional support for his hypothesis. Eijkman's
controlled experiments (with the hens) in
which all the conditions were the same except
that one group was fed on polished rice while
the other was fed on unpolished rice (method of
difference) gave strong support that it was the
lack of what was in the husks which caused beri-
beri in those fed exclusively on rice.

CHAPTER EIGHTEEN -- PATTERNS OF SCIENTIFIC EXPLANATIONS

I 3. Here, in brief, is the explanation:

(1) The American rich have a fear of
expropriation.

(2) Many people in America, other than the
rich, can display the traditional signs
of wealth (luxury cars, etc.). Therefore
the American rich no longer display their
wealth.

There is implicit in the premisses the notion
that fear and the lack of success in displaying
wealth generally lead the wealthy to avoid dis-
playing their wealth (given the existing con-
ditions in some countries today). Since this
probabilistic generality is part of the expli-
cans, the explicandum would not be a deductive
consequence of the explicans. The explanation
is a probabilistic one. It is not a historical
explanation (though a similar explanation in a
different context could be) nor a teleological
explanation. It deals with the reasons the
American rich have (or had) for not displaying
their wealth. It seems to be an instance of a
fairly common kind of sociological explanation
(the reasons such-and-such people have for be-
having in such-and-such a way). Galbraith
cites some of the events which are connected

56

with and, in part, produce the fear considered
in (l). If these had been explained in more
detail, we would have the ingredients of an
empathetic explanation.

II 1. Psuedo-explanation (not testable).

2. Psuedo-explanation (explicans follows from the
explicandum) .

3. Psuedo-explanation (not testable).

4. Not testable in light of the fact that we would
not know how to test it. If it is regarded as
scientific, then it is a psuedo-explanation.

5. Perhaps a non-scientific explanation. If it is
regarded as scientific, it is not testable.

6. Psuedo-explanation (not testable).

CHAPTER NINETEEN — SOME LOGICAL FEATURES OF SCIENCE

II 1. No conflict. Even if saying (long ago) that

the sun sets in the west entailed that the sun
revolves around the earth (which is doubtful),
we do not mean this today.

2. No conflict. He was not "conscious" according
to his definition of the word.

3. No conflict. In this passage what is being
called a "cyclone" is different from what we
ordinarily call a "cyclone."

4. No conflict. "Structure" is being used in such
a way that liquids having an internal archi-
tecture is compatible with the statement
"Liquids are fluid," as we would ordinarily
understand this.

Ill 4. Though the vastness of the universe seems to
produce a feeling of humility and terror in
Russell, Eddington, and Jeans, it does not
produce this feeling in everyone. To para-
phrase a remark by Ramsey, in connection with
"humility": the stars cannot think, love, or

57

lead virtuous lives, and these are the qual-
ities which impress me, not size or distance.
It is those qualities which I myself can take
credit for and which others have to a greater
degree than I which bring on feelings of
humbleness. I can take no credit for weighing
160 lbs.

Do such astronomical facts show that man is
unimportant or insignificant? If it were true
that those who believe that man is important
and not insignificant based their belief on the
assumption that the limits of the universe are
a few thousand feet or that men are to be found
all over the universe, then such facts would
certainly upset their belief. But do those who
believe that man is significant have this as
their reason? Such beliefs come up in reli-
gious and certain philosophical contexts. For
example, that man is important and not insig-
nificant follows from the Christian notion that
man is created in the image of God. And no
matter what the relative size of man to the
.universe may be, such facts are irrelevant to
such beliefs.

58

1

L

i

Carney-
Solutions to exercises in
Fundamentals of logic

101'

.C35

S6