As well as the standard negation of Polish logic, it is possible to define two others: A strong negation, which is ordinarily expressed as “Not Possible”, and a weak negation, which is ordinarily expressed as “Not necessary”. Normally I don’t use them. However, they do have use.
I had mentioned in one of the earlier posts in this series that I had read about a number of controversies on the foundations of mathematics. A Dutch mathematician named L. E. J. Brouwer had in the early 1900s created his own philosophy of mathematics, which he termed intuitionism, in which he rejected the law of the excluded middle. His student Arend Heyting gave an axiomatic basis for this variant of mathematical logic. Although this has never gained the support of most mathematicians, it remains an area of some interest. What I found was that at least one version of the Heyting axioms of Intuitionistic Propositional Logic could be expressed as theorems of Polish logic, if I used a strong negation instead of the standard.
There is also a branch of logic termed “Fuzzy logic”, which assigns numeric degrees of truth to propositions. The truth table are constructed so that a logical “AND” has the value of the highest truth truth value of the propositions being combined, and a logical “OR” has the lowest. The principle obstacle to adoption seems to be indecision on the nature of the conditional. However, if all the degrees between true and false are expressed by the single value U, This is remarkably similar to the Polish logic. The algebra of the Polish logic and the algebra of fuzzy logic both belong to a class known as deMorgan algebras, which is similar to Boolean algebra but lacks the complementation laws, which unsurprisingly are equivalent to the law of the excluded middle and law of bivalence.
The absence of the law of the excluded middle makes the standard mathematical proof by contradiction tricky. This connected to negation introduction or negation elimination rules in natural deduction systems. A possible set of natural deduction rules can be formulated for the Polish logic, but it requires a stronger contradiction: possible and not possible, necessary and not necessary, or even necessary and impossible.
This also has connection to the field known as paraconsistent logic.
As with classical logic, this can also be extended by the use of propositional functions applied to particular objects or sets of objects to predicate logic.
In spite of the versatility of this system, it still isn’t quite enough to capture traditional modal logic. In particular, with only three values, it is not possible to capture the distinction between “contingently true” and “contingently false”. However, it may clarify the way such a system would need to be constructed.
Many of these subjects are rather technical, and I would welcome a discussion with other experts on logic. However, of all the people I know, few are interested in symbolic logic at all, and none of those interested in closely examining my results.
There are a few avenues I have yet to pursue, but for now, this will end my series of musings on logic. Perhaps I will go back to Aristotle and pick up the thread of historical investigation.