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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

■■\

.

'

■

.

■

■

.

•

■ '

■

■

I

.

'