Implication

In Mathématiques and traditional Logique, a proposal P implies logically a proposal Q if and only if the proposal ¬ P ∨ Q is true.

¬ P ∨ Q is written also P ⇒ Q , which is a simple short cut of writing.
In all letters, this is read:
“(not P ) or Q ” is also written “ P implies Q ”.
The symbol “⇒” is called connector of implication. “ P ⇒ Q ” is called a logical implication.

Properties

The Truth table of the implication is given by the table:

Are P , Q and R three proposals.

  • ( P ⇒ Q ) ∧ ( Q ⇒ P ) is written also P ⇔ Q ; it is the logical equivalence.

  • P ⇒ P (the implication is reflexive)
  • (( P ⇒ Q ) ∧ ( Q ⇒ R )) ⇒ ( P ⇒ R ) (transitivity of the implication or law of the syllogism)
  • (¬ ( P ⇒ Q )) ⇔ ( P ∧ (¬ Q )) (negation of an implication)
  • ( P ∧ ( P ⇒ Q )) ⇒ Q (rule of direct deduction or the detachment)
  • ( P ⇒ Q ) ⇔ (¬ Q ⇒ ¬ P ) (law of contraposition)
An implication equivalent to its is contraposée
  • ( P ⇔ Q ) ⇔ (( P ⇒ Q ) ∧ ( Q ⇒ P )) (law of reciprocity)
  • (( P ∨ Q ) ∧ ( P ⇒ R ) ∧ ( Q ⇒ R )) ⇒ R (disjunction of the cases)

Not Associativeness of the implication

Are P , Q and R three proposals.

  • (( P ⇒ Q ) ⇒ R ) ⇎ ( P ⇒ ( Q ⇒ R ))
Indeed the first term states that an implication implies R whereas the second states that P implies an implication.

Let us give a counterexample:

Let us consider the three following proposals:

  • P : (- 1 = 0)
  • Q : (0 = 0)
  • R : (0 = 1)
The proposal P ⇒ Q is true since Q is true, and as R is false, the proposal ( P ⇒ Q ) ⇒ R is false.

The proposal Q is true and the proposal R is false thus the implication ( Q ⇒ R ) is false and as P is false, the implication P ⇒ ( Q ⇒ R ) is true.

We deduce from it that in general the proposals P ⇒ ( Q ⇒ R ) and ( P ⇒ Q ) ⇒ R is not equivalent and thus the implication is not associative.

It is thus impossible for us to write chains of implications of the form:

P1 ⇒ P2 ⇒ P3 ⇒… ⇒ Pn-1 ⇒ Pn
This is why, we lay out in practice, the implications in this way:

what means that implications:

P1 ⇒ P2,…, Pn-1 ⇒ Pn
are true, and we use the transitivity of the implication to show that:
P1 ⇒ Pn.

Difference with equivalence

Here an example of relation of implication: “the weather is nice” ⇒ “I am happy”. This proposal is true if I am always happy when the weather is nice.

Not to confuse with the relation of equivalence which it implies that I would be happy ONLY when the weather is nice.

  • the relation of implication represents the IF (⇒) a sufficient condition in a direction, a requirement in the other: in ⇒ B has, has is a sufficient condition of B , and B is a requirement of has
    - and -

  • the relation of equivalence represents the IF AND ONLY IF (⇔), a requirement and sufficient;
    has ⇔ B is equivalent to ( has ⇒ B ) AND ( B ⇒ has )

to also see: contraposée Property

Notice

In a mathematical theory, the true implications P ⇒ Q shown starting from the axioms are called Théorème S.

To show a theorem, it is to establish that a proposal of the form P ⇒ Q is a true Assertion (in the theory).

To show such theorems, there exist several types of possible reasoning, based on the preceding properties of the implication:

  • the direct deduction
  • the deduction by exclusion (or incompatibility)
  • the reasoning by contraposée (also the Reasoning by the absurdity)

Important remark

In spite of its notation (⇒) which suggests a relation of cause and effect, the logical implication is not chronological like have it a cause and an effect. The Temps does not play of role, and does not even need to be defined, during the examination of a relation of implication. “ P implies Q ” means only that Q can result from P by a logical reasoning.

That also applies if P is false. For example: “(0 = 1) ⇒ (0 = 0)” is true, because one can deduce (0 = 0) of (0 = 1) by cutting off member with member (0 = 1) to (0 = 1). Generally, if P is false then the implication P ⇒ Q is true; and thus all the implications which we will write starting from a proposal distorts will be true ! Starting from the forgery one can show anything.

In practice it will thus be shown that the implication P ⇒ Q is true by showing that if P is true, then Q is true.

In fact the direct Déduction of Q starting from P is represented by the always true implication (Tautologie):

( P ∧ ( P ⇒ Q )) ⇒ Q

So under certain conditions, P is true thus that P ⇒ Q , then the preceding implication shows that Q is true.

In the natural language, to translate that P implies Q, we will say indifferently.

  • P involves Q .
  • P is a sufficient condition of Q .
  • Q is a requirement of P .
  • So that Q is true it is enough that P is true.
  • So that P is true it is necessary that Q is true.

Let us add that other formulations of the French language represent implications:

  • “If… then…”
  • “… thus…”
  • “… from where…”
  • “… thus…”
  • “of…, we deduce that…”
  • “… consequently…”

Others

The truth table of the implication was known as of the ancient Greece, in particular by the stoical : Truth follows truth… Forgery follows the forgery… Forgery follows truth… But of truth, the forgery cannot follow .

Internal bonds

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