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P R E F A C I O

SEGUNDA PARTE

LA LÓGICA DE PRIMER ORDEN

6. INTRODUCCIÓN A LA LÓGICA DE PREDICADOS 147

6.1 INTRODUCCIÓN INFORMAL A LC 148

6.1.1 TÉRMINOS SINGULARES 148

6.1.2 TÉRMINOS GENERALES 149

6.1.3 ENUNCIADOS SINGULARES 150

6.1.4 ENUNCIADOS GENERALES 152

6.2 INTRODUCCIÓN FORMAL AL LENGUAJE LC 153

6.3 SINTAXIS FORMAL DE LC 162

6.4 ENUNCIADOS CATEGÓRICOS 168

6.5 ENUNCIADOS CATEGÓRICOS COMPLEJOS 177

6.6 ENUNCIADOS CON CUANTIFICACIÓN MÚLTIPLE 185

6.7* LA FORMA NORMAL PRENEXA DE UNA FÓRMULA DE LC 193

6.8 LÓGICA CUANTIFICADA CON IDENTIDAD 200

6.9 DESCRIPCIONES DEFINIDAS 206

7. SEMÁNTICA DEL LENGUAJE LC 212

7.1 LA VALUACIÓN E INTERPRETACIÓN DE LAS FÓRMULAS DE LC 213

7.1.1 EL UD 213

7.1.2 LA INTERPRETACIÓN DE LOS PREDICADOS 214

7.1.3 LA INTERPRETACIÓN DE LAS CONSTANTES 217

7.1.4 VALUACIONES NO INTERPRETADAS 218

7.1.5 DEFINICIÓN FORMAL DE UNA VALUACIÓN 220

7.2 REGLAS DE VALUACIÓN DE LC 221

7.2.1 REGLAS PARA FÓRMULAS ATÓMICAS 221

7.2.2 REGLAS PARA FÓRMULAS MOLECULARES 223

7.2.3 REGLAS PARA FÓRMULAS CUANTIFICADAS 225

7.3 MODELOS 233

7.4 PROPIEDADES SEMÁNTICAS DE LAS FÓRMULAS DE LC 239

7.5 EQUIVALENCIA SEMÁNTICA 244

7.6 CONSISTENCIA SEMÁNTICA 247

7.7 VALIDEZ E IMPLICACIÓN SEMÁNTICA 249

7.8 SEMÁNTICA PARA LCI 253

7.9* EL FRAGMENTO MONÁDICO DE LC 255

C O N T E N I D O

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P R E F A C I OC O N T E N I D O

11.2.1 EL SISTEMA T 363

11.2.2 EL SISTEMA KB 364

11.2.3 EL SISTEMA K4 366

11.2.4 EL SISTEMA D 367

11.3 EXTENSIONES COMPUESTAS DE K 370

11.3.1 EL SISTEMA B 370

11.3.2 EL SISTEMA S4 371

11.3.3 EL SISTEMA S5 372

12. DEDUCCIÓN NATURAL MODAL 375

12.1 EL SISTEMA T 375

12.2 LOS SISTEMAS S4 Y S5 385

APÉNDICE

RESPUESTAS A LOS EJERCICIOS PARES 389

BIBLIOGRAFÍA 523

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252

C A P Í T U L O 7

Un conjunto de fórmulas de LC implica semánticamente en LC a

una fórmula P si y sólo si no hay ninguna valuación bajo la cual todos los

miembros de sean verdaderos y P sea falsa.

De nuevo utilizaremos el símbolo “ ” para denotar la implicación semántica.

Podemos reescribir el argumento (39), por ejemplo, en los siguientes términos:

(39a) {~(�y)Dy, (�y)Dy � (�x)Dx)} “(�x)~Dx”

Recordemos que la fórmula implicada está entre comillas porque el signo “ ” no es

un conector veritativo-funcional, sino que hace parte de nuestro metalenguaje. Su

labor no es establecer una relación lógica entre las fórmulas, sino que se limita a

indicarnos, en el metalenguaje, que la fórmula “(�x)~Dx” es una consecuencia se-

mántica del conjunto formado por las fórmulas “~(�y)Dy” y “(�y)Dy � (�x)Dx)”.

Ejercicio 7.7

A. Demuestre que cada uno de los siguientes argumentos es inválido en LC construyen-

do una interpretación en la cual las premisas sean verdaderas y la conclusión sea falsa.

