[Paper Review] There exist consistent temporal logics admitting changes of History

Gavriel Segre|ArXiv.org|Dec 3, 2006

Advanced Database Systems and Queries3 citations

TL;DR

This paper demonstrates that consistent temporal logics can accommodate changes to the past without contradiction, challenging the assumption that causal loops necessarily violate logical consistency. By formalizing a classical propositional logic with temporal operators and defining a consistency condition based on truth-preservation under history changes, the author proves that such logics can be both consistent and allow for retrocausal influence.

ABSTRACT

Introducing his Chronology Protection Conjecture Stephen Hawking said that it seems that there exists a Chronology Protection Agency making the Universe safe for historians. Without taking sides about such a conjecture we show that the Chronology Protection Agency is not necessary in order to make the Universe unsafe for historians but safe for logicians.

Motivation & Objective

• To investigate whether temporal logics can remain consistent even when the past is subject to change, countering the assumption that such changes lead to logical incoherence.
• To challenge the physical dismissal of non-globally hyperbolic spacetimes with localized chronology-violating regions, arguing they should be taken seriously in foundational physics.
• To show that consistency conditions—such as those in Novikov’s conjecture—can be formalized within classical propositional logic without requiring quantum gravity or ad hoc axioms.
• To provide a logical framework where retrocausal influence is allowed, but contradictions are avoided through truth-preservation under history transitions.

Proposed method

• Formalizing a classical propositional logic over a binary alphabet Σ = {0, 1} with standard connectives: ¬, ∧, ∨, →, ↔.
• Introducing temporal operators G (future universal), H (past universal), F (future existential), P (past existential) to model temporal quantification.
• Defining a consistency condition for history changes by requiring that the truth of propositions remains invariant under transitions from one history to another.
• Establishing logical theorems such as double negation, noncontradiction, De Morgan’s laws, and reductio ad absurdum to ensure internal consistency.
• Using the principle of contraposition and modus ponens to derive valid inference rules within the system.
• Applying the framework to spacetime models with chronology-violating sets, showing that logical consistency is preserved even when causal loops exist.

Experimental results

Research questions

• RQ1 Can a consistent temporal logic be constructed that allows for changes to the past without generating logical contradictions?
• RQ2 Is it logically permissible to have retrocausal influence in a system where the past can be altered, provided truth values are preserved under history transitions?
• RQ3 Does the existence of a chronology-violating set in a spacetime necessarily imply logical inconsistency in a formal temporal logic?
• RQ4 Can classical propositional logic, augmented with temporal operators, serve as a foundation for reasoning about time travel without requiring quantum gravity or ad hoc axioms?
• RQ5 What logical conditions must be satisfied to ensure that a history change does not lead to a contradiction in a temporal system?

Key findings

• The paper proves that classical propositional logic with temporal operators can be consistent even when the past is subject to change, provided that truth values are preserved under history transitions.
• Theorems such as noncontradiction (¬(x ∧ ¬x)) and tertium non datur (x ∨ ¬x) hold within the system, confirming classical logical structure is preserved.
• The Scotus Theorem (ex absurdo quodlibet) is formally derived, showing that from a contradiction, any proposition follows—confirming standard classical logic behavior.
• The system satisfies modus ponens and modus tollens, ensuring valid inference rules are preserved under history changes.
• The consistency of the logic under retrocausal influence is maintained via a truth-preservation condition, avoiding paradoxes like the Grandfather Paradox.
• The framework does not require quantum gravity or the Chronology Protection Conjecture to ensure consistency, suggesting classical logic alone can handle causal loops if properly formalized.