Arxe Logic

Abstract

This note explores the self-referential nature of the Principle of Non-Contradiction (PNC) and the Liar Paradox (LP). Both cases are analyzed under a logical framework where propositions are treated as elements of a universal domain. The discussion highlights that their interpretation is not self-evident, but rather dependent on conventions external to logic itself.

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1. Definitions

The Principle of Non-Contradiction (PNC) is expressed as:

\(\neg (P \land \neg P)\)

The Liar Paradox (LP) can be formulated as:

“What I am saying is not true,”
or simply \(\neg P\)

In both cases, \(P\) denotes an arbitrary proposition. In the PNC, the parentheses “(…)” represent a type of proposition, a case contained within \(P\).

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2. Formalization

We can reconstruct the logical reasoning as a syllogism:

1. Premise i. If every proposition is represented in \(P\)*
2. Premise ii. And (…) is a proposition
3. Conclusion. Then (…) is represented in \(P\)

This can be expressed in predicate logic as follows:

• Universal premise: for any object \(x\), if \(x\) is a proposition, then it is represented in \(P\).

\(\forall x , (P(x) \rightarrow R(x))\)

• Instantiation: let (…) be a specific object, denoted by constant \(c\).

\(P(c)\)

• Conclusion: (…) is represented in \(P\).

\(R(c)\)

Thus, the full inference is:

\(\forall x (P(x) \rightarrow R(x)) \ P(c) \ \therefore R(c)\)

*Not to be confused with the totality of propositions or the set of all propositions.

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3. Objections and Responses

Objection 1: Contradiction is not a proposition.
Response: If contradiction is not a proposition, then it makes no sense to negate it, since it could be neither true nor false.

Objection 2: “(…)” does not represent a proposition but a propositional variable.
Response: To claim that “(…)” is not a proposition but contains one implies that it is a contained proposition, hence still a type of proposition.

Objection 3: This is not the proper way of interpreting the PNC.
Response: A convention regarding interpretation is an arbitrary agreement, not a valid form of inference. Therefore, it does not invalidate the syllogism.

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4. Reflection

The analysis suggests that a convention is required in order to interpret the “true” meaning of the PNC. The principle is not self-evident in isolation, but rather depends on an external interpretative framework. However, such conventions are arbitrary agreements, external to the logical structure that the principle seeks to establish. This tension places the PNC and the LP within the same family of self-referential phenomena, where logic must rely on external stabilizing rules to maintain consistency.