Disce IV technicas deductionis necessarias
Summarium:In hac lectione describuntur quattuor technicae deductionis logicae propositionis, ad calculum propositionis rudimentarium hactenus propositum locupletandum. Exponitur regula praesumptionis eiusque coniunctio cum regula monotoniae, item syllogismus hypotheticus et duae viae ad hanc regulam deductionis obtinendam. Explicantur quoque aequivalentiae negationis duplicatae et contrapositivum implicationis.
Propositi Discendi:
Expleta hac lectione discipulus poterit
- Meminisse structuram ratiocinii et simplices exempla.
- Intellegere regulam praesumptionis eiusque relationem ad theorema deductionis.
- Intellegere regulam syllogismi hypothetici eiusque nexum cum modo ponente.
- Adhibere theorema deductionis in logica propositionis.
- Adhibere regulam monotoniae in deductione expressionum.
- Intellegere aequivalentias negationis duplicatae et contrapositivum implicationis in logica propositionis.
- Noscere demonstrationes technicarum deductionis easque in praxi adhibere posse.
INDEX RERUM
REGULA PRAESUMPTIONIS (PRE)
SYLLOGISMUS HYPOTHETICUS (SH)
AEQUIVALENTIAE NEGATIONIS DUPLICATAE (DN)
AEQUIVALENTIA CONTRAPOSITIVI IMPLICATIONIS (CPI)
Iam vidimus qualis sit structura ratiocinii et exempla simplicia. Nunc scientiam istam experiemur ratione utens IV technicis deductionis logicae propositionis. Per hoc non solum ostendemus haec praecepta valere, sed etiam initium faciemus ditandi processum qui calculum propositionis, hactenus rudimentarium, ex eo statu eruat.
Si \alpha, \beta et \gamma sunt expressiones calculi propositionis, tunc fieri potest ut sequentia deductionis praecepta ex fundamentis concludantur:
Regula Praesumptionis (Pre)
Facillima omnium regula deductionis est regula praesumptionis. Haec directe obtinetur applicando reciprocum theoremae deductionis ad theorema \vdash(\alpha\rightarrow\alpha). Si hoc tibi arcano sermone videtur enuntiatum, omnia quae scire debes inveniuntur hic.
\{\alpha\}\vdash \alpha
Cum regula monotoniae coniuncta, haec tibi permittet expressiones utiles intra deductiones tuas inserere.
Syllogismus Hypotheticus (SH)
Syllogismus hypotheticus, sive transitivitas implicationis, species est evolutionis modi ponentis. Eius enuntiatio est haec:
\{(\alpha\rightarrow\beta), (\beta\rightarrow\gamma)\}\vdash (\alpha\rightarrow\gamma)
Multae sunt viae ad hanc regulam deductionis obtinendam, duas mox considerabimus.
Si ex expressionibus ratiocinemur, facile erit ratiocinium sequentem construere:
| (1) | \alpha | ; Praemissa |
| (2) | (\alpha \rightarrow \beta) | ; Praemissa |
| (3) | (\beta\rightarrow \gamma) | ; Praemissa |
| (4) | \beta | ; MP(1,2) |
| (5) | \gamma | ; MP(4,3) |
Ergo \{\alpha,(\alpha\rightarrow\beta),(\beta\rightarrow\gamma)\}\vdash\gamma
Postremo, applicando theorema deductionis ad hanc ultimam expressionem, habetur:
\{(\alpha\rightarrow\beta),(\beta\rightarrow\gamma)\}\vdash(\alpha\rightarrow \gamma)
Alia via ad hanc regulam demonstrandam est ratiocinando ex deductionibus, construendo per praesumptionem et monotoniam. Considera sequentem ratiocinationem ex deductionibus ortam:
| (1) | \{\alpha, (\alpha\rightarrow \beta), (\beta\rightarrow\gamma)\}\vdash \alpha | ; Praesumptio et Monotonia |
| (2) | \{\alpha, (\alpha\rightarrow \beta), (\beta\rightarrow\gamma)\}\vdash (\alpha\rightarrow \beta) | ; Praesumptio et Monotonia |
| (3) | \{\alpha, (\alpha\rightarrow \beta), (\beta\rightarrow\gamma)\}\vdash (\beta\rightarrow\gamma) | ; Praesumptio et Monotonia |
| (4) | \{\alpha, (\alpha\rightarrow \beta), (\beta\rightarrow\gamma)\}\vdash \beta | ; MP(1,2) |
| (5) | \{\alpha, (\alpha\rightarrow \beta), (\beta\rightarrow\gamma)\}\vdash \gamma | ; MP(4,3) |
| (6) | \{(\alpha\rightarrow \beta), (\beta\rightarrow\gamma)\}\vdash (\alpha \rightarrow \gamma) | ; TD(5) |
Hic animadvertendum est utramque demonstrationem identicas esse, solummodo diversis modis expositas. In praxi, inter hos duos stylos alternare potes, secundum quod tibi commodius est.