1. Iap 2. (�x)~Vxa

Ipa � Iaa (�x)~Vxa

3. (�x)(Ax & Bx) 4. (�y)(My � Ny)

(�x)(Ax & Cx) Nb

(�x)(Bx & Cx) Mb

5. (�x)(Ax � Bx) � (�x)Cx 6. ~Sc � (~(�y)Sy � (�y)Sy)

(�x)(Cx � Bx) (�x)Sx

(�x)(~Ax � Bx)

7. (�z)(Kz � Lz) 8. (�w)(�y)(Awy � Bwy)

~(�z)Kz (�w)(�y)(Awy � (Bwy & Byw))

(�z)~Lz

9. (�x)(Hx � Ixx) 10. (�x)Txc � Ac

(�y)Hy (�x)~Txd � Ad

(�y)(�x)(Hy & Iyx) (�x)Txc

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Page 302: Introducción a la Lógica Moderna, Páez

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Page 304: Introducción a la Lógica Moderna, Páez

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-- !- =�">U?;-K-='"-K-&;">V- ?@ABC7D7EF-

-- #- ?;-K-='G-K-&;G>- ?@ABC7D7EF-

-- J- 'G-K-&;G- #-=K2>-

-- M- =�">='"-K-&;">- J-=�$>-

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- <M- - - T==�!>.!-�-L;>- <R-='>-

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- !- =�!>=:!-�-:!>- H-=�$>--

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- 3- :;-�-:;- <-=�2>-

-- H- :;- ?@ABC7D7EF-

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-- #- :;- ?@ABC7D7EF-

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- M- :;- 3I-HOJ-=�2>-

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- 3- =�">=1"-�-/">-- ?@ABC7D7EF-

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- 3- =�!>=�">U=(!-K-T(">-K-T1!"V- ?@ABC7D7EF-

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-- J- =�">U=(;-K-%">-�-1;"V- H-=�2>-

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-- H- =�!>T=:!-K-T0!>- <-=$+>-

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- #- T=TS;-�-0;>- ?@ABC7D7EF-

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34- )567859-U=�!>==:!-K-0!>-�-S!>-K-T=�!>=:!-�-S!>V-�-T=�!>0!#

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- 3- T=�!>=:!-�-S!>- <-=K2>-

- H- =�!>T=:!-�-S!>- 3-=$+>-

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- N- =:;-K-0;>-�-S;- "-=�2>-

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- F- =%HQ- >?@AB6C6DE-

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- !- 9�!:=%!!- G-9�$:-

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- F- =0H- >?@AB6C6DE-

-- G- 9�!:0!- >?@AB6C6DE-

- !- 0C- >?@AB6C6DE-

- J- C-P-H-�-C-P-Q- <-9�2:-

- K- C-P-H- >?@AB6C6DE-

- #- =0Q- >?@AB6C6DE-

- L- =0C- FN-K-9P2:-

- "- 0C- !-9':-

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- 3- =�$>U=L$D-K-=�">=L"D-�-$-Y-">>-K-.$;V- ?@ABC7D7EF-

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34- )567859-=�!>-!-Y-;-

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- H- );- ?@ABC7D7EF-

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- M- =�">U=."-K-L">-K-+;"V- <I-!OJ-=�2>-

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-- "- LI-<-- 3I-#-=!>-

-- N- T.I-<- TLI-<- J-=TK>-

- �( �(

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Page 516: Introducción a la Lógica Moderna, Páez

!"#

$ % & ' ( % & ) * & + * + , - & + % . % $ / 0 / 0 - & + ' * $ % &

,1+2345671+89+6:1+;16;<7<=>1+8:+87+9?9;851+@A

,1+2345671+89+6:1+;16;<7<=>1+8:+87+9?9;851+@A

BA+ C+ DED!DF+ +�FG+"+ H/+++ +

+ #+ D!DFI+"+"+ DD!DFI+"+"+ C+ED G+++ J+ D�FI+"+"+ �FI+"+"+ C+ED G+++ B+ �DDFI+"+"+ #+EKG+++ !+ !DFI+"+ J+EKG+++ L+ "!C+ B+E�G+++ M+ DDFI+C+ B+E�G+++ N+ DFI+C+ !I+L+E!G++ O+ �+ !DFI+"+ #+EDDG+++ C"+ "!C+ J+E�G+++ CC+ FI+C+ J+E�G+++ C#+ DFI+C+ OI+C"+E!G

++ �!

