These are the most basic for 2 reasons. One, they are the most fundamental to the motion of God’s mind itself. God’s knowledge is the content of His mind logic is the motion of God’s mind. If we do not follow these most basic motions, then we no longer think. Second, they form the foundation for all other logics and mathematics.
The best advice I can give for being better at logic, and thinking more like God is practice. Particularly, to practice in the basics of logic, such as the three basic laws, defining terms, having precise and truthful premises, avoiding informal follicles and lastly to make valid inferences. It is not good enough to just read a book on logic, but one must practice over and over until it becomes second nature. This lesson is to practice the most basic, the Law of non-contradiction. For this reason I will post several people teaching on this for your practice of it.
FIRST. The law of contradiction.
Jesus appeals to it in,
Mark 12:37, “Therefore David himself calls Him ‘Lord’; how is He then his Son?” And the common people heard Him gladly.”
If the LoC is not an immutable motion of thinking, then Jesus’ appeal to it would mean David’s son(Jesus) is not David’s Lord and is David’s Lord. They would cancel each other out. There would be no doctrine to affirm or deny. There would be no knowldge, no thinking. Jesus’ point was the David’s promised Son is not merely human, for if so, then he could not be David’s Lord. But since this promised son is the God-man, then it is possible for Him to be both David’s son, physically, but also David’s Lord because He is the eternal Son of God. This means the laws of identity (thus, no category fallacies) and non-contradiction are not violated. Subjects and predicates are put together in proper categories and affirmed in understanding.
Also, in 1 Corinthians 14:7, “Even things without life, whether flute or harp, when they make a sound, unless they make a distinction in the sounds, how will it be known what is piped or played?”
For a word or sound to have intellectual meaning it must not only mean something, it must also not mean something: it must have definite meaning; it must make a distinction from other meanings. Aristotle’s explanation will open up what this means more.
Below is Clark to talk about the meaning behind the Law of noncontradiction. I could just give a symbolic notation for the Law such as ( A is B” and “A is not B”) are exclusive of each other, or give it in Natural Deduction, but on this matter a more in-depth explanation seems to be better to due to the importance of the subject. I prefer to see circle diagrams in my mind for a visual help.
If Oshea is predicated in the category of “man,” then there is no way for the category of non-man to be predicated to Oshea because of the immediate deduction of (obversion), “Oshea is a man,” is “No Oshea is non-man.” That is, to destroy the LoC would be to fail the rest of logic/Logos, it would kill God in essence. God could not affirm, “Jesus is My only Son,” because it could men “Jesus is not My only Son.” Jesus could not affirm that He is the truth, because it could be that Jesus is not the truth. See diagram.
For Doctrine: There are many verses which directly teach or use logic in the Bible. Jesus is the Logos. Jesus appeals the law of non-contradiction. Jesus at times would not quote the O.T. but only use logic to refute His opponents. We are made in His image. Thus, within the doctrine of Systematic Theology, there is a Doctrine of Logic. Christians ought to know this better.
Cannot Deny It!
“I deny the Law of non-Contradiction.” I must use the LoC to say this premise, otherwise, my denying of the LoC would men, “I affirm the Law of -non-Contradiction.”
“Words are not important.” If that is the case, then the words I used to say words are not important are not important.
“Anyone who says anything absolutely is arrogant.” If this is not said absolutely, then it does not apply to me, and so I do not care. But if the person who said it, does say it absolutely(as a dogmatic), then they by definition, are arrogant.
These statements, in order for them to be true, must be false at the same time. But the world I live in is not false. My existence is not false; that is, I must us my existence to deny it, and thus, I prove it.
St. Augustine had a way to show the stupidity of skepticism with the self-authenticating aspect of the LoC, by asking his opponent, “would you please deny your own existence (hint: without using it)?”
This is an important aspect ( must use the LoC to deny it) to remember, because Romans 1 says that the non-Christian is a moron because they cannot rationally deny God’s innate knowledge He put in them.
Romans 1:20-22, 2:14-15 (NLT)
20 For ever since the world was created, people have seen the earth and sky. Through everything God made, they can clearly see his invisible qualities—his eternal power and divine nature. So they have no [rational denese]* for not knowing God.
21 Yes, they knew God, but they wouldn’t worship him as God or even give him thanks. And they began to think up foolish ideas of what God was like. As a result, their minds became dark and confused. 22 Claiming to be wise, they instead became utter fools [Morons].
