The various forms of entailment (e.g. "from P we can deduce Q") is a preordering on propositions. Implication is the propositional form of this preordering.

Since the truth of the preordering relation is already two-valued, in the special case of assigning truth values to propositions in a two-valued logic, the truth value assigned to Q→R is precisely the truth of Q ≤ R.

More generally, the defining property of implication is

P∧Q ≤ R  if and only if  P ≤ Q→R

from which we infer Q→R is the largest proposition amongst all X with the property that X∧Q ≤ R.

For those with category theory experience, you might recognize that if we form the category of all propositions with the arrows being entailment, this says that category is Cartesian closed: P∧Q is the product P×Q and Q→R is the exponential RQ

The various forms of entailment (e.g. "from P we can deduce Q") is a preordering on propositions. Implication is the propositional form of this preordering.

Since the truth of the preordering relation is already two-valued, in the special case of assigning truth values to propositions in a two-valued logic, the truth value assigned to Q→R is precisely the truth of Q ≤ R, and T≤F is the only time ≤ can be false.

More generally, the defining property of implication is

P∧Q ≤ R  if and only if  P ≤ Q→R

from which we infer Q→R is the largest proposition amongst all X with the property that X∧Q ≤ R.

For those with category theory experience, you might recognize this as saying that if we form the category of all propositions with the arrows being entailment, the category is Cartesian closed: P∧Q is the product P×Q and Q→R is the exponential RQ