LA+ C+ DP�E*+�+FG+�+E�*+�+�FGQI+"+"+ H/+++ #+ �E*+�+FGI+"+"+ C+ED�G+++ J+ DE�*+�+�FGI+"+"+ C+ED�G+++ B+ D�*I+"+"+ J+ED�G+++ !+ D�FI+"+"+ J+ED�G+++ L+ !D*I+"+ B+EKG+++ M+ !DFI+"+ !+EKG+++ N+ "!C+ #+E�G+++ O+ *+�+FI+C+"+ #+E�G++

++ C"+ *I+C+ FI+C+ O+E�G+++ CC+ D*I+C+ + + LI+N+E!G++ C#+ �+ DFI+C+ MI+N+E!G

+ + + �!

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Page 517: Introducción a la Lógica Moderna, Páez

!#RH

$ % & ' ( ) * + + $ , % - . - / . - / , 0 $ + . - 1 ( ) 2 ' % .

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X;-C7\@75F[5-8;X@;D7EF9

! %R- %<- b&c-

- -

! !=�&I-%R>-Y-;((a((!=��&I-%R>-Y-<-

- -

-

-

! %R- %<- %3- -

- -

- b?c- bT?c- !

! "=!?I-%R>-Y-;((a(("=!!?I-%R>-Y-<-

-

<R4- <- T=�&-�-��&>I-R-"- 1+-

-- 3- �&I-R-"- <-=T�>-

-- H- T��&I-R-"- <-=T�>-

-- !- !T�&I-R- H-=)>-

- #- R&<- 3-=�>--- J- &I-<- 3-=�>--- M- T�&I-<-"- !I-#-=!>-

-- "- !T&I-<- M-=)>-

- -

"4- <- T=!?-�-!!?>I-R-"- 1+-

-- 3- !?I-R- <-=T�>-

-- H- T!!?I-R-"- <-=T�>- -

-- !- �T!?I-R-"- H-=)>-

-- #- R&<- !-=�>--- J- T!?I-<-"- !-=�>-- M- �T?I-<-"- J-=)>-

-- "- ?I-<- 3I-#-=!>-

- N- <&3- M-=�>--- <R- T?I-3- M-=�>--

%R %<

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Page 518: Introducción a la Lógica Moderna, Páez

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2*-

34- <- T�TLI-R-"- 1+-

-- 3- TLI-R- 1+-

-- H- !TTLI-R- <-=)>-

-- !- R&R- <-='>-

- #- TTLI-R-"- HI-!-=!>-

- - - �(

!4- <- T�2I-R-"- 1+-

- 3- TT2I-R- 1+-

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- !- R&R- <-='>-

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- -

(- /;-ZE6W@X;-5C-@F;-[;@[BXB\d;-5F-5X-C7C[5W;-_4-

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- #- T!=TL-�-T.>I-R-"- H-=T�>-

- J- �T=TL-�-T.>I-R-"- #-=)>-

- M- R&<- J-=�>-- "- T=TL-�-T.>I-<-"- J-=�>-- N- TLI-<- "-=T�>-

- <R- TT.I-<- "-=T�>-

- <<- .-�-LI-<-"- !I-M-=!>-

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Page 519: Introducción a la Lógica Moderna, Páez

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8;X@;D7EFI-X;-A65W7C;-5C-856^;^56;-a-X;-DBFDX@C7EF-Z;XC;9

! bT0I-TSc! %R- %<- b0I-Sc-

- - -

! "=�=0-�-S>I-%R>-Y-;((a(("=��=0-�-S>I-%R>-Y-<-

- -

%R %<

2X-;6\@W5F[B-5C-C5W]F[7D;W5F[5-8]X7^B-5F-5X-C7C[5W;-_L4

34--

34- <- :-�-0I-R-!- 1+-

- 3- T!T!=:-K-T0>I-R-!- 1+-

- H- �TT!=:-K-T0>I-R-!- 3-=)>-

- !- R&<- H-=�>-- #- TT!=:-K-T0>I-<-!- H-=�>-- J- !=:-K-T0>I-<- #-=TT>-

- M- <&R- !-=?>-

- "- :-K-T0I-R-!- JI-M-=!>-

- N- :I-R- "-=K>-

- <R- T0I-R- "-=K>-

-

- <<- T:I-R- 0I-R- <-=�>-

- �( �(

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!4- <- �=0-�-S>I-R-"- 1+-

-- 3- T��=0-�-S>I-R-"- 1+-

-- H- !T�=0-�-S>I-R- 3-=)>-

- !- R&<- <-=�>--- #- 0-�-SI-<-"- <-=�>--- J- T�=0-�-S>I-<-"- HI-!-=!>-

-- M- !T=0-�-S>I-<- J-=)>-

- -

- "- 0I-<- SI-<- #-=�>- -

- N- <&R- <&R- !-=?>-

- <R- T=0-�-S>I-R-"- T=0-�-S>I-R-"- MI-N-=!>-

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- <3- TSI-R- TSI-R- <R-=T�>-

