The 2 Axes of the Logic Spectrum

A foundational model for understanding logical systems along two fundamental dimensions. The first axis runs from Formal Logic (concerned with pure form, syntax, validity regardless of content—math-like reasoning) to Informal Logic (concerned with real-world arguments, fallacies, natural language—how people actually reason). The second axis runs from Classical Logic (bivalent, law of excluded middle, truth-functional—Aristotle to Frege) to Non-Classical Logic (deviations: many-valued, paraconsistent, intuitionistic, fuzzy). These two axes create four basic logical orientations: formal-classical (standard mathematical logic), formal-nonclassical (modal logic, fuzzy logic), informal-classical (critical thinking textbooks, fallacy studies), informal-nonclassical (practical reasoning with uncertainty, everyday fuzzy logic). The model reveals that "logic" isn't one thing—it's a family of tools for different purposes, from pure mathematics to everyday argument evaluation.

The 2 Axes of the Logic Spectrum "You say someone's argument is illogical. The 2 Axes ask: by which logic? Classical formal logic might call it invalid. Informal logic might see it as reasonable in context. Fuzzy logic might give it .73 truth. Same argument, three different verdicts. The axes help you see that 'logic' isn't a single judge—it's a panel, and they don't always agree."

by Dumu The Void February 25, 2026

The 4 Axes of the Logic Spectrum

An expanded model adding two crucial dimensions to the basic framework. Axis 1: Formal-Informal (form vs. content). Axis 2: Classical-Nonclassical (standard vs. alternatives). Axis 3: Deductive-Inductive (certain inference vs. probabilistic inference). Axis 4: Monotonic-Nonmonotonic (adding premises never invalidates conclusions vs. conclusions can be defeated by new information). These four axes create sixteen logical positions. Mathematical logic is formal, classical, deductive, monotonic. Legal reasoning is informal, classical (mostly), inductive (evidence weighs), nonmonotonic (new evidence changes everything). AI reasoning is often formal, nonclassical (fuzzy, probabilistic), inductive, nonmonotonic. The 4 Axes reveal that different domains require different logics—using monotonic deductive logic for legal reasoning would be disastrous.

The 4 Axes of the Logic Spectrum "You think logic is universal. The 4 Axes show otherwise: math logic is monotonic—once proven, always proven. Legal logic is nonmonotonic—new evidence overturns verdicts. Same logic label, completely different behavior. The axes help you see why your 'logical' argument fails in court: you're using the wrong logic for the domain."

by Dumu The Void February 25, 2026

The 6 Axes of the Logic Spectrum

A comprehensive model adding dimensions of truth and inference. Axis 1: Formal-Informal. Axis 2: Classical-Nonclassical. Axis 3: Deductive-Inductive. Axis 4: Monotonic-Nonmonotonic. Axis 5: Bivalent-Many-Valued (two truth values vs. many). Axis 6: Truth-Preserving-Information-Preserving (logic keeps truth vs. logic keeps information). These six axes generate sixty-four logical positions. Relevance logic is formal, nonclassical, deductive, monotonic, bivalent, but demands relevance between premises and conclusion—it preserves relevance, not just truth. Fuzzy logic is formal, nonclassical, can be deductive or inductive, monotonic typically, many-valued (degrees of truth), truth-preserving (of degrees). The 6 Axes reveal that logical systems are designed for different goals—some prioritize certainty, others nuance, others relevance.

The 6 Axes of the Logic Spectrum "You want a logic that handles uncertainty. The 6 Axes ask: uncertainty as degrees of truth (fuzzy) or as probability (inductive)? Many-valued or probabilistic? Both are nonclassical, but they're different nonclassical. The axes help you choose the right tool, not just any tool labeled 'logic for uncertainty.'"

by Dumu The Void February 25, 2026

The 8 Axes of the Logic Spectrum

A detailed model adding dimensions of quantification and modality. Axis 1: Formal-Informal. Axis 2: Classical-Nonclassical. Axis 3: Deductive-Inductive. Axis 4: Monotonic-Nonmonotonic. Axis 5: Bivalent-Many-Valued. Axis 6: Truth-Preserving-Information-Preserving. Axis 7: First-Order-Higher-Order (quantification over individuals vs. over properties/functions). Axis 8: Extensional-Intensional (logic of truth values vs. logic of meanings/possibilities). These eight axes create 256 logical positions. Modal logic (necessity/possibility) is formal, nonclassical (in some classifications), deductive, monotonic, bivalent typically, truth-preserving, can be higher-order, intensional (deals with meanings across possible worlds). The 8 Axes demonstrate that the explosion of logical systems in the 20th century reflects different choices on these fundamental dimensions.

