5 Symmetriae Logicae Propositionalis
Summarium:
Per totam hanc lectionem explorabimus quomodo duplex negatio, syllogismus hypotheticus, contrapositio implicationis, theorema deductionis et definitiones connectorum componantur ad symmetrias logicae propositionalis constituendas. Demonstrationibus claris ac simplicibus, disces quomodo aequivalentias intelligas easque ad tuas quaestiones logicas applices.
Symmetriae in hac lectione tractatae sunt: \downarrow-Symmetria, \vee-Symmetria, \wedge-Symmetria, \leftrightarrow-Symmetria et \veebar-Symmetria. Praeterea illustrantur interactiones inter demonstrationes et quomodo unaquaeque in praecedentibus nitatur ad deductiones futuras simpliciores reddendas. Haec lectio non solum cognitionem profundam Logicae Propositionalis tibi praebebit, sed etiam docebit quomodo demonstrationes priores ad discendi processum optimandum adhibeas.
Propositi Discendi:
Expleta hac lectione, discipulus erit capax
- Memorandi notiones fundamentales logicae propositionalis, ut syllogismum hypotheticum et duplicem negationem.
- Agnoscendi quinque Symmetrias Logicae Propositionalis.
- Intelligendi processum demonstrationis aequivalentiarum symmetriarum.
- Applicandi Praesumptionem, Theorema Deductionis eiusque Reciprocum in demonstrationibus.
- Referendi definitiones connectorum logicorum ad symmetrias.
- Aestimandi momentum unam demonstrationem semel perficiendi eamque in demonstrationibus futuris iterum adhibendi.
- Excolendi facultates analyticas ac criticas in demonstrationibus logicis conficiendis.
INDEX CONTENTORUM
\vee – SYMMETRIA
\downarrow – SYMMETRIA
\wedge – SYMMETRIA
\leftrightarrow – SYMMETRIA
\veebar – SYMMETRIA
OBSERVATIONES FINALES
Consequentia directa syllogismi hypothetici, duplex negationis et contrapositionis implicationis, theorematum deductionis atque definitionum connectorum sunt quinque symmetriae logicae propositionalis quas infra considerabimus.
| (\alpha \downarrow \beta) \dashv\vdash (\beta\downarrow \alpha) | \downarrow-Symmetria |
| (\alpha \vee \beta) \dashv\vdash (\beta\vee \alpha) | \vee-Symmetria |
| (\alpha \wedge \beta) \dashv\vdash (\beta\wedge \alpha) | \wedge-Symmetria |
| (\alpha \leftrightarrow \beta) \dashv\vdash (\beta\leftrightarrow \alpha) | \leftrightarrow-Symmetria |
| (\alpha \veebar \beta) \dashv\vdash (\beta\veebar\alpha) | \veebar-Symmetria |
Demonstrationes harum aequivalentiarum non sunt omnino triviales, sed, dissimiliter a quibusdam demonstrationibus quas iam vidimus, satis simplices sunt. Infra unaquaeque demonstratio in una directione ostenditur; demonstratio directionis inversae fere eadem est et lectori pro exercitio relinquitur.
\vee-Symmetria
| [/latex] | \{(\alpha \vee \beta)\}\vdash (\alpha \vee\beta) | ; Praesumptio |
| [/latex] | \{(\alpha \vee \beta)\}\vdash (\neg \alpha \rightarrow \beta) | ; quia (\alpha \vee \beta) := (\neg \alpha \rightarrow \beta) |
| (3) | \{(\alpha \vee \beta)\}\vdash (\neg \beta \rightarrow \alpha) | ; CPI(2) |
| [/latex] | \boxed{\{(\alpha \vee \beta)\}\vdash ( \beta \vee \alpha)} | ; quia ( \beta \vee \alpha) := (\neg\beta\rightarrow\alpha) |
Ratiocinatio in sensu inverso parvis tantum mutationibus obtinetur incipiendo a praesumptione \{(\beta\vee\alpha)\}\vdash (\beta\vee\alpha)
\downarrow-Symmetria
| [/latex] | \{\neg(\alpha \downarrow \beta)\}\vdash \neg(\alpha \downarrow \beta) | ; Praesumptio |
| [/latex] | \{\neg(\alpha \downarrow \beta)\}\vdash (\alpha \vee \beta) | ; ex (1) quia (\alpha\vee\beta) := \neg(\alpha \downarrow \beta) |
| [/latex] | \{\neg(\alpha \downarrow \beta)\}\vdash (\beta \vee \alpha) | ; \vee-Symmetria |
