In exploring the mysteries of the world, we often encounter contingent propositions. These propositions give us all kinds of possibilities and uncertainties about the world, but do they depend on mathematical consistency?
Suppose we find a contradiction in mathematics, say "0=1". This is a shocking discovery because it directly challenges the foundational principles of mathematics. However, if we follow the explosive principle of classical logic, this contradiction can lead to the truth (and false!) of any accidental (but obviously false) proposition about the world.
How do we deal with this situation? Should we assume that mathematics itself must be consistent just because the world does not appear to be contradictory? This is a complex question because it involves our understanding of logic and consistency.
First, let's be clear: not all people who assert contingent propositions believe that consequential relations are classical. In correlation logic, the resulting relationships are super-consistent, so there is no contradictory evidence of all the evidence. This means that even if a contradiction is found in mathematics, we cannot simply apply the explosive principle of classical logic to derive any proposition about the world.
However, if we assert any proposition and believe that this result is classical, then we will be willing to believe that there is no real contradiction. This attitude can lead us to become a triflalist, that is, focusing too much on the details and neglecting the whole. Thus, we are right in a sense, but this is by no means unique to contingent propositions, nor is it universally applicable.
In addition, there are some counter-examples to prove our point. In the philosophy of logic, not everyone is a logical monist. Some pluralists believe that various systems of logic are good for one end and bad for another. As a result, they may use classical logic in some cases and abandon it in cases where contradictions arise. In fact, adaptive logic is exactly that, compromising both upper and lower bound logic. The upper bound logic is classical, and you can always use the ULL until you find a contradiction, in which case only inferences are allowed in the lower bound logic (LLL), and LLL is usually much weaker than classical logic: it is usually a quasi-consistent system, so once again inferential trivialism is not allowed.
To sum up, we need to be cautious and open to the contradictions in mathematics and the probabilities of the world. We cannot simply apply the explosive principle of classical logic to derive any proposition about the world, as this would lead us to become trivialists. Instead, we should try to understand the advantages and disadvantages of various logical systems and use them when appropriate. At the same time, we should also be aware of the uncertainty and complexity of the world and strive to find better ways to understand and deal with them.