[Irving M. Copi, Carl Cohen] Introduction to Logic(BookFi.org)
[Irving M. Copi] Symbolic Logic

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Transcript of [Irving M. Copi] Symbolic Logic
Preface
Jr., of Wichita State University, Murray Braden of Malcalester College, Lorin
Browning of the College of Charleston, Mario Bunge of McGill University,David R. Dilling of Grace College, Earl Eugene Eminhizer of YoungstownState University, Barry R. Gross of York College, City University of New York,Herbert Guerry of Idaho State University, James N. Hullett of Boston Univer
sity, R. Jennings of Simon Fraser University, Robert W. Loftin of Stetson
University, Warren Matthews of Old Dominion College, Robert W. Murungiof the University of Dubuque, Jean Porte of the Centre National de laRecherche Scientiflque, Samuel A. Richmond of Cleveland State University,Donald Scherer of Bowling Green University, Anjan Shukla of the Universityof Hawaii, Leo Simons of the University of Cincinnati, Frederick Suppe ofthe University of Illinois, Norman Swartz of Simon Fraser University, William
J. Thomas of the University of North Carolina at Charlotte, William C. Wilcoxof the University of Missouri, and Jason Xenakis of Louisiana State University.
I should like to express my thanks to my daughter Margaret Copi and to
Miss Karen Lee for help in reading proof.Most of all I am deeply grateful to my wife for her help and encourage
ment in preparing this new edition.
Contents
5 The Logic of Relations 112
5.1 Symbolizing Relations 112
5.2 Arguments Involving Relations 126
5.3 Some Attributes of Relations 1305.4 Identity and the Definite Description 1365.5 Predicate Variables and Attributes of Attributes 145
S Deductive Systems 152
6.1 Definition and Deduction 152
6.2 Euclidean Geometry 153
6.3 Formal Deductive Systems 157
6.4 Attributes of Formal Deductive Systems 159
6.5 Logistic Systems 161
7 A Propositional Calculus 165
7.1 Object Language and Metalanguage 1657.2 Primitive Symbols and Well Formed Formulas 1677.3 Axioms and Demonstrations 1787.4 Independence of the Axioms 1827.5 Development of the Calculus 1887.6 Deductive Completeness 201
B Alternative Systems and Notations 210
8.1 Alternative Systems of Logic 2108.2 The HilbertAckermann System 211
8.3 The Use of Dots as Brackets 227
8.4 A ParenthesisFree Notation 231
8.5 The Stroke and Dagger Operators 2328.6 The Nicod System 234
xii 9 A FirstOrder Function Calculus 242
9.1 The New Logistic System RS 1 242
9.2 Development of RS I 2489.3 Duality 2549.4 RS I and the ',Natural Deduction' Techniques 2599.5 Normal Forms 2639.6 Completeness of RS I 2709.7 RS I with Identity 280
Contents
Appendix A: Normal Forms and Boolean Expansions 283
Appendix B: The Algebra of Classes 290
Appendix C: The Ramified Theory of Types 300
Solutions to Selected Exercises 311
Special Symbols 337
Index 339
Sec. 1.2] The Nature of Argument
from the others, which are regarded as grOlmds for the truth of that one. In
ordinary usage the word 'argument' also has other meanings, but in logic it
has the technical sense explained. In the following chapters we will use theword 'argument' also in a derivative sense to refer to any sentence or collectionof sentences in which an argument is formulated or expressed. When we dowe will be presupposing that the context is sufficiently clear to ensure that
unique statements are made or unique propositions are asserted by the utter
ance of those sentences.
Every argument has a structure, in the analysis of which the terms 'premiss'and 'conclusion' are usually employed. The conclusion of an argument is that
proposition which is affirmed on the basis of the other propositions of the
argument, and these other propositions which are affirmed as providinggrounds or reasons for accepting the conclusion are the premisses of that
argument.We note that 'premiss' and 'conclusion' are relative tenns, in the sense that
the same proposition can be a premiss in one argument and conclusion in
another. Thus the proposition All men are mortal is premiss in the argument
Introduction: Logic and Language [Ch. 1
complicates our problem. Argwnents formulated in English or any other
natural language are often difficult to appraise because of the vague and
equivocal nature of the words in which they are expressed, the ambiguityof their construction, the misleading idioms they may contain, and their
pleasing but deceptive metaphorical style. The resolution of these difficultiesis not the central problem for the logician, however, for even when they are
resolved, the problem of deciding the validity or invalidity of the argwnentremains.
To avoid the peripheral difficulties connected with ordinary language,workers in the various sciences have developed specialized technical vocabularies. The scientist economizes the space and time required for writing his
reports and theories by adopting special symbols to express ideas which wouldotherwise require a long sequence of familiar words to formulate. This hasthe further advantage of reducing the amount of attention needed, for when
a sentence or equation grows too long its meaning is more difficult to grasp.The introduction of the exponent symbol in mathematics permits the expression of the equation
Arguments Containing Compound Statements [CII. 2
and conclusions have different truth values, by considering all possible ar
rangements of truth values for the statements substituted for the distinctstatement variables in the argument form to be tested. These can be set forthmost conveniently in the form of a truth table, with an initial or guide columnfor each distinct statement variable appearing in the argument form. Thusto prove the validity of the Disjunctive Syllogism form
See. 3.1J Formal Proof of 'aUdi"
Disjunctive Syllogism. From the second and fourth premisses, C :> (D :> E)and C, we validly infer D :> E by Modus Pooens. And finally, from theselast two conclusions (or subconclusions), B :> D and D :> E, we validly inferB :> E by a Hypothetical Syllogism. That its conclusion can be deduced fromits premisses using valid arguments exclusively proves the original argumentto be valid. Here the elementary valid argument forms Modus Pooens (M.P.),Modus Tol/ens (M.T.), Disjunctive Syllogism (D.S.), and Hypothetical Syllogism(H.S.) are used as Rules of Inference by which conclusions are validly deducedfrom premisses.
