# Advancements in formal logic in the 21st century?

I've recently learned a lot about the history of formal logic, from Frege and Pierce in the late 19th century, to Russell, Hilbert, and Quine (et al)'s development of 1st-order logic, to advances with modal/non-classical logics (by Kripke, Hintikka, and Boolos), in the 70s, 80s, and 90s.

However, it seems that the last 20 to 30 years have been the quietest chapter yet. As far as I know, there have been no major kinds of logic thought up recently, no new figures rising to prominence, and no new innovations towards solving existing issues and controversies that continue to plague various modal logics.

Is formal logic a static discipline today, or am I missing something significant?

• Logical pluralism has become a consensus view in the 21st century. That seems like it would qualify as a major "advancement". Jan 31 at 23:11
• Two major new developments in the 21st century logic were Voevodsky's univalent foundations (2006) and Girard's transcendental syntax (2013). Feb 1 at 0:55
• Other areas where logic is still progressing include paraconsistent logics, substructural logics, quantum logic and computation, the integration of quantifier logic with probability theory, modal logics, connexive logic, and the logic of conditionals generally. Feb 1 at 6:47
• @Dcleve I'd argue that the acceptance of opinion about logic does not count as an advancement in logic, in the same way that "[insert software here] sold more copies this year than last year" does not count as an update to that software. P.S. I'm aware this isn't a perfect analogy, but my point is that I am asking about new creations/developments/analyses within the discipline itself, rather than mere changes in opinion.
– Nico
Feb 1 at 19:16

Paraconsistent logic has progressed fairly well. A couple of articles in which this logic is applied:

1. Zach Weber, "Transfinite Cardinals in Paraconsistent Set Theory." Goes over a classically themed set theory with a paraconsistent background logic. Includes funny objects like ℵOn, the cardinality of the set of all ordinal numbers in Weber's theory. Because for better or worse, he's got universal sets, since contradictions arising from them are not so objectionable (or, he does not object so much to those contradictions...), or he has unusual inferential resources for evading some of the contradictions besides.

2. Agudelo and Carnielli, "Quantum Computation Via Paraconsistent Computation." References to dialethic Turing machines.

See also J. Béziau's report on how he became engaged with the question of paraconsistent logic. Béziau is linked to Alessio Moretti, whose "geometrical logic" (not a logic for geometry, but an analysis of geometrical structures that graph-theoretically express parts of logic) has intersected developments in deontic logic.

Depending on how one interprets computer science, that could be a domain that has featured many salient examples of what you're asking for (except the thing about "famous" essayists, maybe). There are then the intersections with AI research to consider. I can't think of any recent specific papers or reports from those quarters, though; I mean, I'm not well-versed in that field. I've also heard about categorical logic and how category theory involves new ways of trying to describe, and argue about, the nature of logic; not sure how recent that all is, though.

Overall, I don't know that the historical conditions, including cultural ones, that lead to "prominent-figure" accounts in an overview of logic's development as an academic subdiscipline, have been quite so in place for a while now. I mean that hero-worship/main-character-syndrome/valiant-entrepreneur motifs in various societies have decreased in popularity, so maybe there's no modern Frege or Russell or whoever, so to say; or maybe those past examples actually weren't directly established as "legends" in their own time, but were faced with detractors, paradoxes in their systems, and so on enough to where they and others doubted them more at those times. More, that is, than many people have been given to doubt the "authority" of those past essayists; the whole theme of a "written canon" has burned to ashes in the air, in a lot of places.

• "including cultural ones, that lead to "prominent-figure" accounts in an overview of logic's development as an academic subdiscipline, have been quite so in place for a while now. I mean that hero-worship/main-character-syndrome/valiant-entrepreneur motifs in various societies have decreased in popularity". I'm not sure if this holds water, since disciplines like math and physics still have their celebrities; Terrence Tao, Andrew Wiles, Michio Kaku, & Kip Thorne, just to name a few. People 100% still can and do gain fame/clout for significant works, even without any "hero-worship motifs".
– Nico
Feb 1 at 19:23
• "Depending on how one interprets computer science, that could be a domain that has featured many salient examples of what you're asking for". Personally, I wouldn't consider computer science to be an answer to my question. One can master almost all common programming languages (JavaScript, Python, C++, etc.) without any knowledge whatsoever or material implication, natural deduction, universal/existential quantifiers, modalities, or really any logical concept other than boolean AND/OR/NOT.
– Nico
Feb 1 at 19:31
• @Nicolino there are celebrities, but are there legends? But again, I suppose that Frege and Russell at the time were not already legends, already the "most frequently referenced" in a relevant discipline (analytic philosophy/logic), either. That kind of question has to wait on becoming more of history, perhaps. Feb 1 at 20:38
• I'm not asking about legends, I was merely wondering if there are any contemporary logicians that are well-known in the field, in a similar way to Tao and Wiles are in math.
– Nico
Feb 2 at 0:07