nomadreid
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- Do/can examples of non-transitive (intransitive) implication/ inference exist?
Usually, transitivity of implication is welded into the axioms and/or rules of inference for a theory. However, I am trying to explain to someone that, except for explosive contradiction, there is nothing "sacred" in logic. I tried to get the general idea across with the example of Euclidean/non-Euclidean geometries, but the target person insists that implication is some sort of absolute. Of course, if the definition of implication includes transitivity, then of course she is right in that transitivity is "baked-in", but I am not certain that all types of implication or inference require it. I would be happy to be corrected on this, of course.
Of course, if the theory is inconsistent, then transitivity is automatic, so I am assuming consistency.
Some first ideas that came to mind (very speculative):
[1] Circular structures ( Rock beats Scissors Scissors beats Paper Rock does not beat Paper.)
[2] A implies* B if A classically implies B over a certain threshold of the time, e.g., implies* is "probably implies"
[3] Incomplete information
[4] perhaps based on a filter that is not an ultrafilter
[5] Flippant: A implies B if A loves B. Definitely not transitive.
[6] A multi-valued logic
Either explanations, or outlines of explanations, or links would be very much appreciated.
Of course, if the theory is inconsistent, then transitivity is automatic, so I am assuming consistency.
Some first ideas that came to mind (very speculative):
[1] Circular structures ( Rock beats Scissors Scissors beats Paper Rock does not beat Paper.)
[2] A implies* B if A classically implies B over a certain threshold of the time, e.g., implies* is "probably implies"
[3] Incomplete information
[4] perhaps based on a filter that is not an ultrafilter
[5] Flippant: A implies B if A loves B. Definitely not transitive.
[6] A multi-valued logic
Either explanations, or outlines of explanations, or links would be very much appreciated.
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