'
A more formal and more concise way of writing out this proof of validityis to list the premisses and the statements deduced from them in one column,with the latter's "justifications" written beside them. In each case the "justification" for a statement specifies the preceding statements from which, and
the Rule of Inference by which, the statement in question was deduced. It, is convenient to put the conclusion to the right of the last premiss, separated
from it by a slanting line which automatically marks all of the statements
above it to be premisses. The formal proof of validity for the given argumentcan be written as
The Method of Deduction
14. Double Negation (D.N.):
15. Transposition (Trans.):
16. Material Implication (Impl.):17. Material Equivalence (Equiv.):
Sec. 3.2)
9. (G v H)'(I v J).'. [(G v H)' 1] v [(G v H)'J]
*10. (K'L) � {M'[(N'O)'P]}.'. (K'L) � {M'[(O.N).P]}
11. {Qv [(R'S)'(Tv U)]}.'. {Qv [(R'S) v (Tv U)]}
12. V � {W � [(X'Y) � Z]}.'. V � ([W'(X'Y)) � Z}
13. [A v (B v C)) v [(D v D) v E). '. [A v (B v C)) v [D v (D v E))
14. (F � G)'{((G � H)'(H � G)) � (H � In.'. (F � G)'{(G = H) � (H � In
*15. J = (((K'L) v M)'[(K'L) v N]}.'. J = {(K'L) v (M'Nn
16. 0 � [(P'Q) = (P'R)].'. 0 � [(P'Q) = (P'R))
17. S = {T � [Uv (T'S)]}.'. 5 = {Tv [Uv (T'S)]}
18. V� {(W � X) v [(Y � Z)v(Z � Y)]}.'. V � {(X � W) v [(Y � Z) v (Z � Y)]}
19. (A'B) � [(C'C) � (C � D)).'. (A'B) � [C � (C � D))
20. (E'F) � [G � (G � H)].'. (E'F) � [(G'G) � H))
See. 3.2)
11. E ::> F
E::>G... E::> (F.G)
12. H ::> (I v J)I
".H::>/13. (K v L) ::> (M. N)
(Mv N) ::> (0 = P)(0 = P) ::> (Q.R)... (Lv K) ::> (R.Q)
14. S ::> T
Sv T
".T
*15. (Uv V)'(Uv W)x::> W
".vvx
16. A ::> (B ::> C)C::> (D. E)... A::> (B::> D)
See. 3.4] Incompleteness of the Nineteen Rules
p q r p=>q q=>r p => r
0 0 0 0 0 00 0 1 0 1 10 0 2 0 2 2
0 1 0 1 0 00 1 1 1 1 10 1 2 1 1 2
0 2 0 2 0 00 2 1 2 0 10 2 2 2 0 2
1 0 0 0 0 01 0 1 0 1 11 0 2 0 2 11 1 0 1 0 01 1 1 1 1 11 1 2 1 1 11 2 0 1 0 01 2 1 1 0 11 2 2 1 0 12 0 0 0 0 02 0 1 0 1 02 0 2 0 2 02 1 0 0 0 02 1 1 0 1 02 1 2 0 1 02 2 0 0 0 02 2 1 0 0 02 2 2 0 0 0
Only in the first, tenth, nineteenth, twentysecond, twentyfifth, twentysixth,and twentyseventh rows do the two premisses 'p => q' and 'q => r' have thevalue 0, and in each of them the conclusion 'p => r' has the value 0 also. Even
larger tables would be needed to show that having the value 0 is hereditarywith respect to Constructive Dilemma and Destructive Dilemma, but theyare easily constructed. (It is not absolutely necessary to construct them,however, because the alternative analytical definitions on page 48 can be usedto show that having the value 0 is hereditary with respect to the Dilemmas,
49as on page 50.)'When we construct threevalued tables to verify that having the value 0
is hereditary with respect to replacement of statements by their logicalequivalents, we notice that although the biconditionals themselves need not
have the value 0, the expressions flanking the equivalence sign necessarilyhave the same value. For example, in the table appropriate to the first ofDe Morgan's Theorems,
The Method of Deduction [th.3
P q p q p.q (p.q) p v q (p.q) = (pv q)0 0 2 2 0 2 2 00 1 2 1 1 1 1 10 2 2 0 2 0 0 01 0 1 2 1 1 1 1
1 1 1 1 1 1 1 1
1 2 1 0 2 0 0 02 0 0 2 2 0 0 02 1 0 1 2 0 0 02 2 0 0 2 0 0 0
See. 3.8] The Strengthened Rule of Conditional Proof
I. (A v B) ::> [(C v D) ::> E] /:. A ::> [(C' D) ::> E]2. A
3. AvB4. (Cv D) ::> E
p: rD8. E
9. (C, D) :J E
10. A::> [(C' D) :J E]
See. 3.9] Shorter Truth Table TechniqueReductio Ad Absurdum Method
This method of proving the validity of an argument is a version of the reductioad absurdum technique, which uses truth value assignments rather than Rulesof Inference.
It is easy to extend the use of this method to the classification of statements
(and statement forms). Thus to certify that Peirce's Law [(p ::> q) :> p] ::> Pis a tautology, we assign it the truth value F, which requires us to assign T
to its antecedent [(p :> q) ::> p] and F to its consequent p. For the conditional
[(p ::> q) ::> p] to be true while its consequent p is false, its antecedent (p :> q)must be assigned the truth value F also. But for the conditional p :> q to be
false, its antecedent p must be assigned T and its consequent q assigned F.
However, we were previously forced to assign F to p, so assuming Peirce'sLaw false leads to a contradiction, which proves it a tautology.
If it is possible to assign truth values consistently to its components on the
assumption that it is false, then the expression in question is not a tautology,but must be either contradictory or contingent. In such a case we attemptto assign truth values to make it true. If this attempt leads to a contradictionthe expression cannot possibly be true and must be a contradiction. But iftruth values can be assigned to make it true and (other) truth values assignedto make it false, then it is neither a tautology nor a contradiction, but is
contingent.The reductio ad absurdum method of assigning truth values is by far the
quickest and easiest method of testing arguments and classifying statements.
It is, however, more readily applied in some cases than in others. If F is
assigned to a disjunction, F must be assigned to both disjuncts, and whereT is assigned to a conjunction, T must be assigned to both conjuncts. Herethe sequence of assignments is forced. But where T is assigned to a disjunctionor F to a conjunction, that assignment by itself does not determine which
disjunct is true or which conjunct is false. Here we should have to experimentand make various 'trial assignments', which will tend to diminish the advantageof the method for such cases. Despite these complications, however, in thevast majority of cases the reductio ad absurdum method is superior to anyother method known.
Propositional Functions and Quantifiers
Then we can use the notation already introduced to rewrite it as
See. 4.2] Proving Validity: Preliminary Quantification Rules
letter 'y' will be used to denote any arbitrarily selected individual. In this
usage the expression '«Py' is a substitution instance of the propositional hmction'«Px', and it asserts that any arbitrarily selected individual has the property«P. Clearly '«Py' follows validly from '(x)«Px' by UI, since what is true of all
individuals is true of any arbitrarily selected individual. The inference is
equally valid in the other direction, since what is true of any arbitrarilyselected individual must be true of all individuals. We augment our list ofRules of Inference further by adding the principle that the universal quantification of a propositional hmction can validly be inferred from its substitution
instance with respect to the symbol 'y'. Since this rule permits the inferenceof general propositions that are universal quantifications, we refer to it as the
'principle of Universal Generalization', and abbreviate it as 'UG'. OUT symbolicexpression for this second quantification rule is
Propositional Functions and Quantifiers [Ch. 4
propositional function with respect to the variable 'x'" or "the universal (orexistential) quantification of a propositional function with respect to thevariable 'y'" and so on.
It should be clear that since '(x)[Fx :J Gx)' and '(y)[Fy :J Gy)' are alternativetranslations of the proposition 'Everything which is an F is also a G', the
universal quantification of 'Fx :J Gx' with respect to 'x' has the same meaningand is logically equivalent to the universal quantification with respect to 'y'of the propositional function which results from replacing all free occurrences
of 'x' in 'Fx :J Gx' by 'y'for the result of that replacement is 'Fy :J Gy'.In the early stages of our work it will be desirable to have at most one
quantification with respect to a given variable in a single proposition. Thisis not strictly necessary, but it is helpful in preventing confusion. Thus thefirst multiply general proposition considered, 'If all dogs are carnivorous thensome animals are carnivorous', is more conviently symbolized as '(x)[Dx :J
Cx] :J (3y)[Ay.Cy)' than as '(x)[Dx :J Cx] :J (3x)[Ax.Cx)', although neitheris incorrect.
It has been remarked that no proposition can contain a free occurrence
of any variable. Hence in symbolizing any proposition we must take care that
every occurrence of every variable used lies within the scope of a quantifierwith respect to that variable. Some examples will help to make the matter
clear. The proposition
Propositional Functions and Quantifiers [Ch. 4
CPv for use in applying EI where (31L)CPIL is '(3y)(Fx = Fy)'. It should beobvious that the �g!m.l��_t ,�,� !n'lalid: it fails for a model containing some thingsthat are F and some things that are not F, which would make the premisstrue, whereas the conclusion is false for every model, being selfcontradictory.
