These definitions result in a partial modal logic which includes the classical results “Not certainly = possibly not” and “not possible = certainly not”, and allows “If certain then true” and “if true then possible”, but not their converses.

When I began working with this scheme, I kept finding intuitively agreeable and familiar consequences falling out of it so easily and naturally that I was amazed to find little mention and less use of it in the standard sources in non-classical logic. ]]>

However, I welcome the advance in expressive power over two-valued logic, even if it doesn’t have the full flexibility of natural language.

There is some similarity to other modal logics, but it’s not identical to any of the Lewis systems. ]]>

With your “strong negation” it’s not clear to me how double-negation gets interpreted as “not impossible” – it seems to me that it should say “it’s impossible that it be impossible” not just “it happens not to be impossible”. Also, I don’t understand how iteration of the “weak negation” gets a strong statement – it seems to me that it should say “it is not confirmed that it is unconfirmed”, which doesn’t seem to imply that it is in fact confirmed. Do you mean the outer negation in both cases to be a classical negation, rather than either weak or strong negation?

Of course, I’m not exactly sure what the details of your system are. The interpretation you’re speaking about with “impossible” and “necessary” truth suggest to me something basically similar to the S4 modal models of intuitionist logic, in which the primary negation is rendered as necessity of the subsidiary negation.

]]>One of them is a strong negation (-), which could be interpreted as “impossible”. Negating this, in turn, gives a weak “not impossible” assertion, so that P implies -(-P), but not vice versa.

With this negation, the rule of bivalence(P v -P) (P is true or impossible) doesn’t necessarily hold, but the corresponding ~(P & ~P), (not both true and impossible) does.

Another is a weak negation, which could be regarded as “unconfirmed”. Negating this gives a strong assertion. Here we have -(-P) implies P, but not vice versa. This has (P v ~P) (P is true or unconfirmed) holding, but not necessarily ~(P v ~P) (not both true and unconfirmed)

Defining negation so that P is equivalent to ~(~P) gives a situation such that neither (P v ~P) nor ~(P & ~P) is necessarily true, but they are equivalent, just as classical two-valued logic would have it.

However, when the tertiam non datur is strongly asserted as a necessary truth, it cripples the ability to speak of the uncertain. Likewise, the ability to speak of the uncertain apparently forces a weakening of the tertiam non datur in some form, losing double negation, truth functionality, or some similar price.

I’m not sure that either natural language or intuition is required to be consistent, but having a formal language to speak about uncertainty seems better than not having one. ]]>

I think it’s quite plausible that natural language will turn out to be too messy and pragmatic to have any particular logic, but that a sort of rigorized scientific language could well turn out to need intuitionist, classical, or possibly relevant logic.

Fuzzy logic and Lukasiewicz logics and various other systems all seem to have applications, but I don’t know of any convincing arguments that they accurately describe something in human language.

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