#05-Formalization

August 14, 2025

➘ #05 Source Article

Language — Domains

$D$ : set of all agents.
$G \in D$ : biblical God.
$x,y \in D$ : variables.

Explanation: The domain $D$ is the universe of individuals we are talking about. $G$ is a constant for God, and $x$ , $y$ can be replaced with any agent in the domain.

Predicates

$\ell(x,y)$ : “ $x$ loves $y$ ” in the ordinary, human-recognizable sense (promotes well-being, avoids harm, maintains care).
$\ell'(x,y)$ : theologically redefined “love” (may include harm).
$P(x,y)$ : “ $x$ promotes $y$ ’s well-being.”
$H(x,y)$ : “ $x$ inflicts avoidable, unproductive harm on $y$ .”
$I(y)$ : “ $y$ is innocent.”

Explanation: These predicates are the vocabulary of the logic. Each represents a simple, testable relationship or property so that more complex theological claims can be built from them.

Bridge Principles for Ordinary “Love”

$\mathbf{A1}:\ \forall x\forall y(\ell(x,y)\rightarrow(P(x,y)\wedge\neg H(x,y)))$

Explanation: In ordinary usage, if one loves another, one promotes their well-being and avoids harming them.

$\mathbf{A2}:\ \forall x\forall y(H(x,y)\rightarrow\neg P(x,y))$

Explanation: If one harms another, one is not promoting their well-being. This reinforces the incompatibility of love and harm in normal discourse.

Event Premise from the Narratives

$\mathbf{B}:\ \exists y(I(y)\wedge H(G,y))$

Explanation: There exists at least one innocent whom God has harmed. This formalizes the biblical accounts (e.g., the Flood, Amalekite massacre) for use in the argument.

Core Derivation — Contradiction under Ordinary “Love”

Goal: $\forall y((I(y)\wedge H(G,y))\rightarrow\neg\ell(G,y))$

Proof Sketch:

From $\mathbf{A1}$ , instantiation gives: $\ell(G,y)\rightarrow(P(G,y)\wedge\neg H(G,y))$ .
From $\mathbf{B}$ , select $y^{\ast}$ with $I(y^{\ast})\wedge H(G,y^{\ast})$ .
Assume $\ell(G,y^{\ast})$ . Then by step 1, $\neg H(G,y^{\ast})$ .
But step 2 asserts $H(G,y^{\ast})$ .
Contradiction: $H(G,y^{\ast})\wedge\neg H(G,y^{\ast})$ .
Therefore $\neg\ell(G,y^{\ast})$ .
Generalize to all $y$ to get the goal.

Explanation: If God harms an innocent, then by the ordinary definition of love, it is impossible to truthfully assert that God loves that person. The harm premise directly contradicts the love definition.

Redefinition Schema and Semantic Inversion

$\mathbf{R1}:\ \forall x\forall y(\ell'(x,y)\leftrightarrow(\ell(x,y)\vee H(x,y)))$

Explanation: Under this redefinition, “love” is satisfied if either ordinary love holds or harm occurs. This allows “love” to be applied even in cases that contradict the original definition.

From $\mathbf{R1}$ and the contradiction result, we can derive:

$\exists y(I(y)\wedge H(G,y)\wedge\ell'(G,y)\wedge\neg\ell(G,y))$

Explanation: There exists at least one innocent harmed by God for whom the redefined “love” applies, even though ordinary love does not. This is semantic inversion — the term now covers cases it previously excluded.

Disjunction-Shaped Main Conclusion

Semantic vacuity definition:

$\mathrm{Vac}(\ell')\ :=\ \exists x\exists y(\ell'(x,y)\wedge H(x,y)\wedge\neg\ell(x,y))$

Main schema:

$(\exists y(I(y)\wedge H(G,y)))\ \rightarrow\ (\forall y((I(y)\wedge H(G,y))\rightarrow\neg\ell(G,y))\ \vee\ \mathrm{Vac}(\ell'))$

Explanation: If any innocent is harmed by God, then either ordinary “love” does not apply (contradiction) or the redefined “love” becomes semantically vacuous — unable to distinguish between love and its opposite.

Notes

The “Core Derivation” and “Semantic Inversion” sections form the argumentative backbone.
The disjunction form condenses the reasoning into a single line for compact presentation.
$\mathbf{A2}$ is optional: it strengthens the bridge between harm and negation of well-being but is not essential for the derivation.

◉ A reader-friendly explanation of the symbolic logic above.

Language and Domains

We’re talking about all possible individuals, which we’ll call “agents.” God is one of these agents, and we use symbols to talk about any agent in general.