-

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Page 520: Introducción a la Lógica Moderna, Páez

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8;X@;D7EFI-X;-A65W7C;-5C-856^;^56;-a-X;-DBFDX@C7EF-Z;XC;9

2X-;6\@W5F[B-5C-C5W]F[7D;W5F[5-8]X7^B-5F-5X-C7C[5W;-_!4

44--

34- <- ��!.I-R-"- 1+-

- 3- T�.I-R-"- 1+-

- H- !T.I-R- 3-=)>-

- !- R&<- <-=�>-- #- �!.I-<-"- <-=�>-- J- T.I-<- HI-!-=!>-

- M- <&3- #-=�>-- "- !.I-3- #-=�>-- N- R&3- !I-M-=&>-

- <R- T.I-3- HI-N-=!>-

-

-

- - - - - - - - - ---

-

! %R- %<- %3- -

- -

- bT.c- bT.I-!.c- !

! !=��!.I-%R>-Y-;((a((!=�.I-%R>-Y-<(-

!4- <- T�=.-K-TL>I-R-"- 1+-

- 3- T=�!.-�-�!L>I-R-"- 1+-

- H- !T=.-K-TL>I-R- <-=)>-

- !- �!.I-R-"- 3-=T�>-

- #- T�!LI-R-"-- 3-=T�>-

- J- !T!LI-R- #-=)>-

- M- R&<- !-=�>-- "- !.I-<- !-=�>-- N- T!LI-<-"- JI-M-=!>-

- <R- �TLI-<-"- N-=)>-

- <<- <&3- <R-=�>-- <3- TLI-3- <R-=�>-- <H- .I-3- "I-<<-=!>-

- <!- R&3- MI-<<-=&>-

- <#- T=.-K-TL>I-3-"- HI-<!-=!>-

-

- <J- T.I-3- TTLI-3- <#-=TK>-

- - �( �(

%R %< %3

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Page 521: Introducción a la Lógica Moderna, Páez

!#RM

$ % & ' ( ) * + + $ , % - . - / . - / , 0 $ + . - 1 ( ) 2 ' % .

54-

34- <- _I-R- 1+-

- 3- T�=_-�-T_>I-R-"- 1+-

- H- !T=_-�-T_>I-R- 3-=)>-

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- #- T=_-�-T_>I-<-"- HI-!-=!>-

- J- T_I-<- #-=T�>-

- M- TT_I-<- #-=T�>-

- - �(

- 2X-;6\@W5F[B-5C-C5W]F[7D;W5F[5-8]X7^B-5F-5X-C7C[5W;-)4-

-

!4- <- T�=1-K-T%>I-R-"- 1+-

- 3- T�=1-�-%>I-R-"- 1+-

- H- !T=1-K-T%>I-R- <-=)>-

- !- !T=1-�-%>I-R- 3-=)>-

- #- R&<- <-=2>-

- J- T=1-K-T%>I-<-"- HI-#-=!>-

- M- T=1-�-%>I-<-"- !I-#-=!>-

- "- 1I-<- M-=T�>-

- N- T%I-<- M-=T�>-

-

- <R- T1I-<- TT%I-<- J-=TK>-

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- 2X-;6\@W5F[B-5C-C5W]F[7D;W5F[5-8]X7^B-5F-5X-C7C[5W;-)4-

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24-

34- <- �T�.I-R-"- 1+-

- 3- TT.I-R- 1+-

- H- R&<- <-=�>-- !- T�.I-<-"- <-=�>-- #- !T.I-<- !-=)>-

- J- <&R- H-=?>-

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Page 522: Introducción a la Lógica Moderna, Páez

#R"

' 2 ? P * 2 ? & . ? - . - / ( ? - 2 Q 2 ' + $ + $ ( ? - P . ' 2 ?