Propositional Functions and Quantifiers
.6_ 1. (3x)(y)[(FxGx) ::> Hy] /:. (3x)[(FxGx) ::> Hx]2. (y)[(FzGz) ::> Hy]3. (FzGz) ::> Hy 2, UI4. (3x)[(FxGz) ::> Hy] 3, EG5. (y)(3x)[(FxGy) ::> Hy] 4, UG
6. (y)(3x)[(FxGy) ::> Hy] 1,25, EI7. (3x)[(FxGx) ::> Hx] 6, UI
7_ 1. (3x)Fx2. (3x)Gx /:. (3x)(FxGx)3. Fy4. Gy5. FyGy 3,4. Conj.6. (3x)(FxGx) 5, EG
7. (3x)(FxGx) 2,46, EI
8. (3x)(FxGx) 1,37, EI
8_ 1. (3x)(3y)[(Fx v Gy) Hy] /... (x)(y)(Fy v Gx)2. (3y)[(Fxv Gy)  Hy]
[3. (Fxv Gx)Hx
_
4. Fx v Gx 3, Simp.5. Fx v Gx 2, 34, EI
6. Fx v Gx 1, 25, EI7. (y)(Fy v Gx) 6, UG8. (x)(y)(Fy v Gx) 7, UG
.9_ 1. (3x)(FxGx)2. (3x)(FxGx) /.'.(3x)(FxFx)
r3.Fx Gy
4. Fx 3, Simp.5. Fx 1, 34, EI
r6.Fx Gx
7. Fx 6, Simp.8. Fx 2,67, EI9. FxFx 5,8, Conj.
10. (3x)(FxFx) 9, EG
10_ 1. (x)[(Fx::> Gx)Ga] /.'. (x)Fx2. (x)[(Fx::> Gx)Gy]3. (Fz::> Gz)Gy 2, UI4. (y)[(Fy ::> Gy)Gy] 3, UG
100 5. (Fu ::> Gu)Gu 4, UI6. Fu ::> Gu 5, Simp.7. Gu  (Fu ::> Gu) 5, Com.8. Gu 7, Simp.9. Fu 6,8, M.T.
10. (x)Fx 9, UG
n. (x)[(Fx::> Gx)Gy] ::> (x)Fx 210, C.P.
Sec.4.5]
7. (x)(Qx::> Rx)(x)(Sx ::> Tx).'. (x)(Rx ::> Sx) ::> (y)(Qy ::> Ty)
8. (3x)Ux ::> (y)[(Uy v Vy) ::> Wy](3x)Ux.(3x)Wx.'. (3x)(Ux. Wx)
9. (3x)Xx ::> (y)(Yy ::> Zy)... (3x)(Xx. Yx) ::> (3y)(Xy. Zy)
See. 4.6J Logical Truths Involving Quantifiers
In demonstrating the logical truth of propositions involving quantifiers, we
shall have to appeal not only to the original list of elementary valid argumentforms and the strengthened principle of Conditional Proof, but to our quantification rules as well. Thus a demonstration of the logical truth of the proposition '(x)Fx ::> (3x)Fx' can be set down as
Propositional Functions and Quantifiers [Ch. 4
expressions written above, the propositions 'Ga' and '(z)Hz', and the propositional functions 'Fy' and 'Gw', although lying within the scopes of the quantifiers '(x)', '(3y)', '(x)' and '(z)', respectively, are not really affected by them.Wherever we have an expression containing a quantifier on the variable /Lwithin whose scope lies either a proposition or a propositional function not
containing any free occurrence of p" the entire expression is logically equivalent to another expression in which the scope of the quantifier on p, does not
extend over that proposition or propositional function. An example Or two
will make this clear. In the following, let 'Q' be either a proposition or a
propositional function containing no free occurrence of the variable 'x', andlet 'Fx' be any propositional function containing at least one free occurrence
of the variable 'x'. Our first logical equivalence here is between the universal
quantification of 'Fx' Q' and the conjunction of the universal quantificationof 'Fx' with 'Q', which is more briefly expressed as
See. 5.1]
1. (x)[Vx::> (3y)Oyx]2. (x)( Oxa ::> Rxa]3. (x)Rxa /...  Va
4. Oza::> Rza 2, UI
5. Rza 3, UI
6. Oza 4,5, M.T.7. (y)Oya 6, UG8. (3y)Oya 7, QN9. Va::> (3y)Oya 1, UI
10.  Va 9, 8, M.T.
tile Logic of Relations
*15. (x){[Wx'(y)[Py :> (3z)(Nz' pxzy)]] :> Ix}16. (x){[Px'(y)(Vxy)] :> (z)(Cx.:)}17. (x){Vx:> (y)[Xy :> (3z)[(lz'Bzx)'Dzy]])18. (x){[Lx'(3y)(Py' Eyx)) :> (z)(Wz :> Tgzx)}19. (x){Px:> (3y)[Py'(3z)(Bxzy)])
*20. (x){Px:> (3y)[Py.(3z)(Bxzy)])21. (x){Px :> (y)[Py :> (z)( Bxzy)]}22. (x){Px :> (y)[Py :> (3z)( Bxzy)]}23. (x)[(Nx'Dx) :> (y)(Lxy :> Myx))24. (x)[Px:> (3y)(Py'Xyx)).(3u)[Pu'(v)(Pv :> Xuv))25. (x){[Qx,(y){[(py'Wyx)'(z)(Kyz)) :> By}] :>
(u){[(Pu'Wux)'(v)(Kuv)) :> Ou}}
The Logic of Relations
4. A wise son maketh a glad father. (Proverbs 10:1)*5. He that spareth his rod hateth his son. (Proverbs 13:24)
6. The borrower is servant to the lender. (Proverbs 22:7)7. Whoso diggeth a pit shall fall therein: and he that rolleth a stone, it will
return upon him. (Proverbs 26:27)8. The fathers have eaten sour grapes, and the children's teeth are set on edge.
(Ezekiel 18:2)9. The foxes have holes, and the birds of the air have nests; but the Son of
man hath not where to lay his head. (Matthew 8:20)10. '" the good that I would I do not; but the evil which I would not, that
I do. (Romans 7:19)
See. 5.2)
7. (x)[Mx::> (y)(Ny ::> Oxy)](x)[Px ::> (y)(Oxy ::> Qy)].' . (3x)(Mx' Px) ::> (y)(Ny ::> Qy)
8. (x)[(Rx' Sx) ::> (3y)(Txy' Uy)](3x)[Vx'Rx'(y)(Txy ::> Vy))(x)(Vx ::> Sx).'. (3x)(Vx' Ux)
9. (x)(Wx::> Xx)(x)[(YX' Xx) ::> Zx](x)(3y)(Yy' Ayx)(x)(y)[(AyX' Zy) ::> Zx]
. '. (x)[(y)(Ayx ::> Wy) ::> Zx]10. (x){[BX'(3y)[Cy'DyX'(3z)(Ez'Fxz))) ::> (3w)Gxwx)
(x)(y)(Hxy ::> Dyx)(x)(y)(Fxy ::> Fyx)(x)(Ix ::> Ex)... (x){Bx::> ([(3y)(Cy'Hxy).(3z)(Iz'Fxz)] ::> (3u)(3w) GxwuJ}
The Logic 0' Relations [Ch. 5
uals by boldface italic capital letters 'A', 'B', 'C', . . . , to prevent their beingconfused with attributes of individuals. With this additional symbolic apparatus we can translate into our notation such propositions as 'Unpunctualityis a fault', and 'Truthfulness is a good quality'. Here we use 'Ux', 'Tx', 'FF',and 'GF' to abbreviate 'x is unpunctual', 'x is truthful', 'F is a fault', and 'Fis good', and symbolize the two stated propositions as 'FU' and 'GT'. More
complex propositions can also be symbolized. The propositions
See. 7.2] Primitive Symbols and Well formed formulas
in R.S. in its intended or normal interpretations is easily seen. First we note
that the intended or normal interpretations of P and PQ are given by thetruth tables
P Pand
P Q P�T F T T T
F T T F F
F T F
F F F
That RS. is adequate to express Ul!), f2(l!),f3(F), and fiP) is proved by actuallyformulating them in RS. The function f2(P) is true when P is true and falsewhen P is false, and is therefore expressible in RS. as P itself. The function
fl (F) is false when P is true and true when P is false, and is therefore expressiblein RS. as  P. The function f3(P) is false nO matter which truth value P assumes,and is therefore expressible in RS. as PP. The function f4(l!) is true in everycase and can therefore be expressed in RS. as the negation of f3(l!), that is,as (Pl!). We have thereby shown that all singulary truth functions are
expressible in RS.