Even Gentiles, who do not have God’s written law, show that they know his law when they instinctively obey it, even without having heard it. 15 They demonstrate that God’s law is written in their hearts
*Strongs Greek. “ἀναπολόγητος [anapologetos /an·ap·ol·og·ay·tos/] adj. From 1 (as a negative particle) and a presumed derivative of 626; GK 406; Two occurrences; AV translates as “without excuse” once, and “inexcusable” once.
1 without defense or excuse. 2 that which cannot be defended, inexcusable.”
This is where we get the word Apologetics. It is like a lawyer in court giving an objective rational defence for something. In context of God writing His laws on our minds–then when we see the world it stimulates this innate knowldge of God to our thoughts– is the logical reason why it is impossible to give a rational defence for suppressing God’s truth. To do so, one must use God’s innately written truths to deny it; thus, they prove it. This is not the Law of contradiction itself, but a method using it to show its undeniablity. Vincent Cheung has do a good job explaining this “method” (don’t confuse this for the apologetic itself, which is the Bible) for Christian apologetics. See his book, Ultimate Questions. Also, Gordon Clark, A Christian View of Men and Things.
Because the Bible speaks about this in context of Christian apologetics then it a doctrine Christian ought to know and practice.
It is an amazing coincidence of history that Plato and Aristotle (384-323 B.C.) lived in the same century and that the latter was the pupil of the former. No other century can boast of such an amount of genius; no other pupil had such a teacher, and no other teacher had such a pupil. Extreme enthusiasm for Kant or Hegel might place the one or the other nearly on a level with Plato or Aristotle, but sober judgment fails to find an equal combination anywhere. Coincidences of history, however, may be of little significance. It is the clash of ideas that is important.
In the last chapter, in the section on the Parmenides, it was stated that Aristotle accepted the objections which Plato raised against his own theory. And there are others also. Accordingly, Aristotle considered the world of Ideas as a useless duplication of this world: useless, quite aside from the apparently intolerable difficulties involved in Platonism, because skepticism and all the troubles inherited from the Presocratics can be satisfactorily removed without its dubious aid. The student should be forewarned that Aristotle is not building on unaltered Platonic foundations; but precisely what Aristotle accepts from Plato and what he rejects, and how he combines and modifies the several factors, is a long and intricate story which makes Aristotle one of the hardest philosophers to understand. Then, too, his dull and methodical style does not cheer the flagging spirit. Plato was a vigorous and stimulating writer; he could combine the subtleties of epistemology, the excitement of politics, and the mathematical awe of astronomy all in one dialogue. The interrelations of the subjects, he constantly keeps before our eyes. Aristotle, on the contrary, carefully devotes one book to Logic, another to Physics, another to Psychology, and so on. This method undoubtedly has advantages, but the interrelations, which still exist, are hidden from view and must be sought out.
The Law of Contradiction
It is most appropriate to begin an account of Aristotle with some reference to his views on logic because the books on logic are logically put first in the corpus, and because his discussion of the fundamental laws of logic – the law of contradiction and the law of excluded middle – though taken from the Metaphysics, Book Gamma, forms a firm connection between earlier philosophy and the body of Aristotelian thought. For if Aristotle rejects the essential principles of Platonism, we should see at once how he will avoid the skepticism of Protagoras. Also it is at this point that the connection between logic and natural philosophy in general can be most clearly seen. For although logic aims to discover the principles on which all true judgment depends, it is not a merely formal science of thinking; but rather, since truth requires a relation to reality, the laws of logic must be not only the laws of thought, but the laws of reality as well.
Logic and Reality
Aristotle introduces the topic by questioning whether logic and reality are the objects of the same science or of two different sciences. In view of the fact that the truths of logic and the principles of reality apply universally and are not restricted to any special field of study, Aristotle concludes that they belong to the same science. The truths of botany or of geometry, on the other hand, do not apply universally: Geometry concerns being in so far as it occupies space, and botany is limited to being as it exhibits nutrition and growth. Yet all the special sciences make common use of the laws of logic because these laws hold for all reality, and not merely for that part of reality that the special science studies. But the special sciences use logic without discussing it. It would be incongruous for a botanist or an astronomer to discuss the nature of truth and the law of contradiction. No doubt some of the Presocratics did so, and their inclusion of this material is perhaps defensible on the ground that they thought they were discussing the whole of reality. But in this they were mistaken; for nature is only one genus of reality, and physics, while it is a kind of wisdom, is not the first kind. Therefore there must be a still more universal science that deals with primary being, and to this science Aristotle sometimes gives the name of First Philosophy. As the botanist or physicist is responsible for the most general principles within his special sphere, principles applying to the particular kind of being that forms the subject matter of that science, so the philosopher must state and explain the principles that apply to being without qualification, to all being without exception, to being qua being – principles that are absolutely universal without any restriction at all. It is therefore the prerogative of philosophy, and not of botany or any other special science, to study the most general principles of all existence.
