Vaild argument froms and rules of equivalence
(PP) Modus Ponendo Ponens (mode that affirms by affirming)
A → B If I let go of the apple that I hold, it is going to fall down
A        I let go of the apple
B Hence, the apple falls down (unless I am in a microgravity environment)
(TT) Modus Tollendo Tollens (mode that denies by denying)
A → B When it rains the street gets wet
B       The street is not wet
A Hence, it has not rained (unless the street is hotter than 100C)
(TP) Modus Tollendo Ponens (Disjunctive Syllogiism)
A v B The computer is either on or off
A     The computer is not on
B Hence, the computer is off (unless it is in sleep mode)
(PT) Modus Ponendo Tollens (mode that affirms by denying)
(A B) Socrates is not at the same time on the Isthmus and in Athens
A             Socrates is in Athens
B Hence, Socrates is not on the Isthmus
(DE) Disjunction elimination    
           
  A → C   With pizza, I will have white wine  
  B → C   With fish, I will have white wine  
  A v B       I will have pizza or fish  
  C   Hence, I will have white wine  
           
(HS) Hypothetical Syllogism (Modus Barbara)
A → B The Divine is omnipotent
B → C This ice cream is divine
A → C Hence, this ice cream is omnipotent (equivocation, semantical ambiguity)
() Transitivity of the biconditional   () Transitivity of the conjunction
               
  A B         A B  
  B C          
  A ↔ C         A C  
() Tautology   () Tautology
A → B A v B
A → B A v B
B B
 
(KD) Constructive Dilemma   (DD) Destructive Dilemma
A → B A → B
C → D C → D
A v C    B v D
B v D   A v C
 
(A) Adjunktion (C) Conjunction
A        A
A v B
    B       
    A B
(A) v-Absorption (S) Simplifikation der Konjunktion
       
  A v (B C) <=> A  
       
A B
  A v (A v B) <=> (A v B)          
(SDD) Simplifikation und Duplikation der Disjunktion () Simplifikation und Duplikation der Konjunktion
A v A <=> A       A A <=> A
   
(Komm) Kommutativitt () Kommutativitt
A v B <=> B v A     A B <=> B A
(Ass) Assoziativitt (Ass) Assoziativitt
   
A v (B v C) <=> (A v B) v C     A (B C) <=> (A B) C
     
(Dist.) Distributivgesetz der Konjunktion   () Distributivgesetz der Konjunktion
               
  A v (B C) <=> (A v B) (A v C)     A (B v C) <=> (A B) v (A C)
               
(DN) Double Negation          
               
  A <=> A            
               
(DM) De Morgan
A v B <=> (A B)     A B <=> (A v B)
   
A v B <=> (A B)     A B <=> (A v B)
               
  A v B <=> (A B)     A B <=> (A v B)
               
  A v B <=> (A B)     A B <=> (A v B)
Widerspruch in der Disjunktion   () Gesetz in der Konjunktion
 
         
  A         A  
               
(def. Impl.) Definition der Implikation () Falsifikation
A → B <=> A v B (A → B) <=> A B
(Kontr.) Kontrapositions (Transpositionen)
A → B <=> B → A     A → B <=> B → A
   
A → B <=> B → A     A → B <=> B → A
(ZNK) Zerlegung und Zusammensetzung einer Implikation mit einer Konjunktion im Nachsatz
   
A → (B C) <=>  (A → B) (A → C)
               
(ZVD) Zerlegung und Zusammensetzung einer Implikation mit einer Disjunktion im Vordersatz
               
  (A v B) → C <=> (A → C) (B → C)      
               
() Gesetz einer Implikation mit einer Disjunktion im Nachsatz    
               
  A → (B v C) <=> (A → B) v C        
               
(Exp, Imp) Exportation, Importation          
               
((A B) → C) <=> (A → (B → C))
               
() Definition des Vordersatzes   () Definition des Nachsatzes
               
  A → B <=> A ↔ (A B)     A → B <=> B ↔ (A v B)
   
(Bi) Biconditional () Assoziativitt der Bisubjunktion
               
  A B <=> B A     (A B) C <=> A ↔ (B ↔ C)
  A B <=> A B          
  A B <=> (A B)      
A B <=> (A → B) (B → A)
A B <=> (A B) v (A B)
(Ed) Exclusive disjunction
               
  A B <=> A B        
  A B <=> B A        
  A B <=> (A ↔ B)        
  A B <=> (A B) v (B A)        
  A B <=> (A v B) (B v A)        
  A B <=> (A → B) (B A)        
  A B <=> A ↔ B        
               
  Ableitungen aus der Exklusion        
               
  A|B <=> B|A        
  A|B <=> A v B        
  A|B <=> A B        
  A|B <=> (A B)        
               
(Raa) Reductio ad absurdum (Indirect proof)
 
A
(Cp) Conditional proof
A
B assumption
conjunction  
B → (A B)

 

Deductions

S per raa

1.