There are, of course, more truth functions of two arguments than of one
argument. These are defined by the following truth tables:
P Q fI(P.Q) P Q f2(P.Q) P Q f3(P.Q) P Q f4(P.Q)T T F T T T T T T T T T
T F T T F F T F T T F T
F T T F T T F T F F T T
F F T F F T F F T F F F
P Q f5(P.Q) P Q f6(P.Q) P Q f7(P.Q) P Q fs(P.Q)T T F T T F T T F T T T
T F F T F T T F T T F F
F T T F T F F T T F T F
F F T F F T F F F F F T
P Q f9(P.Q) P Q fIo(P.Q) P Q fu(P.Q) P Q fdP.Q)T T T T T T T T F T T FT F F T F T T F F T F F 173F T T F T F F T F F T TF F F F F F F F T F F F
P Q fI3(P.Q) P Q f14(P.Q) P Q f15(P.Q) P Q f16(P.Q)T T F T T T T T F T T T
T F T T F F T F F T F T
F T F F T F F T F F T T
F F F F F F F F F F F T
A Propositional Calculus [Ch.7
(P Q) :> P
0 0 0 0 0
0 1 1 0 00 2 2 0 01 1 0 0 1
1 2 1 0 11 2 2 0 12 2 0 0 2
2 2 1 0 22 2 2 0 2
(P :> Q) :> [ (Q R) :> (R P)]0 0 0 0 2 0 0 0 0 2 0 0 00 0 0 0 1 0 1 1 0 1 1 1 0
0 0 0 0 0 0 2 2 0 0 2 2 00 1 1 0 1 1 1 0 1 2 0 0 0
0 1 1 0 0 1 2 1 1 1 1 1 0
0 1 1 0 0 1 2 2 0 0 2 2 0
0 2 2 0 0 2 2 0 2 2 0 0 0
0 2 2 0 0 2 2 1 1 1 1 1 00 2 2 0 0 2 2 2 0 0 2 2 01 0 0 0 2 0 0 0 0 1 0 1 1
1 0 0 0 1 .0 1 1 0 0 1 2 1
1 0 0 0 0 0 2 2 0 0 2 2 11 0 1 0 1 1 1 0 0 1 0 1 1
1 0 1 0 0 1 2 1 0 0 1 2 1
1 0 1 0 0 1 2 2 0 0 2 2 1
1 1 2 0 0 2 2 0 1 1 0 1 1
1 1 2 0 0 2 2 1 0 0 1 2 1
1 1 2 0 0 2 2 2 0 0 2 2 1
2 0 0 0 2 0 0 0 0 0 0 2 2
2 0 0 0 1 0 1 1 0 0 1 2 2
2 0 0 0 0 0 2 2 0 0 2 2 2
2 0 1 0 1 1 1 0 0 0 0 2 2
2 0 1 0 0 1 2 1 0 0 1 2 2
2 0 1 0 0 1 2 2 0 0 2 2 2
1842 0 2 0 0 2 2 0 0 0 0 2' 2
2 0 2 0 0 2 2 1 0 0 1 2 2
2 0 2 0 0 2 2 2 0 0 2 2 2
The characteristic is easily seen to be hereditary with respect to R 1 ofR.S. by consulting the table given for ':>'. In the only row in which bothP and P :> Q have the value 0, Q also has the value O. Hence if the characteristic belongs to one or more wjJs it also belongs to every wjJ deduced fromthem by R 1.
Finally it is readily seen that the characteristic in question does not belong
A Propositional Calculus [Ch.7
P :> (P P)0 0 0 0 01 0 1 0 12 0 2 2 2
(P :> Q) :> [ (Q R) :> (R P)]0 0 0 0 2 0 0 0 0 2 0 0 0
0 0 0 0 2 0 0 1 0 2 1 0 0
0 0 0 0 0 0 2 2 0 0 2 2 00 2 I 0 2 1 0 0 0 2 0 0 00 2 1 0 2 I 0 1 0 2 1 0 00 2 I 0 0 1 2 2 0 0 2 2 00 2 2 0 0 2 2 0 2 2 0 0 0
0 2 2 0 0 2 2 1 2 2 1 0 00 2 2 0 0 2 2 2 0 0 2 2 0
1 0 0 0 2 0 0 0 0 2 0 0 I
I 0 0 0 2 0 0 1 0 2 1 0 1
1 0 0 0 0 0 2 2 0 0 2 2 I
1 2 1 0 2 I 0 0 0 2 0 0 I
1 2 1 0 2 1 0 1 0 2 1 0 1
I 2 I 0 0 1 2 2 0 0 2 2 1
I 2 2 0 0 2 2 0 2 2 0 0 1
1 2 2 0 0 2 2 1 2 2 1 0 1
1 2 2 0 0 2 2 2 0 0 2 2 12 0 0 0 2 0 0 0 0 0 0 2 22 0 0 0 2 0 0 1 0 0 1 2 22 0 0 0 0 0 2 2 0 0 2 2 2
2 0 1 0 2 1 0 0 0 0 0 2 22 0 1 0 2 1 0 1 0 0 1 2 2
2 0 I 0 0 1 2 2 0 0 2 2 2
2 0 2 0 0 2 2 0 0 0 0 2 2
2 0 2 0 0 2 2 1 0 0 1 2 2
2 0 2 0 0 2 2 2 0 0 2 2 2
186The characteristic is easily seen to be hereditary with respect to; R 1 of
R.S. by consulting the table for ':>'. In the only roW in which both P andP :> Q have the value 0, Q also has the value O. Hence if the characteristic
belongs to one or more wffs it also belongs to every wff deduced from them
by R 1.
Finally, it is readily seen that the characteristic in question does not belongto Ax. 2. When P and Q are both assigned the value 1, (P Q) :> P has thevalue 2 rather than 0 for (II) :> 1 is 0 :> I which is 2. Hence Ax. 2 is
independent.
See. 7.4) Independence of the Axioms
To prove the independence of Ax. 3 of RS. we use the same threeelementmodel and the same table for 'P'. The difference lies in the table for 'poQ',which follows, along with the derivative table for 'P :> Q'.
See. 7.5]
DR 15. P = QIP = QDR 16. P = Q, R = 5 I PR = Q5DR 16, COR. P = Q, R = 5 I P v R = Q v 5
See. 7.6] Deductive Completeness
TH. 26. f P = pp (Tautology)TH. 26, COR. fp = pvP (Tautology)TH. 27. f (PQ) = (Pv Q) (De Morg�'s Theorem)
TH. 28. f (Pv Q) = (PQ) (De Morgan's Theorem)TH. 29. f (P::> Q) = (Pv Q) (Material Implication)
*TH. 30. fP(Qv R) = PQv PR (Distribution of '.' over V)TH. 30, COR. f (P v Q)R = PR v QRTH. 31. f (P = Q) = [PQ v PQ] (Material Equivalence)TH. 32. f Pv QR = (Pv Q)(Pv R) (Distribution of V over ,.')
