12. Appendix A: The Axioms of Logic in Scripture

Preface

Chapter 4 of this book affirmed that logic is embedded in the Scripture. Though the footnotes given in that chapter have adequately demonstrated that this is a Biblical doctrine, this appendix will show the exegetical basis for each axiom of logic. For the system known as Christianity there is technically only one necessary axiom: “God’s Word is Truth.” Everything else in our worldview can be logically deduced from this one axiom.³⁹⁴ This gives a unified system of thought that starts with God’s mind and covers all of life.

Each discipline (when considered by itself) also has several starting principles, or axioms (the Biblical word στοιχεια), from which all other truths within that system can be deduced. Since they come from the Bible, we could call these Biblical Axioms, Inspired Axioms, or Great Axioms.

For those who would argue that only the starting axiom should be called an axiom, I would reply that our starting axiom (“God’s Word is Truth”) is derived from the Bible just as surely (see Psalm 119:160; John 17:17) as the axioms of this chapter are. If the entire Bible is our axiomatic starting point, then all the starting points we derive from the Bible for each discipline are also axiomatic for that discipline. Since the term “axiom” is widely used for the starting principles of each discipline, it will communicate better to use the term “axiom” for both the major axiom of the entire system of Christianity (“God’s Word is Truth”) and for the minor axioms that form the foundation for each discipline.

Without the laws of logic it would be impossible to reason. This poses a conundrum for the atheist. The atheist knows that the laws of logic are:

1. Universal (i.e., they apply everywhere)
2. Abstract (i.e., they are immaterial and grasped by thought alone and thus cannot be reduced to his materialistic universe of atoms and genes)
3. Invariant (i.e., they are never changing)
4. Authoritative (i.e., they must be accepted)

Without God, how do you account for such laws of thought? The atheist can’t deny them with consistency since a denial of logic is a denial of truth and reason and he would have to give up the argument with the theist. In arguing against the Christian he must constantly resort to logic. If he affirms that logic is universal, abstract, invariant, and authoritative, he has a hard time justifying it. Where do such laws of logic come from? Who made them authoritative? Why do all civilizations seem to recognize logical arguments? If our genes arose from chance, how can chance account for such universal order in our thinking? You cannot say they are mere conventions of man since they are universally found in men and have not changed over history and are authoritative. They can’t be simply part of the material universe since they are abstract and not material.

The four characteristics of logic listed above are exactly what would be expected by the Christian since logic is in the very mind of God (see John 1:1 λόγος) and since God revealed His logical mind in the Scriptures of truth, and since He made man in His image. The image of God in man is composed of logic, language, ethics, dominion, etc., and therefore we cannot escape from logic, try as we might. God can command us, “Come, let us reason together” (Isa. 1:18), and we can respond with reasoning. There is thus a triangular connection between God (the original in heaven), His revealed Word (objective truth), and His natural revelation within man (subjectively understood truth).

What this chapter is most concerned to demonstrate is that logic is embedded in the Scriptures. Logical principles are found on almost every page of Scripture. One example can be given in Matthew 12:24-30 where we see the following laws of logic at work:

1. Argument from analogy (vv. 25-26)
2. The law of logical or rational inference (v. 26)
3. Reductio ad absurdum (vv. 25-26)
4. Argument from analogy (v. 27)
5. The law of logical or rational inference (vv. 28-29)
6. Argument from analogy (v. 29)
7. The law of contradiction (v. 30)
8. The law of excluded middle (v. 30)

If God uses these axioms of logic, then they are true.

Definitions of terms

In propositional logic we evaluate whether propositions are meaningless, true, or false. Propositions are often represented by the letters p,q,r,s. The logical connections are as follows:

And ˄ ∧ (or some logicians may use &, Kpq, or the middle dot, ⋅)

Or ˅ ∨ (or some logicians may use ||, or Apq )

Not ¬ (or some logicians may use ~, Np, or Fp)

Implies → (or some logicians use ⊃)

Biconditional ↔ (or some logicians use ≡)

Equals is represented by =

Truth is represented by T (or some logicians use Vpq, or ⊤)

False is represented by F (or some logicians use Opq, or ⊥)

All is represented by A

No is represented by E

Some is represented by I

“Some is not” is represented by O

Arguments have rules for logical manipulation. For example:

Given the set of sentences, S, it is valid to infer another sentence, p. This would be written as: 𝑆 ⊢ 𝑝

Modus ponens can be written as P⊃ O, P,∴ O

¬ is evaluated first ∧ and ∨ are evaluated next Quantifiers are evaluated next → is evaluated last.

A listing of the axioms of logic

The most basic laws

1. The law of identity — P is P (or A:A).
2. The law of (non-)contradiction — P is not non-P (or ~[A: ~A] or A : ~A).
3. The law of the excluded middle — Either P or non-P. Or since A:A & ~[A: ~A] then either A or ~A.
4. The law of rational inference — “if A=B, and B=C, then A=C.”
5. Modus ponens — If p then q; p; therefore q.
6. Modus tollens — If p then q; not q; therefore not p.

