The phrase good and necessary consequence was used more commonly several centuries ago to express the idea which we would place today under the general heading of logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

; that is, to reason validly by logical deduction or better, deductive reasoning

Deductive reasoning

Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...

.

Even more particularly, it would be understood in terms of term logic

Term logic

In philosophy, term logic, also known as traditional logic or aristotelian logic, is a loose name for the way of doing logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century...

, also known as traditional logic, or as many today would also consider it to be part of formal logic, which deals with the form (or logical form) of arguments as to which are valid

Validity

In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....

 or invalid.

In this context, we may better understand the word "good" in the phrase "good and necessary consequence" more technically as intending a "valid argument form".

One of the best recognised articulations of the authoritative and morally binding use of good and necessary consequence to make deductions from Scripture can be readily found in probably the most famous of Protestant Confessions of faith, the Westminster Confession of Faith

Westminster Confession of Faith

The Westminster Confession of Faith is a Reformed confession of faith, in the Calvinist theological tradition. Although drawn up by the 1646 Westminster Assembly, largely of the Church of England, it became and remains the 'subordinate standard' of doctrine in the Church of Scotland, and has been...

 (1646), Chapter 1, sec. 6, as well as in many others, including the Heidelberg Catechism

Heidelberg Catechism

The Heidelberg Catechism is a Protestant confessional document taking the form of a series of questions and answers, for use in teaching Reformed Christian doctrine...

, the Belgic Confession

Belgic Confession

The Confession of Faith, popularly known as the Belgic Confession, is a doctrinal standard document to which many of the Reformed churches subscribe. The Confession forms part of the Reformed Three Forms of Unity...

, and the London Baptist Confession (1677/1689).