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Galle's Theorem

Introduction

Galle's Theorem is a mathematical theorem that states that if A, then A.

It establishes the foundational principle of propositional self-implication: every statement necessarily implies itself. The theorem has been described as one of the most immediate consequences of classical logic and serves as a cornerstone for the study of tautological structures.

Basic Form (Galle's Theorem)

Let A be any proposition.

Then A ⇒ A

Proof

Assume that A is true.

Since A is true, it follows that A is true.

Then A is true.

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Corollary 1: Self-Sufficiency of Propositions

Every proposition is a sufficient condition for itself.

Proof

For any proposition A belonging to the set of all propositions P,

∀ A ∈ P, A ⇒ A

by Galle's Theorem.

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Corollary 2: Reflexivity of Implication

The implication relation is reflexive.

Proof

Since every proposition implies itself,

∀ A ∈ P, A ⇒ A

satisfies the reflexive property, and thus

∀ A ∈ P, A ⇒ A so that also A ⇒ A

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