Further derived laws

1. Hypothetical syllogism — If p then q; if q then r; therefore, if p then r.
2. Disjunctive Syllogism (also known as the modus tollendo ponens or MTP) — If P is true or Q is true and P is false, then Q is true.
3. Immediate inferences for an obverse categorical statement.

For example, given “All S are P”, one can make the immediate inference that “No S are non-P” which is the obverse of the given statement.

Or, given “No S are P”, one can make the immediate inference that “All S are non-P” which is the obverse of the given statement.

Or, given “Some S are P”, one can make the immediate inference that “Some S are not non-P” which is the obverse of the given statement.

Or, given “Some S are not P”, one can make the immediate inference that “Some S are non-P” which is the obverse of the given statement.

1. Immediate inference for a contrapositive statement.
2. Constructive dilemma.
3. Destructive dilemma.

There are 256 possible ways to make categorical syllogisms using the A,E,I, and O statement forms (the square of opposition). Of the 256, only 24 are valid forms. Of the 24 valid forms, 15 are unconditionally valid, and 9 are conditionally valid. This introduction will only deal with the fact that the basic axioms from which logic is derived are found in the Scriptures. The online Great Axioms Project will develop this Biblical logic further.

Showing these axioms to be inspired axioms/Biblical axioms/great axioms

First axiom of logic: The Law of Identity

This law states that if a proposition is true, then it is true. Ways of stating this law are, P is P, or if p then p, or A:A or A = A.

1. Gen. 35:11. “And God said unto him, I am God Almighty.” From this we can see that God (A) said, I (A) am (=) God (A), or A = A.
2. Ex. 3:14. God said, I AM who I Am.
3. John 3:6. “That which is born of the flesh is flesh, and that which is born of the Spirit is spirit.” That is, flesh is not spirit and vice versa. If flesh = flesh and spirit = spirit then a = a and b = b.
4. 1 Cor. 15:39-41 states the same axiom repeatedly. Human flesh is human flesh, bird flesh is bird flesh, etc.

Second axiom of logic: The Law of (non-)Contradiction

This law states that a proposition and its negation cannot both be true at the same time and in the same sense. P is not non-P (or ~[A: ~A] or A: ~A)

1. Numb. 23:19. “God is not a man.” God is not something that He is not.
2. Heb. 6:10. “God is not unjust.” God is not something that He is not.
3. Jer. 13:23. An Ethiopian cannot change his skin, nor a leopard his spots. This means that an Ethiopian is an Ethiopian; he is not a non-Ethiopian.
4. Since all truth is in God (Col. 2:3; John 14:6) and since God cannot deny Himself (2 Tim. 2:13), truth will never contradict truth.
5. 2 Cor. 1:18. “But as God is faithful, our word to you was not Yes and No.” The definition of God’s faithfulness is that His word is not both Yes and No at the same time and about the same thing.
6. Isaiah 5:20. “Woe unto them that call evil good, and good evil; that put darkness for light, and light for darkness; that put bitter for sweet, and sweet for bitter!” This implies that A does not equal non-A (A ≠ -A), or as stated in logic, ~[A: ~A] or A: ~A.

Third axiom of logic: The Law of the Excluded Middle

This law states that either a proposition is true, or its negation is true. A cannot equal non-A (or P & non-P) at the same time and in the same sense. So, if we have a proposition P, and if that proposition is false, then the proposition not-P would have to be true. Conversely, if P is true, then not-P is false. Another way of saying it is that a proposition always has the opposite truth value of its negation. Since A:A & ~[A: ~A] then either A or ~A.
a. Matt. 6:24. “No one can serve two masters; for either he will hate the one and love the other, or else he will be loyal to the one and despise the other. You cannot serve God and mammon.”
b. Matt. 7:15-20. All prophetic trees are either good trees or bad trees, and good prophetic trees produce good prophecies 100% of the time and bad prophetic trees produce non-prophecies 100% of the time. There is no in between.
c. 1 Kings 18:21. “How long will you limp between two opinions? If the LORD is God, follow him; if Baal, then follow him.” Either (A) Yehowah is God or (~A) Yehowah is not God; likewise, either (A) Baal is God or (~A) Baal is not God.
d. John 14:6. God is truth, and that which is not of God is thus not truth. Any proposition will either be in line with God’s thinking and thus true, or not aligned with God’s thinking and therefore false.
e. Ex. 16:4. God’s test is whether Israel will walk in His law or not. There is no middle position. Either it is the case that they obey or it is the case that they will not obey.
f. Matt. 12:30. “He who is not with Me is against Me, and he who does not gather with Me scatters abroad.”

Fourth axiom of logic: The law of rational inference — “If A=B, and B=C, then A=C.”

1. The doctrine of the Trinity is proved using this law.
2. Luke 15:52. “For from now on five in one house will be divided; three against two, and two against three.” If 5 [A] = [3+2] [B] and if [3+2] [B] = [2 +3] [C] then 5 [A] = [2+3] [C].