!4- <- �T��!LI-R-"- 1+-

- 3- TT�!LI-R-"- 1+-

- H- �!LI-R-"- 3-=TT>-

- !- R&<- <-=�>-- #- T��!LI-<-"- <-=�>-- J- !T�!LI-<- #-=)>-

- M- <&R- !-=?>-

- "- T�!LI-R-"- JI-M-=!>-

- N- !T!LI-R- "-=)>-

- <R- R&3- H-=�>-- <<- !LI-3- H-=�>-- <3- T!LI-3- NI-<R-=!>-

- - �(

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(

34-

34- <- �T!_I-R-"- 1+-

- 3- T�T_I-R-"- 1+-

- H- !TT_I-R- 3-=)>-

- !- R&<- <-=�>-- #- T!_I-<-"- <-=�>-- J- �T_I-<-"- #-=)>-

- M- <&3- J-=�>-- "- T_I-3- J-=�>-- N- R&3- !I-M-=&>-

- <R- TT_I-3- HI-N-=!>-

- - �(

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Page 523: Introducción a la Lógica Moderna, Páez

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$ % & ' ( ) * + + $ , % - . - / . - / , 0 $ + . - 1 ( ) 2 ' % .

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8;X@;D7EFI-X;C-A65W7C;C-CBF-856^;^56;C-a-X;-DBFDX@C7EF-Z;XC;9

!4- <- �!TSI-R-"- 1+-

-- 3- �0-�-�SI-R-"- 1+-

-- H- TT0I-R-"- 1+-

-- !- 0I-R- H-=TT>-

- #- R&R- <-='>--

- -

-- J- T�0I-R-"- �SI-R-"- 3-=�>-

-- M- !T0I-R- - J-=)>-

-- "- T0I-R- - #I-M-=!>- -

- N- �- R&<- J-=�>--- <R- - SI-<- J-=�>-- <<- - R&3- <-=�>-- <3- - !TSI-3- <-=�>-- <H- 3&3- <<-='>-

- <!- TSI-3- <3I-<H-=!>-

-

- --

- %<- bSc-

-

- b0c- %R- - -

-

- %3- bTSc-

- - - - -

! "=�!TSI-%R>-Y-;=(("=�0-�-�SI-%R>-Y-;((a(("=T0I-%R>-Y-<-

-

44- -

34- <- �!�!&I-R-"- 1+-

--- 3- T&I-R- 1+-

-- H- R&<-- <-=�>--- !- !�!&I-<- <-=�>--- #- <&<- H-='>-

-- J- �!&I-<-"- !I-#-=!>-

-- M- <&3- J-=�>--- "- !&I-3- J-=�>--- N- R&3- HI-M-=&>-

-- <R- 3&R- N-=?>-

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%R

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Page 524: Introducción a la Lógica Moderna, Páez

#<R

' 2 ? P * 2 ? & . ? - . - / ( ? - 2 Q 2 ' + $ + $ ( ? - P . ' 2 ?

!4-- <- !!��?I-R- 1+-

-- 3- T�?I-R-!- 1+-

-- H- !T?I-R- 3-=)>-

-- !- R&R- <-='>-

-- #- !��?I-R- <I-!-=!>-

-- J- ��?I-R-!- !I-#-=!>-

-- M- R&<- J-=�>--- "- �?I-<-!- J-=�>--- N- <&3- "-=�>--- <R- ?I-3- "-=�>--- <<- R&3- MI-N-=&>-

-- <3- T?I-3- HI-<<-=!>-

- - �((

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-

J4- <- !Te-�-!!T'I-R-!- 1+-

-- 3- �'I-R-!- 1+-

-- H- T�eI-R-!-- 1+-

-- !- !TeI-R- H-=)>-

-- #- !=!Te-�-!!T'>I-R- <-=�>-

-- J- R&R- <-='>-

-- M- !Te-�-!!T'I-R-!- #I-J-=!>-

- -

-- "- T!TeI-R- !!T'I-R- M-=�>-

- N- �(- !T'I-R- JI-"-=!>-

- <R- - R&<- 3-=�>-- <<- - 'I-<- 3-=�>-- <3- - T'I-<- NI-<R-=!>-

- �(

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Page 525: Introducción a la Lógica Moderna, Páez

!#<<

$ % & ' ( ) * + + $ , % - . - / . - / , 0 $ + . - 1 ( ) 2 ' % .

