The Logic of the Logos

What John 1:3 Proves and How It Proves It

In a former life (not really, but it was like 25 years ago), I taught formal logic at the community college level for four years. Predicate calculus, truth tables, derivations — the whole works. I also studied theology. These two interests don’t intersect as often as you’d think, but when they do, the results are worth paying attention to.

John 1:3 is one of those intersections.

Here’s the verse, in the ESV: “All things were made through him, and without him was not any thing made that was made.”¹

Most Christians read this and nod along — yes, Jesus is the Creator. But there’s something happening in the logic of this verse that’s sharper than most people realize. John isn’t just asserting that the Word created things. He’s constructing an argument — a tight one — that the Word *cannot* be a created thing, and therefore must be God. And the structure of that argument is formally valid. You can prove it with predicate calculus.

Let’s do exactly that. If you haven’t studied formal logic: (a) why the heck not? and (b) I’ve provided less-geeky translation of the math-looking stuff.

Setting Up the Language

First, we need to formalize our terms. In predicate logic, we define a domain of discourse and assign predicates and constants.

Domain: All things that exist.

Constants:

- w = the Word (the Logos, identified in John 1:14 as the one who “became flesh” — Jesus Christ)

Predicates:

- C(x) = “x is a created thing” (x was made)

- T(x, y) = “x was made through y” (y is the instrumental cause of x’s creation)

That’s all we need. Now let’s translate the verse.

Translating John 1:3

The verse makes two claims. They look like a simple repetition, but they’re not — the second clause is the contrapositive of the first, which is logically significant. John is locking down the argument from both directions.

**John 1:3a — “All things were made through him”:**

∀x(C(x) → T(x, w))

“For all x: if x is a created thing, then x was made through the Word.”

Every created thing, without exception, was made through the Logos. The universal quantifier ∀x is doing real work here. John doesn’t say “many things” or “most things.” He says *all things* — πάντα (*panta*) in the Greek²] There are no exceptions in the domain of created things.

**John 1:3b — “without him was not any thing made that was made”:**

¬∃x(C(x) ∧ ¬T(x, w))

“There does not exist an x such that x is a created thing and x was not made through the Word.”

This is the negation of the existential — there is no created thing that came into being independently of the Word. And notice: this is logically equivalent to Premise 1. The equivalence ∀x(P(x) → Q(x)) ≡ ¬∃x(P(x) ∧ ¬Q(x)) is a standard identity in predicate logic.³John is stating the same truth in two ways, positive and negative, to eliminate any wiggle room.

That’s not literary flair. That’s a logician’s instinct.

The Metaphysical Premise

John 1:3 gives us the two formalized premises above, but the argument needs one more — a metaphysical premise that John’s audience would have taken as self-evident:

**Premise 3 — The Impossibility of Self-Creation:**

¬∃x(T(x, x))

“There does not exist an x such that x was made through itself.”

Nothing can be the instrumental cause of its own coming-into-being. This isn’t just a theological claim — it’s a basic metaphysical principle. For something to cause its own creation, it would need to exist before it existed, which is a contradiction. Aristotle knew this.[⁴The Stoics knew this. Every serious philosopher in the ancient world knew this. John’s readers certainly knew it.

This isn’t a hidden assumption. It’s a logical law dressed in metaphysical clothing.

The Derivation

Now we have everything we need. Here’s the formal proof that the Word is not a created thing, laid out as a derivation:

P1) ∀x(C(x) → T(x, w)) [John 1:3a]

P2) ¬∃x(C(x) ∧ ¬T(x, w)) [John 1:3b — equivalent to P1]

P3) ¬∃x(T(x, x)) [Self-creation is impossible]

Step 1 — Universal Instantiation on Premise 1:

Premise 1 holds for *all* x. That means it holds when x = w. Substitute:

(4) C(w) → T(w, w) [Universal Instantiation, P1, x = w]

“If the Word is a created thing, then the Word was made through the Word.”

This is the critical move. The universal quantifier in John 1:3a means the Word is not exempt from its own scope. If the Word is a created thing — if C(w) is true — then *by John’s own premise*, the Word was made through itself.

