Real Booleans have 4, not 2 values

Four Possible Truth Values: A Nuanced View of Theoretical and Practical Truth

In understanding the distinction between the subjective truth of an assertion in theory and the objective truth of the same exact assertion or claim in reality, we must account for the fact that the objective truth (or Boolean value) of any logical claim—defined by its real-world accuracy—can differ from said claim’s Boolean value (or subjective truth) in theory. Our approach acknowledges that a claim can be true subjectively — in a theoretical sense — but false objectivity — in a practical, real-world context — and vice versa.

True in Theory, True in Reality

Example: 2+2=4 when applied to apples.

Explanation: The arithmetic operation 2+2=4 holds both theoretically (based on mathematical principles) and practically (when counting physical objects like apples).

True in Theory, False in Reality

Example: 2+2=4 when applied to the moons of Mars.

Explanation: The arithmetic operation 2+2=4 is true within the mathematical framework, but when applied to the moons of Mars, it is false because Mars only has 2 moons, not 4 (violating Peano's fifth axiom).

False in Theory, True in Reality

Example: You are doing your math homework and make a mistake—like dividing by 0—but somehow still correctly conclude that you will need 16 3-by-3 feet marble slabs to cover a square room that is 12 feet wide (144 sqft).

Explanation: This scenario is rare but can occur in contexts where theoretical models are constructed differently from empirical observations. A claim might be deemed false within that theoretical model but true based on real-world evidence.

False in Theory, False in Reality

Example: 2+2=5 in any context.

Explanation: The arithmetic operation 2+2=5 is false both theoretically (does not follow mathematical rules) and practically (does not correspond to any empirical observation).

Understanding the Duality

By referring to 2+2=4 as “truth in theory” and 2 moons+2 moons=2 moons as “truth in reality,” we highlight the distinction between theoretical constructs and empirical evidence. This duality allows us to appreciate that the same claim can be evaluated differently based on the context:

Subjective Theoretical Truth: Based on adherence to axioms and logical consistency within a defined framework (e.g., mathematics). In mathematics, all theoretical truth is conditional upon the assumptions (axioms) holding true in reality. If an axiom is violated, theorems will not hold true in reality either.

Objective Practical/Empirical Truth: Based on observable and verifiable facts in the real world, which may have constraints or limitations not present in theoretical models. Independently verifiable for real-world accuracy.

Conclusion

This nuanced view introduces the possibility of four truth values, not just two. It emphasizes the importance of context in evaluating the truth of a claim, recognizing that a statement may hold differently in theory and reality. This approach provides a more comprehensive understanding of truth, accommodating the complexities of both abstract reasoning and practical observation.

This aligns with the principle that all probability is Bayesian. Indeed, real-world truth values are also Bayesian—conditional upon the underlying axioms holding true. This is exactly the reason why mathematical proof is so important. Consider, for example, Fermat’s Last Theorem. Until it was proven in 1994, it remained a hypothesis. The difference between a hypothesis and a theorem is that a hypothesis could turn out to be false, but a theorem cannot. This means that if a theorem contradicts empirical evidence—does not hold true in theory—then we know with absolute certainty that one of the axioms from which the theorem was logically deduced is in reality false. This is not the case with a hypothesis, as it could simply turn out to be incorrect in theory, and this is another way that a hypothesis could turn out to be false in reality.

TNT enforces batch processing. This approach updates the database on a regular, real-time schedule—once per minute, for example—as each node individually appends the same block update to its respective blockchain. Batch processing guarantees with absolute certainty, both in theory and in reality, that all peer-to-peer nodes are equally (or symmetrically) informed about not only current account balances but also all pending transactions (or payment requests), making fraud such as double spending, facilitated by asymmetric information about pending payments, impossible in reality, barring software bugs. It is only when unfettered Internet-based commerce is conducted in the presence of asymmetric information that the possibility of fraud opens up, as described by George Akerlof in "The Market for 'Lemons'." Batch processing—accepting payments only during the day and processing them at night—has been used by all banks since the Italian Renaissance to eliminate fraud, such as the use of counterfeit money (bouncing checks), which can only occur when the parties to a transaction are not symmetrically informed about the current account balance and all pending charges to that account. By taking the time each night to fully synchronize all information across bank branches (or peer-to-peer nodes) while no new checks are being accepted, all nodes are equally (or symmetrically) informed, eliminating any possibility of a negative account balance.