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Implication

p -- > q

Converse

q --> p

Inverse

~ p --> ~ q

Contrapositive

~ q --> ~ p

Logically Equivalent Conditions

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

OR

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

OR

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

p

... q [VALID]

modus ponendo ponens: "the way that affirms by affirming"

p

... 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"

~ q

... ~ p [VALID]

modus tollendo tollens: "the way that denies by denying"

Elimination

p v q

~ p

... q [VALID]

~ p

... q [VALID]

Transitivity

p --> q

q --> r

... p --> r [VALID]

q --> r

... p --> r [VALID]

Generalization

p

... p v q

... p v q

Specialization

p ^ q

... p

... p

Proof by Division into cases

p v q

p --> r

q --> r

... r [VALID]

p --> r

q --> r

... r [VALID]

Conjunction

p

q

... p ^ q

q

... p ^ q

Contradiction

~ p --> C

... p

... p

Converse Error

p --> q

q

... p [INVALID]

q

... p [INVALID]

Inverse Error

p --> q

~ p

... ~ q [INVALID]

~ p

... ~ q [INVALID]

p OR q ===

~ p --> q

DeMorgans Laws

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

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

(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))

(p(T) --> q(F))