Implies

I simply cannot remember which is normalized and which is denormalized. I cannot do coordinate transforms — I consider myself “spatially impared”. And I can never remember the truth table for implies.

p implies q” or “p only if q” has the following truth table:

p | q | p -> q
---+---+--------
T | T |   T
T | F |   F
F | T |   T
F | F |   T

where p and q are propositions.

It is logically equivalent to say:

  • p implies q
  • q is a necessary condition for p
  • p is a sufficient condition of q
  • in order that p be true it is necessary that q be true
  • if p is true then q is true

For example:

    If a person is a father then a person is male.

This statement is of the form p -> q where:

  • p: A person is a father
  • q: A person is male

It is necessary for a person to be male to be a father. Being a father is a sufficient condition for being male. If a person is not a father, nothing can be said about if they are male. Whereas if a person is not male, they may not be a father. This last statement is the contrapositive of the proposition.

p | q | -p | -q | p -> q | -q -> -p
---+---+----+----+--------+----------
T | T |  F |  F |   T    |    T
T | F |  F |  T |   F    |    F
F | T |  T |  F |   T    |    T
F | F |  T |  T |   T    |    T

where represents negation. Since both p -> q and -q -> -p have identical truth tables they are said to be logically equivalent.

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One comment

  1. Thanks you so much for this table. I just couldn’t find any website that explained p implies q in simple English. Please add more examples.


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