Intransitive implication

[nomadreid]

TL;DR

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.

 

[pasmith]

A relation on a set does not need to be transitive. \neq is an example of a relation which is not transitive.

"A implies B" is defined as "it is not the case that A is true and B is false". It follows from Boole's laws that <br /> (A \Rightarrow B) \wedge (B \Rightarrow C) \Rightarrow (A \Rightarrow C) and \Rightarrow is in that sense transitive.

 

[fresh_42]

If we create a relation that we decide to call implication, but it is not transitive, then how would implications be used? We cannot concatenate them to create proofs, and it no longer describes subsets. Hence, A loves B is a non-transitive relation but I wouldn't call it an implication.

 

[nomadreid]

Thanks, pasmith and fresh_42. It appears from your answers that there is no standard model that could satisfy a non-transitive implication: I was contemplating a non-standard model, not the standard set interpretation; for example one in which some gaps in the truth value evaluation function does not allow the interpretation needed to fulfill the standard sentence associated with implication, or a model which has other members which would allow a different interpretation than the standard, or a theory which is forced to allow other sentences..... The non-standard implication would not have to be a useful relation, as it would just serve the point that a tautology is always with respect to a certain class of models, and that the same statement that would be a tautology in one class of models may not so be in another class.

 

[fresh_42]

Thanks, pasmith and fresh_42. It appears from your answers that there is no standard model that could satisfy a non-transitive implication: I was contemplating a non-standard model, not the standard set interpretation; for example one in which some gaps in the truth value evaluation function does not allow the interpretation needed to fulfill the standard sentence associated with implication, or a model which has other members which would allow a different interpretation than the standard, or a theory which is forced to allow other sentences..... The non-standard implication would not have to be a useful relation, as it would just serve the point that a tautology is always with respect to a certain class of models, and that the same statement that would be a tautology in one class of models may not so be in another class.

Well, there are considerations along these lines. I have found a dissertation Paradox, Arithmetic and Nontransitive Logic and a paper Non-transitive counterparts of every Tarskian logic. Maybe they or the references therein can help you further.

 

[FactChecker]

You must distinguish between a strictly logical implication, which is always true, versus a probabilistic implication of tendency, which is likely to be true. In the first case, IMO, transitivity always holds. In the second case, transitivity does not have to hold and it can even give an opposite tendency.

This is an example of the second case.
Suppose there is a container with incoming water that will overflow unless a drainage valve is opened. The drainage valve is only opened when the container is near to overflowing and when it is opened the container will not overflow. The drainage valve is only opened a small fraction of the time it is needed. Then it is true that the drainage valve open implies the container is close to overflowing. It is also true that the container being close to full implies that the container is likely to overflow. But it is false that the drainage valve being open implies that the container is likely to overflow. In fact, just the opposite is true.

 

[nomadreid]

Thanks very much, fresh_42 and Factchecker.

fresh_42: I've looked briefly at the papers, found both of them interesting, and have downloaded them for further study. Downright super! Thanks again!

Factchecker: nice example, and good explanation of the general case. (The probabilistic approach appears to be widely applicable to real-life cases. Making "mostly implies" into "implies" (with appropriate caveats) reminds me --even though the details are very different -- of the approach of putting everything in equivalence classes, using "true almost everywhere".) Thanks again!