Andi Mack | veronica streaming vf | The.Adventures.Of.Stella.Star.1978.720p.BluRay.H264.AAC-RARBG

Appendix

Appendix

APPENDIX BY NORMAN M. MARTIN 1. 9 24 Formula (b’) is indeed unprovable in the system of strict implication, as is indicated. If one uses the follo...

184KB Sizes 3 Downloads 28 Views

Recommend Documents

Appendix
This chapter introduces filter and ultrafilters with some basic lemmas. It deals with the filter D (α) (generated by the

APPENDIX BY NORMAN M. MARTIN

1.

9

24

Formula (b’) is indeed unprovable in the system of strict implication, as is indicated. If one uses the following four-valued matrices (due to Dr William T. Parry)

P

r 1 2 3 4

1 2 3 1 2 3 2 2 4 3 4 3 4 4 4

4 4 4 4 4

-pop 4 3 2

1 1 1

1

3

P
75

APPENDIX

- --

By the principles of adjunction and inference, we obtain:

“0 (f) which is equivalent to

(P“ P )

-

(9) (P
p.q“r:<:po.qowT

th)

. - .- - . -

This is equivalent by definition 11.02 to

0[p q

ti)

r :

( p0 .q 0

-

r)J

“ p . q r : . ( p 0 q 0 r)” is by 12.5 equivalent to “ - ( P O .qO - r ) : q - r : . p 7 ’ ; hence by the principle of substitution of equivalents (L.a.L., p. 125), (i) is equivalent to

- -

( p 0 . q 0 r ) :q r : . < p which by definition 17.01 and the elimination of double negations yields (b”). (j)

N

N

Q 26

2.

If (1 3‘) is translated into the language of Principia Mathematica, it yields : (a’)

p3q.3r :-r :.3.p3-q

Substituting “ p 3 q” for “p” and “r’, for “q” in 2.16, we get: (b’)

p3qQ3r:3:.-r3::(p3q)

By the principle of importation we get tc’)

p3qp.3r::r:.3.-(p3q)

By 2.51 and the principle of the syllogism, we obtain (a’): Translating (13’) into the system of strict implication, we obtain : (d’)

p
<.p

<-q

76

APPENDIX

which is not provable since it takes the value 4 if p takes 1, q 2 and r 4 (Cf. note I). If “ p < gy’ is substituted for “r” in (d’), we get

(4

p
:.<.p<-q

which is inconsistent with Lewis’ system since by use of 12.1, the principle D (L.a.L., p. 182), and 20.01 we can obtain

(3%d [- ( P <

(f’)

-

!I)P

<

-

21

which contradicts 12.9. If ( 13ii) is translated into the language of Principia Mathematica we get: :. 3 . p . - q

p 3 4 . 3 r :-r

(a”)

Since “ p 3 q” is equivalent to (b”)

N

(p.

-

( p . N q)” this is equivalent to

“-

q ) :3 r : .

N

r : .3. p

.

N

q

which is consequence of the correct inference scheme (1’). If (13”) is translated into the system of strict implication we obtain :

p < q . < r : - r :. < p - . g

(c”)

By the use of the matrices mentioned in note 1, if p takes 1, q 2 and r 4, (c”) will take the value 4, and hence is not provable. If we substitute “ N p” for “q” and “ p < p” for 3’’ we get (d”) p

< - p . < .p <

- p :

-

(p <-p)

: . < .p

--

P

This is equivalent to

p<-p.<.p
(e’0

but this together with 20.01 leads to the contradictory result

(3P?2) (P

(f”) 3.

Q

-

-

4 :P * 4 )

2s

(h) and (h‘) are provable in the system of strict implication as

77

APPENDIX

the author seems to suggest. The proof depends on the rearrangement of the terms of the antecedent by 12.5, from the equivalent form

p . p < q : q < r :. < r

(i)

which is provable by the use of 19.6, 11.6 and 11.7. From (h), we can proof (h’) by theorem 12.6. f 30

4.

If (29) is translated into the language of the system of strict implication, we obtain :

(4

p
:-q:.
This is provable with the use of 11.6, substituting “p” and “ N p” for “q” giving (b)

-q

<- p . - p

“-

q” for


One can derive (a) from (b) by the use of 12.43 and 19.6. (b) is an instance of (21) (Cf. note 3). (29’) can be derived from (29) by 12.6. If the inference scheme which can be correlated with propositional form (37) is translated into the language of the system of strict implication, we obtain :

p < q . p < r : p :. < q r This is equivalent to

p.p
75

6.

APPENDIX

$ 34

If (61’) is translated into the language of the system of strict implication, we obtain

p < q . <.r
<.p <-q

If p and q take 1 and r and s take 4, this formula takes the value 4 and hence is unprovable (Cf. note 1).

-

If we substitute ( p p)” for both “p” and “q” and ‘ L pN p” for both “8’and “s”, we obtain: “ N

-(p-p).<.-(p-p):<:p-p.<.p-p:.p-y .<.-(p-p)::<:-(p-p).<.p-p

If we substitute “ p

-

p” for both “p” and “q” in 12.41, we get:

”(p-p).<.-(pNp):<

:p-p.<.p-p

By the use of 12.1, 11.02, and 18.12, we obtain: p-1). <--((P-P) By the principles of adjunction and inference and 12.9, we get:

P-P which is contradictory. 7.

p

37

79

APPENDIX

By the use of the principle of equivalence, 12.3 and 12.15, (a 1) can be shown to be equivalent to (a 2), (a 3) to (a 4), (b) to ( c ) and (d) to (e). Substituting “0N p” for “p” in 12.9, we obtain a formula equivalent to (a 1) ; substituting “0p” for “p” in the same theorem, we get an equivalent of (a 3). By the use of 19.77, 20.01 and principle C (L.a.L., p. 182), we can prove an equivalent of (b). Substituting “ N 0p Q p” for “p“ and “ p p” for “q” in 12.44, we obtain: N

-

N

-op“o“p.<.p-p:<:“(p”p).< (-

0P

0 -PI

By use of 18.41, 18.42, 19.68, and 11.1, we obtain -oP-o--P.<.PNp yielding (e) by 12.9, and the principle of inference. 8.

5

40

Pormula (1) is indeed unprovable; if p takes the value 1 and q the value 2, the formula takes the value 4 (Cf. note 1). 9.

$ 42

If the system of strict implication is used as metascience, formula (1) is correct, but (2) is incorrect. Translated, they read :

(1) is equivalent to

p.p3-q:<-q which is an instance of 14.29. If, in (2), p and p take the value 4, (2) takes the value 4 (Cf. note 1).