p -- > q


q --> p


~ p --> ~ q


~ q --> ~ p

Logically Equivalent Conditions

Original (p --> q) === Contrapositive (~q --> ~p)
Converse (q --> p) === Inverse (~p --> ~q)

~ (p --> q) ===

p ^ ~ q

p ^ ~ q ===

~ (p --> q)

p -- > q (Orig) ===

~ q --> ~ p (Contrapositive)

q --> p (Converse) ===

~ p --> ~ q (Inverse)

~ q --> ~ p (Contrapositive) ===

p -- > q (Orig)

~ p --> ~ q (Inverse) ===

q --> p (Converse)

Conditional: If today is Easter, then tomorrow is Monday (write the Contrapositive)

Contrapositive: If tomorrow is not Monday, then today is not Easter

Negation of p --> q

p ^ ~ q

Conditional: If today is Easter, then tomorrow is Monday (write the Converse)

Converse: If tomorrow is Monday, then today is Easter.

Conditional: If today is Easter, then tomorrow is Monday (write the Inverse)

Inverse: If today is not Easter, the tomorrow is not Monday.

Conditional: If Tom is Ann's father, then Jim is her uncle and Sue is her aunt. (write Contrapositive)

Contrapositive: If either Jim is not Ann's uncle or Sue is not her aunt, then Tom is not her father

Conditional: If X, then Y AND Z

Contrapositive: If EITHER not Y OR not Z, then not X

Conditional: If X, then Y OR Z

Contrapositive: If not Y AND Z, then not X

Modus Ponens

p --> q
... q [VALID]
modus ponendo ponens: "the way that affirms by affirming"

Modus Tollens

p --> q
~ q
... ~ p [VALID]
modus tollendo tollens: "the way that denies by denying"


p v q
~ p
... q [VALID]


p --> q
q --> r
... p --> r [VALID]


... p v q


p ^ q
... p

Proof by Division into cases

p v q
p --> r
q --> r
... r [VALID]


... p ^ q


~ p --> C
... p

Converse Error

p --> q
... p [INVALID]

Inverse Error

p --> q
~ p
... ~ q [INVALID]

p OR q ===

~ p --> q

DeMorgans Laws

~(p ^ q) === ~p v ~q
~(p v q) === ~p ^ ~q

p --> q ===

~ p v q

p --> q ===

~q --> ~p

p <--> q ===

(p --> q) ^ (q --> p)
(p ^ q) v (~p ^ ~q)

Vacuously True

Hypothesis is False

Implication is False

ONLY when hyp (p) is T and conc (q) is F
(p(T) --> q(F))

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