therefore

Therefore is an English adverb used to indicate a conclusion, consequence, or result drawn from preceding statements. It is commonly placed at the start of a clause and often followed by a comma, as in The experiment failed; therefore, we revised the hypothesis. It can also appear within a sentence for emphasis or clarity.

Etymology and form: The word derives from Old English and Middle English compounds meaning 'for that reason' or 'therefore.' It functions as a conjunctive adverb or transition, connecting premises to a conclusion. In formal logic and mathematics, the symbol ∴ is used to denote 'therefore.'

Usage in logic and mathematics: In formal arguments, therefore signals the logical consequence of earlier statements. It is common in proofs, theorem statements, and deductive reasoning to introduce a derived result or corollary. In philosophy and rhetoric, it marks the conclusion that follows from stated assumptions.

Relation to other connectives: Therefore is closely related to words such as thus, hence, and consequently. Subtle differences exist: therefore often emphasizes logical deduction from established premises; thus can be broader or methodological; hence stresses the outcome or effect of a given step.

Usage considerations: In formal writing, 'therefore' can convey a precise, scholarly tone. In everyday writing, alternatives like 'so' or 'thus' may be more natural. Punctuation is typically a comma after the introductory usage, but placement can vary to suit rhythm and style.