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ω-consistent theory:

In mathematical logic, an ω-consistent (or omega-consistent, also called numerically segregative1) theory is a theory (collection of sentences) that is not only (syntactically) consistent (that is, does not prove a contradiction), but also avoids proving certain infinite combinations of sentences that are intuitively contradictory. The name is due to Kurt Gödel, who introduced the concept in the course of proving the incompleteness theorem.

Now in Jain dialectic we have the following statements:

These seven propositions also known as saptabhangi are:

syād-asti: "in some ways it is"
syād-nāsti: "in some ways it is not"
syād-asti-nāsti: "in some ways it is and it is not"
syād-asti-avaktavyaḥ: "in some ways it is and it is indescribable"
syād-nāsti-avaktavyaḥ: "in some ways it is not and it is indescribable"
syād-asti-nāsti-avaktavyaḥ: "in some ways it is, it is not and it is indescribable"
syād-avaktavyaḥ: "in some ways it is indescribable"

I just need a hint to convert the latter sentences to logical form.

closed as unclear what you're asking by Joseph Weissman Jan 9 '16 at 16:21

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    Would it be a hint to suggest you think about which of these seven propositions you would consider to contradict each other? – Michael Dorfman Aug 4 '13 at 13:05
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    I think computability, or computational complexity, is likely to be a better framework for extracting any sort of classical formal expression of these sentences (mapping the properties of "is" and "is not" to answers for instances of a yes/no problem, i.e. to possession of qualities rather than mere existence). Of course, the presentation suggests that you might prefer to simply consider a paraconsistent logic instead, which embraces contradictions rather than rejecting them; but then omega-consistency perhaps becomes a less interesting concept in that context anyway. – Niel de Beaudrap Aug 5 '13 at 9:43
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    I can't see what contribution that omega-consistency can make to Jain thought. Its a technical requirement that Godels requires for his theorem to go though, rather than being something of independent interest on its own. – Mozibur Ullah Aug 5 '13 at 16:47
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Recast in a classical logical framework, you could start with a family M of models for which your truth-statements are interpretable (i.e. they refer to things in those models); if these models are all reasonable approximations of reality (even if they differ in some details), then "in some ways it is" means "there exists m in M s.t. p is true in m". Let D be a decidability function (i.e. D(p,m) implies that there is a proof of p given model m--whether you want to work with different models or different sets of truth statements is a more complex issue than I have time to think through). Then "in some ways it is indescribable" means that "there exists m in M s.t. !D(p,m)" (where ! means "not")--it might be true or it might not, but you can't show it.

So then we have these three fundamental statements:

A: exists `m` in `M` s.t. `p`
B: exists `n` in `M` s.t. `!p`
C: exists `r` in `M` s.t. `!D(p,r)`

And the seven propositions are just an exhaustive combinations of these:

A
B
A & B
A & C
B & C
A & B & C
C

(omitting the trivial case of no statement at all).

I am not sure this exactly maps onto the Jain propositions, but at least it's a start in that direction.

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