See. 7.6) Deductive Completeness
by DR 12 � Qj ::> (Sl' S2)' which is I Qj ::> 5. If to S2 then � Q; ::> 52by the f3case assumption, and hence by DR 13 � Q; :> (S1' S2)' which is
�Qj:> So
Alternative Systems and Notations [Ch.8
P p p Q PvQ P� Q0 2 0 0 0 0
1 1 0 1 0 1
2 0 0 2 0 21 0 0 01 1 0 01 2 1 12 0 0 0
2 1 1 02 2 2 0
To prove Postulate 2 independent we use the threeelement model {O, 1, 2}of which 0 is designated, with the tables:
P p P Q PvQ P�Q0 1 0 0 0 01 0 0 1 0 1
2 2 0 2 0 11 0 0 01 1 1 01 2 1 02 0 0 02 1 1 12 2 1 1
To prove Postulate 3 independent we use {O, 1, 2} with 0 designated andtables:
P P P Q PvQ P�Q0 2 0 0 0 0
1 0 0 1 0 22 1 0 2 0 2
1 0 0 0
2141 1 1 01 2 0 02 0 0 02 1 2 1
2 2 2 0
To prove Postulate 4 independent we use the fourelement model {O, 1,2, 3}with 0 designated and tables:
Set:. 8.2] The HilbertAckermann System
p p p Q PvQ p:J Q0 I 0 0 0 0I 0 0 I 0 I2 3 0 2 0 23 0 0 3 0 3
I 0 0 0I I I 0I 2 2 0I 3 3 02 0 0 02 I 2 32 2 2 02 3 0 33 0 0 0
3 I 3 03 2 0 0
3 3 3 0
Alternative Systems and Notations [Ch.8
P'QP Q PvQ P Q PvQ (Pv Q)0 0 0 5 5 5 0
0 1 0 5 5 5 00 2 3 5 4 5 0
0 3 3 5 1 0 50 4 0 5 0 0 50 5 0 5 0 0 5
1 0 0 5 5 5 0
1 1 0 5 5 5 0
1 2 3 5 4 5 0
1 3 3 5 1 0 5
1 4 0 5 0 0 51 5 0 5 0 0 52 0 3 4 5 5 0
2 1 3 4 5 5 0
2 2 3 4 4 5 0
2 3 3 4 1 0 52 4 3 4 0 0 52 5 3 4 0 0 53 0 3 1 5 0 53 1 3 1 5 0 53 2 3 1 4 0 5
3 3 3 1 1 0 53 4 3 1 0 0 53 5 3 1 0 0 5
4 0 0 0 5 0 54 1 0 0 5 0 54 2 3 0 4 0 5
4 3 3 0 1 0 5
4 4 5 0 0 0 54 5 5 0 0 0 55 0 0 0 5 0 55 1 0 0 5 0 55 2 3 0 4 0 55 3 3 0 1 0 5
5 4 5 0 0 0 5216 5 5 5 0 0 0 5
In this model the three elements 0, 1, 2 are designated. The characteristicof taking only designated values is hereditary with respect to the rule: From
P and _(po Q) to infer Q: and the three H.A. formulations of the R.S. axioms
take only designated values. But for the value 2 for P, we have
Pv P = 2 v 2 = 2 v 4 = 3 which is not a designated value. 2
2See Henry Hiz, "A Warning About Translating Axioms," American Mathematical Monthly,vol. 65 (1958), pp. 613 f.; Thomas W. Scharle, "Are Definitions Eliminable in Fonnal Systems"
See. 8.2J The HilbertAckermann System
THEOREM 16. luxp = (Pv P)
Proof: I. P::> (Pv P) P22. (Pv P) ::> P PI3. [P::> (P v P)][(P v P) ::> P] DR44. P = (Pv P) df.
THEOREM 17. IRA P = (PP)
Proof: I. P = (PvF) Th.I62. p = (PvP) DR83. P = P Th.ll4. P = (PvF) DR9
5. P = (PP) df.
DR 10. Q ::> R lux (Pv Q) ::> (R v P)
Proof: I. Q::>R premiss2. (Pv Q) ::> (Pv R) DR23. (Pv R) ::> (R v P) P34. (Pv Q) ::> (R vp) DR 1
DR II. P::> Q,R::> 8 Iux(PvR)::> (Qv8)
Proof: I.R::>8 premiss2. (P v R) ::> (8 v P) DR 10
3. P::> Q premiss4. (8 v P) ::> (Q v 8) DR 10
5. (Pv R) ::> (Q v S) DR 1 (2, 4)
DR 12. P = Q,R = 8 Iux(PvR) = (Qv8)
Proof: I.P = Q premiss2. (P::> Q)( Q ::> P) df.3. P::> Q DR6
4. Q ::> P DR7 2215. R = S premiss6. (R ::> 8)(S ::> R) df.
7. R ::> S DR6
8. S ::> R DR7
9. (PvR)::> (QvS) DR 11 (3, 5)10. (Q v S) ::> (Pv R) DR 11 (4, 8)II. [(PvR)::> (QvS)][(QvS)::> (PvR)] DR4
12. (PvR) = (QvS) df.
See. 8.2] The HilbertAckermann System
THEOREM 18. \Hx(Pv Q) = (Qv P)
Proof: 1. (PvQ):> (QvP) P32. (QvP) => (PvQ) P3
3. [(Pv Q) => (Q v P)][(Q v p) => (Pv Q)] DR4
4. (Pv Q) = (Q vp) elf.
DR 13. P:> Q,P:> R luxp => (QR)
Proof: I.P=>Q premiss2. Q:> P DR5
3. P => R premiss4. R :> P DR5
5. (Q v R) => (PvP) DRll
6. (Pv p) :> (Qv R) DR5
7. (PI') :> (QR) elf.8. P:> (QR) MT I, Cor., Th. 17
THEOREM 19. hu [P v (QR)] => [(P v Q)(P v R)]
Proof 1. (QR) :> Q Th.142. [Pv (QR)] => (Pv Q) DR2
3. (QR) => R Th.154. [Pv (QR)] => (Pv R) DR2
5. [Pv (QR)J => [(Pv Q)(Pv R)] DR 13
DR 14. P => (Q:> R) huQ => (P => R)
Proof: 1. P => (Q => R) premiss2. Pv (Q v R) df.3. [Pv(QvR)] => [Qv(PvR)] Th.74. Qv(PvR) R'I
5. Q :> (P :> R) df.
DR 15. P => (Q :> R) IRA (PQ) :> R223
Proof: 1. P => (Q :> R) premiss2. Pv (Q v R) df.3. (Pv Q) v R MT I, Cor., Th. 12
4. (PvQ)vR MT I, Cor., Th. II
5. (PQ) :> R df.
See. 8.2] The HilbertAckermann System
({3) Here the Metatheorem is assumed to be true for any k < n disjuncts PI'P 2 , . . . , P k . Now consider Q and R each constructed out of exactly n (> 1)disjWlcts PI' P 2 ,. .., Pn' Q is S v T and R is X v Y.
Each of the wffs Sand T contains at least one of the wffs Pi (1 :::; i :::; n).We can assume that PI is a disjunct of S, because if not we can use Th. 18and MT I, Cor. to obtain iHx Q = (S v T) where S now does contain PI as
a disjunct.Because T contains at least one of P 2 , P 3 , . . . , Pn as a disjunct, S contains
fewer than n of the disjuncts Pi' Hence either S is PI and tHA Q = (PI V T),or by the {3case assumption fIci S = (Sl V S'), where S' is a wff that contains
all the disjuncts of S except Pl' In the latter case, by MT I, Cor. we have
Altemative Systems and Notations [Ch. 8
The Polish notation has the obvious advantage of dispensing with all specialpunctuation marks, for the order in which its symbols are written suffices to
make any formula unambiguous.
See. 8.5] The Stroke and Dagger Operators
its standard interpretation is to deny that either of the formulas P or Q is
true, which is the same as affirming that they are both false. It is defined bythe truth table
Sec. 8.6] The Nicod System
is Theorem 8 with Q in place of P, with PIP.I.P:.I:.Q/P.I.P:I:QIP.I.P in
place of Q and of R, and with Q:.I:.Q/P.I.P:I:QIP.I.P::I::Q:.I:.QIP.I.P: I :Q I P.I.P in place of S. Line 16 is the result of applying the Nicod Ruleto lines 12 and 15. Line 17 is the result of applying the Nicod Rule to
lines 14 and 16. Line 18 is Theorem 10 with QIP.I.P:I:QIP.I.P in placeof S. Line 19 is Theorem 3 with QI Q in place of S, and with Q:.I :.QIPI.P:I :QIP.I.P::I ::Q:.I :.QIP.I.P: I :QIP.I.P in place of P. Line 20 is the resultof applying the Nicod Rule to lines 18 and 19. Line 21 is Theorem 8 with
Q:.I:.QIP.I.P:I:QIP.I.P::I::Q:.I:.QIP.I.P:I:QIP.I.P in place of P and of S,and with Q in place of R. Line 22 is the result of applying the Nicod Ruleto lines 20 and 21. Line 23 is the result of applying the Nicod Rule to lines17 and 22. Line 24 is Theorem 4 with Q:.I:.QIP.I.P:I:QIP.I.P::j::Q:.\:.QIP.I.P:I:Q/P.I.P in place of P. Line 25 is the result of applying theNicod Rule to lines 23 and 24.
See. B.6J The Nicod System
DR 1. P, PIP.I.PIP:/:QIQ IN Q (R' 1 of H.A.)
