Agreement and disagreement in a non-classical world

Area of Science:

• Epistemology
• Quantum mechanics
• Probability theory

Background:

• Aumann's Agreement Theorem (1976) states Bayesian agents with common priors cannot have common knowledge of differing posterior probabilities.
• Classical probability theory governs these results, but does not apply to non-classical domains like quantum mechanics.

Purpose of the Study:

• To investigate the epistemic properties of non-classical domains using signed probabilities.
• To determine if agents in non-classical domains can 'agree to disagree' or have common certainty of differing probabilities.

Main Methods:

• Employing signed probabilities to model non-classical epistemic scenarios.
• Extending the principles of Bayesian agreement to quantum-inspired probability frameworks.

Main Results:

• In non-classical domains, it cannot be common knowledge that two agents assign different probabilities to an event.
• Unlike classical domains, common certainty of differing probabilities is possible in non-classical settings.
• If common certainty of differing probabilities is possible, it cannot be common certainty if communication is enabled, even if unused.

Conclusions:

• The inability to 'agree to disagree' extends to non-classical domains, but with nuanced differences regarding common certainty.
• Signed probabilities offer a framework for understanding epistemic limitations and possibilities in quantum-like systems.
• Communication protocols play a crucial role in epistemic agreement, even in non-classical contexts.