This is a handy reference to all of the rule of

logical inference I could dig up. A fun guide for all you

logicians.

**Modus Ponens**
If A, then B.

A.

Therefore B.

**Modus Tollens**
If A, then B.

Not B.

Therefore not A.

**Hypothetical Syllogism**
If A, then B.

If B, then C.

Therefore if A, then C.

**Disjunctive Syllogism**
P or Q.

Not P.

Therefore Q.

**Conjunction**
A.

B.

Therefore A and B.

**Simplification**
A and B.

Therefore A.

**Addition**
A.

Therefore A or B.

Here, they start to get more complex, so I will use some basic notation. X -> Y means if X, then Y. ~X means not X. :: means is the same as. iff is "if and only if", for any sentence if x iff y, then y iff x. It's a mutal dependency thing, dig? FORALL is the upsidedown "A", FORALLx means, guess what? For all members of x. FOR SOME is the backwards "E". FOR SOME x means there is at least one member of x that the statement to follow is true about.

**Tautology**
A :: (A or ~A)

A :: (A AND A)

**Double Negation**
~~A :: A

**Transposition**
A -> B :: ~B -> ~A

**Material Implication**
A -> B :: ~A or B

**Material Equivalence**
A iff B :: ((A -> B) AND (B -> A))

A iff B :: ((A AND B) or (~A AND ~B))

**Constructive Dilemma**
(A -> B) AND (C -> D)

A or C.

Therefore B or D.

**Commutativity**
(A or B) :: (B or A)

(A AND B) :: (B AND A)

**Associativity**
(A or (B or C)) :: ((A or B) or C)

(A AND (B AND C)) :: ((A AND B) or C)

**Distribution**
(A AND (B or C)) :: ((A AND B) or (A AND C)

(A or (B AND C)) :: ((A or B) AND (A or C)

**Exportation**
((A AND B) -> C) :: (A -> (B -> C))

**Universal Instantation**
FORALLx

All instances of x can be replaced by any name.

**Existential Instantation.**
FOR SOMEx

All instances of x can be replaced by a new name (cannot have been previously mentioned in a proof).

May these rules serve well your

predicate logic adventures.