"classical implications" is a correct and usable phrase in written English.
It usually refers to the traditional or ancient connotations associated with something. For example, you could say: "The classical implications of his story suggest that it is from ancient mythology."
He then showed how to define classical implication as $NTpNq$, formulated this definition using a variable functor as the implication and proved that in this version classical bivalent logic based on $C$ and $N$ is contained in intuitionistic logic, provided that detachment is confined to $C$-$N$ fonlylas only.
Alongside this 'syntactic operation' sense of connective explained above for the primitive connectives and extended to derived connectives presently another use of the term is indispensable: to refer to a connective with particular logical behaviour, as when people want to compare the behaviour of intuitionistic and classical implication.
In the particular case of rational relations, the associated set of defaults may be chosen to be minimal; it then consists of a set of formulae, totally ordered through classical implication, with cardinality equal to the height of the given relation.
For this purpose, we start out with the definition of regular belief, which covers a wide range of belief models, from the very weak to those closed under classical logical implication.
A straightforward but logically less interesting possibility is to define implication from conjunction and negation (or disjunction and negation) using the corresponding tautology of classical logic; such implications are called S-implications.
That is, classical and intuitionistic implications collapse in the combined logic.
In turn, the collapsing problem is a limit case of interactions: the interderivability between classical and intuitionistic implications (nothing else than two interaction laws between different connectives) is also spontaneously created by the combination process.
The modifications of body posture in classical dance had implications on the average aesthetic response of our contemporary non-expert participants.
Using the classical equivalence with implication once more, (ii) is defined iff it holds that if there was no trouble, then there is a knave.
The major significance of Einstein's theory is its radical conceptual departure from classical theory and its implications for further growth in physical thought.
