the GNU version of the Collaborative International Dictionary of English
ad. By deduction; by way of inference; by consequence.
The Century Dictionary and Cyclopedia
By deduction; in consequence of a general principle.
By diduction or separation; inferentially.
Word Usage
"Its elementary counterpart is the theorem that the equational theories on a free algebra F (V), defined as the deductively closed sets of equations that use variables from V, are exactly its substitutive congruences."