"4- <- �!%-�-��PI-R-"- 1+-

-- 3- T�PI-R-"- 1+-

-- H- TT!%I-R-"- 1+-

-- !- !%I-R- H-=TT>-

-- #- !TPI-R- 3-=)>-

-- J- !=�!%-�-��P>I-R- <-=�>-

-- M- R&R- <-='>-

-- "- �!%-�-��PI-R-"- JI-M-=!>-

-

-- N- T�!%I-R-"- ��PI-R-"- "-=�>-

- <R- !T!%I-R- - N-=)>-

- <<- T!%I-R-- - MI-<R-=!>-

-- <3- �(- R&<-- N-=�>--- <H- - �PI-<-"- N-=�>--- <!- - <&3- <H-=�>--- <#- - PI-3- <H-=�>--- <J- - R&3- <3I-<!-=&>

-- <M- - TPI-3- #I-<J-=!>-

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Page 526: Introducción a la Lógica Moderna, Páez

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$ % & ' ( % & ) * & + * + , - & + % . % $ / 0 / 0 - & + ' * $ % &

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%1+23456789:+7;+;76<89=>267897+?<1=@:+78+71+;=;9762+&!A

"BA+ "+ !!%+�+!!CD+B+"+ E/+++ #+ �FCD+B+"+ E/+++ G+ FF!%D+B+"+ E/+++ H+ !%D+B++ G+IFFJ++++ !+ !I!!%+�+!!CJD+B++ "+I�J+++ K+ B!B+ "+I$J+++ L+ !!%+�+!!CD+B+"+ !D+K+I!J++ M+ B!"+ #+I�J+++ N+ FCD+"+ #+I�J+

++ "B+ F!!%D+B+"+ !!CD+B+ L+I�J+++ ""+ �F!%D+B+"+ + "B+IOJ+++ "#+ B!#++ + ""+I�J+++ "G+ F!%D+#+"+ + ""+I�J+++ "H+ �F%D+#+"+ + "G+IOJ+++ "!+ #!G+ + "H+I�J+++ "K+ F%D+G+ + "H+I�J+++ "L+ B!G+ + "#D+"!+I)J+++ "M+ %D+G+ + HD+"L+I!J+++ "N+ �++ !CD+B+ KD+"B+I!J+++ #B+ CD+"+ MD+"N+I!J+

"#A+ "+ �!FPD+B+!+ E/+++ #+ �/+�+�PD+B+!+ E/+++ G+ FF/D+B+!+ E/+++ H+ /D+B+ G+IFFJ+++ !+ B!"+ "+I�J+++ K+ !FPD+"+ "+I�J+++ L+ !+I�/+�+�PJD+B+ #+I�J+++ M+ B!B+ "+I$J+++ N+ �/+�+�PD+B+!+ LD+M+I!J+

++ "B+ F�/D+B+!+ �PD+B+!+ N+I�J+++ ""+ !F/D+B+ + "B+IOJ+++ "#+ F/D+B+ + MD+""+I!J+++ "G+ �++ B!#++ "B+I�J+++ "H+ + PD+#+ "B+I�J+++ "!+ + "!B+ !+I&J+++ "K+ + "!#+ "GD+"!+I)J+++ "L+ + FPD+#+ KD+"K+I!J+

++ + �! !

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Page 527: Introducción a la Lógica Moderna, Páez

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<!- <- ��&-�-!�PI-R-"- 1+-

-- 3- TT�=�&-K-T�P>I-R-"- 1+-

-- H- �=�&-K-T�P>I-R-"- 3-=TT>-

-- !- !=��&-�-!�P>I-R- <-=�>-

-- #- R&R-- <-='>-

-- J- ��&-�-!�PI-R-"- !I-#-=!>-

- M- R&<- H-=�>--- "- �&-K-T�PI-<-"- H-=�>--- N- �&I-<-"-- "-=K>-

-- <R- T�PI-<-"- "-=K>-

-- <<- <&3-- N-=�>--- <3- &I-3-- N-=�>--

-- <H- T��&I-R-"- !�PI-R- J-=�>-

-- <!- !T�&I-R- - <H-=)>-

-- <#- T�&I-R-"-- - #I-<!-=!>-

-- <J- !T&I-R- - <#-=)>-

-- <M- R&3- - MI-<<-=&>-

-- <"- T&I-3- - <JI-<M-=!>

-- <N- �(- !TPI-<-- <R-=)>-

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Page 532: Introducción a la Lógica Moderna, Páez

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- <- �U=/-�-1>-K-=/-�-e>V- ?@ABC7D7EF-

- 3- �P-�-T�e- ?@ABC7D7EF-

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