Aequivalentiae Negationis Duplicatae (DN)
Aequivalentiae negationis duplicatae exprimunt notionem intuitivam secundum quam negatio duplicata enuntiationis aequivalet ipsi enuntiationi. Hoc, symbolice scriptum, sic se habet:
\alpha\dashv\vdash\neg\neg\alpha
Videamus nunc demonstrationem:
| (1) | \vdash (\neg\neg \alpha \rightarrow (\neg\neg\neg\neg \alpha \rightarrow\neg\neg\alpha)) | ; A1 |
| (2) | \vdash ((\neg\neg\neg\neg\alpha \rightarrow \neg\neg\alpha)\rightarrow(\neg\alpha \rightarrow \neg\neg\neg\alpha)) | ; A3 |
| (3) | \vdash ((\neg\alpha \rightarrow \neg\neg\neg\alpha)\rightarrow(\neg\neg\alpha \rightarrow \alpha)) | ; A3 |
| (4) | \vdash ((\neg\neg\neg\neg\alpha \rightarrow \neg\neg\alpha)\rightarrow(\neg\neg\alpha \rightarrow \alpha)) | ; SH(2,3) |
| (5) | \{\neg\neg \alpha \} \vdash (\neg\neg\neg\neg \alpha \rightarrow\neg\neg\alpha) | ; RTD(1) |
| (6) | \{\neg\neg \alpha \} \vdash ((\neg\neg\neg\neg\alpha \rightarrow \neg\neg\alpha)\rightarrow(\neg\neg\alpha \rightarrow \alpha)) | ; Monotonia(4) |
| (7) | \{\neg\neg \alpha \} \vdash (\neg\neg\alpha \rightarrow \alpha) | ; MP(5,6) |
| (8) | \{\neg\neg \alpha \} \vdash \alpha | ; RTD(7) |
Ergo \{\neg\neg \alpha \} \vdash \alpha
Ut demonstrationem in alteram partem conficiamus, hac ipsa demonstratione uti possumus, eam aptando per simplicem substitutionem, unde hoc obtinetur:
\{\neg\neg \neg \alpha \} \vdash \neg \alpha
Et ex hoc argumentum in alteram partem componimus:
| (1) | \{\neg\neg \neg \alpha \} \vdash \neg \alpha | ; Quod modo probatum est |
| (2) | \vdash(\neg\neg \neg \alpha\rightarrow \neg \alpha) | ; TD(1) |
| (3) | \vdash((\neg\neg \neg \alpha\rightarrow \neg \alpha) \rightarrow(\alpha \rightarrow\neg\neg\alpha)) | ; A3 |
| (4) | \vdash(\alpha \rightarrow\neg\neg\alpha) | ; MP(2,3) |
| (5) | \{\alpha\}\vdash\neg\neg\alpha | ; RTD(4) |
Ergo \{\alpha \} \vdash \neg\neg \alpha
Denique, ex his duabus demonstrationibus sequitur \alpha \dashv\vdash \neg\neg \alpha .
Aequivalentia Contrapositivi Implicationis (CpI)
Hoc respondet sequentibus aequivalentibus enuntiationibus
(\alpha \rightarrow \beta) \dashv\vdash (\neg\beta \rightarrow \neg\alpha)
(\neg\alpha\rightarrow\beta)\dashv\vdash (\neg\beta\rightarrow\alpha)
(\alpha\rightarrow\neg\beta) \dashv\vdash (\beta\rightarrow\neg\alpha)
Demonstratio huius primae relationis hoc modo perficitur:
Ex una parte, obtinetur directe ex tertio axiomatico:
| (1) | \vdash ((\neg\beta\rightarrow \neg\alpha) \rightarrow (\alpha \rightarrow\beta)) | ; A3 |
| (2) | \{(\neg\beta\rightarrow \neg\alpha)\}\vdash (\alpha \rightarrow \beta) | ; RTD(1) |
Ergo \{(\neg\beta\rightarrow \neg\alpha)\}\vdash (\alpha \rightarrow \beta)
Et in alteram partem, demonstratio elici potest ex ratiocinatione sequenti:
| (1) | \neg\neg\alpha \dashv \vdash \alpha | ; DN |
| (2) | \vdash (\neg\neg \alpha \rightarrow \alpha) | ; TD(1) |
| (3) | \neg\neg\beta \dashv \vdash \beta | ; DN |
| (4) | \vdash (\beta \rightarrow \neg\neg \beta) | ; TD(3) |
| (5) | \{(\alpha \rightarrow \beta)\}\vdash (\neg\neg \alpha \rightarrow \alpha) | ; Mon(2) |
| (6) | \{(\alpha \rightarrow \beta)\}\vdash (\alpha \rightarrow \beta) | ; Pre |
| (7) | \{(\alpha \rightarrow \beta)\}\vdash (\neg\neg \alpha \rightarrow\beta) | ; SH(5,6) |
| (8) | \{(\alpha \rightarrow \beta)\} \vdash (\beta \rightarrow \neg\neg \beta) | ; Mon(4) |
| (9) | \{(\alpha \rightarrow \beta)\}\vdash (\neg\neg \alpha \rightarrow \neg\neg \beta) | ; SH(7,8) |
