
Deduction (from Latin deducere, â€˜to lead fromâ€™) is a form of logic in philosophy and mathematics. In philosophy, a deductive argument is such that if all its premises are true, then its conclusion must be true. It is impossible for all the premises of a deductive argument to be true and yet its conclusion be false. The following argument is deductive: Gabriel is a man; all men are mortal; therefore, Gabriel is mortal. This argument is deductive because if its premises are true, then its conclusion must be true. Its premises may not be true, of course: Gabriel may be an angel rather than a man. But if all its premises are true, then its conclusion cannot be false.
Deduction forms the lifeblood of mathematics, which in its essence consists of various sets of rules (axioms) and the deductions which can be made from them. In a sense, it is impossible to find new information by deduction: all the deductions that can be made are implicit in the original axioms. Mathematics seeks to find all the deductions that can be made, especially those which are not obvious consequences of the axioms. (A recent result, the classification of the finite simple groups, is so nonobvious that its proof takes thousands of pages of argument.)
Deduction in mathematics can be formalized, but it is usually done in an informal, more accessible manner. (This will still use some symbolic language for clarity, at least to the specialist: symbolic language is what makes it inaccessible to the nonspecialist.) AJ SMcL
See also axiomatization; formalism; induction.Further reading N.W. Gowar, Basic Mathematical Structures; , M. Sainsbury, Logical Forms. 