| [/latex] | \vdash (\neg(\alpha \downarrow \beta) \rightarrow (\beta \vee \alpha)) | ; TD(3) |
| [/latex] | \vdash (\neg(\beta \vee \alpha) \rightarrow (\alpha \downarrow \beta)) | ; CPI(4) |
| [/latex] | \vdash ((\beta \downarrow \alpha) \rightarrow (\alpha \downarrow \beta)) | ; ex (5) quia (\beta\vee\alpha) := \neg(\beta \downarrow \alpha) |
| [/latex] | \boxed{\{(\beta \downarrow \alpha) \} \vdash (\alpha \downarrow \beta)} | ; RTD(6) |
Denique, ratiocinando in sensu inverso, deducitur conclusio contraria incipiendo a praesumptione \{\neg(\beta\downarrow\alpha)\}\vdash \neg(\beta\downarrow\alpha)
\wedge-Symmetria
| [/latex] | \{(\alpha \wedge \beta)\} \vdash (\alpha \wedge \beta) | ; Praesumptio |
| [/latex] | \{(\alpha \wedge \beta)\} \vdash (\neg\alpha \downarrow \neg\beta) | ; ex (1) quia (\alpha \wedge \beta) := (\neg\alpha \downarrow \neg\beta) |
| [/latex] | \{(\alpha \wedge \beta)\} \vdash (\neg\beta \downarrow \neg\alpha) | ; \downarrow-Symmetria (2) |
| [/latex] | \boxed{\{(\alpha \wedge \beta)\} \vdash ( \beta \wedge \alpha)} | ; ex (3) quia (\beta \wedge \alpha) := (\neg\beta \downarrow \neg\alpha) |
Sicut priore exemplo, si in sensu inverso ratiocinaris, deducitur cum minimis variationibus incipiendo a praesumptione \{( \beta \wedge \alpha)\}\vdash ( \beta \wedge \alpha)
\leftrightarrow-Symmetria
| [/latex] | \{(\alpha \leftrightarrow \beta)\} \vdash (\alpha \leftrightarrow \beta) | ; Praesumptio |
| [/latex] | \{(\alpha \leftrightarrow \beta)\} \vdash ((\alpha \rightarrow \beta) \wedge (\alpha \rightarrow \beta)) | ; ex (1) quia (\alpha \leftrightarrow \beta) := ((\alpha \rightarrow \beta) \wedge (\beta \rightarrow \alpha)) |
| [/latex] | \{(\alpha \leftrightarrow \beta)\} \vdash ((\beta \rightarrow \alpha) \wedge (\alpha \rightarrow \beta) ) | ; \wedge-Symmetria(2) |
| [/latex] | \boxed{\{(\alpha \leftrightarrow \beta)\} \vdash (\beta \leftrightarrow \alpha)} | ; ex (3) quia (\beta \leftrightarrow \alpha) := ((\beta \rightarrow \alpha) \wedge (\alpha \rightarrow \beta)) |
Sicut in exemplo praecedenti, incipitur a praesumptione \{( \beta \leftrightarrow \alpha)\}\vdash ( \beta\leftrightarrow \alpha)
\veebar-Symmetria
| [/latex] | \{(\alpha \leftrightarrow \beta)\} \vdash (\alpha \leftrightarrow \beta) | ; Praesumptio |
| [/latex] | \{(\alpha \leftrightarrow \beta)\} \vdash ( \beta \leftrightarrow \alpha) | ; \leftrightarrow-Symmetria(1) |
| [/latex] | \vdash ((\alpha \leftrightarrow \beta) \rightarrow ( \beta \leftrightarrow \alpha)) | ; TD(2) |
| [/latex] | \vdash (\neg ( \beta \leftrightarrow \alpha) \rightarrow \neg (\alpha \leftrightarrow \beta)) | ; CPI(3) |
| [/latex] | \{\neg ( \beta \leftrightarrow \alpha)\} \vdash \neg (\alpha \leftrightarrow \beta) | ; RTD(4) |
| [/latex] | \boxed{\{ ( \beta \veebar \alpha)\} \vdash (\alpha \veebar \beta)} | ; ex (5) quia ( \beta \veebar \alpha) := \neg(\beta \leftrightarrow \alpha) et (\alpha \veebar \beta) := \neg (\alpha \leftrightarrow \beta) |
Ut in omnibus casibus praecedentibus, sufficit praesumptionem in sensu inverso probare \{(\beta \leftrightarrow \alpha)\} \vdash (\beta \leftrightarrow \alpha) ad deductionem in eodem sensu obtinendam.
Observationes Finales
Res cui lector attendere debet est ordo in quo hae quinque symmetriae logicae propositionalis demonstrandae electae sunt. Animadvertendum est unamquamque ita factam esse ut aliqua ex demonstrationibus antea peractis utatur. Hoc declarat rationem qua demonstrationes faciendas est: semel perficiuntur (et numquam iterum!); postea, propositum tuum sit ut demonstrationes anteriores ad usum afferas ad futuras deductiones simpliciores reddendas.