Pi is wrong! But so what? It is neither new, nor complicated enough to count as real math! And suggestions that or might be even better show that it not clear-cut either.
I recently invested sufficient energy into some logical questions to make real progress. But while telling friends and fellow logicians about it, I realized how irrelevant all my results and conclusions will be. I will have to publish them in an appropriate journal nevertheless, since they continue David Ellerman’s work on partition logic. Not publishing them would be dismissive towards the value of David Ellerman’s work, and he really invested lots of effort into it, and believes in its value. I won’t talk about those results here, since I don’t know how it would impact my ability to publish them.
Let’s have some fun instead, stay extremely close to classical logic, and still demonstrate a logic without truth. And let’s get back to Gerhard Gentzen.
But is only true if is a total function. Especially is only true (even for specific ) if is total, since otherwise
Let’s also prove one of the other claims:
Note that . So the preserved operations are “and”, “or”, and “xor” if interpreted from a logical perspective. It would be nice if implication were preserved too. This seems hopeless, since is always true, and truth is not preserved. But we do have , which means that the external implication given by the order is preserved. So we should be able to turn this into a logic, where internal truth is not preserved under context morphisms.
Gerhard Gentzen’s sequent calculus
The sequent calculus is a proof calculus with significant practical and theoretical advantages compared to more obvious proof calculus systems. It works with sequents . The propositions (and ) could be logical formulas like (the 4-th Peano axiom). They can also be interpreted as subsets of some universe set , which is sufficient for understanding the basics of the sequent calculus. Then the sequent itself is interpreted as .
|Left structural rules||Right structural rules|
Here stand for arbitrary finite sequences of propositions. The structural rules may be relatively boring. The following global rules are slightly more interesting
None of the rules up to now has used any logical constant or connective. They can be verified directly for the subset interpretation. The following logical rules can only be verified after the (intended) interpretation of the logical connectives has been fixed.
|Left logical rules||Right logical rules|
One possible interpretation for these connectives in terms of subsets would be , , , , and .
But it may be more instructive to see an interpretation where one of the classical logical rules is violated. So let us use instead, where is the interior operator of some topological space. The propositions (and ) are interpreted as open subsets in this case. The rule is violated now, and has to be replaced by the rule . This gives us the intuitionistic sequent calculus, which exactly characterizes the valid conclusions of intuitionistic logic.
To see that is violated, let , correspond to , , and . Above the line we have , which is true. Below the line we have , which is false.
An evil twin of sequent calculus
Note that implication satisfies or rather . Let us replace implication by minus. Note that minus satisfies with . Then we get the following two rules instead of and .
Of course, we also remove and from the language together with the rule . This sequent calculus is still as sound and complete as the original sequent calculus. But we no longer reason about implication, but only about minus. Some sort of implication is still present in , but it is no longer mirrored internal in the language of the logic itself. So this is our logic without truth.
I don’t really know (or understand) whether this sort of context morphism has any sort of relevance, and whether that logic without truth occurs anywhere in the real world. Is there any relation to the fact that it is easier to deny the relevance or truth of a given conclusion than to prove that it is important and true? What I like about that logic is the asymmetry between implication and falsehood, because I wanted to find naturally occurring asymmetries in mathematical hierarchies and logic. Even for the results that I do want to publish, I have the same problem that I don’t really understand the relevance of the corresponding context morphisms, whether there even should be context morphisms, and whether my proposed context morphisms are the correct ones.
That post initially also contained a logic without falsehood, or rather a logic where falsehood is not used. But it started to get long, and this post is already long enough. Not sure whether this was really a good idea, since the explanation of the sequence calculus was also intended to better understand how such a logic with a reduced set of logical constants and connectives still maintains its main features. Maybe I will manage to create another blog post from the removed material. Or maybe nobody including myself cares anyway, as already indicated at the beginning. Or maybe I should better use my time to finish the paper about the results I wrote about at the beginning, and submit them to…