**Step 2 — Instantiation on Premise 3:**

From Premise 3, we know that nothing was made through itself. Instantiate this to extract the relevant case:

(5) ¬T(w, w) [From P3: ¬∃x(T(x, x)), therefore ¬T(w, w)]

“The Word was not made through the Word.”

Self-creation is impossible, and the Word is no exception.

Step 3 — Modus Tollens:

Now we have a conditional and the negation of its consequent. This is textbook modus tollens (If a then b: Not b: Therefore Not a) — one of the most basic valid inference forms in all of logic:

(4) C(w) → T(w, w)

(5) ¬T(w, w)

Which allows us to conclude:

(C) ¬C(w) [Modus Tollens, (4), (5)]

In English: The Word is not a created thing.

That’s it. Three premises, three steps, one conclusion. The argument is deductively valid — the conclusion follows necessarily from the premises. And if the premises are true (which they are, if you accept the inspiration of John 1:3 and the impossibility of self-creation), then the conclusion is not just valid but *sound*.

What Follows from ¬C(w)

We’ve proven that the Word is not a created thing. But John doesn’t stop there — and neither should we. The argument has a second stage.

If we accept an exhaustive dichotomy — that everything which exists is either created or uncreated — we can add one more premise:

P4) ∀x(¬C(x) → U(x)) [Everything uncreated is eternal/divine]

Where U(x) = “x is uncreated” (and therefore eternal, self-existent — the defining attributes of God in classical theism). This isn’t smuggling anything in. In the metaphysical framework shared by Jews, Greeks, and Christians in the first century, the division between Creator and creation was exhaustive. There is no third category. You’re either made, or you’re the Maker.

(6) ¬C(w) [Proven above]

(7) ¬C(w) → U(w) [Universal Instantiation, P4, x = w]

(C) U(w) [Modus Ponens, (6), (7)]

In English: The Word is uncreated — eternal, self-existent, divine.

Which is exactly what John says in the very next breath: “the Word was God” (John 1:1c).⁵ Verse 3 isn’t restating verse 1. It’s *proving* verse 1. The claim that the Word is God in verse 1 is supported by the logical argument embedded in verse 3.

The Complete Derivation

For the sake of clarity, here’s the whole thing in one block:

PREMISES:

P1) ∀x(C(x) → T(x, w)) All created things were made through the Word

P2) ¬∃x(C(x) ∧ ¬T(x, w)) No created thing was made apart from the Word

P3) ¬∃x(T(x, x)) Nothing is the instrument of its own creation

P4) ∀x(¬C(x) → U(x)) Everything not created is uncreated/divine

DERIVATION:

(4) C(w) → T(w, w) Universal Instantiation on P1, x = w

(5) ¬T(w, w) From P3 (instantiated to w)

(6) ¬C(w) Modus Tollens on (4), (5)

(7) ¬C(w) → U(w) Universal Instantiation on P4, x = w

(C) U(w) Modus Ponens on (6), (7)

CONCLUSION:

The Word (w) is uncreated and divine. ∎

Why This Matters: The Arian Problem

This isn’t just an academic exercise. This argument has been at the center of one of the most important theological debates in Christian history.

In the fourth century, Arius argued that the Son was the first and greatest of God’s creations — made before all other things, and then the instrument through which everything else was made. “There was a time when he was not,” as the slogan attributed to Arius goes.⁶ The Council of Nicaea (325 AD) rejected this, and the Johannine Prologue — including the argument embedded in John 1:3 — was part of the biblical case against him.⁷

The same debate is alive today. Jehovah’s Witnesses hold an essentially Arian position — that Jesus is a created being, the first thing God made, through whom all other things were subsequently created. Their New World Translation handles John 1:3 by rendering it: “All things came into existence through him, and apart from him not even one thing came into existence⁸ So far, so close to the Greek. But the key move comes in their broader theology: they introduce the category of “all *other* things” (see their rendering of Colossians 1:16–17, where they insert the word “other” four times despite its absence in the Greek) to exempt Jesus from the domain of “all things.”⁹

Here’s the problem. The logic won’t let you do that.

