Petitio Principii

About

Assuming the truth of the conclusion within the premises of an argument. The conclusion is used as its own justification. The argument may appear formally valid, but it provides no independent support for its claim.

Taxonomy - Structural / Presuppositional fallacy, circular justification. Contrasts with valid deductive inference with independent premises.

Informal Example - "The law is just because it is the law."

Minimal Symbolic Form

P ⊢ P

Where:

• P = the conclusion being argued for
• ⊢ = inferential justification

Shows that the conclusion is asserted without external support.

Logical Analysis

Structural Circularity

(P → P) ∧ P ⊢ P

The inference is formally valid but epistemically empty.

Missing Necessary Condition

A valid argument requires at least one premise independent of the conclusion:

∃Q (Q ⊢ P ∧ Q ≠ P)

There must exist a premise Q that supports P without presupposing P.

Causal DAG Representation

Fallacious Assumed Model

P → P

Correct Requirement

A valid justificatory structure must be acyclic:

Q → P

Where Q is not equivalent to P.

The fallacy is a closed causal or justificatory loop, offering no explanatory power.

Modal/Logical Formalization

In epistemic modal terms:

□P ⊢ P

Where:

□P is assumed rather than justified

Or more explicitly:

□P ⇏ □P

Assuming necessity does not generate necessity.

Epistemic Representation

𝔼(P) ⇏ □(P)

Where:

• 𝔼(P) = evidence that already presupposes P
• □ = justified belief

Evidence that assumes the conclusion cannot justify the conclusion.

Countermodels

Tautological Loop

P ⊢ P

Provides no new information.

Reworded Circularity

P ⊢ (P′ ⊢ P) Where P′ ≡ P.

Hidden Circular Definition

A → B ∧ B → A

Each term derives meaning only from the other.

Historical Background

Petitio principii is identified explicitly by Aristotle in Prior Analytics and Sophistical Refutations as a failure of demonstration, where the conclusion is implicitly assumed rather than proved.

Medieval scholastic logicians formalized the term to describe arguments that presuppose what they claim to establish, particularly in theological and metaphysical contexts.

In early modern logic manuals, the fallacy became standard under the English label begging the question (originally meaning "raising the question," later misunderstood).

Modern logic treats the fallacy as a defect of justification rather than validity: the argument may be deductively sound while being epistemically worthless.