The most certain of all principles is the law of contradiction, for it is impossible to be mistaken about it. It is not an hypothesis, a tentative by which to rise to something more general, for a principle which everyone must have who knows anything about being cannot be so characterized. The principle is this: The same attribute cannot attach and not attach to the same thing in the same respect. Or, otherwise, contrary attributes cannot belong to the same subject at the same time. This principle, be it noted again, is stated not merely as a law of thought, but primarily as a law of being. The ontological form is basic; the purely logical is derivative: It becomes a law of thought because it is first a law of being. If anyone should object to the law of contradiction and should assert, as Heraclitus is supposed to have done, that contrary attributes attach to the same thing, it would be necessary to conclude that he cannot believe what he says. For if we have shown that the number three cannot be both odd and even, and that a stone cannot be both heavy and light, and so on, then it follows that no one can think that three is both odd and even, even though he verbally makes such an assertion. Anyone who pretended to believe that contrary attributes attach to the same subject would be affirming two contrary opinions at the same time; and these two opinions would be, as it were, two contrary attributes attaching to him as a subject. But this is what the law of contradiction makes impossible.
Not only has the Heraclitean coexistence of contraries been maintained, but there are some writers who, thinking that the above derivation of psychological from ontological impossibility is circular, demand that the law of contradiction be formally demonstrated [deduced]. This demand, however, evinces their ignorance. The demonstration [deduction] of a proposition, such as any theorem in geometry, is completed only when it is referred to the axioms. If the axioms in turn required demonstration [deduction], the demonstration [deduction] of the proposition with which we began would remain incomplete, at least until the axioms could be demonstrated [deduced]. But if the axioms rest on prior principles, and if these too must be demonstrated [deduced] – on the assumption that every proposition requires demonstration – the proof of our original theorem would never be finished. This means that it would be impossible to demonstrate [deduce] anything, for all demonstration [deduction] depends on indemonstrable [non-deducible] first principles. Every type of philosophy must make some original assumptions. And if the law of contradiction is not satisfactory, at least these Heracliteans fail to state what principle they regard as more so. Nonetheless, though the law of contradiction is immediately evident and is not subject to demonstration, there is a negative or elenctic[indirect] argument that will reduce the opposition to silence.
The negative method avoids the charge of begging the question, for it is the opponent and not oneself who makes the assertion. Of course, this depends on the opponent’s willingness to say something. The proof aims to show the opponent who attacks the law of contradiction that so soon as he says anything at all, he is recognizing the principle. If he should say nothing, we have neither an opponent nor an objection to face. Nor need we insist that he make some tricky admission that plays into our hands. All that is required is that he say something significant for himself and for us, for this is the presupposition of every understanding between two persons, or even of one person’s understanding himself. Let the opponent then say something: that three is an odd number or that Socrates is a man. It will always be of the form, x is y. Now, in the first place, the word is has a definite meaning and does not mean is not. Therefore, Protagoras was mistaken when he said that everything is and is not.1 But perhaps the argument will be clearer if we consider the x and the y.
In any sentence the predicate, the y, must have a single, definite meaning; and when we say that x is y, or that Socrates is a man, we are asserting of Socrates the meaning of man, whatever it may be – two-footed animal, perhaps. Thus we assert something definite. The remark that words have several meanings will not damage this contention, provided the meanings are limited in number. Suppose the word man had ten different meanings: It would be possible to invent ten different terms so that each term would stand for a single meaning; and once more the predicate and the assertion as a whole would be definite. If, however, terms had an infinite number of meanings, then all reasoning would come to an end. For if a word is to convey a significance, it must not only mean something, it must also not mean something. If it had all the meanings of all the terms in the dictionary, it would be useless in speech. Therefore, if terms had an infinite number of meanings, no term would have one meaning; and not to have one meaning is to have no meaning; but if words have no meaning, it is impossible to argue with other people or even to reason privately within oneself. If we do not think one thing, we think nothing; but if we can think of one thing, then we can assign to it a single unambiguous term. On this basis it is impossible that being a man should mean precisely not being a man, or that perception should be non-perception, or that a wind should be both y and not-y. And this is in reality a justification of the law of contradiction.
