Meaning "to involve something unstated as a logical consequence" first recorded c. 1400; that of "to hint at" is from 1580s. Related: Implied; implying. The distinction between imply and infer is in "What do you imply by that remark?" but, "What am I to infer from that remark?" Or, as Century Dictionary puts it, "An action implies ability or preparation, but involves consequences."
"to reason, from two judgments to infer a third," 1640s, from Latin ratiocinatus, past participle of ratiocinari "to reckon, compute, calculate; to deliberate, meditate; to reason, argue, infer" (see ratiocination). "Now rare in serious use" [OED]. Related: Ratiocinant; ratiocinative; ratiocinatory.
early 15c., "infer, predict, form (an opinion or notion) upon probabilities or slight evidence," from conjecture (n.) or from verbs in Medieval Latin and Old French. Middle English had also the parallel forms conjecte (n.), conjecten (v.). Related: Conjectured; conjecturing.
1856, "capable of being withdrawn," especially from one's taxes or income, with -ible + Latin deducere "lead down, derive" (in Medieval Latin, "infer logically"), from de "down" (see de-) + ducere "to lead" (from PIE root *deuk- "to lead"). As a noun, "amount of a loss which must be borne by the policy-holder in an insurance claim," by 1927. The older adjective is deducible (1610s).
1640s, "derivative" (a sense now obsolete); from 1660s in logic, "consisting of deduction; based on inference from accepted principles," from Latin deductivus"derivative," from deduct-, past-participle stem of deducere "to lead down, derive" (in Medieval Latin, "infer logically"), from de "down" (see de-) + ducere "to lead" (from PIE root *deuk- "to lead"). Related: Deductively.