The 8 Axes of the Logic Spectrum "You think logic is just propositional calculus. The 8 Axes show that's one tiny point: formal, classical, deductive, monotonic, bivalent, truth-preserving, first-order, extensional. Modal logic changes intensional. Fuzzy logic changes many-valued. Nonmonotonic logic changes monotonic. The axes map the entire universe of logic—and you're still in the first galaxy."

by Dumu The Void February 25, 2026

The 12 Axes of the Logic Spectrum

An ultra-fine-grained model adding dimensions of paraconsistency, relevance, and computation. Building on the 8 Axes, we add: Axis 9: Explosive-Paraconsistent (contradiction entails everything vs. contradictions can be contained). Axis 10: Relevant-Irrelevant (premises must be relevant to conclusion vs. relevance not required). Axis 11: Computational-Noncomputational (logic has effective decision procedure vs. undecidable). Axis 12: Static-Dynamic (logic of static propositions vs. logic of change/action). These twelve axes generate 4096 logical positions. Paraconsistent logic is formal, nonclassical, deductive, monotonic or nonmonotonic, bivalent or many-valued, truth-preserving, any order, extensional or intensional, paraconsistent (non-explosive), can be relevant or not, often decidable, static typically. Dynamic logic is formal, nonclassical, deductive, monotonic, bivalent, truth-preserving, higher-order, intensional, explosive, relevant-ish, decidable often, dynamic (explicitly about change). The 12 Axes reveal that logical pluralism isn't optional—different problems require different logical tools, and the axes help you find the right one.

The 12 Axes of the Logic Spectrum "You want a logic for contradictions in legal reasoning. The 12 Axes ask: explosive (standard logic) would destroy everything. Paraconsistent contains them. Relevant ensures the contradiction matters. Dynamic handles changing laws. Twelve axes, twelve design choices. Your 'simple logic' is just the one you're used to—not the one you need."

by Dumu The Void February 25, 2026

The 16 Axes of the Logic Spectrum

The ultimate model, adding the final dimensions of context, psychology, and metaphysics. Building on the 12 Axes, we add: Axis 13: Context-Independent-Context-Dependent (logic applies everywhere vs. context matters). Axis 14: Psychological-Ideal (logic describes how people think vs. how they should think). Axis 15: Ontologically-Neutral-Committed (logic assumes nothing about reality vs. logic has metaphysical implications). Axis 16: Unitary-Pluralist (one true logic vs. many logics for many purposes). These sixteen axes generate 65,536 potential positions—enough to capture every logical system ever conceived. The 16 Axes of the Logic Spectrum reveal that logic is not a single discipline but a multidimensional space of choices about how to reason, what reasoning is for, and what reasoning assumes. The 16 Axes don't tell you which logic is correct—they give you a language for understanding what any logic claims, what it's good for, and where it might fail. They are the map of the space of valid inference—the periodic table of reason itself.

The 16 Axes of the Logic Spectrum "You want the one true logic. The 16 Axes ask: which one? The one that's formal or informal? Classical or nonclassical? Deductive or inductive? Monotonic or nonmonotonic? Bivalent or many-valued? Truth-preserving or information-preserving? First-order or higher-order? Extensional or intensional? Explosive or paraconsistent? Relevant or irrelevant? Computational or noncomputational? Static or dynamic? Context-independent or context-dependent? Psychological or ideal? Ontologically neutral or committed? Unitary or pluralist? Sixteen questions, and until you answer them, 'one true logic' is just a slogan. The axes don't give you the answer—they force you to ask the questions that any real logic must answer. And that's the most logical thing of all."

Classical logic chooses formal, classical, deductive, monotonic, bivalent, truth-preserving, first-order, extensional, explosive, irrelevant (classical doesn't require relevance), computational (for propositional), static, context-independent, ideal, ontologically-neutral (claims to be), unitary (claims to be the one true logic). Relevance logic disagrees on relevance and maybe paraconsistency. Fuzzy logic disagrees on bivalence. Nonmonotonic logic disagrees on monotonicity.

by Dumu The Void February 25, 2026

The 2 Axes of the Knowledge Spectrum

A foundational model for understanding the nature of knowledge along two fundamental dimensions. The first axis runs from A Priori Knowledge (knowledge independent of experience—math, logic, conceptual truths) to A Posteriori Knowledge (knowledge dependent on experience—empirical facts, scientific observations). The second axis runs from Propositional Knowledge (knowing that—facts, information) to Procedural Knowledge (knowing how—skills, abilities, practices). These two axes create four basic knowledge types: a priori-propositional (mathematical truths), a priori-procedural (knowing how to reason), a posteriori-propositional (scientific facts), a posteriori-procedural (knowing how to ride a bike). The model reveals that "knowledge" isn't one thing—it's a family of cognitive achievements with different sources and different forms.

The 2 Axes of the Knowledge Spectrum "You say you know it. The 2 Axes ask: know that or know how? Know from reason or from experience? Knowing that 2+2=4 is very different from knowing how to ride a bike. Same word, different kinds. The axes help you see that 'knowledge' covers a lot of territory—and treating all knowledge like math is a category error."

by Dumu The Void February 25, 2026