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\T7       42$SU_"y#$"$g֚6<,CN!s":$"$IVQ5$p$"$d?:) 0AA@8)*  g4LdLdZ ׯ0ppp@ <4BdBd8vS0l -g4dd\ ׯ0p@ ppʚ;ʚ;<4dddd8S08r0___PPT10 pp2___PPT9/ 0? %O =V4Corso di logica matematica <  Prima lezione $  "Introduzione: Antinomie logiche e semantiche. Antinomia di Russel. A l insieme di tutti gli insiemi X che non hanno se stessi come elemento. r (($K Antinomia semantica  Un uomo dice  IO STO MENTENDO . Se egli mente dice la verit; se dice il vero, mente. :^Y( `'5La frase scritta sulla diapositiva successiva falsa06#P P P 6(4La frase scritta sulla diapositiva precedente vera05#P PP 5 Uno dei compiti della logica matematica quello di determinare il corretto uso dei simboli e delle loro combinazioni per accertare che cosa si pu dimostrare usando assiomi e regole di inferenza. L(,((  GIl calcolo proposizionale: connettivi proposizionali e Tavole di verit,H-@   Consideriamo solo combinazioni vero-funzionali , nelle quali la verit o la falsit della nuova proposizione determinata dalla verit o falsit delle proposizioni che concorrono a formarla. ph (((  Negazione P 6A A V F F V l<<< <<<  Congiunzione H0  A B AB V V V V F F F V F F F F `z $((`($&E6  pForme enunciative. 1)Tutte le lettere enunciative, eventualmente con indice numerico, sono forme enunciative (A, B, C, A1, B2...); 2)SE e sono forme enunciative, allora lo sono anche (), (), (), () e (). Sono forme enunciative solo quelle espressioni determinate per mezzo della 1) e della 2).<h&F   0Z 9 Ciascuna forma enunciativa determina una funzione di verit che pu essere rappresentata graficamente da una tavola di verit per la forma enunciativa. (0 6A B C (A) ((A)B) (((A)B)C) V V V F V V F V V V V V V F V F F V F F V V V V V V F F V F F V F V V F V F F F F V F F F V V F  --------&LH Tautologie$ < d Una forma enunciativa una tautologia se e solo se la sua funzione di verit ha solo il valore V. T`d`,, e*Se una tautologia , si dice che implica logicamente oppure che una conseguenza logica di . Se una tautologia, si dice che e sono logicamente equivalenti. Le tavole di verit costituiscono una procedura effettiva che ci permette di determinare se una forma enunciativa una tautologia. Una forma enunciativa una contraddizione se la sua funzione di verit ha solo il valore F. T  ? 0 Proposizione 1.1 B  Se e () sono tautologie, allora anche una tautologia. ?666666666 B Proposizione 1.2 <  Se una tautologia contenente come lettere enunciative A1, A2,...,An, e si ottiene da per sostituzione di A1, A2,...,An con, rispettivamente, forme enunciative 1, 2,..., n, allora una tautologia((6(( (( (( ((((((( (( (( ( (((( ((( ((( ( ((((@F5S!Proposizione 1.3 < t Se 1 deriva da 1 per sostituzione di a una o pi occorrenze di , allora (() (11)) una tautologia . Quindi, se e sono logicamente equivalenti, lo sono anche 1 e 1. (( ( ((((((((( (((((((((((((-(((((( "Proposizione 1.4 <  Ogni funzione di verit generata da una forma enunciativa in cui occorrono i connettivi , , . dZ,,,,,,, f x1 x2 x3 F(x1,x2,x3) V V V V ABC F V V F V F V V ABC F F V V ABC V V F F F V F F V F F F F F F V ABC (ABC) (ABC) (ABC) (ABC). 8    *>@8$Forma normale disgiuntiva0 Una forma enunciativa scritta come disgiunzioni di congiunzioni di lettere enunciative o delle loro negazioni in forma normale disgiuntiva. Da quanto si visto, ogni forma enunciativa pu essere scritta in forma normale disgiuntiva2t$$_$ PR2 0` ff33f` ffD3f` MMM` f3fD"Yf` f3f6f3f` 3f̙f>?" dd@,|?" dd@   " @ ` n?" dd@   @@``@n?" dd@  @@``PR    @ ` ` p>>  ''&(    H2,$@ 0>   0e0e     ?B)CDElFt @ 5% 8c8c     ?1 0u0@Ty2 NP'p<'p @A)BCD|E?||c $)6N_i> _4% #j +  y  p  d(78@ #  3 0e0e     ?BCnDEF @ 5% 8c8c     ?1 0u0@Ty2 NP'p<'p @A)BCD|E?||//  MoK0YVS_^tT, [v] T5  G r C  ]smm#h g:Y/x|!  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