The difference between Formal system and Logic

Formal system as a noun (logic):

The grouping of a formal language and a set of inference rules and/or axioms.

Logic as an adjective:

logical

Logic as a noun (uncountable):

A method of human thought that involves thinking in a linear, step-by-step manner about how a problem can be solved. Logic is the basis of many principles including the scientific method.

Logic as a noun (philosophy, logic):

The study of the principles and criteria of valid inference and demonstration.

Logic as a noun (uncountable, mathematics):

The mathematical study of relationships between rigorously defined concepts and of mathematical proof of statements.

Logic as a noun (countable, mathematics):

A formal or informal language together with a deductive system or a model-theoretic semantics.

Logic as a noun (uncountable):

Any system of thought, whether rigorous and productive or not, especially one associated with a particular person.

Logic as a noun (uncountable):

The part of a system (usually electronic) that performs the boolean logic operations, short for logic gates or logic circuit.

Logic as a verb (intransitive, pejorative):

To engage in excessive or inappropriate application of logic.

Logic as a verb (transitive):

To apply logical reasoning to.

Logic as a verb (transitive):

To overcome by logical argument.