FOL formulas as a ring

fol formulae as a ring under entailment

In logic, particularly in First-Order Logic (FOL), we can define structures that behave like algebraic rings under certain operations. While FOL formulas aren't literally a ring in the strict algebraic sense (like integers with addition and multiplication), there is an abstract analogy that treats formulas as forming an algebraic structure under logical entailment.

Here's an outline of how FOL formulae can be seen as forming a ring-like structure under entailment:

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🔹 Preliminaries

Let’s consider:

• A set of first-order logic sentences: $\mathcal{L}$

• A consequence relation $\models$ (semantic entailment)

• The equivalence class $[\phi]$ of a formula $\phi$, defined by:

  $$ [\phi] = \{ \psi \in \mathcal{L} \mid \phi \models \psi \text{ and } \psi \models \phi \} $$

  (This is logical equivalence: $\phi \equiv \psi$)

This quotient structure $\mathcal{L} / {\equiv}$ consists of logical equivalence classes.

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🔹 Ring-Like Operations

Define two operations on equivalence classes of formulas:

1. Addition-like (⊕):

   $$ [\phi] \oplus [\psi] := [\phi \leftrightarrow \psi] $$

   This operation captures the symmetric difference of truth — a kind of logical XOR.

2. Multiplication-like (⊗):

   $$ [\phi] \otimes [\psi] := [\phi \land \psi] $$

   This is conjunction of formulas.

These are well-defined on equivalence classes.

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🔹 Ring Axioms (Mod 2 ring analogy)

Under these operations:

• $\oplus$ is commutative and associative
• $\otimes$ distributes over $\oplus$
• $[\bot]$ (falsehood) acts as the additive identity
• $[\top]$ (tautology) can serve as a multiplicative identity
• $[\phi] \oplus [\phi] = [\bot]$ (each element is its own additive inverse — like in $\mathbb{F}_2$)

Thus, the set of logical equivalence classes forms a commutative ring (more specifically, a Boolean ring).

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🔹 Summary: FOL Formulas as a Boolean Ring

• The set of FOL formulas modulo logical equivalence under entailment forms a Boolean ring.

• Operations are:

  • $\phi \oplus \psi := \phi \leftrightarrow \psi$
  • $\phi \otimes \psi := \phi \land \psi$

• This works in a quotient space $\mathcal{L} / {\equiv}$, not on raw formulas.

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Would you like a concrete example, or an algebraic formalization of this construction?