The Sophists, both of antiquity and of the present, ignoring the ontological basis of this argument, attempt the reply that what one person calls a man, another may call a mouse and not a man. Hence the same thing would be both man and not-man. But this is elementary ambiguity. The question is not whether a subject can be man and not-man in name, but whether it can be so actually or ontologically. If man and not-man mean two different things, as was indicated above, and if man means two-footed animal, it follows that anything that is a man must be a biped. But if this must be so, i.e., if this is necessary, the contrary is impossible: It is impossible that the subject should not be a two-footed animal, and hence the same subject cannot possibly be both man and not-man.
Denial of Substance
Further to refute his opponents, Aristotle plunges into logical and ontological complexities that will try the most ambitious student. Those who argue against the law of contradiction must also deny substance and reality. To explain how this is so and why it is absurd requires reference to the theory of categories, later to be explained. To anticipate, however, it may be briefly stated that a category is a predicate; or, more precisely, the ten categories are the ten types of possible predicates. For example, of Socrates it may be said that he is a man, he is ugly, he is wise, he is short, he is heavy, and perchance he is a musician. But of these, the predicate man holds a favored position. Heavy and musical are accidental predicates; that is, it is not necessary or essential to being a man that one should be heavy or musical: There are men who are frail and unmusical. These predicates and other accidental predicates fall under the categories of quality, quantity, relation, or others. But the predicate man, when one says that Socrates is a man, is no accident: Man is what Socrates essentially is. The predicate man falls under the category of substance or reality. And the category of substance is basic because there can be no quality or quantity unless there is a substance that it is the quality of.
The Sophistic opponents of logic, however, do away with substance, for they must say that all attributes are accidents, and that no subject is essentially man. The line of reasoning behind this is as follows. To be essentially and substantially man is incompatible with being not-man or not being man, for when we say that Socrates is essentially man, we are designating his substance; and to designate a thing’s substance, essence, or reality is to deny that it is essentially or really something else. The skeptical relativists must say, therefore, that nothing can be defined, and that all attributes are accidental. But if all predication is accidental, there will be no reality of which the predication is made, and predication would be endless. This, however, is impossible, because, far from being endless, not more than two terms can be joined in accidental predication. We may say that the musician is blond or even that the blond is musical; but the accidental conjunction of blond and musical is possible only because they are both accidents of the same reality – Socrates perhaps. In the absence of an underlying subject of which both are predicates, blond could not be predicated of musical nor musical of blond. Now, when we say that Socrates is musical or that Socrates is blond, the predicate is not related to its subject as in the previous predications, for, while blond and musical were equally accidents of an underlying reality, Socrates and musical are not thus on the same level as accidents of some third subject. Socrates is not a predicate at all, and hence there cannot be an infinite series of predicates: Every series must end with a reality.
As this section of Aristotle is somewhat subtle, and as its importance has been seen in Plato’s refutation of Protagoras, it will be well to elaborate a little. Aristotle may be willing to admit that the law of contradiction as stated does not hold for accidental predication. The musician can be white; yet since white is “not-musical,” the musician can be “not-musical.” But with substantial predication, the case is different. Suppose we ask the opponent if A is a man. He could answer, Yes, but he is also white and musical, and these are not-man; hence, A is man and not-man. This answer is correct to the extent that a subject may have an indefinite number of accidents; but so understood the answer is beside the point. Our original question was, Is A essentially a man? If the opponent ignores the “essentially,” as he did in the answer just given, he should list all the accidents – all, and that includes the negative as well as the positive ones. He should therefore say A is man, musical, white, not-green and therefore blue, not-ship and therefore house. For, if it is true that man is not-man, as the opponent claimed just above, it is all the more true that man is not-ship; but since house is not-ship and since on this theory contrary accidents attach, the man must be both a house and also a ship. Such a list of accidents would be infinite. Yet, if the opponent begins to list these accidents, he ought to continue with them. Let him give all or none. There is no reason for specifying only three or four. From which it follows that if he begins and continues, he will take so long that we shall be spared the trouble of answering him. In other words, if the opponent depends on accidental predication, if he repudiates the distinction between substantial and accidental predication, discussion ends. On this theory no predicate is definitive, and the metaphysical implication is that reality does not exist.