If you accept that the Word was created — C(w) is true — then by Premise 1, T(w, w) must also be true: the Word was made through himself. That’s self-creation. And self-creation is incoherent, not just theologically but logically and metaphysically. The Arian has to either deny Premise 1 (and thereby deny the clear statement of John 1:3a), deny Premise 3 (and affirm the metaphysical absurdity of self-creation), or accept the conclusion.

There’s no fourth option. That’s how deductive arguments work. If the premises are true, the conclusion is inescapable. The universal quantifier ∀x in John 1:3 doesn’t have an asterisk. There’s no footnote that says “except the Word himself.” The whole point of πάντα is its absolute scope.

Arius and his modern heirs aren’t just making a theological error. They’re making a logical one.

John the Theologian — and Logician

What I find remarkable is that all of this is packed into a single verse. Twenty-three words in English. John didn’t write out a formal proof — he didn’t need to. His audience could follow the logic intuitively. But the structure is there, waiting to be unpacked, and it holds up under the most rigorous analysis we can throw at it.

Theologians have long called John’s Gospel the most theologically sophisticated of the four. I’d add that it’s also the most logically sophisticated. The Prologue alone is a masterclass in compressed argumentation — every clause load-bearing, every word doing double duty.

John 1:1 makes the claim: the Word was God. John 1:3 provides the proof. And the proof is deductively valid.

Not bad for a fisherman from Galilee.¹⁰

1. John 1:3, English Standard Version (ESV). The ESV’s “was not any thing made that was made” preserves the structure of the Greek more closely than many modern translations.

2. The Greek text reads: Πάντα δι᾽ αὐτοῦ ἐγένετο, καὶ χωρὶς αὐτοῦ ἐγένετο οὐδὲ ἕν ὃ γέγονεν. The word πάντα (panta) is the neuter plural of πᾶς (pas), meaning “all” or “every” — used here without qualification or restriction.

3. This equivalence is derived via De Morgan’s laws applied to quantifiers, combined with the material conditional equivalence. Any standard logic textbook covers this.

4. Aristotle’s argument against self-causation appears in Metaphysics XII and Physics VIII. Nothing can actualize its own potentiality, since it would need to be both actual and potential in the same respect at the same time — a violation of the law of non-contradiction.

5. John 1:1c (ESV): “and the Word was God.” Note that verse 1 declares the Word’s divinity; verse 3 provides the logical grounds for it. The Prologue’s structure moves from assertion (v. 1) to evidence (v. 3).

6. The slogan attributed to Arius is “there was once when he was not.” Arius held that the Son was created before time but was not eternal. The substance of the claim is well attested in early church sources.

7. The Council of Nicaea (325 AD) condemned Arius’s teaching. The creed affirmed that the Son is “begotten, not made” and “of one substance with the Father.” Athanasius drew heavily on the Johannine Prologue in his arguments.

8. John 1:3, New World Translation (2013). The NWT rendering of John 1:3 itself is not significantly different from standard translations. The theological divergence comes from their handling of parallel passages like Colossians 1:16-17.

9. The NWT renders Colossians 1:16–17 as: “because by means of him all *other* things were created in the heavens and on the earth... All *other* things have been created through him and for him. Also, he is before all *other* things, and by means of him all *other* things were made to exist.” The word “other” (appearing four times) has no basis in the Greek text, which uses simply πάντα (*panta*) — the same word as John 1:3. The insertion restricts the scope of “all things” to exclude Jesus, which is precisely the move the logic of John 1:3 does not permit. See [CARM, “Colossians 1:16–17 and the Jehovah’s Witnesses”](https://carm.org/jehovahs-witnesses/col-116-17-all-other-things-were-created-by-him-and-the-jehovahs-witnesses/).

10. Tradition identifies the author of the Fourth Gospel as John the son of Zebedee, a Galilean fisherman (Mark 1:19–20). Modern scholarship debates the question, but the Reformed tradition has generally affirmed the traditional authorship.