Proof: Line 1 is Theorem 9 with P in place of S, and with PIP in placeof P. Line 2 is the premiss P. Line 3 is the result of applying the Nicod Ruleto lines 1 and 2. Line 4 is the premiss PI P.I.PI P: I:Q I Q. Line 5 is the result
of applying the Nicod Rule to lines 3 and 4.
Sec. 9.1] The New Logistic System RS I
3. Infinitely many capital letters from the first part of the alphabet, withand without subscripts, having righthand superscripts '1', '2', '3'. . . .
Sec. 9.3]
1. W: p.QW6: PvQ
2. W: (x)(Pv Q)W6: (3x)(P._Q)
3. W: (y)(3z)[Pv(QvR.S)]W6: (3y)(z)[P.Q.(R v S)]
See. 9.5) Nonnal Forms
Now the prenex normal form of (3t){ G [D(t) ::> D(t)]} is the formula oftype R that was desired, for it is closed, is in prenex normal form, beginswith an existential quantifier, and I F if and only if I R. Where G is (Qxl)(Qx2) . . . (Qxn)G', then
I (3tH G. [D(t) ::> D(t)]} or
I (3tH [(Qxl)(Qx2) . . . (Qxn)G'J  [D(t) D(t)]}
See. 9.6) Completeness of RS I
I F or I F. This kind of completeness is not desirable either, for on theirnormal interpretations, each of the following wffs
Sec. 9.6] Completeness of RS l
function calculus RS I is complete if and only if every valid cwff is provablein it as a theorem. To say that every valid cwffis provable in RS 1 as a theorem
is, by transposition, to say that for any cwff S, if S is not a tlieorem then S
is not valid.In defining the terms 'valid' and 'satisfiable' we remarked that for any wff
Seither 5 is valid or S is satisfiable. Hence to say that S is not valid is
to say that S is satisfiable. And so we can say that RS I is complete if and
only if, for any cwff S, if S is not a theorem then S is satisfiable. We can
establish this result by introducing a characteristic cp such that both
Normal Forms and Boolean Expansions
Then we rearrange the terms by simply commuting or interchanging thesecond and third disjuncts, to get
Normal Forms and Boolean Expansions
Expansion represent all possible assignments of truth values to its variables.
Since the 2" disjuncts represent all possible assignments of truth values to its
variables, at least one of them must be true. And since it asserts only thatat least one of its disjuncts is true, any disjunctive Boolean Expansion contain
ing n variables and 2" disjuncts is tautologous. This point is made in somewhatdifferent terms in Section 7.6 and again in Section 8.2.
It was pointed out in Chapter 2 that a truthfunctional argument is validif and only if its corresponding conditional statement (whose antecedent is
the conjunction of the argument's premisses and whose consequent is the
argument's conclusion) is a tautology. Since counting the number of disjunctsof its disjunctive Boolean Expansion permits us to decide whether or not a
given form is a tautology, this provides us with an alternative method of
deciding the validity of arguments. Thus the argument form p v q, p .'. qis proved valid by constructing the disjunctive Boolean Expansion of its
corresponding conditional [(p v q)'p] ::> q, and observing that the number ofits disjuncts is 2 2
.
Since the negation of a tautology is a contradiction, an argument is validif and only if the negation of its corresponding conditional is a contradiction.Hence another method of deciding the validity of an argument is to formthe conjunctive Boolean Expansion of the negation of its correspondingconditional and count the number of its conjuncts. H it contains n distinctvariables and has 2" conjuncts, then the argument is valid; otherwise it isinvalid.
The Algebra 01 Classes
a and {3: I = a{3 U ap U af3 U ap, and the product of any two of those four
products is the empty class. Similarly, any n classes will divide the universalclass into 2 11 subclasses which are exclusive and exhaustive. The class expressionwhich symbolizes such a division of the universal class, it should be observed,is a disjunctive Boolean Expansion. A disjunctive Boolean Expansion containingn different simple class terms designates the universal class if it is the sum
of 2 11 distinct products (where a mere difference in the order of their terms
does not make two products distinct). Disjunctive Boolean Expansions thus
provide us with a method for deciding whether or not any class expressiondesignates the universal class regardless of what classes are designated by the
simple class terms which it contains. Given any class expression, we need onlyconstruct its disjunctive Boolean Expansion and count the number of productsof which it is the sum.
A conjunctive Boolean Expansion is a product of distinct sums of simpleclass terms or their complements, where any simple class term which occurs
anywhere in the expression will occur exactly once in every sum. By De
Morgan's Theorem and the other equivalences already mentioned, the com
plement of any disjunctive Boolean Expansion can be transformed into a
conjunctive Boolean Expansion which involves the same simple class terms
and which is the product of as many sums as the disjunctive Boolean Expansionis the sum of products. Since the complement of 1 is 0, a conjunctive Boolean
Expansion containing n different simple class terms designates the empty classif it is the product of 2 11 distinct sums. Hence we have a method for decidingwhether or not any class expression designates the empty class regardless ofwhat classes are designated by the simple class terms which it contains.
The notations introduced thus far permit the symbolization of the A andE subjectpredicate propositions. The E proposition: No a is {3, asserts thatthe classes a and {3 have no members in common, which means that their
product is empty. The E proposition is therefore symbolized as
Appendix B
The 0 proposition: Some a is not fJ, asserts that there is at least one memberof a which is not a member of fJ, i.e., that the product of a and P is not
empty. In symbols, the 0 proposition is expressed as
The Algebra of Classes
not only to validate immediate inferences involving categorical propositions,but is capable of validating categorical syllogisms also.
The symbol' C'
for class inclusion is often used in working with the algebraof classes. The expression 'a C p' asserts that all members of a, if any, are
also members of p, and is used as an alternative symbolization of the A
proposition: All a is p. It can be defined in terms of the symbols alreadyintroduced in various ways: either as ap = 0 or as ap = a or as a U p = por as a U p = 1, all of which are obviously equivalent. The relation C is
reflexive and transitive (see pages 131132) and has the (transposition) propertythat if a C p then pea. The latter is an immediate consequence of double
negation and commutation when 'a C p' is rewritten as 'ap = 0' and 'P C ii'
is rewritten as 'p& = 0'. Its reflexiveness is obvious when 'a C a' is rewrittenas 'aa = 0', and its transitivity has already been established in our algebraicproof of validity for categorical syllogisms containing only W1iversal propositions.
The algebra of classes can be set up as a formal deductive system. Sucha system is called a Boolean Algebra, and a vast number of alternative postulatesets for Boolean Algebra have been proposed. One of them can be set forthas follows.
Special W1defined primitive symbols:
The Algebra of Classes
(A U f) #: 1. And if it is logically true that n #: I, then it is not logicallytrue that n = I, from which it follows that the wff which designates IT is
not a provable theorem in R.S. Since we have an effective criterion for
distinguishing between theorems and nontheorems of R.S., we have therein
an effective criterion for recognizing logically true equations and inequalitiesof class algebra.
The preceding discussion should suffice to indicate the intimacy of the
connection between the algebra of classes and the propositional calculus.
The Ramified Theory of Types
relation, so that's desigflates tf>' is symbolized as 'sDes<P', we begin with thedefinition
The Ramified Theory of Types .
according to the ramified theory of types, say that Bob has all of AI's goodqualities, which would ordinarily be symbolized as
Appendix C
but can say instead either that 'None of the firstorder propositions uttered
by Smith tends to incriminate him', or that 'None of the secondorder propositions uttered by Smith tends to incriminate him', or etc. We would partiallysymbolize the second of these alternative propositions as
The Ramified Theory of Types
from which definition all of the usual attributes of the identity relation can
be deduced. But that definition violates the ramified theory of logical types,since in it reference is made to all functions of type 1. Were we to replaceit by the definition
Appendix C
presence of the semantical terms the paradoxes do not seem to be derivableeven with the aid of the Axiom of Reducibility.lO
It may not be out of place here to indicate briefly how the 'levels of
language' method of avoiding the semantical paradoxes ll is remarkably similarto the ramified type theory's hierarchy of orders. 12 Confining our remarks to
the Grelling paradox, we note that it does not arise in an object language(like the extended function calculus, for example) when we assume that thereare in it no symbols which designate symbols. Nor does it arise in the meta
language of that object language. Since the metalanguage contains synonymsfor all symbols of the object language and 'names for all symbols of the objectlanguage, as well as its own variables and the name relation (which we write
as 'Des'), the symbol 'Het' can be defined in it. By definition:
The Ramified Theory of Types
In the first place, the metametalanguage contains two symbols for the name
relation, 'Des l
'
and 'Des z '.13 The first of these is the metametalanguage'ssynonym for the name relation in the metalanguage. The full sentence.