Now, to repeat a thought previously stated near the beginning of this analysis of the law of contradiction: This analysis or “proof” is a negative or elenctic one. It is not a demonstration based on more original principles. A careless reading might conclude that the law is demonstrated from the principle that every word must have a single meaning. But the truth of the matter is quite the reverse. Aristotle is saying rather that every word must have a single meaning because the principle of contradiction holds. He is applying the law to this particular case. And the particular case is chosen for the purpose of showing that an opponent cannot carry through his own theory. He becomes tangled in an infinite regress and must drop out of the argument. Therefore, if anyone, including the opponent, wishes to argue, reason, discuss, or say anything meaningful, he must presuppose the law of contradiction. Hence, this law is not demonstrated from some higher principle, but Aristotle shows that it must be presupposed by anyone who wishes to speak intelligibly.
The inanity of skeptical relativism was hinted at in the remark above that the musician must be a not-ship and therefore a house. This has a further ontological implication. If contradictory statements are true of the same subject at the same time, evidently all things will be the same thing. Socrates will be a ship, a house, as well as a man; but then Crito too will be a ship, a house, and a man. But if precisely the same attributes attach to Crito that attach to Socrates, it follows that Socrates is Crito. Not only so, but the ship in the harbor, since it has the same list of attributes too, will be identified with this Socrates-Crito person. In fact, everything will be everything. Therefore, everything will be the same thing. All differences among things will vanish and all will be one. Such is the metaphysical nonsense to be derived from Protagoras or anyone else who denies the law of contradiction.
As a Christian deductionalist it is important to consider logic and theology proper. Thus, here is Clark to talk about John 1:1. Clark’s board focus is, “the Logic was God,” but particularly he focuses on the law of non-contradiction and the motion of God’s thinking.
Gordon Clark. He explains how the LOGOS is the Divine Nature itself. Thus, not only is logic a doctrine taught from the Bible, it is a Ultimate Question of life for man.
Logic Is God
It is to be hoped that these remarks on the relation between God and truth will be seen as pertinent to the discussion of logic. In any case, the subject of logic can be more clearly introduced by one more Scriptural reference. The well-known prologue to John’s Gospel may be paraphrased, “In the beginning was Logic, and Logic was with God, and Logic was God…. In logic was life and the life was the light of men.”
This paraphrase-in fact, this translation-may not only sound strange to devout ears, it may even sound obnoxious and offensive. But the shock only measures the devout person’s distance from the language and thought of the Greek New Testament. Why it is offensive to call Christ Logic, when it does not offend to call him a word, is hard to explain. But such is often the case. Even Augustine, because he insisted that God is truth, has been subjected to the anti-intellectualistic accusation of “reducing” God to a proposition. At any rate, the strong intellectualism of the word Logos is seen in its several possible translations: to wit, computation, (financial) accounts, esteem, proportion and (mathematical) ratio, explanation, theory or argument, principle or law, reason, formula, debate, narrative, speech, deliberation, discussion, oracle, sentence, and wisdom.
Any translation of John 1:1 that obscures this emphasis on mind or reason is a bad translation. And if anyone complains that the idea of ratio or debate obscures the personality of the second person of the Trinity, he should alter his concept of personality. In the beginning, then, was Logic.
That Logic is the light of men is a proposition that could well introduce the section after next on the relation of logic to man. But the thought that Logic is God will bring us to the conclusion of the present section. Not only do the followers of Bernard entertain suspicions about logic, but also even more systematic theologians are wary of any proposal that would make an abstract principle superior to God. The present argument, in consonance with both Philo and Charnock, does not do so. The law of contradiction is not to betaken as an axiom prior to or independent of God. The law is God thinking.
For this reason also the law of contradiction is not subsequent to God. If one should say that logic is dependent on God’s thinking, it is dependent only in the sense that it is the characteristic of God’s thinking. It is not subsequent temporally, for God is eternal and there was never a time when God existed without thinking logically. One must not suppose that God’s will existed as an inert substance before he willed to think.
As there is no temporal priority, so also there is no logical or analytical priority. Not only was Logic the beginning, but Logic was God. If this unusual translation of John’s Prologue still disturbs someone, he might yet allow that God is his thinking. God is not a passive or potential substratum; he is actuality or activity. This is the philosophical terminology to express the Biblical idea that God is a living God. Hence logic is to be considered as the activity of God’s willing.