P → Q

1.

T → S

2.

2.

Q v P → T

3.

P

s 2

3.

U v R → Q

4.

Q

pp 1,3

4.

5.

S

assumption

6.

T

tt 1,5

1.

P v (Q R)

7.

(Q v P)

tt 2,6

2.

R P → S

8.

Q P

dm 7

3.

9.

Q

s 8

4.

Q R

tp 1,3

10.

(U v R)

tt 3,10

5.

R

s 4

11.

U R

dm 10

6.

R P

k 3,5

12.

U

s 11

7.

S

pp 2,6

13.

U

tt 4,9

14.

U U

a 5,13

15.

S

raa 14

1.

S → P Q

2.

(Q P)

3.

1.

P Q R

4.

(S)

tt 1,2

2.

S v P Q

5.

S

dn 4

3.

S v Q

6.

S v T

tp 2,3

4.

P Q → R

7.

T

tp 5,6

5.

 

6.

(S Q) (P → Q)

zvd 2

7.

S Q

s 6

1.

Q P R

8.

S v Q

def impl

2.

Q

9.

(S v Q) (S v Q)

k 3.8

3.

10.

(S S) v Q

dist 9

4.

P R

pp 1,2

11.

Q

wd 9

5.

(P v R)

dm 4

12.

Q v R

a 11

6.

S

tt 5,3

13.

(Q R)

dm 12

14.

P

tt 1,13

15.

P Q

k 11,14

1.

S P R

16.

R

pp 4,15

2.

P v R

17.

Q R

k 11,16

3.

S v Q

18.

(Q v R)

dm 17

4.

 

19.

S

tt 5,18

5.

(P R)

dm 2

6.

S

tt 1,5

7.

Q

tp 3,6

U

8.

Q v P

a 7

1.

R (P → T)

9.

R

pp 8

2.

(S v T)

3.

(P R) → Q v R

4.

U

5.

S T

dm 2

1.

S → Q T

6.

S

s 5

2.

Q v T

7.

T

s 5

3.

S v R → P

8.

R

s 1

4.

9.

Q → U

pp 4, 8

5.

(Q T)

dm 2

10.

P → T

s 1

6.

S

tt 1,5

11.

P

tt 7, 10

7.

S v R

a 6

12.

P R

k 8, 11

8.

P

pp 3,7

13.

Q v R

pp 3, 12

9.

P S

k 8

14.

Q

tp 8, 13

10.

U

pp 4,9

15.

U

pp 9, 14


the truth table

P

Q

P P

P Q

P Q

P Q

P | Q

P Q

T

T

T

T

T

T

F

T

T

F

T

T

T

F

T

F

F

T

T

T

F

T

T

F

F

F

T

F

T

T

T

T

P

Q

P P

P Q

P Q

P Q

P ↓ Q

P Q

F

F

F

T

F

F

F

F

F

T

F

F

T

F

F

T

T

F

F

F

F

T

F

T

T

T

F

F

F

F

T

F

logical connectives

- Negation

P

not

reverses the values

- Conjunction

P Q

and (as well ... as ...)

T both values are T

  - Disjunction

P   Q

... or as well ... (Adjunktion)

T at least one value is T

- Implication

P Q

if ... then ... (Subjunktion)

F der ant. is T and the succ. is F

- Replikation

P Q

F the ant. is F and the succ. is T

 - Equivalence

P Q

if ..., and only if ... (Bisubjunktion)

T both values are T or both are F

<=> / = / : / def - Definition

the definition is correct all values are T

/ >―< - Exclusive Disj.

(P Q)

either ... or ...

T one value is T and the other is F

| - Sheffer stroke

(P Q)

not and (nicht both)

F both values are T

↓ - Peirce arrow

(P Q)

neither ... nor (both not)

T both values are T

>― - Postsection

P Q

the one without the other

―< - Presection

P Q

the other without the one

Contradiction

P P

(antilogy)

P Q

unless

logical Axioms

Tautology

P P

Identity

P P

law of noncontradiction

(P P)

(P P)

P | P

ex falso sequitur quodlibet

(P P) Q

(P P) Q

verum sequitur ex quodlibet

P (Q Q)

P (Q P)

Tertium non datur

P P

Clavius' Law

(P P) P

Duns Scotus' Law

P (P Q)

Petrus Hispanus' Law

P Q → P

more

P P Q

P Q (P Q)

Peirce's law

((P Q) P) P

P (P Q)

Trivialisierung

(P P) Q

( P P) P

disjunktive Erweiterung

(P (P Q)

quivalente Abschwchung der K.

(P    Q) (P Q)

implikative Abschwchung der K.

(P  Q) (P Q)

disjunktive Abschwchung der K.

(P  Q) (P Q)

replikative Abschwchung der K.

(P    Q) (P Q)