The Ramified Theory of Types
This is very like the theory of orders, because the contradiction is evaded
by arranging that certain symbols of the metametalanguage are defined over
certain ranges. Thus 'Des 1'
is defined over a narrower range than 'Des2', and'Het 1
'
is defined over a narrower range than 'Het 2 '; Des 1 and Het 1 beingsatisfied only by symbols of the object language, Des2 and Het 2 being satisfiedonly by symbols of the metalanguage, which is a wider and more inclusivelanguage. Not only is the levels of language theory remarkably analogous tothe theory of orders, but where each metalanguage is conceived as actuallycontaining the object language with which it deals,14 it can be identified withthe Russellian theory of orders as applied to symbols rather than to thefunctions they denote.
In spite of the indicated similarities, there are fundamental differencesbetween the two. Most significant is that unlike the ramified type theory, thelevels of language device for avoiding the paradoxes does not jeopardize thederivation of any parts of classical mathematics, so that no need arises forany analogue to the reducibility axiom.
Solutions 10 Selected Exercises on Pages 111124
8. 1. (3x)(Fx v Q) 1. (3x)(Fx v Q)2. FxvQ 2. (x)(Fx v Q) 1,QN3. Q 3. (Fx v Q) 2, UI4. Fx 2,3,D.S. 4. Fx'Q 3, DeM.5. (3x)Fx 4,EG 5. Fx 4, Simp.6. Q ::> (3x)Fx 35, C.P. 6. (x)Fx 5,UG7. Q v (3x)Fx 6, Impl., D.N. 7. (3x)Fx 6,QN8. (3x)Fx v Q 7, Com. 8. Q 4, Simp.9. (3x)Fx v Q 1, 28, EI 9. (3x)Fx' Q 7,8, Conj.
10. (3x)(Fx v Q) ::> 10. [(3x)Fx v Q] 9,DeM.
[(3x)Fx v Q] 19, C.P. 11. (3x)(Fx v Q) ::>
[(3x)Fx v Q] 110, C.P.12. [(3x)Fx v Q] ::>
(3x)(Fx v Q) 11, Trans.
12. 1. (3x)(Fx v Gx)2. (x)(Fx v Gx) I,QN3. (Fx v Gx) 2, UI4. Fx'Gx 3,DeM.5. Fx 4, Simp.6. Gx 4, Simp.7. (x)Fx 5,UG8. (x)Gx 6,UG9. (x)Fx'(x)Gx 7, 8, Conj.
10. (3x)Fx' (3x)Gx 9,QN11. [(3x)Fx v (3x)Gx] 10, De M.
12. (3x)(Fx v Gx) ::> [(3x)Fx v (3x)Gx] 111, C.P.13. [(3x)Fx v (3x)Gx] ::> (3x)(Fx v Gx) 12, Trans.14. (3x)(Fx v Gx)15. Fy v Gy16. (3x)Fx17. (x)Fx 16,QN18. Fy 17, UI19. Gy 15, 18, D.S.20. (3x)Gx 19,EG21. (3x)Fx::> (3x)Gx 1620, C.P.22. (3'.\Fx v (3x)Gx 21, Impl., D.N.
23. (3x)Fx v (3x)Gx 14, 1522, EI
32224. (3x)(Fx v Gx) ::> [(3x)Fx v (3x)Gx] 1423, C.P.25. {13}'{24} 13, 24, Conj.26. [(3x)Fx v (3x)Gx] = (3x)(Fx v Gx) 25, Equiv.
Solutions to Selected Exercises on Page 145
Exercises on page 145:
2. 1. (3x){Px.Sx.(y)((Py.Sy) ::> x = y].Lx} /..
. (x)((Px. �x) ::> Lx]2. Pz. Sz3. Px.Sx.(y)((Py.Sy) ::> x = y].Lx4. (y)((Py.Sy) ::> x = y] 3, Simp.5. (Pz. Sz) ::> x = Z 4, UI
6. x = Z 5, 2, M.P.7. Lx 3, Simp.8. Lz. 6, 7, Id.
9. Lz 1,38, EI
10. (Pz. Sz) :> L::; 29, C.P.11. (x)[(Px. Sx) ::> Lx] 1O,UG
4. 1. (3x){Px.(y)[(Py.x ¥' y) ::> Fxy].Sx} /:. (y)((Py. Sy) :> (3x)(Px. Fxy)]2. Py.Sy3. Px.(y)((Py.x ¥' y) ::> Fxy].Sx4. Sx 3, Simp.5. Sy 2, Simp.6. x¥'y 4,5,Id.7. Py 2, Simp.8. Py.x ¥' y 7, 6, Conj.9. (y)[(Py.x ¥' y) ::> Fxy] 3, Simp.
10. (Py.x ¥' y) :> Fxy 9, UI11. Fxy 10,8, M.P.12. Px 3, Simp.13. Px.Fxy 12, 11, Conj.14. (3x)(Px Fxy) 13,EG15. (3x)(Px. Fxy) 1,314, EI
16. (Py.Sy)::> (3x)(PxFxy) 215, C.P.17. (y)((Py. Sy) :> (3x)(Px. Fxy)] 16, UG
6. 1. (x){Fx::> (y)[(Fy.Lxy) :> SxyJ}/:. (3x){Fx.(y)((Fy.x ¥' y) :> LxyJ} :> (3x){Fx(y)[(Fy.x ¥' y) ::> SxyJ}
2. (3x){Fx.(y)((Fy.x ¥' y) ::> LxyJ}3. Fx(y)[(Fy.x ¥' y) :> Lxy]4. Fx 3, Simp.5. (y)[(Fy.x ¥' y) :> Lxy] 3, Simp.6. Fx :> (y)[(Fy. Lxy) ::> Sxy] 1, UI7. (y)((Fy.Lxy) :> Sxy] 6,4, M.P.8. (Fy.Lxy) :> Sxy 7,UI9. (Fy.x ¥' y) :> Lxy 5,UI 327
10. Lxy :> (Fy ::> Sxy) 8, Com., Exp.11. (Fy.x ¥' y) ::> (Fy ::> Sxy) 9, 10, H.S.12. (Fy.x ¥' y.Fy) ::> Sxy 11, Exp.13. (Fy.x ¥' y) :> Sxy 12, Com., Taut.14. (y)((Fy.x ¥' y) ::> Sxy] 13, UG15. Fx(y)((Fy.x ¥' y) ::> Sxy] 4,14, Conj.16. (3x){Fx.(y)((Fy.x ¥' y) ::> SxyJ} 15,EG17. (3x){Fx.(y)((Fy.x ¥' y) ::> SxyJ} 2,316, EI
18. 2 ::> 17 217, C.P.
Solutions to Selected Exercises on Pages 150170
Exercises on pages 150151:
I. 3. (x)(y)[x ¥' y :J (3F)(Fx'Fy)]6. (3x){Fxd'(y)(Fyd :J x = y)'(G)[(GxFG) :J Gd]'(H)[(HxVH) :J HdJ}9. (x){[Mx(F)(VF:J Fx)] :J Vx}'(3x)[MxVx(3F)(VF'Fx)]
II. 2. 1. (3x)(3F)Fx 1. (3F)(3x)f.x���� ����3. Fx 3. Fx4. (3x)Fx 3, EG 4. (3F)Fx 3, EG5. (3F)(3x)Fx 4, EG 5. (3x)(3F)Fx 4, EG
6. (3F)(3x)Fx 2, 35, EI 6. (3x)(3F)Fx 2, 35, EI
7. (3F)(3x)Fx 1, 26, EI 7. (3x)(3F)Fx 1, 26, EI
8. 1 :J 7 17, C.P. 8. 1 :J 7 17, C.P.