Although Aristotle’s theology is no better (and perhaps worse) than his epistemology, he used a phrase to describe God, which, with a slight change, may prove helpful. He defined God as “thought-thinking-thought.” Aristotle developed the meaning of this phrase so as to deny divine omniscience. But if we are clear that the thought which thought thinks includes thought about a world to be created-in Aristotle God has no knowledge of things inferior to him-the Aristotelian definition of God as “thought-thinking-thought” may help us to understand that logic, the law of contradiction, is neither prior to nor subsequent to God’s activity.
This conclusion may disturb some analytical thinkers. They may wish to separate logic and God. Doing so, they would complain that the present construction merges two axioms into one. And if two, one of them must be prior; in which case we would have to accept God without logic, or logic without God; and the other one afterward. But this is not the presupposition here proposed. God and logic are one and the same first principle, for John wrote that Logic was God. At the moment this much must suffice to indicate the relation of God to logic. 
H.W.B. Joseph gives a short summary of the Three Laws of Logic.
The Three Laws of Thought.
The connection between questions about our thinking, and what we must think things to be, is excellently shown in the so-called Laws of Thought. These are certain very general principles exemplified in all thinking, and some have supposed the task of Logic to consist merely in developing their implications. They are known as the Law of Identity, the Law of Contradiction, and the Law of Excluded Middle. The Law of Identity may be formulated by saying that whatever is, is ‘ : or symbolically, that A is A’ ; the Law of Contradiction, that ‘ a thing cannot both be and not be so and so ‘ that ‘ contradictory propositions cannot both be true ‘, or that A cannot be B and not be B ; the Law of Excluded Middle, that is a thing either is or is not so and so ‘, that contradictory propositions cannot both be false ‘, or that ‘ A either is or is not B ‘. In other words, if we think about anything, then (1) we must think that it is what it is; (2) we cannot think that it at once has a character and has it not; (3) We must think that it either has it or has it not.
Now though these are called laws of thought, and in fact, we cannot think except in accordance with them, yet they are really statements which we cannot but hold true about things. We cannot think contradictory propositions, because we see that a thing cannot have at once and not have the same character; and the so-called necessity of thought is really the apprehension of a necessity in the being of things. This we may see if we ask what would follow, were it a necessity of thought only; for then, while e.g. I could not think at once that this page is and is not white, the page itself might at once be white and not be white. But to admit this is to admit that I can think the page to have and not have the same character, in the very act of saying that I cannot think it; and this is self-contradictory. The Law of Contradiction then is metaphysical or Ontological.
So also, is the Law of Identity. It is because what is must be determinately what it is, that I must so think. That is why we find a difficulty in admitting the reality of absolute change, change when nothing remains the same; for then we cannot say what it is which changes.
The Law of Excluded Middle is so far different as a disjunctive proposition expresses doubt, and doubt belongs to the mind, not to things. But to deny that this page need either be or not be white is to deny that it need be anything definite; determinateness involves the mutual exclusiveness of determinate characters, which is the ground of negation; and that is a statement about things. In other words, unless the primary Laws of Thought were Laws of Things, our thought would be doomed by its very nature to misapprehend the nature of things.
Vincent Cheung explains the concept of how a contradiction cancels the two propositions out. Thus, in theology would be there would be no doctrine to affirm or deny. There would be no salvation, no God or no man to affirm or deny.
For any proposition that affirms X, the proposition that contradicts it is one that affirms not-X. This is what a contradiction means. Any proposition that affirms one thing is by necessity also a denial of its opposite. To affirm X is to deny not-X, and to affirm not-X is to deny X. To keep this simple, let us assume that Y = not-X, so that the opposite of X is Y. Thus to affirm X is to deny Y, and to affirm Y is to deny X. Or, X = not-Y, and Y = not-X. Then, since to affirm a proposition is to deny its opposite, to affirm X and Y at the same time is the equivalent of affirming not-Y and not-X. Thus to affirm two contradictory propositions is in reality to deny both. But to affirm both not-Y and not-X is also to affirm X and Y, which again means to deny Y and X. And so the whole operation becomes meaningless. The upshot is that it is impossible to affirm two contradictory propositions at the same time.
