...

To draw a conclusion for my question whether it is logically possible that 5+5=11:

Since it is logically possible that what we hold to be logically possible is the outcome of a higher consciousness - or let's be traditional and call it a Cartesian genius malignus - restricting our epistemic access to the realm of actual logical possibilities, the claim that it is logically impossible that 5+5=11 is logically possible is unfounded.

But does that mean that it is logically possible that 5+5=11?

We can't make this claim either, because in order to do so we would have to exclude the logical possibility that e.g. the genius malignus G that restricts our access to the realm of logical possibilities might himself be subjected to another genius malignus G* restricting G's access to the realm of logical possibilities.

As long as the logical possibility of G being deceived by G* cannot be excluded, G might be wrong in thinking that he is deceiving us about the logical possibility of 5+5=11.

So, provided that there is no third genius malignus G** fooling around with G* - our conviction that it is logically impossible that 5+5=11 might be not only

(a) induced by the genius malignus G

but also

(b) *true* due to G's being misled by G*.

Conclusion:

(1) We cannot positively claim that 5+5=11 is logically impossible because we cannot exclude the logical possibility of being misled in our judgement by a genius malignus G.

(2) We cannot positively claim that 5+5=11 is logically possible because we cannot exclude the logical possibility that G is misled by another genius malignus G*.

Still,

(3) We have the strong intuition that 5+5=11 is in some sense impossible (provided that we use the decimal system, the common semantics of mathematical symbols etc.)

The question is: Is this impossibility

*logical* or merely

*mathematical*?

Another question is: Does the concept of logical possibility make sense after all in view of skeptical arguments like the ones sketched above or does it rather reduce to some kind of common sense/intuitive or

*pragmatic possibility*?

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