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In the 1900s, Hilbert published a list of 23 (later 24) unsolved problems in mathematics, which sparked increased research into each of them and the subsequent resolution of several of these problems. This question concerns a similar sort of intention to that of Hilbert, except on the topic of logic.
What, at present, are the major unsolved problems for logic and the philosophy of logic?
One should keep in mind that the meaning of "logic" changed over the last century, and is now more confined to formal logic, although it is broader than deductive or mathematical logic in the narrow sense. The interface between the formal and the informal, formalization, formal semantics, is also included. But this is not what Kant meant by "transcendental logic", Hegel by "Science of Logic" or Husserl by "Logical investigations", that sense of logic is closer to what is now called epistemology.
As for the outstanding problems I'll name three major ones, semantic paradoxes, vague predicates, and constructing formal semantics of natural languages. The first two have ancient roots, the third is a more recent project dating back to the linguistic turn of mid-20th century, all three are subject of active research.
The oldest semantic paradox is the Liar, "Epimenides the Cretean says that all Creteans are liars", "the least number that cannot be described in less than 12 words", the Berry paradox, is another example. Semantic paradoxes usually play on self-reference and are the reason for the difficulties with formal languages within which one can talk about truth of their own sentences, this leads to inconsistency. Nonetheless, our natural language is apparently of this sort. Standard approaches introduce artificial hierarchies of languages, such as the classical Tarski hierarchy or Kripke hierarchy, which involves truth gaps. However, even Kripke's version can not deal with the Strengthened Liar, and the talk of neither true nor false sentences is made in an external language, as Kripke put it, "ghost of the Tarski hierarchy is still with us". Another approach involves paraconsistent logics which allow some degree of inconsistency, but many find swallowing them too steep a price to pay.
The idea of vague predicates goes back to the sorites paradox (ascribed to the same Eubulides of Miletus as the Liar): one grain of wheat does not make a heap, adding a grain does not make a non-heap into a heap, therefore, there are no heaps. At first it seems easy to resolve the paradox by admitting borderline cases, but even if we admit a grey area in between heaps and non-heaps, semi-heaps, we face the same problem with non-heaps and semi-heaps. It appears necessary to give up either modus ponens or induction, and both are costly. Truth gaps and paraconsistency were deployed, and obviously fuzzy logic. The problem with fuzzy logic is that there is no obvious bridge between fuzzy truth values and the traditional ones, indeed building one is just the sorites in disguise. A peculiar approach is epistemicism, there is a sharp boundary between heaps and non-heaps, it says, but we do not know (and shall never know) what it is. This has some formal benefits, but it is hardly illuminating, as Field pointed out it confuses factual but unknown with non-factual.
In a sense, constructing semantics of natural languages is a catch-all umbrella mega-problem, of which the above two are small parts. Its crystallization is probably due to Dummet's efforts in 1970-s. The idea is to build "ideal language" with good formal properties that tracks our linguistic intuitions as closely as possible. For a comprehensive recent review and radical approach that disposes with the classical predicate calculus see Ben-Yami's Logic & Natural Language.
Other aspects include dealing with the conditional, the classical material conditional is clearly not the intuitive one, and other formal candidates, e.g. strong or relevance conditionals, have their issues too. The dispute between Frege's descriptivist and Kripke's referential semantics of names so far ended in impasse, both capture intuitive aspects of language but do not lend themselves to a neat synthesis. Modal logic, and its interpretations, both formal and informal, also remain controversial despite Kripke's efforts. Deontic logic, the logic of imperatives, offers its own host of intractable difficulties, see Hansen's Imperatives and Deontic Logic for a review.
