Logic is the science of necessary inference. An inference is the drawing of a conclusion from premises by logical methods. The adjective "necessary" in necessary inference or necessary consequence means that the conclusion is inescapable. An argument consists of one or more propositions in support of another proposition. The propositions in support of the other proposition are premises; the proposition supported by the other propositions is the conclusion. The familiar argument about Socrates illustrates the relation between premises and conclusion in deductive argument.
A proposition is a declarative sentence in which the predicate is affirmed or denied of the subject. A proposition is the meaning of a declarative sentence. Propositions are either true or false. We indicated above that propositions (and only propositions) are the elements (premises and conclusions) of an argument. Sentences that express commands, pose questions, or convey exhortations are neither true nor false, and therefore are never elements of an argument. Gordon Clark puts it this way:
Of course in English Rhetoric there are questions that are intended as propositions. They are called rhetorical questions. But logically they are propositions. A question that is intended as a question is neither true nor false. It can play no part in an argument.,(Logic, p. 30 PB)
Consider the following:
Arguments divide into two classes: deductive arguments and inductive arguments. This classification amounts to two different claims. The premises of Inductive Arguments claim to provide incomplete or partial reasons in support of a conclusion. The premises of Deductive Arguments claim to provide conclusive reasons for a conclusion. With Inductive Argument the conclusion is said to be either probable or improbable. In Deductive Argument, the conclusion follows necessarily or it does not. That is to say, the conclusion is either a necessary consequence of the premises or it is not a necessary consequence of the premises. Another way of stating the same thing: a Deductive Argument consists of a conclusion deduced from premises. The deduction of conclusions from premises is at the heart of logic.
The phrases, necessary consequence and necessary implication, mean necessary inference. The use of one or the other phrase depends on the emphasis. If one stresses that the premises imply a conclusion, one speaks of necessary implication. If one stresses the conclusion resulting from premises, one speaks of necessary consequence. Either phrase points to a claim of necessary inference between premises and conclusion in a Deductive Argument. If the conclusion of a Deductive Argument is a necessary consequence of the premises, then the argument is valid; otherwise, invalid. Using other words: If the premises of a Deductive Argument necessarily imply the conclusion, then the argument is valid; otherwise, invalid.
Logic is the study of the relation between premises and conclusion in Deductive Arguments. If the conclusion follows from premises necessarily (that is, the conclusion is unavoidable), then the argument enjoys valid status; if not (that is, the conclusion can be avoided), then the argument is invalid. By this definition, every deductive argument is either valid or invalid. If a deductive argument is valid and all of its propositions are True, then the argument is said to Sound; otherwise Unsound.
On p. 149 of Religion, Reason, and Revelation Gordon Clark challenges anyone to state his position (theory) without making use of the law of contradiction. Indeed, the necessity of the other laws of logic, the law of identity and the law of excluded middle, are equally necessary for meaning and rational discourse. Absent the laws of logic, the result is nonsense, and if uttered, it is gibberish. (See Gordon H. Clark. Religion, Reason, and Revelation, The Trinity Foundation, Unicoi, TN 37692)
John Robbins wrote in a Trinity Review:
In the act of speaking God reveals his rationality: The laws of speech are the laws of logic. The rules of grammar are derivative from the principles of logic. For a word – any word, human or divine – to mean something (and every word of God means something, for God does not talk nonsense), that word must also not-mean something else. When God says, “Let there be light,” light does not mean dark; or bees, or matter; let does not mean do not let, write, or rent; be does not mean buy, destroy, or eat. … “[I]n the beginning,” does not mean AD 2000 or even one second after the beginning. This is the logical law of contradiction: Not both A and not-A. If sounds and written symbols do not obey this fundamental rule of logic, they are mere noises in the air or mere scribbling on the paper; they are not words; they are not speech. God can and does speak because, as John tells us, God is Logic. (Trinity Review, #309b, Nov-Dec 2012, p.4)
To repeat then, “… [T]he law of contradiction…requires that a given word must not only mean something, it must also not mean something. The term dog must mean dog, but also it must not mean mountain, and mountain must not mean metaphor. Each term must refer to something definite, and at the same time there must be some objects to which it does not refer.” (Clark, p. 149)
Suppose the word mountain meant dog, and Bible, and books, and library. Suppose it meant everything. If it did, it would mean nothing. If the word mountain meant everything, one could write a book of any length by writing: mountain, mountain, mountain, mountain, mountain.... It could mean the dog flew up the mountain. It could also mean that World War III will be a nuclear war. In short, it could mean whatever one might imagine.
The point should be clear: One cannot write a book or speak a sentence that means anything without using the law of contradiction. Logic is an innate necessity, not an arbitrary convention that may be discarded at will. (Clark, p.150)
For those who argue that the laws of logic are mere linguistic conventions, let them construct an argument that refutes each law. As Clark, Robbins, and others have said: To refute the laws of logic requires that one use them, thereby demonstrating their necessity for all meaningful, rational discourse.
In all our conversation and writing the forms of logic are indispensable: without them discussion on every subject would cease. (Gordon H. Clark. A Christian View of Men and Things, Trinity Foundation, Unicoi, TN, p. 308)