From c20ceb3d1951fbd1706a0ddd64af1eea2089679b Mon Sep 17 00:00:00 2001 From: Robert Jeutter Date: Thu, 29 Apr 2021 17:11:43 +0200 Subject: [PATCH] Vorlesung 2 --- Assets/Logik-Deduktionsbeispiel.png | Bin 0 -> 15902 bytes Assets/Logik-Disjunktionselimination.png | Bin 0 -> 14002 bytes Assets/Logik-Falsumeinführung.png | Bin 0 -> 3478 bytes Assets/Logik-Implikationseinführung.png | Bin 0 -> 7973 bytes ...ogik-Implikationseleimination-beispiel.png | Bin 0 -> 29560 bytes Assets/Logik-Implikationselimination.png | Bin 0 -> 7239 bytes Assets/Logik-Konjunktionselimination.png | Bin 0 -> 8446 bytes Assets/Logik-Negationseinführung.png | Bin 0 -> 6771 bytes Assets/Logik-Negationselimination.png | Bin 0 -> 4779 bytes Assets/Logik-beispiel-1.png | Bin 0 -> 25860 bytes Assets/Logik-beispiel-2.png | Bin 0 -> 28427 bytes Assets/Logik-beispiel-3.png | Bin 0 -> 11316 bytes Assets/Logik-beispiel-4.png | Bin 0 -> 27873 bytes Assets/Logik-deduktionsbaum.png | Bin 0 -> 2398 bytes Assets/Logik-konjunktionseinführung.png | Bin 0 -> 5916 bytes Assets/Logik-reductio-ad-absurdum.png | Bin 0 -> 7057 bytes Logik und Logikprogrammierung.md | 207 ++++++++++++++++++ 17 files changed, 207 insertions(+) create mode 100644 Assets/Logik-Deduktionsbeispiel.png create mode 100644 Assets/Logik-Disjunktionselimination.png create mode 100644 Assets/Logik-Falsumeinführung.png create mode 100644 Assets/Logik-Implikationseinführung.png create mode 100644 Assets/Logik-Implikationseleimination-beispiel.png create mode 100644 Assets/Logik-Implikationselimination.png create mode 100644 Assets/Logik-Konjunktionselimination.png create mode 100644 Assets/Logik-Negationseinführung.png create mode 100644 Assets/Logik-Negationselimination.png create mode 100644 Assets/Logik-beispiel-1.png create mode 100644 Assets/Logik-beispiel-2.png create mode 100644 Assets/Logik-beispiel-3.png create mode 100644 Assets/Logik-beispiel-4.png create mode 100644 Assets/Logik-deduktionsbaum.png create mode 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eine Deduktion gibt mit Hypothesen aus $\Gamma$ und Konklusion $\varphi$. Wir sagen "$\varphi$ ist eine syntaktische Folgerung von $\Gamma$". +> +> Eine Formel $\varphi$ ist ein Theorem, wenn $\varnothing\vdash\varphi$ gilt. + +### Bemerkung +$\Gamma\vdash\varphi$ sagt (zunächst) nichts über den Inhalt der Formeln in $\Gamma\cup\{\varphi\}$ aus, sondern nur über die Tatsache, dass $\varphi$ mithilfe des natürlichen Schließens aus den Formeln aus $\Gamma$ hergeleitet werden kann. + +Ebenso sagt "$\varphi$ ist Theorem" nur, dass $\varphi$ abgeleitet werden kann, über "Wahrheit" sagt dieser Begriff (zunächst) nichts aus. + +### Satz +Für alle Formeln $\varphi$ und $\psi$ gilt $\{\lnot(\varphi\vee\psi)\}\vdash\lnot\varphi\wedge\lnot\psi$. + +Beweis: Wir geben eine Deduktion an... +- $\{\lnot\varphi\wedge\lnot\psi\}\vdash\lnot(\varphi\vee\psi)$ + ![](Assets/Logik-beispiel-1.png) +- $\{\lnot\varphi\vee\lnot\psi\}\vdash\lnot(\varphi\wedge\psi)$ + ![](Assets/Logik-beispiel-2.png) +- $\{\varphi\vee\psi\} \vdash \psi\vee\varphi$ + ![](Assets/Logik-beispiel-3.png) + +### Satz +Für jede Formel $\varphi$ ist $\lnot\lnot\varphi\rightarrow\varphi$ ein Theorem. + +Beweis: Wir geben eine Deduktion mit Konklusion $\lnot\lnot\varphi\rightarrow\varphi$ ohne Hypothesen an... + +### Satz +Für jede Formel $\varphi$ ist $\varphi\vee\lnot\varphi$ ein Theorem. + +Beweis: Wir geben eine Deduktion mit Konklusion $\varphi\vee\lnot\varphi$ ohne Hypothesen an... + +Bemerkung: Man kann beweisen, dass jede Deduktion der letzten beiden Theoreme die Regel (raa) verwendet, sie also nicht "intuitionistisch" gelten. + +### Satz +$\{\lnot(\varphi\wedge\psi)\}\vdash\lnot\varphi\vee\lnot\psi$ + +![](Assets/Logik-beispiel-4.png) +