| (10) | \vdash (\neg\neg \alpha \rightarrow \neg\neg \beta) \rightarrow (\neg \beta \rightarrow \neg \alpha ) | ; A3 |
| (11) | \{(\alpha \rightarrow \beta)\}\vdash ((\neg\neg \alpha \rightarrow \neg\neg \beta) \rightarrow (\neg \beta \rightarrow \neg \alpha )) | ; Mon(10) |
| (11) | \{(\alpha \rightarrow \beta)\}\vdash (\neg \beta \rightarrow \neg \alpha ) | ; SH(10;11) |
Ergo \{(\alpha \rightarrow \beta)\}\vdash (\neg \beta \rightarrow \neg \alpha )
Quapropter, ex duobus ratiociniis superioribus sequitur:
(\alpha \rightarrow \beta) \dashv\vdash (\neg \beta \rightarrow \neg \alpha )
Ad demonstrandam secundam, fieri possunt haec duo ratiocinia:
| (1) | \beta \dashv\vdash \neg\neg\beta | ; DN |
| (2) | \neg\neg\neg\alpha \dashv\vdash \neg\alpha | ; DN |
| (3) | \vdash (\beta \rightarrow \neg\neg\beta) | ; TD(1) |
| (4) | \vdash (\neg\neg\neg\alpha \rightarrow \neg\alpha) | ; TD(2) |
| (5) | \{(\neg\alpha \rightarrow \beta)\}\vdash (\neg\alpha \rightarrow \beta) | ; Pre |
| (6) | \{(\neg\alpha \rightarrow \beta)\}\vdash (\beta \rightarrow \neg\neg\beta) | ; Mon(3) |
| (7) | \{(\neg\alpha \rightarrow \beta)\}\vdash (\neg\neg\neg\alpha \rightarrow \neg\alpha) | ; Mon(4) |
| (8) | \{(\neg\alpha \rightarrow \beta)\}\vdash (\neg\alpha \rightarrow \neg\neg\beta) | ; SH(5,6) |
| (9) | \{(\neg\alpha \rightarrow \beta)\}\vdash (\neg\neg\neg\alpha \rightarrow \neg\neg\beta) | ; SH(7,8) |
| (10) | \vdash (\neg\neg\neg\alpha \rightarrow \neg\neg\beta) \rightarrow (\neg\beta \rightarrow \neg\neg\alpha) | ; A3 |
| (11) | \{(\neg\alpha \rightarrow \beta)\}\vdash ((\neg\neg\neg\alpha \rightarrow \neg\neg\beta) \rightarrow (\neg\beta \rightarrow \neg\neg\alpha)) | ; Mon(10) |
| (12) | \{(\neg\alpha \rightarrow \beta)\}\vdash (\neg\beta \rightarrow \neg\neg\alpha) | ; MP(9,11) |
| (13) | \neg\neg \alpha \dashv \vdash \alpha | ; DN |
| (14) | \vdash (\neg\neg \alpha\rightarrow \alpha) | ; TD(13) |
| (15) | \{(\neg\alpha \rightarrow \beta)\} \vdash (\neg\neg \alpha\rightarrow \alpha) | ; Mon(14) |
| (16) | \{(\neg\alpha \rightarrow \beta)\} \vdash(\neg\beta \rightarrow \alpha) | ; SH(12,15) |
Ergo \{(\neg\alpha \rightarrow \beta)\} \vdash(\neg\beta \rightarrow \alpha)
Nunc demonstratio in sensu inverso perficienda est. Id fieri potest per ratiocinationem sequentem:
| (1) | \alpha \dashv \vdash \neg\neg\alpha | ; DN |
| (2) | \vdash (\alpha \rightarrow \neg\neg\alpha) | ; TD(1) |
| (3) | \{(\neg\beta\rightarrow\alpha)\}\vdash (\neg\beta\rightarrow\alpha) | ; Pre |
| (4) | \{(\neg\beta\rightarrow\alpha)\}\vdash (\alpha \rightarrow \neg\neg\alpha) | ; Mon(2) |
| (5) | \{(\neg\beta\rightarrow\alpha)\}\vdash (\neg\beta\rightarrow\neg\neg\alpha) | ; SH(3,4) |
| (6) | \vdash (\neg\beta\rightarrow\neg\neg\alpha)\rightarrow (\neg\alpha \rightarrow \beta) | ; A3 |
| (7) | \{(\neg\beta\rightarrow\alpha)\}\vdash ((\neg\beta\rightarrow\neg\neg\alpha)\rightarrow (\neg\alpha \rightarrow \beta)) | ; Mon(6) |
| (8) | \{(\neg\beta\rightarrow\alpha)\}\vdash (\neg\alpha \rightarrow \beta) | ; MP(5,7) |
Ergo \{(\neg\beta\rightarrow\alpha)\}\vdash (\neg\alpha \rightarrow \beta)
Denique, ex his duabus ratiocinationibus concluditur (\neg\beta\rightarrow\alpha) \dashv \vdash (\neg\alpha \rightarrow \beta) , quod erat demonstrandum.
Ultima aequivalentia exercitio relinquetur. Ad eam demonstrandam, duabus iam datis demonstrationibus uti potes ut duce. Haec optima est via ad artem deductionis bene callendam.