6. 1. (x)(y)(z)[(Rxy'Ryz) :J Rxz]'(x)Rxx2. (x)(y)(z)[(Rxy' Ryz) :J Rxz] 1, Simp.3. (y)(z)[(Rxy' Ryz) :J Rxz] 2, UI4. (z)[(Rxy' Ryz) :J Rxz] 3, UI
5. (Rxy'Ryx) :J Rxx 4, UI
6. (x)Rxx 1, Simp.7. Rxx 6, UI
8. (Rxy' Ryx) 5, 7, M.T.
9. Rxy v Ryx 8, De M.10. Rxy ::> Ryx 9, Impl.11. (y)(Rxy :J Ryx) 10, UG12. (x)(y)(Rxy :J Ryx) 11, UG
13. {I} :J 12 112, C.P.14. (R){13} 13, UG
10. 1. (x)(y)[(x = y) = (F)(Fx = Fy)] /.'. (x)(x = x)2. Fx
3. Fx 2, D.N.
4. Fx 3, D.N.
5. Fx :J Fx 24, C.P.6. (Fx :J Fx)'(Fx :J Fx) 5, Taut.
7. Fx = Fx 6, Equiv.8. (F)(Fx = Fx) 7, UG9. (y)[(x = y) = (F)(Fx = Fy)] 1, UI
10. (x = x) = (F)(Fx = Fx) 9, UI
11. [(x = x) :J (F)(Fx = Fx)]'328 [(F)(Fx = Fx) :J (x = x)] 10, Equiv.
12. (F)(Fx
Fx) :J (x = x) 11, Simp.nx=x ��M�14. (x)(x = x) 13, UG
Solutions to Selected Exercises on Pages 227257
THEOREM 1. IP;; (P v Q) :> (Q v P)
Proof: 1. [Q:> (QvP)]:> {(PvQ):> [Pv(QvP)]}2. Q:> (Q v P)3. (Pv Q) :> (Pv (Q v P)]4. [Pv (Qv P)] :> [Qv (Pv P)]5. (Pv Q) :> [Q v (Pv P)]6. [(PvP):> P]:> {(Qv(PvP)]:> (QvP)}7. (P v P) :> P
8. [Qv(PvP)]:>(Qvp)9. (PvQ):> (QvP)
Solutions to Selected Exercises on Page 296
1. 20. If a n /3 :;6 0 and /3 n  y = 0, then any :;6 0
Completenessdeductive, 160161, 182, 201,
203207, 215, 225227, 270280
expressive, 159, 170
functional, 170, 172175,232233of the method of deduction, 56,
208209of the 'natural deduction' apparatus,
280
Compound statement, 89, 71, 170
Conclusion, 3, 5, 179Condition
necessary, 16
sufficient, 16
Conditional
corresponding, 28, 51, 289
proof, 5053, 55, 5657, 5861, 201,250, 259
statement, 1416, 28, 5052, 5657,68, 170171, 289
Conj., 32
Conjunct, 8
Conjunction, 89, 70, 112, 297
principle of, 32, 101, 192, 195, 198,200
Conjunctive normal form, 225227, 286Boolean, 288289, 293
Connective, 9, 13
Consequent, 14
Consistency, 6263, 79n., 159161, 182,247248, 275279
Constant
individual, 64, 243, 276, 278
predicate, 243
propositional, 242
Constructive dilemma, 32, 49, 50, 200
Contingent, 26, 6263
Contradiction, 26, 5355, 57, 61, 63,148149, 159, 166167, see also
Appendix C
Contradictory, 910, 26, 6263, 67, 69,294
co�traries, 67, 69
Conventionof association to the left, 171172,
245
governing '', 11, 171, 244
governing '.', 171,244
Fallacyof affirming the consequent, 22
of denying the antecedent, 2223
Feigl, H., 309n.
Firstorder
function, 302, 305function calculus, see Chapter 9
proposition, 303304
Form
argument, 1823, 28, 289Boolean normal, 287289, 292293
elementary valid argument, 3132
normal, 263270, 286, 287
prenex normal, 263265Skolem normal, 267270
specificof an argument, 19, 23, 32
specificof a statement, 25
statement, 25, 2728of valid categorical syllogism,
294295Formal
criterion, 163deductive system, 157161, 295,
297
definition, 147
equivalence, 305nature of validity, 18
proof of validity, 3032, 3740, 61,8990
truth, 2526
Formula, 158159, 163, 167, 245
associated propositional, 247248of type R, 267well formed, 162164, 168169, 245
Free occurrence of a variable, 84, 108,246
Freeing of bound variables, 90; see also
EI, UI
Frege, G., 142n., 149n., 188, 282n.Function
binary, 174
calculus, 210, see also Chapter 9
, dyadic, 174
monadic, 172
order of, 302303
propositional, 65, 8384, 8993, 114
singulary, 172
ternary, 174
triadic, 174
L.S., 188
Lambtla, 274
Langford, C. H., 23On.
Language, 56, 163164, 165167levels of, 165167, 306309
object, 165167, 306309
Syntax, 165, 100
Least upper bound, 305
Lee, Karen, ix
Leibniz, 137Levels of language, 165167,306309Lewis, C. I., 230n.
Liar paradox, 166; see also Appendix CLimited
generality, 120
scope of an assumption, 6061, 75,90,9697
Lincoln, 112, 122, 136, 141
Lobachevsky, 156, 210
Loftin, Robert W., ix
Logicdefinition, 1
science of, 152
study of, 1
symbolic, 57task of deductive, 3
Logicalanalogy, 1819
equivalence, 2728, 37
proof, 162
sum, 202, 203, 290
truth, 79n., 104111, 116, 271, 273
types, 148149; see also Appendix C
Logistic system, 161164, 295; see also
Chapters 7, 8, 9 passim.Lower function calculus, 242n.
Lukasiewicz, J., lOOn., 188, 210, 231
Luke, 71
Proof (cont.)of functional completeness, 172175,
232233of functional incompleteness,
176177
of incompleteness of rules, 4750
of independence of axioms, 160,182187
indirect, 5356, 57, 61, 201
of invalidity, 4546, 7881in R S., 190192reductio ad absurdum, 5356,57,61,
6263
shorter, 101of tautology, 5657of validity, 3032, 3740
versus demonstration, 190192
Proposition, 2, 5; see also Statement
categorical, 290, 293294
general, 6470, 78
multiply general, 8387
negative, 65, 67
numerical, 140141orders of, 303304
particular, 67
relational, 112122
singly general, 83
singular, 6465, 84
subjectpredicate, 6770, 290
universal, 67
Propositionalcalculus, 164; see also Chapters 7
and 8
constant, 242
function, 65, 8384, 8993, 114
symbol, 167, 242, 272
variable, 168, 242
Proverbs, 14, 71, 126
Protasis, 14
Psalms, 14, 71, 125
Psi, 69, 9091
Psychology, 12
Punctuation, 1112, 227232
Pythagoras, 153154
Strongdisjunction, 10, 170
induction, 176177
Subclass, 292293
Subcontraries, 67, 69
Subject term, 64, 84,290
Subjectpredicate propositions, 6770,118, 290
Subset, 274
Substitution instance, 18, 25, 3132, 65,66, 68, 8485, 90
Substitution rule for functional
variables, 269n.
'Sum, logical, 202, 203, 290
Summand, 203
Superlative, 143
Suppe, Frederick, ix
Swartz, Norman, ix
Syllogismcategorical, 294295
disjunctive, 1011, 1920, 32, 200
hypothetical, 21, 22, 32, 4849, 194,200
Symmetrical relations, 130
Syntactical, 163164, 165
variable, 244
Syntax language, 165, 166, 178
Systemdeductive, see Chapter 6
logistic, 161164, 295; see also
Chapters 7, 8, 9 passim