Fifth axiom of logic: Modus ponens — If p then q; p; therefore q.³⁹⁵

1. Matt. 8:2-3. If Jesus is willing [p] then he can cleanse a man of leprosy [then q]. Jesus said that he was willing [p]; therefore he was healed [therefore q].
2. Prov. 23:13-14. If a person engages in biblical discipline [p] then the child will not die spiritually [q]. The imperative of biblical discipline when carried out [p] results in delivering from the second death or hell [therefore q].
3. If people clearly perceive God, then they are obligated to glorify Him as God and be thankful. People do clearly perceive God. Therefore, they are obligated to glorifying Him as God and be thankful. P⊃ O, P,∴ O (modus ponens)

Sixth axiom of logic: Modus tollens — If p then q; not q; therefore not p.³⁹⁶

1. James 2:17-18. If a person has living faith [p] then he will have works [q]. When there are no works [not q]; there is no living faith [not p].
2. 1 John 2:19. If these professing believers were truly of us [p] then they would have continued with us [then q]; they left us [not q], therefore they proved themselves to not be of us [therefore not p].
3. 1 Cor. 15:13,20. If there is no resurrection [p] then Christ is not risen [then q]; Christ is risen [not q]; therefore there is a resurrection [not p].

Almost all logical laws can be derived from the above most basic ones. Here is a sampling of other laws that are helpful to be spelled out:

Seventh axiom of logic: Hypothetical syllogism — If p then q; if q then r; therefore, if p then r.

There are many examples of hypothetical syllogisms in the Scripture. Here are two:

1. Rom. 4:1-3. If Abraham was justified by works then he had reason to boast. Abraham had no reason to boast. Therefore, Abraham was not justified by works.
2. Rom. 8:31 If God is for us, then none can be against us. God is for us. Therefore, none can be against us.

Eighth axiom of logic: Disjunctive Syllogism (also known as the modus tollendo ponens or MTP) — If P is true or Q is true and P is false, then Q is true.

When we are told that only one of two statements is true and then are told which one is not true then we can know for certain the remaining statement is true. This is called a disjunctive syllogism. Either A or B. Not A. Therefore B.

1. Rom. 3:27. With regard to righteousness before God either one works and boasts or they do not work and do not boast. People cannot work and boast. Therefore they do not work and do not boast.
2. Rom. 5:5-8. Dying for worthy people is a demonstration of love. Dying for unworthy people is an even greater demonstration of love. Christ died for unworthy people. Therefore His death was a greater demonstration of love.
3. Rom. 11:5-6. Either it is of works or it is of grace. It is not of works. Therefore it is of grace.

Ninth axiom of logic: Immediate inference

This is an inference that can be made from only one statement or proposition. If the statement, “All frogs are green” were true (forget that it is not), then we could logically make the immediate inference that no frogs are not green. There are three kinds of immediate inference:

Converse: If we say “No S are P” we can make the immediate inference that “No P are S,” which is the converse of the statement.

1. Prov. 14:4. “Where no oxen are, the trough is clean.” No oxen-occupied stalls are clean. We can infer that clean stalls are not oxen-occupied.

Obverse can take four forms:

1. All S is P ≡ No S is non-P
2. No S is P ≡ All S is non-P
3. Some S is P ≡ Some S is not non-P
4. Some S is not P ≡ Some S is non-P
5. Mark 2:22 is an example of obverse. “And no one puts new wine into old wineskins; or else the new wine bursts the wineskins, the wine is spilled, and the wineskins are ruined. New wine must be put into new wineskins.” No new wine is old-skin-safe. All new wine is non-old-skin-safe.

Contrapositive has two forms:

1. “All S are P” means that we can make the immediate inference that “All non-P are non-S,” which is the contrapositive of the given statement.
2. “Some S are P” means that we can make the immediate inference that “Some non-P are not non-S,” which is the contrapositive of the statement.
3. James 2:17-18. This gives the truth of both an implication and its contrapositive — faith implies works, and no works implies no faith.

Tenth axiom of logic: Constructive Dilemma

The constructive dilemma is an argument that states that if P implies Q and R implies S and either P or R is true, then either Q or S has to be true. In sum, if two conditionals are true and at least one of their antecedents is, then at least one of their consequents must be too.

1. Rom. 3:19-20. Our options for righteousness are (1) we do not keep the moral law and are guilty and (2) we keep the moral law and are just. Those who do not keep the moral law are obviously guilty. Those trying to keep the moral law are guilty too because they do not actually keep it. Therefore, with both options, we are guilty. The problem of this dilemma is that either way we are doomed. Starting in Romans 3:21 Paul will present a third option that solves the dilemma — Christ imputes His righteousness to us.

This chapter is a cursory introduction into the topic of logic in order to demonstrate that logic is imbedded into the Bible and the Bible cannot be understood apart from logic. Thus, any interpretations that violate the rules of logic should be rejected. The Biblical Blueprints website will (Lord willing) develop these and other axiomatic system in much more detail.