To affirm the proposition, “Adam is a man” (X), is to deny the contradictory proposition, “Adam is not a man” (Y, or not-X). Likewise, to affirm the proposition, “Adam is not a man” (Y), is to deny the contradictory proposition, “Adam is a man” (X). Now, to affirm both “Adam is a man” (X) and “Adam is not a man” (Y) is only to deny both propositions in reverse order. That is, it is equivalent to denying “Adam is not a man” (Y) and “Adam is a man” (X). But then we are back to affirming the two propositions in reverse order again. When we affirm both, we deny both; when we deny both, we affirm both. Therefore, there is no intelligible meaning in affirming two contradictory propositions. It is to say nothing and to believe nothing.
To illustrate, it is clear that divine sovereignty and human freedom contradict each other.If God controls everything, including man’s thoughts, then man is not free from God. If man is free from God in any sense or to any degree, then God does not control everything. Yet some theologians claim that the Bible teaches both divine sovereignty and human freedom, and so they insist that we must affirm both. However, since to affirm divine sovereignty is to deny human freedom, and to affirm human freedom is to deny divine sovereignty, then to affirm both only means to reject both divine sovereignty (in the form of an affirmation of human freedom) and human freedom (in the form of an affirmation of divine sovereignty). But to deny both means to affirm both in reverse order, and to affirm both means to deny both in reverse order again.
The necessary result is that the person who claims to believe both divine sovereignty and human freedom believes neither. In claiming to believe all of the Bible, he in fact believes none of it. In this example, since the Bible affirms divine sovereignty and denies human freedom, there is no contradiction – not even an apparent one. On the other hand, when non-Christians allege that the incarnation of Christ entails a contradiction, the Christian does not have the option to deny either the divinity or the humanity of Christ. Rather, he must formulate the doctrine as the Bible teaches it, and show that there is no contradiction. The same applies for the doctrine of the Trinity. In any case, if a person claims that he sees contradictions in the Bible, this means that he does not – he cannot – believe the Bible.
A popular response is that these are only apparent contradictions; that is, the doctrines only seem like contradictions to the mind of men, but they are in perfect harmony in the mind of God. This answer is futile. There is no difference between an apparent contradiction and an actual contradiction when it comes to affirming it. It remains that to affirm one thing is to deny the other at the same time, so that to affirm both is to deny both, and that to deny both is to affirm both again. Thus the person who affirms an apparent contradiction really affirms nothing and denies nothing. Whether the contradiction is only an apparent one is irrelevant. As long as it appears real to the person, it is real enough.
Moreover, how can a person distinguish between an apparent contradiction from an actual contradiction? He can never know that a contradiction is only an apparent one.
Unless he knows how to resolve the apparent contradiction, it will appear the same to him as an actual contradiction. And if he knows that a contradiction is only an apparent one, then he has already resolved it, and the term contradiction no longer applies. If we must tolerate apparent contradictions, then we must tolerate all contradictions. We often challenge non-Christian views on the basis that they contradict themselves. But if we tolerate apparent contradictions, then there is nothing to prevent non-Christians from claiming that the contradictions in their worldviews are only apparent ones.
 Obversion works because of the Law of Noncontradiction and excluded middle.
 Clark will be giving Aristotle’s explanation for the law of noncontradiction with his own commentary woven throughout. In General it seems Clark agreed with Aristotle’s indirect defense of the LoC because Clark used some similar remarks about the LoC and in defending the coherence of the Scripture against non-Christian worldviews in, “A Christian View of Men and Things,” see last chapter. And in his essay, “God and Logic.”
 Gordon is using the term “indemonstrable” as “non-deducible.” He states there are indirect arguments for logical tautologies. These are more like indrect methods to show. Similar to how a reductio ad adsurdum argument is an indirect use of denying the consequent.
 Gordon Clark. Thales To Dewey – A History of Philosophy. The Trinity Foundation. 2000. Pg.86-92. Chapter on Aristotle.
 – Added by Author. Emphasis added by author.
 H.W.B. Joseph. 1906. An introduction to LOGIC. Pg.13
Read the book here. https://archive.org/details/introductiontolo00jose/page/n6
 The doctrine of divine sovereignty will be discussed and applied throughout this book. Also see Vincent Cheung, Commentary on Ephesians and The Author of Sin.
 The doctrine of compatibilism teaches that man is not free from God, but that man is still free in a sense. However, unless the kind of freedom under consideration is freedom from God, it is irrelevant, since the topic concerns God’s control over man. See Vincent Cheung, The Author of Sin.