What if the best reason to reject classical logic isn’t that it’s wrong everywhere, but that it unnecessarily rules out true theories by banning both truth-value gaps and truth-value gluts?
My links: https://linktr.ee/frictionphilosophy.
1. Guest
Jc Beall holds the O’Neill Family Chair of Philosophy at the University of Notre Dame and his research primarily focuses on logic and the philosophy of logic.
2. Interview Summary
JC Beall lays out a “minimalist” picture of logic: logic is the topic-neutral consequence/validity relation that governs the sparse core vocabulary shared by every true theory (negation, disjunction, quantifiers, etc.). From that starting point, he argues there’s no principled reason to rule out either gaps (sentences that are neither true nor false) or gluts (both true and false): once you recognize truth and falsity as basic statuses, four combinations are on the table, and standard truth/falsity conditions for negation don’t automatically exclude the “both” or the “neither.” On this basis he defends a “simple argument” for going sub-classical: you don’t lose any true theories (since theory-specific consequence can still add stronger principles when needed—e.g., mathematics may behave classically), but you do keep live candidate theories that classical logic would dismiss as impossible. Transcript:
A big middle section engages objections associated with Timothy Williamson. Beall pushes back on the idea that “stronger logic = more explanatory,” noting that a consequence relation can be maximally strong (even trivial) while being explanatorily useless; so “more valid inferences” doesn’t, by itself, buy genuine explanation. He also rejects the charge that sub-classical logic is less unifying: logic already unifies all theories by governing the common topic-neutral vocabulary, and demanding extra classical principles as “unification” risks begging the question. On the “simplicity” worry (more semantic statuses looks more complex), his reply is basically: if reality really exhibits more than the classical two statuses in some domains, then recognizing that isn’t a theoretical vice—it’s just accuracy; and if someone thinks those statuses aren’t real possibilities, they owe an argument, not an appeal to aesthetic economy.
Later, the discussion turns to truth and paradox: Beall distinguishes the logical operators (“it’s true that” / “it’s false that”) from truth as a predicate, and he motivates “transparent truth” by requiring that “P” and “P is true” be everywhere intersubstitutable—mirroring the operator’s behavior. But once truth is introduced as a predicate in a rich grammar, it becomes “creative” and generates unintended byproducts (“spandrels”) like liar-style sentences, which then have to be handled rather than wished away. The interview closes by sampling applications and extensions: exploratory work on whether future-tense discourse might be gap-like or glut-like, and a set of theological cases (e.g., omnipotence/stone-style puzzles) where a gappy treatment can be a serious live option—even though Beall also thinks some central Christian theology doctrines are better modeled with gluts, and he notes ongoing disagreements about how unified the overall picture should be.
3. Interview Chapters
00:00 - Introduction
00:42 - Logic and subclassical logic
13:44 - Arguing for subclassical logic
23:36 - Criticism: less explanatory?
27:18 - Criticism: less unifying?
34:02 - Criticism: less simple?
42:20 - Criticism: unmotivated?
49:23 - Truth as a transparent logical property
53:34 - Truth as an emergent property
54:44 - Logical vocabulary
59:16 - Spandrels of truth
1:04:27 - Future contradictions
1:13:12 - God responsible for logical consequence?
1:16:35 - God of the “gaps”
1:17:46 - Issues with classical accounts of omnipotence
1:21:34 - Trinity as contradictory?
1:23:19 - The Contradictory Christ
1:30:49 - Kenotic view
1:33:11 - Inflated metaphysics
1:37:21 - Natures not inconsistent?
1:42:49 - Theological significance
1:49:20 - Conclusion
