The difference between Formal system and Theory
When used as nouns, formal system means the grouping of a formal language and a set of inference rules and/or axioms, whereas theory means a description of an event or system that is considered to be accurate.
check bellow for the other definitions of Formal system and Theory
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Formal system as a noun (logic):
The grouping of a formal language and a set of inference rules and/or axioms.
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Theory as a noun:
A description of an event or system that is considered to be accurate.
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Theory as a noun (obsolete):
Mental conception; reflection, consideration.
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Theory as a noun (sciences):
A coherent statement or set of ideas that explains observed facts or phenomena and correctly predicts new facts or phenomena not previously observed, or which sets out the laws and principles of something known or observed; a hypothesis confirmed by observation, experiment etc.
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Theory as a noun (uncountable):
The underlying principles or methods of a given technical skill, art etc., as opposed to its practice.
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Theory as a noun (mathematics):
A field of study attempting to exhaustively describe a particular class of constructs.
Examples:
"Knot theory classifies the mappings of a circle into 3-space."
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Theory as a noun:
A hypothesis or conjecture.
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Theory as a noun (countable, logic):
A set of axioms together with all statements derivable from them; or, a set of statements which are . Equivalently, a formal language plus a set of axioms (from which can then be derived theorems).
Examples:
"A theory is consistent if it has a model."