I'm generally confused about how to work this out, these are the questions that have been given to me, and I just don't know where to start on them:
- Consider the set of natural deduction rules, is the set of natural deduction without the rule RAA sound, and complete?
How do I go about figuring this out? check if the set of rules are still sound and complete?
- State whether this new rule:
--
⊥
is sound, and complete? How would I also go about figuring this out? I'm guessing it wouldn't be sound because true doesn't evaluate to false.
I think your question is more directly addressed here: https://en.wikipedia.org/wiki/Paraconsistent_logic . Some laws of logic are weakened or dismissed yet allow for non-trivial conclusions.
Foundational bases for classical logic can be uprooted, like the idea that statements must either be true or false: Deviant Logic
Variations on the Law of Excluded Middle have implications in programming.
In addition to varying the foundational rules of logic themselves, you can vary commonly accepted axiomsrr. Non-Euclidean Geometry was discovered when Gauss and others rejected the parallel principle and were able to develop consistent theorems.